CONTENTS S.No. Chapter Page 1. Real Numbers 3 2. Polynomials 9 3. Pair of Linear Equations in two Variables 15 4. Similar Triangles 24 5. Trigonometry 36 6. Statistics 45 Sample Paper 56 2 [X вЂ“ Maths] CHAPTER 1 REAL NUMBERS KEY POINTS 1. EuclidвЂ™s division lemma : For given positive integers вЂ�aвЂ™ and вЂ�bвЂ™ there exist unique whole numbers вЂ�qвЂ™ and вЂ�rвЂ™ satisfying the relation a = bq + r, 0 п‚Ј r < b. 2. EuclidвЂ™s division algorithms : HCF of any two positive integers a and b. With a > b is obtained as follows: Step 1 : Apply EuclidвЂ™s division lemma to a and b to find q and r such that a = bq + r , 0 п‚Ј r < b. Step 2 : If r = 0, HCF (a, b) = b Step 3 : if r п‚№ 0, apply EuclidвЂ™s lemma to b and r. 3. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorized) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur. p , q п‚№ 0 to be a rational number, such that the prime q factorization of вЂ�qвЂ™ is of the form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which is terminating. 4. Let x пЂЅ 5. Let x пЂЅ 6. пѓ–p is irrational, where p is a prime. A number is called irrational if it p cannot be written in the form q where p and q are integers and q п‚№ 0. p , q п‚№ 0 be a rational number, such that the prime factorization q of q is not of the form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating. 3 [X вЂ“ Maths] MULTIPLE CHOICE QUESTIONS 1. 2. 5 Г— 11 Г— 13 + 7 is a (a) prime number (b) composite number (c) odd number (d) none Which of these numbers always ends with the digit 6. (a) 4n (b) 2n (c) 6n (d) 8n where n is a natural number. 3. 4. 5. 6. 7. For a, b (a п‚№ b) positive rational numbers ____ (a) Rational number (c) пЂЁ a пЂ b пЂ© 2 пЂЁ a пЂ« b пЂ©пЂЁ a пЂ b пЂ© is a (b) irrational number (d) 0 If p is a positive rational number which is not a perfect square then пЂ3 p is (a) integer (b) rational number (c) irrational number (d) none of the above. All decimal numbers areвЂ“ (a) rational numbers (b) irrational numbers (c) real numbers (d) integers In Euclid Division Lemma, when a = bq + r, where a, b are positive integers which one is correct. (a) 0 < r п‚Ј b (b) 0 п‚Ј r < b (c) 0 < r < b (d) 0 п‚Ј r п‚Ј b Which of the following numbers is irrational number (a) 3.131131113... (b) 4.46363636... (c) 2.35 (d) b and c both 4 [X вЂ“ Maths] 21 8. The decimal expansion of the rational number 7 п‚ґ 2 пЂі п‚ґ 5 4 will terminate after ___ decimal places. 9. 10. 11. 12. 13. 14. (a) 8 (b) 4 (c) 5 (d) never HCF is always (a) multiple of L.C.M. (b) Factor of L.C.M. (c) divisible by L.C.M. (d) a and c both The product of two consecutive natural numbers is always. (a) an even number (b) an odd number (c) a prime number (d) none of these Which of the following is an irrational number between 0 and 1 (a) 0.11011011... (b) 0.90990999... (c) 1.010110111... (d) 0.3030303... pn = (a Г— 5)n. For pn to end with the digit zero a = __ for natural no. n (a) any natural number (b) even number (c) odd number (d) none. A terminating decimal when expressed in fractional form always has denominator in the form of вЂ” (a) 2m3n, m, n п‚і 0 (b) 3m5n, m, n п‚і 0 (c) 5n 7m, m, n п‚і 0 (d) 2m5n, m, n п‚і 0 пѓ¦3 пѓ¶пѓ¦3 пѓ¶ пѓ§пѓЁ вЂ“ 5 пѓ·пѓё пѓ§пѓЁ пЂ« 5 пѓ·пѓё is 4 4 (a) An irrational number (b) A whole number (c) A natural number (d) A rational number 5 [X вЂ“ Maths] 15. 16. If LCM (x, y) = 150, xy = 1800, then HCF (x, y) = (a) 120 (b) 90 (c) 12 (d) 0 Which of the following rational numbers have terminating decimal expansion? (a) (c) 91 2100 (b) 343 2 3 5 п‚ґ2 п‚ґ7 (d) 3 64 455 29 73 SHORT ANSWER TYPE QUESTIONS 17. Solve 50. What type of number is it, rational or irrational. 18. Find the H.C.F. of the smallest composite number and the smallest prime number. 19. If a = 4q + r then what are the conditions for a and q. What are the values that r can take? 20. What is the smallest number by which 5 пЂ 3 be multiplied to make it a rational no? Also find the no. so obtained. 21. What is the digit at unitвЂ™s place of 9n? 22. Find one rational and one irrational no. between 23. State EuclidвЂ™s Division Lemma and hence find HCF of 16 and 28. 24. State fundamental theorem of Arithmetic and hence find the unique factorization of 120. 25. Prove that 26. Prove that 5 пЂ 18 п‚ґ 1 2пЂ 5 3 and 5. is irrational number. 2 3 is irrational number. 7 6 [X вЂ“ Maths] 27. Prove that 28. Find HCF and LCM of 56 and 112 by prime factorisation method. 29. Why 17 + 11 Г— 13 Г— 17 Г— 19 is a composite number? Explain. 30. Check whether 5 Г— 6 Г— 2 Г— 3 + 3 is a composite number. 31. Check whether 14n can end with the digit zero for any natural number, n. 32. If the HCF of 210 and 55 is expressible in the form 210 Г— 5 + 55y then find y. 2 пЂ« 7 is not rational number. LONG ANSWER TYPE QUESTIONS 33. Find HCF of 56, 96 and 324 by EuclidвЂ™s algorithm. 34. Show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. 35. Show that any positive odd integer is of the form 6q + 1, 6q + 5 where q is some integer. 36. Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q. 37. Prove that the product of three consecutive positive integers is divisible by 6. 38. Show that one and only one of n, n + 2, n + 4 is divisible by 3. 39. Two milk containers contains 398 l and 436 l of milk. The milk is to be transferred to another container with the help of a drum. While transferring to another container 7l and 11l of milk is left in both the containers respectively. What will be the maximum capacity of the drum. ANSWERS 1. b 2. c 3. a 4. c 5. c 6. b 7. a 8. b 7 [X вЂ“ Maths] 9. b 10. a 11. b 12. b 13. d 14. d 15. c 16. c 17. 30, rational 18. 2 19. r, q whole no. 0 п‚Ј r < 4 20. пЂЁ 23. 4 24. 2 Г— 2 Г— 2 Г— 3 Г— 5 28. HCF = 56, LCM = 112 30. Yes 31. No 32. HCF (210, 55) = 5, 5 пЂ« пЂ© 3 , 2 21. 1 5 = 210 Г— 5 + 55y пѓћ y = вЂ“ 19 33. 4 34. a = 3q + r 35. a = 6q + r 38. n = 3q + r 39. 17 l 8 [X вЂ“ Maths] CHAPTER 2 POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively. 2. A quadratic polynomial in x with real coefficient is of the form ax2 + bx + c, where a, b, c are real numbers with a п‚№ 0. 3. The zeroes of a polynomial p(x) are precisely the xвЂ“coordinates of the points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero of polynomial p(x) if p(a) = 0. 4. A polynomial can have at most the same number of zeros as the degree of polynomial. 5. For quadratic polynomial ax2 + bx + c (a п‚№ 0) Sum of zeros пЂЅ пЂ b a Product of zeros пЂЅ 6. c . a The division algorithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that : p(x) = g(x).q (x) + r(x), g(x) п‚№ 0 where r(x) = 0 or degree of r(x) < degree of g(x). MULTIPLE CHOICE QUESTIONS 1. A real no. пЃЎ is a zero of the polynomial f(x) if (a) f(пЃЎ) > 0 (b) f(пЃЎ) = 0 (c) f(пЃЎ) < 0 (d) none 9 [X вЂ“ Maths] 2. 3. 4. 5. 6. 7. 8. 9. The zeroes of a polynomial f(x) are the coordinates of the points where the graph of y = f(x) intersects (a) x-axis (b) y-axis (c) origin (d) (x, y) If пЃў is 0 zero of f(x) then ____ is one of the factors of f(x) (a) (x вЂ“ пЃў) (b) (x вЂ“ 2пЃў) (c) (x + пЃў) (d) (2x вЂ“ пЃў) If (y вЂ“ a) is factor of f(y) then ___ is a zero of f(y) (a) y (b) a (c) 2a (d) 2y The graph of a linear polynomial will meet the x-axis at вЂ“ points, (a) 1 (b) 2 (c) 3 (d) None Cubic polynomial x = f(y) cuts y-axis at atmost (a) one point (b) two points (c) three points (d) four points Polynomial x2 + 1 has ___ zeroes (a) only one real (b) no real (c) only two real (d) one real and the other non-real. What type of graph will вЂ“x2 + 3x вЂ“ 4 represents (a) straight line (b) upward parabols (c) down ward parabola (d) none If degree of polynomial f(x) is вЂ�nвЂ™ then maximum number of zeroes of f(x) would be вЂ“ (a) n (b) 10 2n [X вЂ“ Maths] (c) 10. 11. 12. 13. n + 1 (d) n вЂ“ 1 If 2 is a zero of both the polynomials, 3x2 + ax вЂ“ 14 and 2x вЂ“ b then a вЂ“ 2b = ___ (a) вЂ“2 (b) 7 (c) вЂ“8 (d) вЂ“7 If zeroes of the polynomial ax2 + bx + c are reciprocal of each other then (a) a = c (b) a = b (c) b = c (d) a = вЂ“ c The zeroes of the polynomial h(x) = (x вЂ“ 5) (x2 вЂ“ xвЂ“6) are (a) вЂ“2, 3, 5 (b) вЂ“2, вЂ“3, вЂ“5 (c) 2, вЂ“3, вЂ“5 (d) 2, 3, 5 Graph of y = ax2 + bx + c intersects x-axis at 2 distinct points if (a) b2 вЂ“4ac > 0 (b) b2 вЂ“ 4ac < 0 (c) b2 вЂ“4ac = 0 (d) none SHORT ANSWER TYPE QUESTIONS 14. If пЃЎ and пЃў are the zeroes of the polynomial 2x2 вЂ“ 7x + 3. Find the sum of the reciprocal of its zeros. 15. If пЃЎпЂ¬пЂ пЃў are the zeroes of the polynomial p(x) = x2 вЂ“ a (x + 1) вЂ“ b such that (пЃЎ + 1) (пЃў + 1) = 0 then find value of b. 16. If пЃЎпЂ¬пЂ пЃўпЂ are the zeroes of the polynomial x2 вЂ“ (k + 6) x + 2 (2k вЂ“ 1). Find 1 k if пЃЎ пЂ« пЃў пЂЅ пЃЎпЃў. 2 17. If (x + p) is a factor of the polynomial 2x2 + 2px + 5x + 10 find p. 18. Find a quadratic polynomial whose zeroes are 5 пЂ 3 2 and 5 пЂ« 3 2 . 19. If пЂЁ пЂ© пЂЁ пЂ© 1 and вЂ“ 2 are respectively product and sum of the zeroes of a quadratic 5 polynomial. Find the polynomial. 11 [X вЂ“ Maths] 20. If one of the zero of the polynomial g(x) = (k2 + 4) x2 + 13x + 4k is recoprocal of the other, find k. 21. If пЃЎпЂ¬пЂ пЃў be the zeroes of the quadratic polynomial 2 вЂ“ 3x вЂ“ x2 then find the value of пЃЎ + пЃў (1 + пЃЎ). 22. Form a quadratic polynomial, one of whose zero is пЂЁ 2 пЂ« sum of zeroes is 4. 23. If sum of the zeroes of kx2 + 3k + 2x is equal to their product. Find k. 24. If one zero of 4x2 вЂ“ 9 вЂ“ 8kx is negative of the other, find k. 5 пЂ© and the LONG ANSWER TYPE QUESTIONS 25. Find the quadratic polynomial whose zeroes are 2 and вЂ“3. Verify the relation between the coefficients and the zeroes of the polynomial. 26. If one zero of the polynomial (a2 + a) x2 + 13x + 6a is reciprocal of the other, find value (s) of a. 27. вЂ“5 is one of the zeroes of 2x 2 + px вЂ“ 15. Quadratic polynomial p(x2 + x) + k has both the zeroes equal to each other. Then find k. 28. What should be subtracted from the polynomial 2x3 + 5x2 вЂ“ 14x + 10 so that the resultant polynomial is a multiple of (2x вЂ“ 3). 29. If f(x) = 2x4 вЂ“ 5x3 + x2 + 3x вЂ“ 2 is divided by g(x), the quotient is q(x) = 2x2 вЂ“ 5x + 3 and r(x) = вЂ“ 2x + 1 find g(x). 30. If (x вЂ“ 2) is one of the factors of x3 вЂ“ 3x2 вЂ“ 4x + 12, find the other zeroes. 31. If пЃЎ and пЃў are the zeroes of the polynomial x2 вЂ“ 5x + k such that пЃЎ вЂ“ пЃў = 1, find the value of k. 32. If пЃЎпЂ¬пЂ пЃў are zeroes of quadratic polynomial 2x2 + 5x + k, find the value of 21 k, such that (пЃЎпЂ пЂ«пЂ пЃў)2 вЂ“ пЃЎпЃў = . 4 33. Obtain all zeroes of x4 вЂ“ x3 вЂ“7x2 + x + 6 if 3 and 1 are zeroes. 34. If the two zeroes of the polynomial 2x4 вЂ“ 2x3 вЂ“ px2 + qx вЂ“ 6 are вЂ“1 and 2, find p and q. 12 [X вЂ“ Maths] 35. If пЂЁ2 пЂ« 3 пЂ© and пЂЁ 2 пЂ 3 пЂ© are two zeroes of x4 вЂ“ 4x3 вЂ“ 8x2 + 36x вЂ“ 9 find the other two zeroes. 36. What must be subtracted from 8x4 + 14x3 вЂ“ 2x2 + 7x вЂ“ 8 so that the resulting polynomial is exactly divisible by 4x2 + 3x вЂ“ 2. 37. When we add p(x) to 4x4 + 2x3 вЂ“ 2x2 + x вЂ“ 1 the resulting polynomial is divisible by x2 + 2x вЂ“ 3 find p(x). 38. Find a and f if (x4 + x3 + 8x2 + ax + f) is a multiple of (x2 + 1). 39. If the polynomial 6x4 + 8x3 + 17x2 + 21x + 7 is divided by 3x2 + 1 + 4x then r(x) = (ax + b) find a and b. 40. пѓ¦ Obtain all the zeroes of 2x4 вЂ“ 2x3 вЂ“ 7x2 + 3x + 6 if пѓ§ x п‚± пѓ§ пѓЁ 3 2 пѓ¶ пѓ·пѓ· are two пѓё factors of this polynomial. 41. If (x3 вЂ“ 3x + 1) is one of the factors of the polynomial x5 вЂ“ 4x3 + x2 + 3x вЂ“ 1, find the other two factors. 42. What does the graph of the polynomial ax2 + bx + c represents. What type of graph will it represent (i) for a > 0, (ii) for a < 0. What happens if a = 0? ANSWERS 1. b 2. a 3. a 4. b 5. a 6. c 7. b 8. c 9. a 10. d 11. a 12. a 13. a 14. 13 1 1 7 пЂ« пЂЅ пЃЎ пЃў 3 [X вЂ“ Maths] 15. 1 16. k = 7 17. p = 2 18. x2 вЂ“ 10x + 7 19. x 21. k = вЂ“ 5 23. пЂ 25. x2 + x вЂ“ 6 27. p пЂЅ 7, k пЂЅ 29. g(x) = x2 вЂ“ 1 30. вЂ“2, 3 31. k = 6 32. k = 2 33. вЂ“2, вЂ“1 34. 1, вЂ“3 35. В± 3 36. 14x вЂ“ 10 37. 61x вЂ“ 65 38. r(x) = 0 2 пЂ« 2x пЂ« 1 5 20. 2 22. x2 вЂ“ 4x вЂ“ 1 2 3 24. 0 26. 5 7 4 28. 7 пѓћ пЂЁ a пЂ 1пЂ© x пЂ« пЂЁ f пЂ 7 пЂ© пЂЅ 0 пѓ№ пѓє пѓћ a пЂЅ 1 and f пЂЅ 7 пѓ» 39. r (x) = x + 2 = ax + b пѓћ a = 1 and b = 2 40. 2, пЂ1 п‚± 41. (x вЂ“ 1), (x + 1) 42. A curve (parabola), upward parabola, downward parabola, straight line. 3 2 14 [X вЂ“ Maths] CHAPTER 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES KEY POINTS 1. The most general form of a pair of linear equations is : a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 Where a1, a2, b1, b2, c1, c2 are real numbers and a12 + b12 п‚№ 0, a22 + b22 п‚№ 0. 2. 3. The graph of a pair of linear equations in two variables is represented by two lines; (i) If the lines intersect at a point, the pair of equations is consistent. The point of intersection gives the unique solution of the equation. (ii) If the lines coincide, then there are infinitely many solutions. The pair of equations is consistent. Each point on the line will be a solution. (iii) If the lines are parallel, the pair of the linear equations has no solution. The pair of linear equations is inconsistent. If a pair of linear equations is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 (i) (ii) a1 b п‚№ 1 пѓћ the pair of linear equations is consistent. (Unique a2 b2 solution). a1 b c пЂЅ 1 п‚№ 1 пѓћ the pair of linear equations is inconsistent a2 b2 c2 (No solution). 15 [X вЂ“ Maths] (iii) a1 b c пЂЅ 1 пЂЅ 1 пѓћ the pair of linear equations is dependent and a2 b2 c2 consistent (infinitely many solutions). MULTIPLE CHOICE QUESTIONS 1. 2. 3. 4. 5. 6. Every linear equation in two variables has ___ solution(s). (a) no (b) one (c) two (d) infinitely many a1 b c пЂЅ 1 пЂЅ 1 is the condition for a2 b2 c2 (a) intersecting lines (b) parallel lines (c) coincident lines (d) none For a pair to be consistent and dependent the pair must have (a) no solution (b) unique solution (c) infinitely many solutions (d) none of these Graph of every linear equation in two variables represent a ___ (a) point (b) straight line (c) curve (d) triangle Each point on the graph of pair of two lines is a common solution of the lines in case of ___ (a) Infinitely many solutions (b) only one solution (c) no solution (d) none of these The pair of linear equations x = y and x + y = 0 has (a) no common solution (b) infinitely many solutions (c) unique solution (d) none 16 [X вЂ“ Maths] 7. 8. 9. 10. 11. 12. 13. One of the common solution of ax + by = c and y-axis is _____ (a) пѓ¦ cпѓ¶ пѓ§ 0, b пѓ· пѓЁ пѓё (b) пѓ¦ bпѓ¶ пѓ§ 0, c пѓ· пѓЁ пѓё (c) пѓ¦c пѓ¶ пѓ§ b , 0пѓ· пѓЁ пѓё (d) cпѓ¶ пѓ¦ пѓ§ 0, пЂ b пѓ· пѓЁ пѓё For x = 2 in 2x вЂ“ 8y = 12 the value of y will be (a) вЂ“1 (b) 1 (c) 0 (d) 2 The pair of linear equations is said to be inconsistent if they have (a) only one solution (b) no solution (c) infinitely many solutions. (d) both a and c On representing x = a and y = b graphically we get ____ (a) parallel lines (b) coincident lines (c) intersecting lines at (a, b) (d) intersecting lines at (b, a) A motor cyclist is moving along the line x вЂ“ y = 2 and another motor cyclist is moving along the line x + y = 2. Tell whether the y (a) move parallel (b) move coincidently (c) collide somewhere (d) none For 2x + 3y = 4, y can be written in terms of x asвЂ” (a) y пЂЅ 4 пЂ« 2x 3 (b) y пЂЅ 4 пЂ 3x 2 (c) x пЂЅ 4 пЂ 3y 2 (d) y пЂЅ 4 пЂ 2x 3 For what value of p, the pair of linear equations 2x + py = 8 and x + y = 6 has a unique solution x = 10, y = вЂ“4. (a) вЂ“3 (b) 3 (c) 2 (d) 6 17 [X вЂ“ Maths] 14. 15. 16. The point of intersection of the lines x вЂ“ 2y = 6 and y-axis is (a) (вЂ“3, 0) (b) (0, 6) (c) (6, 0) (d) (0, вЂ“3) Graphically x вЂ“ 2 = 0 represents a line (a) parallel to x-axis at a distance 2 units from x-axis. (b) parallel to y-axis at a distance 2 units from it. (c) parallel to x-axis at a distance 2 units from y-axis. (d) parallel to y-axis at a distance 2 units from x-axis. If ax + by = c and lx + my = n has unique solution then the relation between the coefficients will be ____ (a) am п‚№ lb (b) am = lb (c) ab = lm (d) ab п‚№ lm SHORT ANSWER TYPE QUESTIONS 17. Form a pair of linear equations for : If twice the sonвЂ™s age is added to fatherвЂ™s age, the sum is 70. If twice the fatherвЂ™s age is added to the sonвЂ™s age the sum is 95. 18. Amar gives 9000 to some athletes of a school as scholarship every month. Had there been 20 more athletes each would have got 160 less. Form a pair of linear equations for this. 19. Write a pair of linear equations which has the unique solution x = вЂ“1 and y = 4. How many such pairs are there? 20. What is the value of a for which (3, a) lies on 2x вЂ“ 3y = 5. 21. Find the number of solutions for the following pair of linear equations x + 2y вЂ“ 8 = 0 22. 2x + 4y = 16 Dinesh is walking along the line joining (1, 4) and (0, 6), Naresh is walking along the line joining (3, 4,) and (1,0). Represent on graph and find the point where both of them cross each other. 18 [X вЂ“ Maths] 23. Solve the pair of linear equations x вЂ“ y = 2 and x + y = 2. Also find p if p = 2x + 3 24. Check graphically whether the pair of linear equations 3x + 5y = 15, x вЂ“ y = 5 is consistent. Also check whether the pair is dependent. 25. For what value of p the pair of linear equations (p + 2) x вЂ“ (2 p + 1)y = 3 (2p вЂ“ 1) 2x вЂ“ 3y = 7 has unique solution. 26. Find the value of K so that the pair of linear equations : (3 K + 1) x + 3y вЂ“ 2 = 0 (K2 + 1) x + (kвЂ“2)y вЂ“ 5 = 0 is inconsistent. 27. Given the linear equation x + 3y = 4, write another linear equation in two variables such that the geometrical representation of the pair so formed is (i) intersecting lines (ii) parallel lines (iii) coincident lines. 28. Solve x вЂ“ y = 4, x + y = 10 and hence find the value of p when y = 3 x вЂ“p 29. Determine the values of a and b for which the given system of linear equations has infinitely many solutions: 2x + 3y = 7 a(x + y) вЂ“ b(x вЂ“ y) = 3a + b вЂ“ 2 30. The difference of two numbers is 5 and the difference of their reciprocals is 31. 1 . Find the numbers 10 Solve for x and y : пЂЁ x пЂ« 1пЂ© пЂЁ y пЂ 1пЂ© пЂЁ x пЂ 1пЂ© пЂЁ y пЂ« 1пЂ© пЂ« пЂЅ 8; пЂ« пЂЅ 9 2 3 3 19 2 [X вЂ“ Maths] 32. Solve for x and y : 2x + 3y = 17 2x + 2 вЂ“ 3y+1 = 5. 33. Solve for x and y 139x пЂ« 56 y пЂЅ 641 56x пЂ« 139 y пЂЅ 724 34. Solve for x and y 5 1 пЂ« пЂЅ 2 x пЂ« y x пЂ y 15 5 пЂ пЂЅ пЂ2 x пЂ« y x пЂ y 35. Solve the following system of equations graphically x + 2y = 5, 2x вЂ“ 3y = вЂ“ 4. Write the coordinates of the point where perpendicular from common пѓ¦ 1 пѓ¶ solution meets x-axis. Does the point пѓ§пѓЁ вЂ“ , 1пѓ·пѓё lie on any of the lines? 2 36. Draw the graph of the following pair of linear equations x + 3y = 6 and 2x вЂ“ 3y = 12 Find the ratio of the areas of the two triangles formed by first line, x = 0, y = 0 & second line x = 0, y = 0. LONG ANSWER TYPE QUESTIONS 37. Solve for x and y 1 12 1 пЂ« пЂЅ 2 2 пЂЁ 2x пЂ« 3 y пЂ© 7 пЂЁ 3x пЂ 2y пЂ© 7 4 пЂ« пЂЅ 2 for 2x + 3y п‚№ 0 and 3x вЂ“ 2y п‚№ 0 пЂЁ 2x пЂ« 3 y пЂ© пЂЁ 3 x пЂ 2 y пЂ© 20 [X вЂ“ Maths] 38. Solve for p and q p пЂ«q p пЂq пЂЅ 2, пЂЅ 6, p п‚№ 0, q п‚№ 0. pq pq 39. 40. On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeer gains Rs. 2000. But if he sells the T.V. at 10% gain & the fridge at 5% loss, he gains Rs. 1500 on the transaction. Find the actual price of the T.V. and the fridge. 2 x пЂ« 3 y пЂЅ 2, 4 x пЂ 9 y пЂЅ пЂ1 ; x п‚№ 0, y п‚№ 0 41. If from twice the greater of two numbers, 20 is subtracted, the result is the other number. If from twice the smaller number, 5 is subtracted, the result is the greater number. Find the numbers. 42. In a deer park the number of heads and the number of legs of deer and visitors were counted and it was found that there were 39 heads and 132 legs. Find the number of deers and visitors in the park, using graphical method. 43. A two digit number is obtained by either multiplying the sum of the digits by 8 and adding 1; or by multiplying the difference of the digits by 13 and adding 2. Find the number. How many such nos. are there. 1 In an examination one mark is awarded for every correct answer and 4 mark is deducted for every wrong answer. A student answered 120 questions and got 90 marks. How many questions did he answer correctly? 44. 45. A boatman rows his boat 32 km upstream and 36 km down stream in 7 hours. He can row 40 km upstream and 48 km downstream in 9 hours. Find the speed of the stream and that of the boat in still water. 46. In a function if 10 guests are sent from room A to B, the no. of guests in room A and B are same. If 20 guests are sent from B to A, the no. of guests in A is double the no. of guests in B. Find no. of guests in both the rooms in the beginning. 47. In a function Madhu wished to give Rs. 21 to each person present and found that she fell short of Rs. 4 so she distributed Rs. 20 to each and 21 [X вЂ“ Maths] found that Rs. 1 were left over. How much money did she gave and how many persons were there. 48. A mobile company charges a fixed amount as monthly rental which includes 100 minutes free per month and charges a fixed amount there after for every additional minute. Abhishek paid Rs. 433 for 370 minutes and Ashish paid Rs. 398 for 300 minutes. Find the bill amount under the same plan, if Usha use for 400 minutes. ANSWERS 1. d 2. c 3. c 4. b 5. a 6. c 7. a 8. a 9. b 10. c 11. c 12. d 13. b 14. d 15. b 16. a 17. FatherвЂ™s age x years, SonвЂ™s age = y years x + 2y = 70, 2x + y = 95 18. No. of athletes = x, No. of athletes increased = y 19. Infinite 20. 21. Infinite 22. (2, 2) 23. (2, 0) P = 7 24. No 25. p п‚№ 4 26. k пЂЅ пЂ1, k п‚№ 28. (7, 3), 18 29. a = 5, b = 1 30. вЂ“5, вЂ“10 or 10, 5 31. (7, 13) 1 3 19 2 22 [X вЂ“ Maths] 32. (3, 2) 33. (2, вЂ“1) 34. (3, 2) 35. (1, 2), (1, 0), yes 36. (6, 0), 1 : 2 37. (2, 1) 38. пѓ¦ 1 1пѓ¶ пѓ§пЂ 2, 4пѓ· пѓЁ пѓё 39. 40. (4, 9) 41. 15, 10 42. 27, 12 43. 41 or 14(2) 44. 96 45. 2 km/hr, 10km/hr. 46. 100, 80 47. Rs. 101, 5 48. 1пѓ№ пѓ© пѓЄ Rs. 298, Rs. 2 пѓє Rs. 448 пѓ« пѓ» 23 20000, 100000 [X вЂ“ Maths] CHAPTER 4 SIMILAR TRIANGLES KEY POINTS 1. Similar Triangles : Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are proportional. 2. Criteria for Similarity : in пЃ„ABC and пЃ„DEF (i) AAA Similarity : пЃ„ABC ~ пЃ„DEF when пѓђA = пѓђD, пѓђB = пѓђE and пѓђC = пѓђF (ii) SAS Similarity : пЃ„ABC ~ пЃ„DEF when (iii) 3. AB AC пЂЅ and пѓђB пЂЅ пѓђE DE DF SSS Similarity :пЂ пЃ„ABC ~ пЃ„DEF , AB AC BC пЂЅ пЂЅ . DE DF EF The proof of the following theorems can be asked in the examination : (i) Basic Proportionality Theorem : If a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio. (ii) The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. (iii) Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 24 [X вЂ“ Maths] (iv) Converse of Pythagoras Theorem : In a triangle, if the square of one side is equal to the sum of the squares of the other two sides then the angle opposite to the first side is a right angle. MULTIPLE CHOICE QUESTIONS 1. 2. пЃ„ABC ~ пЃ„DEF. If DE = 2 AB and BC = 3cm then EF is equal to _______. (a) 1.5 cm (b) 3 cm (c) 6 cm (d) 9 cm In пЃ„DEW, AB || EW if AD = 4 cm, DE = 12cm and DW = 24 cm then the value of DB = ____ (a) 4 cm (b) 8 cm (c) 12 cm (d) 16 cm 3. A D Q Q c e O O B f b a C E d F In the figure the value of cd = ________ 4. (a) ae (b) af (c) bf (d) be If the corresponding medians of two similar triangles are in the ratio 5 : 7, then the ratio of their sides is : (a) 25 : 49 (b) 7 : 5 (c) 1 : 1 (d) 5 : 7 25 [X вЂ“ Maths] 5. 6. The area of two isosceles triangles are in the ratio 16 : 25. The ratio of their corresponding heights isвЂ” (a) 5 : 4 (b) 3 : 2 (c) 4 : 5 (d) 5 : 7 In the figure, пЃ„ABC is similar to ______ 16 cm B A 53В° 53В° cm 36 cm 24 C D 7. (a) пЃ„BDC (b) пЃ„DBC (c) пЃ„CDB (d) пЃ„CBD пЃ„AMB ~ пЃ„CMD. Also 2ar (пЃ„AMB) = ar (пЃ„CMD) the length of MD is (a) (c) 8. (b) 2 MB 2 MB If пЃ„ABC ~ пЃ„QRP and (d) ar ar 2 MD 2 MD пЂЁ пЃ„ABC пЂ© 9 пЂЅ , пЂЁ пЃ„PQR пЂ© 4 AB = 18 cm, BC = 15 cm what will be PR. (a) 10 cm (b) 9 cm (c) 4 cm (d) 18 cm 26 [X вЂ“ Maths] 9. 10. 11. In пЃ„ABC, D and E are points on side AB and AC respectively such that DE || BC and AD : DB = 3 : 1. If EA = 3.3 cm then AC = (a) 1.1 cm (b) 4.4 cm (c) 4 cm (d) 5.5 cm ABC and BDE are two equilateral triangles such that D is the midpoint of BC. Ratio of the areas of triangles ABC and BDE isвЂ” (a) 2 : 1 (b) 1 : 2 (c) 4 : 1 (d) 1 : 4 In пЃ„ABC, DE || BC. In the figure, the value of x is ______ A x x+3 D E x+1 x+5 B 12. C (a) 1 (b) вЂ“1 (c) 3 (d) вЂ“3 In пЃ„ABC, пѓђB = 90В°, BE is the perpendicular bisector of AC then ar пЂЁ пЃ„BEC пЂ© пЂЅ _______ ar пЂЁ пЃ„ABC пЂ© 13. (a) 1 2 (b) 2 1 (c) 4 1 (d) 1 4 The altitude of an equilateral triangle, having the length of its side 12cmis 27 [X вЂ“ Maths] 14. 15. (a) 12 cm (b) 6 2 cm (c) 6 cm (d) 6 3 cm A right angled triangle has its area numerically equal to its perimeter. The length of each side is an even number & the hypotenuse is 10 cm what is the perimeter. (a) 26 (b) 24 (c) 30 (d) 16 If in an isosceles right-angled triangle the length of the hypotenuse is 10 cm then the perimeter of the triangle is (a) 5 2 cm (c) 10 пЂЁ пЂ© 2 пЂ« 1 cm (b) 2 5 cm (d) 10 пЂЁ пЂ© 2 пЂ 1 cm SHORT ANSWER TYPE QUESTIONS 16. In figure пЃ„ABC ~ пЃ„APQ. If BC = 8 cm, PQ = 4cm AC = 6.5 cm, AP = 2.8 cm, find AB and AQ. B P A Q C 1 1 CD . Prove that CA2 = AB2 + BC 2 3 2 17. In пЃ„ABC, ADпЃћBC and BD пЂЅ 18. In the figure name the similar triangle. 28 [X вЂ“ Maths] 18 cm 10 cm 15 cm A P 47В° Q cm 12 47В° B C 19. An isosecles triangle ABC is similar to triangle PQR. AC = AB = 4 cm, RQ = 10 cm and BC = 6 cm. What is the length of PR? What type of triangle is пЃ„PQR ? 20. In the adjoining figure, find AE. A E 8 cm 4 cm 6 cm B 21. 3 cm C In пЃ„PQR, DE || QR and DE пЂЅ D ar пЂЁ пЃ„PQR пЂ© 1 QR. Find . 4 ar пЂЁ пЃ„PDE пЂ© P D E Q R 29 [X вЂ“ Maths] 22. is the value of 23. AB BC 1 пЂЅ пЂЅ then what PQ QR 2 In triangles ABC and PQR if пѓђB = пѓђQ and PR ? AC ABC is a right angled пЃ„ at B, AD and CE are two medians drawn from A and C respectively. If AC = 5 cm and AD пЂЅ 24. 3 5 cm. Find CE. 2 In the adjoining figure DE || BC. What is the value of DE. A 10 cm D E 2c m B C 3 cm LONG ANSWER TYPE QUESTIONS 25. In the figure, find SR if пѓђQPR = пѓђPSR. PR = 6 cm and QR = 9 cm P 6 Q S cm R 9 cm 26. In the following figure, DE || AC and 30 BE BC пЂЅ . Prove that DC || AP EC CP [X вЂ“ Maths] A D B E P C 27. Two similar triangles ABC and PBC are made on opposite sides of the same base BC. Prove that AB = BP. 28. In a quadrilateral ABCD, пѓђB = 90В°, AD2 = AB2 + BC2 + CD2. Prove that пѓђACD = 90В°. D C B A 29. In figure DE || BC, DE = 3 cm, BC = 9 cm and ar (пЃ„ADE) = 30 cm2. Find ar (trap. BCED). A D E 3 cm B C 9 cm 30. Amit is standing at a point on the ground 8m away from a house. A mobile network tower is fixed on the roof of the house. If the top and 31 [X вЂ“ Maths] bottom of the tower are 17m and 10m away from the point. Find the heights of the tower and house. BC пЂЅ AB In a right angled triangle ABC, right angle at B , 32. In a right angled triangle PRO, PR is the hypotenuse and the other two sides are of length 6cm and 8cm. Q is a point outside the triangle such that PQ = 24cm RQ = 26cm. What is the measure of пѓђQPR? 33. In пЃ„ABC, P and Q are points on AB and AC respectively such that PQ||BC. Prove that the median AD bisects PQ. 34. PQRS is a trapezium. SQ is a diagonal. E and F are two points on parallel sides PQ and RS respectively. Line joining E and F intersects SQ at G. Prove that SG Г— QE = QG Г— SF. 35. Two poles of height a metres and b metres are apart. Prove that the height of the point of intersection of the lines joining the top of each pole ab to the foot of the opposite pole is given by metres. a пЂ« b D 3. Find AB . AC 31. B O bm am h C x L y A 36. Diagonals of a trapezium PQRS intersect each other at the point O, PQ||RS and PQ = 3 RS. Find the ratio of the areas of triangles POQ & ROS. 37. In a rhombus, prove that four times the square of any sides is equal to the sum of squares of its diagonals. 38. ABCD is a trapezium with AE || DC. If ABD is similar to пЃ„BEC. Prove that AD = BC. 32 [X вЂ“ Maths] 39. In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite to the first side is a right triangle. 40. If BL and CM are medians of a triangle ABC right angled at A. Prove that 4 (BL2 + CM2) = 5 BC2. 41. ABCD is a rectangle in which length is double of its breadth. Two equilateral triangles are drawn one each on length and breadth of rectangle. Find the ratio of their areas. 42. Amar and Ashok are two friends standing at a corner A of a rectangular garden. They wanted to drink water. Amar goes due north at a speed of 50m/min and Ashok due west at a speed of 60m/min. They travel for 5 minutes. Amar reaches the tap and drink water. How far (minimum distance) is Ashok from the tap now. C B A 43. If two triangles are equiangular, prove that the ratio of the corresponding sides is same as the ratio of the corresponding altitudes. 44. In figure, if AD пЃћ BC and BD DA пЂЅ , prove that пЃ„ABC is a right triangle. DA DC A B D 33 C [X вЂ“ Maths] 45. In figure DE || BC and AD : DB = 5 : 4. Find ar пЃ„ DEF ar пЃ„ CFB A E D F C B ANSWERS 1. c 2. b 3. a 4. d 5. c 6. d 7. a 8. a 9. b 10. c 11. c 12. d 13. d 14. b 15. c 16. AB = 5.6 cm, AQ = 3.25 cm 18. пЃ„APQ ~ пЃ„ABC 19. 20 cm , isosceles triangle 3 20. 5пѓ–5 cm 21. 16 : 1 22. 1 2 23. 1 пЂЁ 25 вЂ“ 6 5 пЂ© cm 4 34 [X вЂ“ Maths] 24. 2.5 cm 25. 4 cm 29. 240 cm2 30. 9m, 6m 31. 1 2 32. 90В° 36. 9 : 1 41. 4 : 1 42. 50 61 m 43. 5 10 cm 45. 25 81 35 [X вЂ“ Maths] CHAPTER 5 TRIGONOMETRY KEY POINTS 1. Trigonometric Ratios : In пЃ„ABC, пѓђB = 90В° for angle вЂ�AвЂ™ Perpendicular Hypotenuse sin A пЂЅ Base Hypotenuse cos A пЂЅ tan A пЂЅ Perpendicular Base cot A пЂЅ Base Perpendicular sec A пЂЅ cosec A пЂЅ 2. Hypotenuse Base Hypotenuse Perpendicular Reciprocal Relations : sin пЃ± пЂЅ 1 cosec пЃ± , cosec пЃ± пЂЅ 1 sin пЃ± cos пЃ± пЂЅ 1 sec пЃ± , sec пЃ± пЂЅ 1 cos пЃ± 36 [X вЂ“ Maths] tan пЃ± пЂЅ 3. 1 tan пЃ± , cot пЃ± пЂЅ , cot пЃ± пЂЅ Quotient Relations : tan пЃ± пЂЅ 4. 1 cot пЃ± sin пЃ± cos пЃ± cos пЃ± sin пЃ± Indentities : sin2 пЃ± + cos2 пЃ± = 1 пѓћ sin2 пЃ± = 1 вЂ“ cos2 пЃ± and cos2 пЃ± = 1 вЂ“ sin2 пЃ± 1 + tan2 пЃ± = sec2 пЃ± пѓћ tan2 пЃ± = sec2 пЃ± вЂ“ 1 and sec2 пЃ± вЂ“ tan2 пЃ± = 1 1 + cot2 пЃ± = cosec2 пЃ± пѓћ cot2 пЃ± = cosec2 пЃ± вЂ“ 1 and cosec2 пЃ± вЂ“ cot2 пЃ± = 1 5. Trignometric Ratios of Some Specific Angles : пѓђA 0В° 30В° 45В° 60В° 90В° sin A 0 1 2 1 3 2 1 cos A 1 3 2 1 1 2 0 tan A 0 cosec A Not defined sec A 1 cot A Not defined 2 1 3 2 1 3 2 2 2 3 2 2 2 3 3 37 1 1 3 Not defined 1 Not defined 0 [X вЂ“ Maths] 6. Trigonometric Ratios of Complementary Angles sin (90В° вЂ“ пЃ±) = cos пЃ± cos (90В° вЂ“ пЃ±) = sin пЃ± tan (90В° вЂ“ пЃ±) = cot пЃ± cot (90В° вЂ“ пЃ±) = tan пЃ± sec (90В° вЂ“ пЃ±) = cosec пЃ± cosec (90В° вЂ“ пЃ±) = sec пЃ± MULTIPLE CHOICE QUESTIONS Note : In the following questions 0В° п‚ЈпЂ пЃ± п‚Ј 90В° 1. 2. 3. 4. If x = a sin пЃ± and y = a cos пЃ± then the value of x2 + y2 is _______ (a) a (b) a2 (c) 1 (d) 1 a The value of cosec 70В° вЂ“ sec 20В° is _____ (a) 0 (b) 1 (c) 70В° (d) 20В° If 3 sec пЃ± вЂ“ 5 = 0 then cot пЃ± = _____ (a) 5 3 (b) 4 5 (c) 3 4 (d) 3 5 (b) 1 (d) 2 2 If пЃ± = 45В° then sec пЃ± cot пЃ± вЂ“ cosec пЃ± tan пЃ± is (a) (c) 0 2 38 [X вЂ“ Maths] 5. 6. 7. 8. If sin (90 вЂ“ пЃ±) cos пЃ±пЂ = 1 and пЃ±пЂ is an acute angle then пЃ±пЂ = ____ (a) 90В° (b) 60В° (c) 30В° (d) 0В° The value of (1 + cos пЃ±) (1 вЂ“ cos пЃ±) cosec2пЃ±пЂ = _____ (a) 0 (b) 1 (c) cos2 пЃ± (d) sin2 пЃ± пЃ„TRY is a right-angled isosceles triangle then cos T + cos R + cos Y is _____ (a) 2 (c) 1пЂ« (c) 10. 2 2 (d) 1пЂ« 1 2 If sec пЃ±пЂ + tan пЃ± = x, then sec пЃ±пЂ = (a) 9. 2 (b) x 2 пЂ«1 (b) пЂ1 2x (d) x x 2 x x 2 пЂ«1 2x 2 пЂ1 x пѓ¦пЃ° пѓ¶ пѓ¦пЃ° пѓ¶ The value of cot пЃ± пЂ sin пѓ§ пЂ пЃ± пѓ· cos пѓ§ пЂ пЃ± пѓ· is _______ 2 2 пѓЁ пѓё пѓЁ пѓё (a) cot пЃ± cos2 пЃ± (b) cot2пЃ± (c) cos2 пЃ± (d) tan2 пЃ± If sin пЃ±пЂ вЂ“ cos пЃ±пЂ = 0, 0 п‚ЈпЂ пЃ±пЂ п‚ЈпЂ 90В° then the value of пЃ±пЂ is _____ (a) cos пЃ± (b) 45В° (c) 90В° (d) sin пЃ± 39 [X вЂ“ Maths] sin пЃ± 11. (a) sin пЃ± cos пЃ± (d) tan пЃ± (a) sec2пЃ±пЂ + tan2пЃ± (b) sec пЃ±пЂ вЂ“ tan пЃ± (c) sec2пЃ± вЂ“ tan2пЃ± (d) sec пЃ±пЂ + tan пЃ± In an isosceles right-angled пЃ„ABC, пѓђB = 90В°. The value of 2 sin A cos A is _____ (a) 1 (b) 1 (c) If 2 sin 20п‚° пЂ« sin 70п‚° пЂЁ 1 2 (d) 2 2 2 2 2 cos 69п‚° пЂ« cos 21 п‚° 15. sin пЃ± 1 пЂ« sin пЃ± is equal to 1 пЂ sin пЃ± 12. 14. (b) cot пЃ± (c) 13. can be written as 2 1 пЂ sin пЃ± пЂЅ пЂ© 2 sec 60п‚° then K is ______ K (a) 1 (b) 2 (c) 3 (d) 4 If tan пЃ± пЂЅ 1 7 2 , then 2 cosc пЃ± пЂ sec пЃ± 2 2 пЂЅ cosec пЃ± пЂ« sec пЃ± (a) 3 4 (b) 5 7 (c) 3 7 (d) 1 12 40 [X вЂ“ Maths] SHORT ANSWER TYPE QUESTIONS 3 , write the value of cos P. 5 16. In пЃ„PQR, пѓђQ = 90В° and sin R пЂЅ 17. If A and B are acute angles and sin A = cos B then write the value of A + B. 18. If 4 cot пЃ± = 3 then write the value of tan пЃ± + cot пЃ±пЂ® 19. Write the value of cot2 30В° + sec2 45В°. 20. Given that 16 cot A = 12, find the value of 21. If пЃ±пЂ = 30В° then write the value of sin пЃ±пЂ + cos2 пЃ±. 22. If 1 пЂ tan пЃ± пЂЅ 23. Find the value of пЃ±пЂ¬ if 24. If пЃ± and пЃ¦ are complementary angles then what is the value of 2 sin A пЂ« cos A пЂ® sin A пЂ cos A 2 then what is the value of пЃ±. 3 3 tan 2пЃ± пЂ 3 пЂЅ 0. cosec пЃ± sec пЃ¦ вЂ“ cot пЃ± tan пЃ¦ 25. If tan (3x вЂ“ 15В°) = 1 then what is the value of x. 26. If sin 5пЃ± = cos 4пЃ±, where 5пЃ± and 4пЃ± are acute angles. Find the value of пЃ±пЂ® LONG ANSWER TYPE QUESTIONS 27. Simplify : tan2 60В° + 4 cos2 45В° + 3 (sec2 30В° + cos2 90В°) 28. Evaluate пѓ¦ cos 58п‚° пѓ¶ 2пѓЁ пЂ sin 32п‚° пѓё 29. пѓ¦ cos 38п‚°cosec 52п‚° пѓ¶ 3пѓЁ tan 15п‚° tan 60п‚° tan 75п‚° пѓё Prove that cosec4 пЃ± вЂ“ cosec2 пЃ± = cot2 пЃ± + cot4 пЃ±. 41 [X вЂ“ Maths] 30. If sin пЃ± + sin2 пЃ± = 1 then find the value of cos2 пЃ± + cos4 пЃ±пЂ® 31. If sin 2пЃ± = cos (пЃ± вЂ“ 36В°), 2пЃ± and пЃ± вЂ“ 26В° are acute angles then find the value of пЃ±. 32. If sin (3 x + 2y) = 1 and cos пЂЁ 3x пЂ 2y пЂ© пЂЅ 3 , where 0 п‚Ј (3x + 2y) п‚ЈпЂ 90В° 2 then find the value of x and y. 33. If sin (A + B) = sin A cos B + cos A sin B then find the value of (a) sin 75В° (b) cos 15В° cos A cos A пЂ« пЂЅ cos A, A п‚№ 45п‚°. 1 пЂ tan A 1 пЂ cot A 34. Prove that 35. Prove that 36. Find the value of sec пЃ± пЂ 1 пЂ« sec пЃ± пЂ« 1 sec пЃ± пЂ« 1 пЂЅ 2cosec пЃ± sec пЃ± пЂ 1 sin2 5В° + sin2 10В° + sin2 15В° + .... + sin2 85В° 37. Prove that tan пЃ± пЂ« sec пЃ± пЂ 1 cos пЃ± пЂЅ . tan пЃ± пЂ sec пЃ± пЂ« 1 1 пЂ sin пЃ± 38. If 2 sin пЂЁ 3 x пЂ 15 пЂ© пЂЅ sin 2 пЂЁ 2x 3 then find the value of пЂ« 10 пЂ© пЂ« tan 2 пЂЁx пЂ« 5пЂ©. 39. Find the value of sin 60В° geometrically. 40. 1 Let p = tan пЃ± + sec пЃ± then find the value of p пЂ« p . 41. Find the value of 2 2 пЂ tan пЃ± cot пЂЁ 90п‚° пЂ пЃ± пЂ© пЂ« sec пЃ± cosec пЂЁ 90 п‚° пЂ пЃ± пЂ© пЂ« sin 35 п‚° пЂ« sin 55 п‚° tan 10п‚° tan 20 п‚° tan 30 п‚° tan 70 п‚° tan 80 п‚° 42 [X вЂ“ Maths] cos пЃЎ cos пЃЎ пЂЅ m and пЂЅ n show that (m2 + n2) cos2 пЃў = n2. cos пЃў sin пЃў 42. If 43. Prove that cos 1В° cos 2В° cos 3В°.........cos 180В° = 0. 44. sin пЃ± пЂ« cos пЃ± sin пЃ± пЂ cos пЃ± 2 sec пЃ± Prove that sin пЃ± пЂ cos пЃ± пЂ« sin пЃ± пЂ« cos пЃ± пЂЅ . 2 tan пЃ± пЂ 1 45. If A, B, C are the interior angles of a triangle ABC, show that 2 A A пѓ¦B пЂ« C пѓ¶ пѓ¦B пЂ« C пѓ¶ sin пѓ§ cos пЂ« cos пѓ§ sin пЂЅ 1. пѓ· пѓ· 2 2 пѓЁ 2 пѓё пѓЁ 2 пѓё ANSWERS 1. b 2. a 3. c 4. a 5. d 6. b 7. a 8. b 9. a 10. b 11. d 12. d 13. a 14. d 15. a 16. cos P пЂЅ 17. 90В° 18. 25 12 19. 5 20. 7 21. 5 4 22. 30В° 23. 30В° 24. 1 25. x = 20. 26. 10В° 43 3 5 [X вЂ“ Maths] 27. 9 28. 1 30 . 1 31 . 42В° 32 . x = 20, y = 15 33. 3 пЂ«1 2 2 , 3 пЂ«1 2 2 , take A = 45В°, B = 30В° 36. 17 2 38. 13 12 40. 2 sec пЃ± 41. 2 3 44 [X вЂ“ Maths] CHAPTER 6 STATISTICS KEY POINTS 1. The mean for grouped data can be found by : пѓҐ fixi . пѓҐ fi (i) The direct method пЂЅ X пЂЅ (ii) The assumed mean method X пЂЅ a пЂ« пѓҐ fidi , пѓҐ fi where di = xi вЂ“a. (iii) The step deviation method X пЂЅ a пЂ« 2. пѓҐ fiui пѓҐ fi п‚ґ h, where u i пЂЅ xi пЂ a . h The mode for the grouped data can be found by using the formula : f1 пЂ f 0 пѓ© пѓ№ mode пЂЅ l пЂ« пѓЄ пѓє п‚ґh пѓ« 2f 1 пЂ f 0 пЂ f 2 пѓ» l = lower limit of the modal class. f1 = frequency of the modal class. f0 = frequency of the preceding class of the modal class. f2 = frequency of the succeeding class of the modal class. h = size of the class interval. Modal class - class interval with highest frequency. 45 [X вЂ“ Maths] 3. The median for the grouped data can be found by using the formula : n 2 пЂ Cf пѓ№ median пЂЅ l пЂ« пѓ© п‚ґh пѓЄпѓ« пѓєпѓ» f l = lower limit of the median class. n = number of observations. Cf = cumulative frequency of class interval preceding the median class. f = frequency of median class. h = class size. 4. Empirical Formula : Mode = 3 median - 2 mean. 5. Cumulative frequency curve or an Ogive : (i) Ogive is the graphical representation of the cumulative frequency distribution. (ii) Less than type Ogive : (iii) (iv) вЂў Construct a cumulative frequency table. вЂў Mark the upper class limit on the x = axis. More than type Ogive : вЂў Construct a frequency table. вЂў Mark the lower class limit on the x-axis. To obtain the median of frequency distribution from the graph : вЂў Locate point of intersection of less than type Ogive and more than type Ogive : Draw a perpendicular from this point on x-axis. вЂў The point at which it cuts the x-axis gives us the median. 46 [X вЂ“ Maths] MULTIPLE CHOICE QUESTIONS 1. 2. 3. 4. 5. 6. 7. Mean of first 10 natural numbers is (a) 5 (b) 6 (c) 5.5 (d) 6.5 If mean of 4, 6, 8, 10, x, 14, 16 is 10 then the value of вЂ�xвЂ™ is (a) 11 (b) 12 (c) 13 (d) 9 The mean of x, x + 1, x + 2, x + 3, x + 4, x + 5 and x + 6 is (a) x (b) x + 3 (c) x + 4 (d) 3 The median of 2, 3, 2, 5, 6, 9, 10, 12, 16, 18 and 20 is (a) 9 (b) 20 (c) 10 (d) 9.5 The median of 2, 3, 6, 0, 1, 4, 8, 2, 5 is (a) 1 (b) 3 (c) 4 (d) 2 Mode of 1, 0, 2, 2, 3, 1, 4, 5, 1, 0 is (a) 5 (b) 0 (c) 1 (d) 2 If the mode of 2, 3, 5, 4, 2, 6, 3, 5, 5, 2 and x is 2 then the value of вЂ�xвЂ™ is (a) 2 (b) 3 (c) 4 (d) 5 47 [X вЂ“ Maths] 8. The modal class of the following distribution is Class Interval Frequency 9. 10. 11. 12. 13. 10вЂ“15 15вЂ“20 20вЂ“25 25вЂ“30 30вЂ“35 4 7 12 8 2 (a) 30вЂ“35 (b) 20вЂ“25 (c) 25вЂ“30 (d) 15вЂ“20 A teacher ask the students to find the average marks obtained by the class students in Maths, the student will find (a) mean (b) median (c) mode (d) sum The empirical relationship between the three measures of central tendency is (a) 3 mean = mode + 2 median (b) 3 median = mode + 2 mean (c) 3 mode = mean + 2 median (d) median = 3 mode вЂ“ 2 mean Class mark of the class 19.5 вЂ“ 29.5 is (a) 10 (b) 49 (c) 24.5 (d) 25 Measure of central tendency is represented by the abscissa of the point when the point of intersection of вЂ�less than ogiveвЂ™ and вЂ�more than ogiveвЂ™ is (a) mean (b) median (c) mode (d) None of these The median class of the following distribution is Class Interval : Frequency : 0вЂ“10 10вЂ“20 20вЂ“30 30вЂ“40 40вЂ“50 50вЂ“60 60вЂ“70 4 4 8 10 12 8 4 48 [X вЂ“ Maths] 14. 15. 16. 17. 18. 19. 20. (a) 20вЂ“30 (b) 40вЂ“50 (c) 30вЂ“40 (d) 50вЂ“60 The mean of 20 numbers is 17, if 3 is added to each number, then the new mean is (a) 20 (b) 21 (c) 22 (d) 24 The mean of 5 numbers is 18. If one number is excluded then their mean is 16, then the excluded number is (a) 23 (b) 24 (c) 25 (d) 26 The mean of first 5 prime numbers is (a) 5.5 (b) 5.6 (c) 5.7 (d) 5 The sum of deviations of the values 3, 4, 6, 8, 14 from their mean is (a) 0 (b) 1 (c) 2 (d) 3 If median = 15 and mean = 16, then mode is (a) 10 (b) 11 (c) 12 (d) 13 The mean of 11 observations is 50. If the mean of first six observations is 49 and that of last six observations is 52, then the sixth observation is (a) 56 (b) 55 (c) 54 (d) 53 The mean of the following distribution is 2.6, then the value of вЂ�xвЂ™ is Variable 1 2 3 4 5 Frequency 4 5 x 1 2 49 [X вЂ“ Maths] (a) 24 (b) 3 (c) 8 (d) 13 SHORT ANSWER TYPE QUESTIONS 21. The mean of 40 observations was 160. It was detected on rechecking that the value of 165 was wrongly copied as 125 for computing the mean. Find the correct mean. 22. Find вЂ�xвЂ™ if the median of the observations in ascending order 24, 25, 26, x + 2, x + 3, 30, 31, 34 is 27.5. 23. Find the median of the following data. x : 10 12 14 16 18 20 f : 3 5 6 4 4 3 24. Find the value of вЂ�pвЂ™, if mean of the following distribution is 7.5 Variable : 3 5 7 9 11 13 Frequency : 6 8 15 p 8 4 25. Find the mean of the following distribution. x : 12 16 20 24 28 32 f : 5 7 8 5 3 2 26. The Arithmetic Mean of the following frequency distribution is 53. Find the value of P. Class Interval : Frequency : 0вЂ“20 20вЂ“40 40вЂ“60 60вЂ“80 80вЂ“100 12 15 32 P 13 50 [X вЂ“ Maths] 27. From the cumulative frequency table, write the frequency of the class 20вЂ“30. Marks Number of Student Less than 10 1 Less than 20 14 Less then 30 36 Less than 40 59 Less than 50 60 28. Following is a cumulative frequency curve for the marks obtained by 40 students. Find the median marks obtained by the student. 29. The following вЂ�more than ogiveвЂ™ shows the weight of 40 students of a class. What is the lower limit of the median class. 51 [X вЂ“ Maths] LONG ANSWER TYPE QUESTIONS 30. The mean of the following frequency distribution is 62.8 and the sum of all the frequencies is 50. Find the values of x and y. Class Interval : Frequency : 31. 0вЂ“20 20вЂ“40 40вЂ“60 60вЂ“80 80вЂ“100 100вЂ“120 5 x 10 y 7 8 The following frequency distribution gives the daily wages of a worker of a factory. Find mean daily wage of a worker. Daily Wage (in ) Number of Workers More than 300 0 More than 250 12 More than 200 21 More than 150 44 More than 100 53 More than 50 59 More than 0 60 52 [X вЂ“ Maths] 32. The median of the following frequency distribution is 28.5 and sum of all the frequencies is 60. Find the values of x and y. Class Interval : 0вЂ“10 10вЂ“20 20вЂ“30 30вЂ“40 40вЂ“50 50вЂ“60 5 x 20 15 y 5 Frequency : 33. Find the mean, median and mode of the following : Class Interval : Frequency : 34. 35. 0вЂ“10 10вЂ“20 20вЂ“30 30вЂ“40 40вЂ“50 50вЂ“60 60вЂ“70 6 8 10 15 5 4 2 The following frequency distribution shows the marks obtained by 100 students in a school. Find the mode. Marks Number of Students Less than 10 10 Less than 20 15 Less than 30 30 Less than 40 50 Less than 50 72 Less than 60 85 Less than 70 90 Less than 80 95 Less than 90 100 Draw вЂ�less thanвЂ™ and вЂ�more thanвЂ™ ogives for the following distribution Marks : 0вЂ“10 10вЂ“20 20вЂ“30 30вЂ“40 40вЂ“50 50вЂ“60 60вЂ“70 70вЂ“80 80вЂ“90 90вЂ“100 No. of Students : 5 6 8 10 15 9 8 7 7 5 Also find median from graph. 36. A survey regarding the heights (in cm) of 50 students of class x of a school was conducted and the following data was obtained. 53 [X вЂ“ Maths] Height (in cm) : 120вЂ“130 130вЂ“140 140вЂ“150 150вЂ“160 160вЂ“170 Total 2 8 12 20 8 50 No. of Students : Find the mean, median and mode of the above data. 37. The mode of the following distribution is 65. Find the values of x and y, if sum of the frequencies is 50. Class Interval : 0вЂ“20 20вЂ“40 40вЂ“60 60вЂ“80 80вЂ“100 100вЂ“120 120вЂ“140 6 8 x 12 6 y 3 Frequency : 38. During the medical checkup of 35 students of class вЂ�XвЂ™ their weights recorded as follows : Weight (in kg.) : 38вЂ“40 40вЂ“42 42вЂ“44 44вЂ“46 46вЂ“48 48вЂ“50 50вЂ“52 3 2 4 5 14 4 3 Number Students : Find mean, median and mode of the above data. 39. The weekly observations on cost of living index in a city for the year 2008-2009 are given below : Cost of Living Index : 140вЂ“150 150вЂ“160 160вЂ“170 170вЂ“180 180вЂ“190 No. of Weeks : 5 10 20 9 190вЂ“200 Total 2 52 6 Find the mean weekly cost of living index. 40. Calculate the mode from the following table Class Interval : 0вЂ“5 5вЂ“10 10вЂ“15 15вЂ“20 20вЂ“25 25вЂ“30 30вЂ“35 35-40 40-45 Frequency : 20 24 32 28 20 16 34 10 8 54 [X вЂ“ Maths] ANSWERS 1. c 2. b 3. b 4. a 5. b 6. c 7. a 8. b 9. a 10. b 11. c 12. b 13. c 14. a 15. d 16. b 17. a 18. d 19. a 20. c 21. 161 22. x = 25 23. 14.8 24. p = 3 25. 20 26. 28 27. 22 28. 40 29. 147.5 30. x = 8, y = 12 32. x = 8, y = 7 35. 47.3 (Approx) 31. 182.50 33. Mean = 30, Median = 30.67, Mode = 33.33 34. 41.82 36. Mean = 149.8 cm, Median = 151.5 cm, Mode = 154 cm 37. x = 10, y = 5. 38. Mean = 45.8, Median = 46.5, Mode = 47.9 39. 166.3 40. 55 13.33 [X вЂ“ Maths] SAMPLE QUESTION PAPER - I MATHEMATICS, SA - 1 Time allowed : 3 hours Maximum Marks : 90 General Instructions 1. All questions are compulsory. 2. The question paper consists of 34 questions divided into four sections A, B, C and D. Section A comprises of 8 questions of 1 mark each. Section B comprises of 6 questions of 2 marks each. Section C comprises of 10 questions of 3 marks each and Section D comprises of 10 questions of 4 marks each. 3. Question numbers 1 to 8 in Section A are multiple choice questions where you are to select one correct option out of the given four. 4. There is no overall choice. However, internal choices has been provided in 1 question of two marks, 3 questions of three marks each and 2 questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculator is not permitted. SECTION A Question numbers 1 to 8 carry one mark each. For each question, four alternative choices have been provided of which only one is correct. You have to select the correct choice. 1. 2. The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is : (a) 65 (b) 875 (c) 1750 (d) 13 The zeroes of a quadratic polynomial x2+ 99x +127 are : 56 [X вЂ“ Maths] 3. (a) both positive (b) (c) one positive one negative (d) In the given figure XY QR and both negative both equal PX 1 пЂЅ then : XQ 2 P Y X Q 4. 5. 6. 7. R (a) XY = QR (b) XY= 1 QR 3 (c) (XY)2 = (QR)2 (d) XY= 1 QR 2 If sin(60В°+A) = 1, then the value of cosec A is : (a) 1 (b) 0 (c) 1 2 (d) 2. The H.C.F. of the smallest composite number and the smallest prime number is : (a) 0 (b) 1 (c) 8 (d) 2. The value of k for which the pair of equations kx вЂ“ 5y = 2; 6x + 2y = 7 has no solution, is : (a) 15 (b) вЂ“15 (c) 10 (d) 3 If sec2 пЃ±пЂ . ( 1 + sin пЃ±) . (1 вЂ“ sin пЃ±) = k, then the value of k is : (a) 0 (b) 1 (c) вЂ“1 (d) 2 57 [X вЂ“ Maths] 8. Consider the following frequency distribution Class 0вЂ“5 6-11 12-17 18-23 24-29 Frequency 13 10 15 8 11 The upper limit of the median class is : (a) 17 (b) 18 (c) 17.5 (d) 18.5. SECTION B Question number 9 to 14 carry two marks each. 9. Using Euclid's division algorithm, find the H.C.F. of 255 and 867. 10. Find a quadratic polynomial whose zeroes are 5 + пѓ–2 and 5 вЂ“пЂ пѓ–2. 11. In пЃ„ABC, D and E are points on the sides AB and AC respectively, such that DE || BC. If AD = 3x + 19, DB = x + 3, AE = 3x + 4, EC = x, find x. 12. If 3 tan пЃ± = 4, then evaluate : 13. If the sum of the squares of zeroes of the polynomial 5x2 + 3x + k is 3 sin пЃ± пЂ« 2cos пЃ± 3 sin пЃ± вЂ“ 2cos пЃ± 11 , 25 find the value of k. 14. The following table shows the cumulative frequency distribution of marks of 670 students in an examination : Marks below 10 below 20 below 30 below 40 below 50 below 60 below 70 Number of Students 10 50 130 270 440 570 670 Construct a frequency distribution table for the above data. OR 58 [X вЂ“ Maths] Find the mode for the following frequency distribution : Class Frequency 0-5 5-10 10-15 15-20 20-25 25-30 30-35 10 15 30 80 40 20 5 SECTION C Question number 15 to 24 carry three marks each. 15. If the areas of two similar triangles are equal, then prove that the triangles are congruent. 16. Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients of x2 пЂ« 17. 1 x вЂ“2 6 257 in the form 2m Г— 5n 5000 where m, n are non-negative integers. Hence, write its decimal expansion, without actual divison. Write the denominator of the rational number OR Show that any positive even integer is of the form 6q, 6q + 2 or 6q + 4, where q is some integer. 3 , and 0В° < (A + B) п‚ЈпЂ 90В°, 2 A > B, find A and B and hence find cos 2B. 18. If sin (A + B) = 1 and cos (A вЂ“ B) пЂЅ 19. Divide 5x3 вЂ“ 13x2 + 21x вЂ“ 14 by 3 вЂ“ 2x + x2 and verify the division lemma. 20. The sum of two numbers is 8. If their sum is four times their difference, find the numbers. OR Solve for x and y. ax + by = a вЂ“ b; bx вЂ“ ay = a + b 59 [X вЂ“ Maths] 21. Find the mean of the following distribution : Class Frequency 22. 0вЂ“10 10вЂ“20 20вЂ“30 30вЂ“40 40вЂ“50 7 10 15 8 10 In the given figure, пЃ„ABC is right angled at A; AD пЃћ BC, BD = 4 and DC = 9 cm. Find x . C 9 cm D 4 cm A B OR Diagonals of a trapezium PQRS intersect each other at the point O. PQ || RS and PQ = 3RS. Find the ratio of the areas of triangles POQ and ROS. 23. sin пЃ± sin пЃ± Prove that : cot пЃ± пЂ« cosec пЃ± пЂЅ 2 пЂ« cot пЃ± вЂ“ cosec пЃ± 24. Find the median of the following distribution : Class Frequency 5вЂ“14 15вЂ“24 25вЂ“34 35вЂ“44 45вЂ“54 55вЂ“64 6 11 21 23 14 5 SECTION D Question number 25 to 34 carry four marks each. 25. Prove that пѓ–5 is an irrational number. Hence show that 3 + 2 пѓ–5 is irrational. 26. Represent the following pair of equations graphically and write the coordinates of points where the lines intersect y-axis : x + 3y = 6; 2x вЂ“ 3y = 12. 60 [X вЂ“ Maths] 27. Prove that : tan пЃ± cot пЃ± пЂ« пЂЅ 1 пЂ« sec пЃ± . cosec пЃ±. 1 пЂ cot пЃ± 1 пЂ tan пЃ± 28. The median of the following data is 32.5 : Class Frequency 0вЂ“10 x 10вЂ“20 5 20вЂ“30 9 30вЂ“40 12 40вЂ“50 y 50вЂ“60 3 60вЂ“70 2 Total 40 Find the values of x and y. 29. Find all the zeroes of the polynomial 4x4 вЂ“ 20x3 + 23x2 + 5x вЂ“ 6, if two of its zeroes are 2 and 3. OR 8 men and 12 boys can finish a piece of work in 10 days while 6 men and 8 boys can finish the same work in 14 days. Find the number of days taken by one man alone to complete the work and also one boy alone to complete the work. 30. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of the corresponding sides. OR Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 31. Evaluate : 2(cos4 60В°+ sin4 30В°) вЂ“ (tan2 60В°+cot2 45В°) + 3 sec2 30В°. 61 [X вЂ“ Maths] 32. In an isosceles пЃ„ABC with AB =AC, BD is the perpendicular drawn from the vertex B to the side AC. Prove that BD2 вЂ“ CD2 = 2CD.AD. 33. Evaluate : sec пЃ± .cosec пЂЁ90 п‚° вЂ“ пЃ±пЂ© вЂ“ tan пЃ±. cot пЂЁ90 п‚° вЂ“ пЃ±пЂ© пЂ« sin 2 55 п‚° пЂ« sin 2 35 п‚° tan10п‚° .tan20 п‚°.tan60 п‚°.tan70 п‚°.tan80 п‚° 34. The frequency distribution of marks obtained by 53 students out of 100 in a certain examination is given below : Marks No. of Students 0вЂ“10 5 10вЂ“20 3 20вЂ“30 4 30вЂ“40 3 40вЂ“50 3 50вЂ“60 4 60вЂ“70 7 70вЂ“80 9 80вЂ“90 7 90вЂ“100 8 Draw a вЂ�less that typeвЂ™ ogive for the given data. 62 [X вЂ“ Maths] SAMPLE QUESTION PAPER - II MATHEMATICS, SA - 1 Time allowed : 3 hours Maximum Marks : 90 General Instructions 1. All questions are compulsory. 2. The question paper consists of 34 questions divided into four sections A, B, C and D. Section A comprises of 8 questions of 1 mark each. Section B comprises of 6 questions of 2 marks each. Section C comprises of 10 questions of 3 marks each and Section D comprises of 10 questions of 4 marks each. 3. Question numbers 1 to 8 in Section A are multiple choice questions where you are to select one correct option out of the given four. 4. Use of calculator is not permitted. SECTION вЂ“ A 1. 2. 3. In Euclid Division Lemma when a = bq + r, where a, b are positive integer, choose the correct option : (a) 0 < r п‚Ј b (b) 0 п‚Ј r пЂј b (c) 0 < r пЂј b (d) 0 п‚Ј r п‚Ј b What type of graph of the polynomial f(x) = вЂ“x2 + 3x вЂ“ 4 represents ; (a) Straight the (b) upward parabola (c) downward parabola (d) none of these In the corresponding sides of two similar triangles are in the ratio 5 : 7, then the ratio of their perimeter is (a) 2 : 7 (b) 5 : 7 (c) 5 : 2 (d) 7 : 5 63 [X вЂ“ Maths] 4. 5. If x = 7 sin пЃ±, y = 2 2 7 cos пЃ± then the value of x + y is : (a) 0 (b) 1 (c) 7 (d) 1 7 The decimal expansion of the rational number 21 7 п‚ґ 23 п‚ґ 54 will terminate after .... decimal places. 6. 7. 8. (a) 3 (b) 4 (c) 5 (d) never The point of intersection of the lines x вЂ“ 2y = 6 and y-axis is (a) (вЂ“3, 0) (b) (0, 6) (c) (6, 0) (d) (0, вЂ“3) The value of cos 1В° cos 2В° cos 3В° .... cos 171В° is (a) 1 (b) (c) 0 (d) вЂ“1 1 2 If the mode of observations 2, 3, 5, 4, 2, 6, 3, 5, 5, 2 and x is 2, then the value of x is : (a) 2 (b) 3 (c) 4 (d) 5 SECTION вЂ“ B 9. Whether 6n can end with the digit 0 for any natural number. Give reason for your answer. 10. If the product of the zeroes of the polynomial ax2 вЂ“ 6x вЂ“ 6 is 4, find the value of a. 11. If пЃ„ABC ~ пЃ„QRP, ar пЂЁ пЃ„ABC пЂ© 9 пЂЅ , AB = 18 cm, BC = 15 cm, find the ar пЂЁ пЃ„PQR пЂ© 4 value of PR. 64 [X вЂ“ Maths] 12. If 16 cot A = 12 then find the value of sin A пЂ« cos A . sin A вЂ“ cos A 13. Find a quadratic polynomial whose zeroes are 2 and вЂ“3, verify the relation between the coefficients and zeroes of the polynomial. 14. Find x, if the median of the observations in ascending order 24, 25, 26, x + 2, x + 3, 30, 31, 34 is 27.5. SECTION вЂ“ C 15. A girl Rita of height 90 cm is walking away from the base of a lamp post at a speed of 1.2 m/sec. If the lamp is 3.6 m above the aground, find the length of her shadow after 4 seconds? 16. Find the other zeroes of the polynomial for x4 вЂ“ 20x3 + 23x2 + 5x вЂ“ 6, if two of its zeroes are 2 and 3. 17. Use EuclidвЂ™s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. 18. If sin (3x + 2y) = 1 and cos пЂЁ 3x пЂ 2 y пЂ© пЂЅ 3 when 0 п‚Ј 3x + 2y п‚Ј 90В° then 2 find the value of x and y. 19. Find the largest positive integer that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively. 20. Solve for x and y, when x п‚№ В±y 30 пЂґ4 пЂ« пЂЅ 10 x пЂy x пЂ«y 40 55 пЂ« пЂЅ 13 x пЂy x пЂ«y 21. The mean of the following frequency distribution is 62.8 and the sum of all the frequencies is 50. Find the values of x and y. Class interval Frequency 0вЂ“20 20вЂ“40 40вЂ“60 60вЂ“80 80вЂ“100 100вЂ“120 5 x 10 y 7 8 65 [X вЂ“ Maths] 22. The diagonals of a trapezium PQRS intersect each other at the point O, PQ||RS and PQ = 3 RS, find the ratio of the areas of пЃ„POQ and пЃ„ROS. 23. If A, B and C are interior angles of a пЃ„ABC, then show that A пѓ¦B пЂ« Cпѓ¶ sin пѓ§ пЂЅ cos пѓЁ 2 пѓ·пѓё 2 24. Find the median weight of the following data : Weight in (kg.) 38вЂ“40 40вЂ“42 42вЂ“44 44вЂ“46 46вЂ“48 48вЂ“50 50вЂ“52 No. of Students 3 2 4 5 14 4 3 SECTION вЂ“ D 25. Prove that 5 is an irrational number and hence prove 2 вЂ“пЂ irrational number. 5 is an 26. Draw the graph of the following pair of linear equations x + 3y = 6 and 2x вЂ“ 3y = 12. Find the ratio of the areas of the two triangles formed by first line, x = 0, y = 0 and second line x = 0, y = 0. 27. Prove that sin пЃ± вЂ“ cos пЃ± пЂ« 1 1 пЂЅ sin пЃ± пЂ« cos пЃ± пЂ 1 sec пЃ± вЂ“ tan пЃ± 28. Change the distribution to a more than type distribution and draw its ogive Class Interval Frequency 50-55 55-60 60-65 65-70 70вЂ“75 75вЂ“80 2 8 12 24 38 16 29. On dividing 4x3 вЂ“ 8x2 + 8x + 1 by a polynomial d(x), the quotient and remainder are (2x2 вЂ“ 3x + 2) and (x + 3) respectively, find d(x). 30. State and prove converse of pythagoras theorem. 31. Prove the following identity : cos A 1 пЂ« sin A пЂ« пЂЅ 2 sec A. 1 пЂ« sin A cos A 66 [X вЂ“ Maths] 32. пЃ„ABC is right angled at C. Let BC = a, CA = b, AB = c, CD пЃћ AB, CD = p. Prove that 1 (i) 33. cp = ab (ii) p 2 пЂЅ 1 a 2 пЂ« 1 b2 . Evaluate : 5 cos2 60п‚° пЂ« 4 sec 2 30п‚° вЂ“ tan2 45п‚° sin2 30п‚° пЂ« cos2 30п‚° 34. In a city, the number of old aged citizens lived in an old age home is as given below Age (in years) No. of People 50вЂ“55 10 55вЂ“60 12 60вЂ“65 17 65вЂ“70 13 70вЂ“75 16 75вЂ“80 22 (a) Find mean of the above data? (b) Why the old people are neglected in the society and what are the various steps to be taken to improve the status of old people in the society? (c) What value is depicted in this question? ANSWERS 1. b 2. c 3. b 4. c 5. b 6. d 7. c 8. a 10. a = вЂ“3/2 11. 67 10 cm [X вЂ“ Maths] 12. 7 13. x2 + x вЂ“ 6 14. x = 25 15. 1.6 m 16. В± 18. x = 20, y = 15 19. 17 20. x = 8, y = 3 21. x = 8, y = 12 22. 9 : 1 24. 46.5 kg 26. 1 : 2 29. d(x) = 2x вЂ“ 1 33. 67 12 34. (a) 66.8 1 2 (b) Generation gap, Lack of social and moral values. (c) Impart social and moral values in the society. 68 [X вЂ“ Maths] CONTENTS S.No. Chapter Page 1. Quadratic Equations 70 2. Arithmetic Progression 78 3. Coordinate Geometry 85 4. Some Applications of Trigonometry 92 5. Circle 100 6. Constructions 114 7. Mensuration 117 8. Probability 134 Sample Papers 141 69 [X вЂ“ Maths] CHAPTER 1 QUADRATIC EQUATIONS KEY POINTS 1. The equation ax2 + bx + c = 0, a п‚№ 0 is the standard form of a quadratic equation, where a, b and c are real numbers. 2. A real number пЃЎ is said to be a root of the quadratic equation ax2 + bx + c = 0, a п‚№ 0. If aпЃЎ2 + bпЃЎ + c = 0, the zeros of quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same. 3. If we can factorise ax2 + bx + c = 0, a п‚№ 0 into product of two linear factors, then the roots of the quadratic equation can be found by equating each factors to zero. 4. The roots of a quadratic equation ax2 + bx + c = 0, a п‚№ 0 are given by пЂb п‚± 5. 6. 7. 2 b пЂ 4ac , provided that b2 вЂ“ 4ac п‚і 0. 2a A quadratic equation ax2 + bx + c = 0, a п‚№ 0, has ___ (a) Two distinct and real roots, if b2 вЂ“ 4ac > 0. (b) Two equal and real roots, if b2 вЂ“ 4ac = 0. (c) Two roots are not real, if b2 вЂ“ 4ac < 0. A quadratic equation can also be solved by the method of completing the square. (i) a2 + 2ab + b2 = (a + b)2 (ii) a2 вЂ“ 2ab + b2 = (a вЂ“ b)2 Discriminant of the quadratic equation ax2 + bx + c = 0, a п‚№ 0 is given by D = b2 вЂ“ 4ac. 70 [X вЂ“ Maths] MULTIPLE CHOICE QUESTIONS 1. 2. 3. 4. 5. 6. 7. The number of non zero roots of x2 + 6x = 0 are (a) 0 (b) 1 (c) 2 (d) does not exist. Maximum number of solutions of a quadratic equation are : (a) 0 (b) 1 (c) 2 (d) 3 If the equation x2 вЂ“ (2 + m) x + (вЂ“ m2 вЂ“ 4m вЂ“ 4) = 0 has coincident roots, then (a) m = 0, m = 1 (b) m = 2, m = 2 (c) m = вЂ“ 2, m = вЂ“ 2 (d) m = 6, m = 1 If one root of x2 вЂ“ ax вЂ“ 4 = is negative of the other then the value of a is: (a) 0 (b) 1 (c) 2 (d) 4 (b) x2 + 1 = (x + 3)2 (d) x пЂ« Which is a quadratic equation? 1 пЂЅ 2 x (a) x пЂ« (c) x (x + 2) 1 . x If the roots of a quadratic equation are 2 and 3, then the equation is: (a) x2 + 5x + 6 = 0 (b) x2 + 5x вЂ“ 6 = 0 (c) x2 вЂ“ 5x вЂ“ 6 = 0 (d) x2 вЂ“ 5x + 6 = 0 Roots of the equations x2 вЂ“ 3x + 2 = 0 are (a) 1, вЂ“2 (b) вЂ“1, 2 (c) вЂ“1, вЂ“2 (d) 1, 2 71 [X вЂ“ Maths] 8. 9. 10. 11. 12. 13. 14. If the roots of a quadratic equation are equal, then discriminant is (a) 1 (b) 0 (c) greater than 0 (d) less than zero. If one root of 2x2 + kx + 1 = 0 is вЂ“ 1 , then the value of вЂ�kвЂ™ is 2 (a) 3 (b) вЂ“3 (c) 5 (d) вЂ“5 The sum of the roots of the quadratic 5x2 вЂ“ 6x + 1 = 0 is (a) 6 5 (c) пЂ 5 6 (b) 1 5 (d) пЂ 1 5 The product of the roots of the quadratic equation 2x2 + 5x вЂ“ 7 = 0 is (a) 5 2 (c) пЂ 5 2 (b) пЂ (d) 7 2 7 2 If the roots of the quadratic 2x2 + kx + 2 = 0 are equal then the value of вЂ�kвЂ™ is (a) 4 (b) вЂ“4 (c) В± 4 (d) В± 16 If the roots of x2 + px + 12 = 0 are in the ratio 1 : 3 then p = _____ (a) В± 2 (b) пЃ8 (c) пЃ 4 (d) 8 If the sum and product of roots of a quadratic equation are пЂ 7 5 and 2 2 respectively, then the equation is 72 [X вЂ“ Maths] 15. (a) 2x2 + 7x + 5 = 0 (b) 2x2 вЂ“ 7x + 5 = 0 (c) 2x2 вЂ“ 7x вЂ“ 5 = 0 (d) 2x2 + 7x вЂ“ 5 = 0 Which constant must be added or subtracted to solve the equation 9x 2 пЂ« 3 x пЂ 4 2 пЂЅ 0 by the method of completing the square (a) 1 8 (b) 1 64 (c) 1 16 (d) none SHORT ANSWER TYPE QUESTIONS 16. If one root of the equation x2 + 7x + k = 0 is вЂ“2, then find the value of k and the other root. 17. If the roots of 4x2 + 3px + 9 = 0 are real and distinct, find the value of p? 18. For what value of m the sum of the roots of mx2 + 6x + 4m = 0 be equal to the product of the roots ? 19. The product of two consecutive odd integers is 63. Represent this in form of a quadratic equation. 20. Find the roots of the equation : x пЂ« 21. Find the value of c for which one roots of 4x2 вЂ“ 2x + (c вЂ“ 4) = 0 is reciprocal of the other. 22. Divide 51 in to two parts such that their product is 378. 23. Find вЂ�kвЂ™ so that (k вЂ“ 12) x2 + 2 (k вЂ“ 12) x + 2 = 0 has equal roots. (k п‚№ 12). 24. If (вЂ“5) is a root of the equation 2x2 + px вЂ“ 15 = 0 and the equation p(x2 + x) + k = 0 has equal roots, find values of p and k. 25. Find the value of K, so that the difference of the roots of x2 вЂ“ 5x + 3 (k вЂ“ 1) is 11. 73 1 1 пЂЅ 4 , x п‚№ 0. x 4 [X вЂ“ Maths] 26. The difference of two numbers is 5 and the difference of their reciprocals 1 . Find the numbers. is 10 27. If the roots of the equation (b вЂ“ c)x2 + (c вЂ“ a) x + (a вЂ“ b) = 0 are equal, then prove that 2b = a + c. 28. Find the nature of the roots of the following quadratic equations. If roots are real, find them. 29. (a) 5x2 вЂ“ 3x + 2 = 0. (b) 2x2 вЂ“ 9x + 9 = 0. Sum of two numbers is 15, if sum of their reciprocals is 3 . Find the 10 numbers. 30. Solve the following quadratic equations пѓ¦ 2x пЂ« 3 пѓ¶ пѓ¦ x пЂ 3 пѓ¶ 2пѓЁ пЂ 25 пѓЁ пЂЅ 5 x пЂ 3 пѓё 2x пЂ« 3 пѓё x п‚№ 3, x п‚№ пЂ 31. 1 1 1 1 пЂ« пЂ« пЂЅ , a, b, x п‚№ 0 and x п‚№ вЂ“ пЂЁ a пЂ« b пЂ© a b x aпЂ«b пЂ«x 32. 4 3x 33. ab x2 + (b2 вЂ“ ac) x вЂ“ bc = 0. 34. x пЂ1 x пЂ 3 10 пЂ« пЂЅ , x п‚№ 2, x п‚№ 4. x пЂ2 x пЂ4 3 35. 1 1 11 пЂ пЂЅ , x п‚№ пЂ4, x п‚№ 7. x пЂ« 4 x пЂ 7 30 2 3 2 пЂ« 5x пЂ 2 3 пЂЅ 0. 36. Shikha and Mona each wish to grow 100 m2 rectangular vegetable garden. They used 30m wire each for fencing three sides of the garden letting compound wall of their houses act as the fourth side. Find the possible sides of their gardens. 37. A shop keeper buys a number of pens for 80. If he had bought 4 more pens for the same amount, each pen would have cost him 1 less. How many pens did he buy? 74 [X вЂ“ Maths] 38. A two digit number is such that the product of the digits is 35, when 18 is added to the number, the digits inter change their places. Find the number. 39. Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46, find the integers. 40. A motor boat whose speed is 9 km/h in still water goes 12 km down stream and comes back in a total time 3 hours. Find the speed of the stream. 41. A train travels 360 km at uniform speed. If the speed had been 5 km/hr more it would have taken 1 hour less for the same journey. Find the speed of the train. 42. The hypotenuse of right angled triangle is 6cm more than twice the shortest side. If the third side is 2 cm less than the hypotenuse, find the sides of the triangle. 43. By a reduction of 2 per kg in the price of tomato. Anita can purchase 2 kg tomato more for 224. Find the original price of tomato per kg. 44. 6500 were divided equally among a certain number of students. Had there been 15 more students, each would have got 30 less. Find the original number of students. 45. Two trains leave a Railway Station at the same time. The first train travels due west and the second train due North. Speed of first train is 5 km/h faster than the second train.If after two hours they are 50 km apart, find the average speed of the faster train. 46. A girl is twice as old as her sister. Four years hence, the product of their ages will be 160. Find their present ages. 47. Two years ago a manвЂ™s age was three times the square of his sonвЂ™s age. Three years hence his age will be four times his sonвЂ™s age. Find their present ages. 48. In a cricket match against Sri Lanka, Sehwag took one wicket less than twice the number of wickets taken by Unmukt. If the product of the number of wickets taken by these two is 15, find the number of wickets taken by each. 49. A takes 10 days less than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days. Find the time taken by B to finish the work alone. 75 [X вЂ“ Maths] 50. 8 minutes. If one pipe 11 takes 1 minute more than the other to fill the cistern, find the time in which each pipe would fill the cistern alone. Two pipes running together can fill a cistern in 2 ANSWERS 1. b 2. c 3. c 4. a 5. a 6. d 7. d 8. b 9. a 10. a 11. b 12. c 13. b 14. a 15. b 16. k = 10, second root = вЂ“ 5 17. p < вЂ“4 or p > 4 18. вЂ“ 19. x2 + 2x вЂ“ 63 = 0 20. 4, 21. c = 8 22. 9, 42 23. k = 14 24. 7, 26. 10, 5 27. Hint : For equal roots D = 0. 29. 5, 10 7 4 3 2 1 4 25. k = вЂ“7 28. (a) Not real roots. (b) Roots are real, 3, 76 3 . 2 [X вЂ“ Maths] 30. 32. 6, 1 3 2 , пЂ 4 3 5 2 31. вЂ“a, вЂ“b 33. c b , пЂ b a 34. 5, 36. 5 Г— 20, 10 Г— 10 37. 16 38. 57 39. 4, 5, 6 40. 3 km/hr. 41. 40 km/hr. 42. 26 cm, 24 cm, 10 cm 43. Rs. 16 44. 50 45. 20 km. 46. 12, 6 47. 29 yrs., 5 yrs. 48. 3, 5 49. 30 days. 50. 5, 6 min 35. 1, 2 77 [X вЂ“ Maths] CHAPTER 2 ARITHMETIC PROGRESSION KEY POINTS 1. Sequence : A set of numbers arranged in some definite order and formed according to some rules is called a sequence. 2. Progression : The sequence that follows a certain pattern is called progression. 3. Arithmetic Progression : A sequence in which the difference obtained by substracting any term from its proceeding term is constant throughout, is called an arithmetic sequence or arithmetic progression (A.P.). The general form of an A.P. is a, a + d, a + 2d, ..... (a : first term d : common difference). 4. General Term : If вЂ�aвЂ™ is the first term and вЂ�dвЂ™ is common difference in an A.P., then nth term (general term) is given by an = a + (n вЂ“ 1) d . 5. Sum of n Terms of An A.P. : If вЂ�aвЂ™ is the first term and вЂ�dвЂ™ is the common difference of an A.P., then sum of first n terms is given by Sn пЂЅ n пЃ»2a пЂ« пЂЁ n пЂ 1пЂ© d пЃЅ 2 If вЂ�lвЂ™ is the last term of a finite A.P., then the sum is given by Sn пЂЅ 6. n пЃ»a пЂ« l пЃЅ . 2 (i) If an is given, then common difference d = an вЂ“ anвЂ“1. (ii) If sn is given, then nth term is given by an = sn вЂ“ snвЂ“1. (iii) If a, b, c are in A.P., then 2b = a + c. (iv) If a sequence has n terms, its rth term from the end = (n вЂ“ r + 1)th term from the beginning. 78 [X вЂ“ Maths] MULTIPLE CHOICE QUESTIONS 1. 2. 3. Three numbers in A.P. have sum 24. The middle term isвЂ” (a) 6 (b) 8 (c) 3 (d) 2 If nth term of on A.P. is 2n + 7, then 7th term of the A.P. is (a) 15 (b) 21 (c) 28 (d) 25 If the sum of n terms of an A.P. is 5 2 3n n пЂ« , then sum of its 10 terms 2 2 is 4. 5. 6. 7. (a) 250 (b) 230 (c) 225 (d) 265 If nth term of the A.P. 4, 7, 10, ________ is 82, then the value of n is (a) 29 (b) 27 (c) 30 (d) 26 If a, b and c are in A.P. then a пЂ«c 2 (a) a пЂЅ b пЂ«c 2 (b) b пЂЅ (c) c пЂЅ a пЂ«b 2 (d) b = a + c 12th term of the A.P. x вЂ“ 7, x вЂ“ 2, x + 3 is (a) x + 62 (b) x вЂ“ 48 (c) x + 48 (d) x вЂ“ 62 Common difference of A.P. 8 (a) 1 2 3 , 8 , 8 , ________ is 8 8 8 1 8 (b) 79 1 1 8 [X вЂ“ Maths] (c) 8. 9. 10. 11. 12. 13. 14. 8 1 8 (d) 1 nth term of the A.P. вЂ“5, вЂ“2, 1, ________ is (a) 3n + 5 (b) 8 вЂ“ 3n (c) 8n вЂ“ 5 (d) 3n вЂ“ 8 If nth term of an A.P. is 5 вЂ“ 3n, then common difference of the A.P. is (a) 2 (b) вЂ“3 (c) вЂ“2 (d) 3 If 5, 2k вЂ“ 3, 9 are in A.P., then the value of вЂ�kвЂ™ is (a) 4 (b) 5 (c) 6 (d) вЂ“5 Sum of first 10 natural numbers is (a) 50 (b) 55 (c) 60 (d) 65 9th term from the end of the A.P. 7, 11, 15, _______ 147 is (a) 135 (b) 125 (c) 115 (d) 110 If the sum of n terms of an A.P. is n2, then its nth term is (a) 2n вЂ“ 1 (b) 2n + 1 (c) n2 вЂ“ 1 (d) 2n вЂ“ 3 The sum of 3 numbers in A.P. is 30. If the greatest number is 13, then its common difference is (a) 4 (b) 3 (c) 2 (d) 5 80 [X вЂ“ Maths] 15. The sum of 6th and 7th terms of an A.P. is 39 and common difference is 3, then the first term of the A.P. is (a) 2 (b) вЂ“3 (c) 4 (d) 3 LONG ANSWER TYPE QUESTIONS 16. Is 17. Find an A.P. whose 2nd term is 10 and the 6th term exceeds the 4th term by 12. 18. Which term of the A.P. 41, 38, 35... is the first negative term? Find the term also. 19. Nidhi saves Rs. 2 on day 1, Rs. 4 on day 2, Rs. 6 on day 3 and so on. How much money she save in month of Feb. 2011? 20. Find the number of terms in an A.P. whose first term and 6th term are 3 and 23 respectively and sum of all terms is 406. 21. How many two digits numbers between 6 and 102 are divisible by 6. 22. If sn the sum of first n terms of an A.P. is given by sn = 3n2 вЂ“ 4n, then find its nth term and common difference. 23. The sum of 4th and 8th terms of an A.P. is 24 and sum of 6th and 10th terms is 44. Find A.P. 24. Find the sum of odd positive integers between 1 and 199. 25. How many terms of the A.P. 22, 20, 18, _____ should be taken so that their sum is zero? 26. 4k + 8, 2k2 + 3k + 6, 3k2 + 4k + 4 are the angles of a triangle. These form an A.P. Find value of k. 27. If 11 times of 11th term is equal to 17 times of 17th term of an A.P. find its 28th term. 28. Find an A.P. of 8 terms, whose first term is 2, 8, 18, 32, ______ an A.P.? If yes, then find its next two terms. 81 1 17 . and last term is 2 6 [X вЂ“ Maths] 29. The fourth term of an A.P. is equal to 3 times the first term and the seventh term exceeds twice the third term by 1. Find the first term and common difference of the A.P. 30. Find the middle term of the A.P. 20, 16, 12, ......, вЂ“176. 31. If 2nd, 31st and last terms of on A.P. are 31 1 13 , and пЂ respectively. 4 2 2 Find the number of terms in the A.P. 32. Find the number of terms of the A.P. 57, 54, 51, ______ so that their sum is 570. Explain the double answer. 33. The sum of three numbers in A.P. is 24 and their product is 440. Find the numbers. 34. Find the sum of the first 40 terms of an A.P. whose nth term is 3 вЂ“ 2n. 35. In an A.P., the first term is 2, the last term is 29 and the sum of the terms is 155. Find common difference вЂ�dвЂ™. 36. If nth term of an A.P. is 4, common difference is 2 and sum of n terms is вЂ“14, then find first term and the number of terms. 37. Find the sum of all the three digits numbers each of which leaves the remainder 3 when divided by 5. 38. The sum of first six terms of an A.P. is 42. The ratio of the 10th term to the 30th term is 1 : 3. Find first term and 11th term of the A.P. 39. The sum of n terms of two A.P.вЂ™s are in the ratio 3n + 8 : 7n + 15. Find the ratio of their 12th terms. 40. The eight term of on A.P. is half the second term and the eleventh term exceeds one-third of its fourth term by 1. Find a15. 41. The sum of first 8 terms of an A.P. is 140 and sum of first 24 terms is 996. Find the A.P. 42. The digits of a three digits positive number are in A.P. and the sum of digits is 15. On subtracting 594 from the number the digits are interchanged. Find the number. 43. A picnic group for Shimla consists of students whose ages are in A.P., the common difference being 3 months. If the youngest student Neeraj 82 [X вЂ“ Maths] is just 12 years old and the sum of ages of all students is 375 years. Find the number of students in the group. 44. The sum of first 20 terms of an A.P. is one third of the sum of next 20 terms. If first term is 1, then find the sum of first 30 terms. 45. The sum of first 16 terms of an A.P. is 528 and sum of next 16 terms is 1552. Find the first term and common difference of the A.P. 46. Kriti, starts a game and scores 200 points in the first attempt and she increases the points by 40 in each attempt. How many points will she score in the 30th attempt? 47. In an A.P. the sum of first ten terms is вЂ“150 and the sum of its next ten terms is вЂ“550. Find the A.P. 48. The first and the last term of an A.P. are 4 and 81 respectively. If common difference is 7. Find the number of terms and their sum. 49. The sum of 5th and 9th terms of an A.P. is 8 and their product is 15. Find the sum of first 28 terms of the A.P. 50. Pure and Ashu live in two different villages 165 km apart. They want to meet each other but there is no fast means of transport. Puru travels 15km the first day, 14 km the second day, 13 km the third day and so on. Ashu travels 10 km the first day, 12 km the second dry, 14 km the third day and so on. After how many days will they meet. ANSWERS 1. b 2. b 3. d 4. b 5. b 6. c 7. a 8. d 9. b 10. b 11. b 12. c 13. a 14. b 15. d 16. 83 Yes, 50, 72 [X вЂ“ Maths] 17. 4, 10, 16, ............... 18. 15th term, вЂ“1 19. Rs. 812 20. 14 21. 15 22. 6n вЂ“ 7, Common difference = 6 23. вЂ“13, вЂ“8, вЂ“3, 2 ............... 24. 9800 25. 23 26. 0, 2 27. 0 28. 29. First term = 3, common difference = 2 30. вЂ“76, вЂ“80 31. 59 32. 19 or 20, {20th term is zero} 33. 5, 8, 11 34. вЂ“1520 35. 3 36. First term = вЂ“ 8, 1 5 7 , , , ............... 2 6 6 Number of terms = 7 37. 99090 38. First term = 2, 11th term = 22 39. 7 : 16 40. 3 41. 7, 10, 13, 16, ............... 42. 852 43. 25 students 44. 900 45. First term = 3, Common difference = 4 46. 1360 47. 3, вЂ“1, вЂ“5 ............... 49. 217, 7 d пЂЅ п‚± пЃ» 1 2 пЃЅ 48. 12, 510 50. 6 days 84 [X вЂ“ Maths] CHAPTER 3 CO-ORDINATE GEOMETRY KEY POINTS 1. T he length of a line segm ent joining A and B is the distance between two points A (x1, y1) and B (x2, y2) is пѓ–{(x2 вЂ“ x1)2 + (y2 вЂ“ y1)2}. 2. The distance of a point (x, y) from the origin is пѓ–(x2 + y2). The distance of P from x-axis is y units and from y-axis is x-units. 3. The co-ordinates of the points p(x, y) which divides the line segment joining the points A(x1, y1) and B(x2, y2) in the ratio m1 : m2 are пѓ¦ m 1x 2 пЂ« m 2 x 1 m 1y 2 пЂ« m 2 y 1 пѓ¶ , пѓ§ m пЂ«m пѓ· m1 пЂ« m 2 пѓЁ пѓё 1 2 we can take ratio as k : 1, k пЂЅ 4. m1 . m2 The mid-points of the line segment joining the points P(x1, y1) and Q(x2, y2) is пѓ¦ x1 пЂ« x2 y1 пЂ« y2 пѓ¶ , пѓ§ пѓ· 2 2 пѓЁ пѓё 5. The area of the triangle formed by the points (x1, y1), (x2, y2) and (x3, y3) is the numeric value of the expressions 1 пѓ© x пЂЁ y пЂ y 3 пЂ© пЂ« x 2 пЂЁ y 3 пЂ y 1 пЂ© пЂ« x 3 пЂЁ y 1 пЂ y 2 пЂ© пѓ№пѓ» . 2пѓ« 1 2 6. If three points are collinear then we can not draw a triangle, so the area will be zero i.e. |x1(y2 вЂ“ y3) + x2 (y3 вЂ“ y1) + x3(y1 вЂ“ y2)| = 0 85 [X вЂ“ Maths] MULTIPLE CHOICE QUESTIONS 1. 2. 3. 4. 5. 6. 7. P is a point on x axis at a distance of 3 unit from y axis to its left. The coordinates of P are (a) (3, 0) (b) (0, 3) (c) (вЂ“3, 0) (d) (0, вЂ“3) The distance of point P (3, вЂ“2) from y-axis is (a) 3 units (b) (c) вЂ“2 units (d) 2 units 13 units The coordinates of two points are (6, 0) and (0, вЂ“8). The coordinates of the mid point are (a) (3, 4) (b) (3, вЂ“4) (c) (0, 0) (d) (вЂ“4, 3) If the distance between (4, 0) and (0, x) is 5 units, the value of x will be (a) 2 (b) 3 (c) 4 (d) 5 The coordinates of the point where line x y пЂ« пЂЅ 7 intersects y-axis are a b (a) (a, 0) (b) (0, b) (c) (0, 2b) (d) (2a, 0) The area of triangle OAB, the coordinates of the points A (4, 0) B (0, вЂ“7) and O origin, is (a) 11 sq. units (b) 18 sq. units (c) 28 sq. units (d) 14 sq. units пѓ¦ 11 пѓ¶ пѓ¦ 2 пѓ¶ The distance between the points P пѓ§ пЂ , 5 пѓ· and Q пѓ§ пЂ , 5 пѓ· is пѓЁ 3 пѓё пѓЁ 3 пѓё (a) 6 units (b) 4 units (c) 3 units (d) 2 units 86 [X вЂ“ Maths] 8. 9. 10. 11. 12. 13. 14. x y пЂ« пЂЅ 1 intersects the axes at P and Q, the coordinates of 2 4 the mid point of PQ are The line (a) (1, 2) (b) (2, 0) (c) (0, 4) (d) (2, 1) The coordinates of vertex A of пЃ„ABC are (вЂ“4, 2) and point D(2, 5), D is mid point of BC. The coordinates of centroid of пЃ„ABC are (a) (0, 4) (b) 7пѓ¶ пѓ¦ пѓ§ пЂ1, 2 пѓ· пѓЁ пѓё (c) 7пѓ¶ пѓ¦ пѓ§ пЂ2, 3 пѓ· пѓЁ пѓё (d) (0, 2) The distance between the line 2x + 4 = 0 and x вЂ“ 5 = 0 is (a) 9 units (b) 1 unit (c) 5 units (d) 7 units The distance between the points (5 cos 35В°, 0) and (0, 5 cos 55В°) is (a) 10 units (b) 5 units (c) 1 unit (d) 2 units The points (вЂ“4, 0), (4, 0) and (0, 3) are the vertices of a : (a) right triangle (b) Isosceles triangle (c) equilateral triangle (d) Scalene triangle The perimeter of triangle formed by the points (0, 0), (2, 0) and (0, 2) is (a) 4 units (b) 6 units (c) 6 2 units (d) 4 пЂ« 2 2 units AOBC is a rectangle whose three vertices are A (0, 3), O (0, 0), B (5, 0). The length of its diagonal is : (a) (c) 5 units 34 units 87 (b) 3 units (d) 4 units [X вЂ“ Maths] 15. If the centroid of the triangle formed by (9, a), (b, вЂ“4) and (7, 8) is (6, 8) then (a, b) is (a) (4, 5) (b) (5, 4) (c) (5, 2) (d) (3, 2) SHORT ANSWER TYPE QUESTIONS 16. Find the value of a so that the point (3, a) lies on the line represented by 2x вЂ“ 3y = 5. 17. A line is drawn through a point P(3, 2) parallel to x-axis. What is the distance of the line from x-axis? 18. What is the value of a if the points (3, 5) and (7, 1) are equidistant from the point (a, 0)? 19. пѓ¦ b aпѓ¶ Prove that the points пЂЁ 0, 9 пЂ© , пѓЁ , пѓё and (b, 0) are collinear. 2 2 20. AB is diameter of circle with centre at origin. What are the coordinates of B if coordinates of A are (3, вЂ“4)? 21. A (3, 2) and B (вЂ“2, 1) are two vertices of пЃ„ABC, whose centroid G has 1пѓ¶ пѓ¦5 coordinates пѓЁ , вЂ“ пѓё . Find the coordinates of the third vertex C of пЃ„ABC. 3 3 22. For what value of p, are the points (вЂ“3, 9), (2, p) and (4, вЂ“5) collinear? 23. Find the relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5). 24. Find the coordinates of point P if P and Q trisect the line segment joining the points A(1, вЂ“2) and B (вЂ“3, 4). 25. Find x if the distance between the points (x, 2) and (3, 4) be 26. Find the area of triangle whose vertices are (1, вЂ“1), (вЂ“3, 5) and (2, вЂ“7). 27. Find a point on y-axis which is equidistant from the points (вЂ“2, 5) and (2, вЂ“3). 28. The mid point of the line segment joining the points (5, 7) and (3, 9) is 88 8 units. [X вЂ“ Maths] also the mid point of the line segment joining the points (8, 6) and (a, b). Find a and b. 29. Find the coordinates of the point which divides the line segment joining the points (1, 3) and (2, 7) in the ratio 3 : 4. 30. Find the value(s) of x for which the distance between the points P (2, вЂ“3) and Q (x, 5) is 10 units. 31. The point K (1, 2) lies on the perpendicular bisector of the line segment joining the points E (6, 8) and F (2, 4). Find the distance of the point K from the line segment EF. 32. The vertices of пЃ„ABC are A (вЂ“1, 3), B (1, вЂ“1) and C (5, 1). Find the length of the median drawn from the vertex A. 33. Find the distance between the points A (a, b) and B (b, a) if a вЂ“ b = 4. 34. Three vertices of a parallelogram taken in order are (вЂ“3, 1), (1, 1) and (3, 3). Find the coordinates of fourth vertex. 35. Triangle ABC is an isosceles triangle with AB = AC and vertex A lies on y-axis. If the coordinates of B and C are (вЂ“5, вЂ“2) and (3, 2) respectively then find the coordinates of vertex A. 36. If A (3, 0), B (4, 5), C (вЂ“1, 4) and D (вЂ“2, вЂ“1) are four points in a plane, show that ABCD is a rhombus but not a square. 37. Find the coordinates of a point which is 38. The area of a triangle with vertices (6, вЂ“3), (3, K) and (вЂ“7, 7) is 15 sq. unit. Find the value of K. 39. Find the abscissa of a point whose ordinate is 4 and which is at a distance of 5 units from (5, 0). 40. A point P on the x-axis divides the line segment joining the points (4, 5) and (1, вЂ“3) in certain ratio. Find the coordinates of point P. 41. In right angled пЃ„ABC, пѓђB = 90В° and AB пЂЅ 34 unit. The coordinates of points B, C are (4, 2) and (вЂ“1, y) respectively. If ar (пЃ„ABC) = 17 sq. unit, then find the value of y. 89 3 of the way (3, 1) to (вЂ“2, 5). 4 [X вЂ“ Maths] 42. If A (вЂ“3, 2) B (x, y) and C (1, 4) are the vertices of an isosceles triangle with AB = BC. Find the value of (2x + y). 43. If the point P (3, 4) is equidistant from the points A (a + b, b вЂ“ a) and B (a вЂ“ b, a + b) then prove that 3b вЂ“ 4a = 0. 44. The vertices of quadrilateral ABCD are A (вЂ“5, 7), B (вЂ“4, 5), C (вЂ“1, вЂ“6) and D (4, 5). Find the area of quadrilateral ABCD. 45. If midpoints of sides of a пЃ„PQR are (1, 2), (0, 1) and (1, 0) then find the coordinates of the three vertices of the пЃ„PQR. 46. The line segment joining the points A (2, 1) and B (5, вЂ“8) is trisected at the points P and Q such that P is nearer to A. If P is also lies on line given by 2x вЂ“ y + k = 0, find the value of K. 47. The line segment joining the points (3, вЂ“4) and (1, 2) is trisected at the пѓ¦5 пѓ¶ point P and Q. If the coordinates of P and Q are (p вЂ“2) and пѓ§ , q пѓ· пѓЁ3 пѓё respectively, find the values of p and q. 48. If A (вЂ“5, 7), B (вЂ“4, вЂ“5), C (вЂ“1, вЂ“6) and D (4, 5) are vertices of quadrilateral ABCD. Find the area of quadrilateral ABCD. 49. If P (x, y) is any point on the line joining the points A(a, 0) and B (0, b), x y пЂ« пЂЅ 1. then show that a b 50. If the points (x, y), (вЂ“5, вЂ“2) and (3, вЂ“5) are collinear, prove that 3x + 8y + 31 = 0. ANSWERS 1. c 2. a 3. b 4. b 5. c 6. d 7. c 8. a 9. a 10. d 11. b 12. b 90 [X вЂ“ Maths] 13. d 14. c 15. c 16. a пЂЅ 17. 2 units 18. a = 2 20. (вЂ“3, 4) 21. C (4, вЂ“4) 22. p = вЂ“ 1 23. x вЂ“ y = 2 24. пѓ¦ 1 пѓ¶ пѓ§пЂ 3 , 0пѓ· пѓЁ пѓё 25. x = 1, 5 26. 5 sq. unit 27. (0, 1) 28. a = 0, b = 10 29. 30. 4, вЂ“8 31. 5 units 32. 5 units 33. 34. (вЂ“1, 3) 35. (0, вЂ“2) 37. пѓ¦ 3 пѓ¶ пѓ§пЂ 4 , 4пѓ· пѓЁ пѓё 38. K пЂЅ 39. 2, 8 40. 41. вЂ“1 42. 1 44. 72 sq. unit 45. Coordinates of the vertices are (2, 1), (0, 3), (0 вЂ“1) 46. K = вЂ“ 8 48. 7289 units 47. 91 1 3 пѓ¦ 10 33 пѓ¶ пѓ§ 7 , 7 пѓ· пѓЁ пѓё 4 2 units 21 13 пѓ¦ 17 пѓ¶ пѓ§ 8 , 0пѓ· пѓЁ пѓё p пЂЅ 7 ,q пЂЅ 0 3 [X вЂ“ Maths] CHAPTER 4 SOME APPLICATIONS OF TRIGONOMETRY KEY POINTS 1. Line of Sight : The line of sight is the line drawn from the eyes of an observer to a point in the object viewed by the observer. 2. Angle of Elevation : The angle of elevation is the angle formed by the line of sight with the horizontal, when it is above the horizontal level i.e. the case when we raise our head to look at the object. 3. Angle of Depression : The angle of depression is the angle formed by the line of sight with the horizontal when it is below the horizontal i.e. case when we lower our head to look at the object. MULTIPLE CHOICE QUESTIONS 1. 2. 3. The length of the shadow of a man is equal to the height of man. The angle of elevation is (a) 90В° (b) 60В° (c) 45В° (d) 30В° The length of the shadow of a pole 30m high at some instant is 10 3 m. The angle of elevation of the sun is (a) 30В° (b) 60В° (c) 45В° (d) 90В° Find the angle of depression of a boat from the bridge at a horizontal distance of 25m from the bridge, if the height of the bridge is 25m. 92 [X вЂ“ Maths] 4. 5. 6. (a) 45В° (b) 60В° (c) 30В° (d) 15В° The tops of two poles of height 10m and 18m are connected with wire. If wire makes an angle of 30В° with horizontal, then length of wire is (a) 10m (b) 18m (c) 12m (d) 16m From a point 20m away from the foot of the tower, the angle of elevation of the top of the tower is 30В°. The height of the tower is (a) 20 3 m (b) (c) 40 3 m (d) 20 3 40 3 m m The ratio of the length of a tree and its shadow is 1 : 1 elevation of the sun is 7. (a) 30В° (b) 45В° (c) 60В° (d) 90В° 3 . The angle of A kite is flying at a height of 50 3 m above the level ground, attached to string inclined at 60В° to the horizontal, the length of string is 8. (a) 100 m (b) 50 m (c) 150 m (d) 75 m In given fig. 2 the perimeter of rectangle ABCD is D C 10 m 30В° A B Fig. 2 93 [X вЂ“ Maths] 9. 10. (a) 40 m (b) 20 пЂЁ 3 пЂ«1 m пЂ© (c) 60 m (d) 10 пЂЁ 3 пЂ«1 m пЂ© A tree is broken at a height of 10 m above the ground. The broken part touches the ground and makes an angle of 30В° with the horizontal. The height of the tree is (a) 30 m (b) 20 m (c) 10 m (d) 15 m If the shadow of a tree is пЂ± 3 times the height of the tree, then find the angle of elevation of the sun. (a) 30В° (b) 45В° (c) 60В° (d) 90В° C A пЃЎ B Fig. 3 11. In given fig. 4, D is mid point of BC, пѓђCAB = пЃЎ1 and пѓђDAB = пЃў2 then tan пЃЎ1 : tan пЃў2 is equal to 94 [X вЂ“ Maths] A пЃЎ1 C пЃў2 D B Fig. 4 12. (a) 2 : 1 (b) 1 : 2 (c) 1 : 1 (d) 1 : 3 In given fig. 5, tan пЃ± пЂЅ 8 if PQ = 16 m, then the length of PR is 15 (a) 16 m (b) 34 m (c) 32 m (d) 30 m P пЃ± R Q Fig. 5 13. The height of a tower is 50 m. When angle of elevation changes from 45В° to 30В°, the shadow of tower becomes x metres more, the value of x is (a) 50 m (b) 95 50 пЂЁ пЂ© 3 пЂ1 m [X вЂ“ Maths] (c) 14. (d) 50 3 m 50 3 m The angle of elevations of a building from two points on the ground 9m and 16m away from the foot of the building are complementary, the height of the building is (a) 18 m (b) 16 m (c) 10 m (d) 12 m LONG ANSWER TYPE QUESTIONS 15. A pole of height 5m is fixed on the top of the tower. The angle of elevation of the top of the pole as observed from a point A on the ground is 60В° and the angle of depression of the point A from the top of the tower is пЂЁ 45В°. Find the height of tower. Take 3 пЂЅ 1.732 пЂ© 16. From a point on the ground the angle of elevations of the bottom and top of a water tank kept on the top of the 30m high building are 45В° and 60В° respectively. Find the height of the water tank. 17. The shadow of a tower standing on the level ground is found to be 60m shorter when the sunвЂ™s altitude changes from 30В° to 60В°, find the height of tower. 18. The angle of elevation of a cloud from a point пЃ¬ metres above a lake is пЃЎ and the angle of depression of its reflection in the lake is пЃў, prove that 2пЃ¬ sec пЃЎ the distance of the cloud from the point of observation is tan пЃў пЂ tan пЃЎ . 19. The angle of elevation of a bird from a point on the ground is 60В°, after 50 seconds flight the angle of elevation changes to 30В°. If the bird is flying at the height of 500 3 m. Find the speed of the bird. 20. The angle of elevation of a jet fighter plane from a point A on the ground is 60В°. After a flight of 15 seconds, the angle of elevation changes to 30В°. If the jet is flying at a speed of 720 km/h. Find the constant height at which the jet is flying. 21. пЂЁ Take пЂ© 3 пЂЅ 1.732 . From a window 20m high above the ground in a street, the angle of elevation and depression of the top and the foot of another house opposite 96 [X вЂ“ Maths] side of the street are 60В° and 45В° respectively. Find the height of opposite house. 22. An aeroplane flying at a height of 1800m observes angles of depressions of two points on the opposite bank of the river to be 60В° and 45В°, find the width of the river. 23. The angle of elevation of the top of the tower from two points A and B which are 15m apart, on the same side of the tower on the level ground are 30В° and 60В° respectively. Find the height of the tower and distance of point B from he base of the tower. пЂЁ Take 3 пЂЅ 1.732 пЂ© 24. The angle of elevation of the top of a 10m high building from a point P on the ground is 30В°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flag staff from P is 45В°. Find the length of the flag staff and the distance of the building from point P. 25. The angle of elevation of a bird from a point 12 metres above a lake is 30В° and the angle of depression of its reflection in the lake is 60В°. Find the distance of the bird from the point of observation. 26. The shadow of a vertical tower on level ground increases by 10 mtrs. When sunвЂ™s attitude changes from 45В° to 30В°. Find the height of the tower, upto one place of decimal пЂЁ 3 пЂЅ 1.73 пЂ© . 27. A man on a cliff observes a boat at an angle of depression of 30В°, which is approaching the shore to point вЂ�AвЂ™ immediately beneath the observer with a uniform speed, 12 minutes later, the angle of depression of the boat is found to be 60В°. Find the time taken by the boat to reach the shore. 28. A man standing on the deck of a ship, 18m above the water level observes that the angle of elevation and depression of the top and the bottom of a cliff are 60В° and 30В° respectively. Find the distance of the cliff from the ship and height of the cliff. 29. A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60В°. When he moves 40m away from the bank he finds the angle of elevation to be 30В°. Find the height of the tree and the width of the river. 30. An aeroplane, when 300 m high, passes vertically above another plane at an instant when the angle of elevation of two aeroplanes from the same point on the ground are 60В° and 45В° respectively. Find the vertical distance between the two planes. 97 [X вЂ“ Maths] 31. The angle of depression of the top and bottom of a 10m tall building from the top of a tower are 30В° and 45В° respectively. Find the height of the tower and distance between building and tower. 32. A boy standing on a horizontal plane, finds a bird flying at a distance of 100m from him at an elevation of 30В°. A girl, standing on the roof of 20m high building, finds the angle of elevation of the same bird to be 45В°. Both the boy and girl are on the opposite sides of the bird. Find the distance of bird from the girl. 33. A man standing on the deck of a ship, which is 10m above the water level observes the angle of elevation of the top of the hill as 60В° and the angle of depression of the base of the hill is 30В°. Calculate the distance of the hill from the ship and the height of the hill. 34. The angle of elevation of a building from two points P and Q on the level ground on the same side of the building are 36В° and 54В° respectively. If the distance of the points P and Q from the base of the building are 10m пЂЁ and 20m respectively, find the height of the building. Take 2 пЂЅ 1.414 пЂ© ANSWERS 1. c 2. b 3. a 4. d 5. b 6. c 7. a 8. b 9. a 10. c 11. a 12. b 13. b 14. d 15. 6.83 m 16. 17. 30 3 m 19. 20 m/sec. 20. 2598 m 21. 98 30 20 пЂЁ пЂЁ пЂ© 3 пЂ1 m пЂ© 3 пЂ«1 m [X вЂ“ Maths] пЂЁ пЂ© 22. 600 3 пЂ« 23. Height = 12.97 m, distance = 7.5 m 24. Length of flag staff пЂЅ 10 25. 24 3 m 26. 13.6 mts. 27. 18 minutes 28. 18 3 m, 72 m 29. Height = 34.64 m, Width of the river = 20 m. 30. 1000 3 пЂ 31. Height пЂЅ 5 3 пЂ« 32. 30 m 33. Distance пЂЅ 10 3m , Height of the hill = 40 m 34. 14.14 m пЂЁ 3 m пЂЁ пЂ© 2 пЂ 1 m, Distance of the building пЂЅ 10 3 m. пЂ© 3 m пЂЁ пЂЁ пЂ© 3 m , distance пЂЅ 5 3 пЂ« 99 пЂ© 3 m [X вЂ“ Maths] CHAPTER 5 CIRCLE KEY POINTS 1. Tangent to a Circle : It is a line that intersects the circle at only one point. 2. There is only one tangent at a point of the circle. MULTIPLE CHOICE QUESTIONS 1. In the given fig. 1 PQ is tangent then пѓђPOQ + пѓђQPO is equal to Q P o Fig. 1 2. (a) 120В° (b) 90В° (c) 80В° (d) 100В° If PQ is a tangent to a circle of radius 5cm and PQ = 12 cm, Q is point of contact, then OP is (a) 13 cm (b) (c) 7 cm (d) 100 17 cm 119 cm [X вЂ“ Maths] 3. In the given fig. 2 PQ and PR are tangents to the circle, пѓђQOP = 70В°, then пѓђQPR is equal to Q 70В° P o R Fig. 2 4. (a) 35В° (b) 70В° (c) 40В° (d) 50В° In the given fig. 3 PQ is a tangent to the circle, PQ = 8 cm, OQ = 6 cm then the length of PS is Q o P S Fig. 3 5. (a) 10 cm (b) 2 cm (c) 3 cm (d) 4 cm In the given fig. 4 PQ is tangent to outer circle and PR is tangent to inner circle. If PQ = 4 cm, OQ = 3 cm and OR = 2 cm then the length of PR is 101 [X вЂ“ Maths] Q P o R Fig. 4 6. (a) 5 cm (b) (c) 4 cm (d) 21 cm 3 cm In the given fig. 5 P, Q and R are the points of contact. If AB = 4 cm, BP = 2 cm then the perimeter of пЃ„ABC is A B P c Q R o Fig. 5 (a) 12 cm (b) 8 cm (c) 10 cm (d) 9 cm 102 [X вЂ“ Maths] 7. In the given fig. 6 the perimeter of пЃ„ABC is A 3c m R Q 2 cm B C P 5 cm Fig. 6 8. 9. (a) 10 cm (b) 15 cm (c) 20 cm (d) 25 cm The distance between two tangents parallel to each other to a circle is 12 cm. The radius of circle is (a) 13 cm (b) 6 cm (c) 10 cm (d) 8 cm In the given fig. 7 a circle touches all sides of a quadrilateral. If AB = 6 cm, BC = 5 cm and AD = 8 cm. Then the length of side CD is D C B A Fig. 7 (a) 6 cm (b) 8 cm (c) 5 cm (d) 7 cm 103 [X вЂ“ Maths] 10. 11. In a circle of radius 17 cm, two parallel chords are drawn on opposite sides of diameter. The distance between two chords is 23 cm and length of one chord is 16 cm, then the length of the other chord is (a) 34 cm (b) 17 cm (c) 15 cm (d) 30 cm In the given fig. 8 P is point of contact then пѓђOPB is equal to B o 40В° P A Fig. 8 12. (a) 50В° (b) 40В° (c) 35В° (d) 45В° In the given fig. 9 PQ and PR are tangents to the circle with centre O, if пѓђQPR = 45В° then пѓђQOR is equal to Q o 45В° P R Fig. 9 (a) 90В° (b) 110В° (c) 135В° (d) 145В° 104 [X вЂ“ Maths] 13. In the given fig. 10 O is centre of the circle, PA and PB are tangents to the circle, then пѓђAQB is equal to A o Q 40В° P B Fig. 10 14. (a) 70В° (b) 80В° (c) 60В° (d) 75В° In the given fig. 11 пЃ„ABC is circumscribed touching the circle at P, Q and R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, then value of BC is A 4 P R 6 B Q C Fig. 11 (a) 6 cm (b) 14 cm (c) 10 cm (d) 18 cm 105 [X вЂ“ Maths] 15. In the given fig. 12 пЃ„ABC is subscribing a circle and P is mid point of side BC. If AR = 4 cm, AC = 9 cm, then value of BC is equal to A R B Q P C Fig. 12 (a) 10 cm (b) 11 cm (c) 8 cm (d) 9 cm QUESTIONS 16. In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length. 17. An incircle is drawn touching the equal sides of an isosceles triangle at E and F. Show that the point D, where the circle touches the third side is the mid point of that side. 18. The length of tangent to a circle of radius 2.5 cm from an external point P is 6 cm. Find the distance of P from the nearest point of the circle. 19. TP and TQ are the tangents from the external point of a circle with centre O. If пѓђOPQ = 30В°, then find the measure of пѓђTQP. 20. In the given fig. 13 AP = 4 cm, BQ = 6 cm and AC = 9 cm. Find the semi perimeter of пЃ„ABC. 106 [X вЂ“ Maths] A 9c m m 4c R P C Q 6 cm B Fig. 13 21. In the given fig. 14 OP is equal to the diameter of the circle with centre O. Prove that пЃ„ABP is an equilateral triangle. A P o B Fig. 14 22. In the given fig. (15) a semicircle is drawn outside the bigger semicircle. Diameter BE of smaller semicircle is half of the radius BF of the bigger semicircle. If radius of bigger semicircle is 4пѓ–3 cm. Find the length of the tangent AC from A on a smaller semicircle. B D E F A C Fig. 15 107 [X вЂ“ Maths] 23. A circle is inscribed in a пЃ„ABC having sides AB = 12 cm, BC = 8 cm and AC = 10 cm find AD, BE, CF. C F A E B D Fig. 16 24. On the side AB as diameter of a right angled triangle ABC a circle is drawn intersecting the hypotenuse AC in P. Prove that PB = PC. 25. Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that пѓђAPB = 2 пѓђOAB A O P B Fig. 17 26. If an isosceles triangle ABC in which AB = AC = 6 cm is inscribed in a circle of radius 9 cm, find the area of the triangle. 27. In the given fig. (18) AB = AC, D is the mid point of AC, BD is the diameter of the circle, then prove that AE = 1/4 AC. 108 [X вЂ“ Maths] E B A D C Fig. 18 28. In the given fig. (19) radii of two concentric circles are 5 cm and 8 cm. The length of tangent from P to bigger circle is 15 cm. Find the length of tangent to smaller circle. Q o P R Fig. 19 29. An incircle is drawn touching the sides of a right angled triangle, the base and perpendicular of the triangle are 6 cm and 2.5 cm respectively. Find the radius of the circle. 30. In the given fig. (20) AB = 13 cm, BC = 7 cm. AD = 15 cm. Find PC. A R B S C o 4 cm P Fig. 20 Q D 109 [X вЂ“ Maths] 31. In the given fig. (21) find the radius of the circle. A 23 cm 29 cm 5 cm R B S C o r Q P D Fig. 21 32. In the given fig. (22) if radius of circle r = 3 cm. Find the perimeter of пЃ„ABC. A C 3пѓ–5 cm o 3пѓ–5 cm B Fig. 22 33. A circle touches the side BC of a пЃ„ABC at P and AB and AC produced at Q and R respectively. Prove that AQ is half the perimeter of пЃ„ABC. 34. In the given fig. (23) XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that XA + AR = XB + BR. 110 [X вЂ“ Maths] P A O R X B Q Fig. 23 35. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. 36. Prove that in two concentric circles the chord of the larger circle which touches the smaller circle is bisected at the point of contact. 37. In the given fig. (24) PQ is tangent and PB is diameter. Find the value of x and y. P y A o x y Q 35В° B Fig. 24 38. In the given fig. (25) AC is diameter of the circle with centre O and A is point of contact, then find x. C x o B 40В° P A Q Fig. 25 111 [X вЂ“ Maths] 39. Prove that the length of tangents, drawn from an external point to a circle are equal. 40. In the given fig. (26) PA and PB are tangents from point P. Prove that KN = AK + BN. A K o C P N B Fig. 26 41. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which is tangent to the smaller circle. 42. In the given fig. (27) PA and PB are tangents to the circle with centre O. Prove that OP is perpendicular bisector of AB. A o P B Fig. 27 43. In the given fig. (28) PQ is chord of length 6 cm of the circle of radius 6 cm. TP and TQ are tangents. Find пѓђPTQ. o P T Q Fig. 28 112 [X вЂ“ Maths] ANSWERS 1. b 2. a 3. c 4. d 5. b 6. a 7. c 8. b 9. d 10. d 11. a 12. c 13. a 14. b 15. a 16. 30В° 18. 4 cm 19. 60В° 20. 15 cm 22. 12 cm 23. AD = 7 cm, BE = 5 cm, CF = 3 cm 26. 8пѓ–2 cm2 28. 2пѓ–66 cm 29. 1 cm. 30. 5 cm 31. 11 cm. 32. 32 cm 37. x = 35В°, y = 55В° 38. 40В° 41. 8 cm 43. 120В° 113 [X вЂ“ Maths] CHAPTER 6 CONSTRUCTIONS KEY POINTS 1. Construction should be neat and clean and as per scale given in question. 2. Steps of construction should be provided only to those questions where it is mentioned. QUESTIONS 1. Draw a line segment AB = 7 cm. Take a point P on AB such that AP : PB = 3 : 4. 2. Draw a line segment PQ = 10 cm. Take a point A on PQ such that PA 2 пЂЅ . Measure the length of PA and AQ. PQ 5 3. Construct a пЃ„ABC in which BC = 6.5 cm, AB = 4.5 cm and пѓђACB = 60В°. Construct another triangle similar to пЃ„ABC such that each side of new triangle is 4. 4 of the corresponding sides of пЃ„ABC. 5 Draw a triangle XYZ such that XY = 5 cm, YZ = 7 cm and пѓђXYZ = 75В°. Now construct a пЃ„ABC ~ пЃ„XYZ with its sides 3 times of the corresponding 2 sides of пЃ„XYZ. 5. Construct an isoscales triangle whose base is 8 cm and altitude 5 cm and then construct another triangle whose sides are 3 times the corresponding 4 sides of the given triangle. 114 [X вЂ“ Maths] 6. Draw an isosceles пЃ„ABC with AB = AC and base BC = 7 cm and vertical angle is 120В°. Construct пЃ„ABВґCВґ ~ пЃ„ABC with its sides 1 1 times of the 3 corresponding sides of пЃ„ABC. 7. Draw пЃ„PQR in which пѓђQ = 90В°, PQ = 6 cm, QR = 8 cm. Construct пЃ„PвЂ�QRВґ ~ пЃ„PQR with its sides equal to 2/3rd of corresponding sides of пЃ„PQR. 8. Construct a right angled triangle in which base is 2 times of the perpendicular. Now construct a triangle similar to it with base 1.5 times of the original triangle. 9. Draw an equilateral triangle PQR with side 5cm. Now construct пЃ„PQВґRВґ PQ 1 пЂЅ . such that PQВґ 2 10. Draw a circle of radius 4 cm with centre O. Take a point P outside the circle such that OP = 6cm. Draw tangents PA and PB to the circle. Measure the lengths of PA and PB. 11. Draw a line segment AB = 8 cm. Taking AB as diameter a circle is drawn with centre O. Now draw OPпЃћAB. Through P draw a tangent to the circle. 12. Draw a circle of radius OP = 3 cm. Draw пѓђPOQ = 45В° such that OQ = 5 cm. Now draw two tangents from Q to given circle. 13. Draw a circle with centre O and radius 3.5 cm. Draw two tangents PA and PB from an external point P such that пѓђAPB = 45В°. What is the value of пѓђAOB + пѓђAPB. 14. Draw a circle of radius 4 cm. Now draw a set of tangents from an external point P such that the angle between the two tangents is half of the central angle made by joining the points of contact to the centre. 15. Draw a line segment AB = 9 cm. Taking A and B as centres draw two circles of radius 5 cm and 3 cm respectively. Now draw tangents to each circle from the centre of the other. 16. Draw a circle of radius 3.5 cm with centre O. Take point P such that OP = 6 cm. OP cuts the circle at T. Draw two tangents PQ and PR. Join Q to R. Through T draw AB parallel to QR such that A and B are point on PQ and PR. 115 [X вЂ“ Maths] 17. Draw a circle of diameter 7 cm. Draw a pair of tangents to the circle, which are inclined to each other at an angle of 60В°. 18. Draw a circle with centre O and radius 3.5 cm. Take a horizontal diameter. Extend it to both sides to point P and Q such that OP = OQ = 7 cm. Draw tangents PA and QB one above the diameter and the other below the diameter. 116 [X вЂ“ Maths] CHAPTER 7 MENSURATION (Continued) SURFACE AREAS AND VOLUMES KEY POINTS 1. c = 2пЃ°r where c п‚® circumference of the circle пЃ° be taken as 22/7 or 3.14 (app.) and вЂ�rвЂ™ be the radius of the circle. 2. Area of circle = пЃ°r2 where вЂ�rвЂ™ is the radius of the circle. 3. Area of Semi circle пЂЅ 4. Area enclosed by two concentric circles пЃ°r 2 . 2 = пЃ°пЂ (R2 вЂ“ r2) r R = пЃ° (R + r) (R вЂ“ r ); R > r where вЂ�RвЂ™ and вЂ�rвЂ™ are radii of two concentric circles. 5. The arc length вЂ�lвЂ™ of a sector of angle вЂ�пЃ±вЂ™ in a circle of radius вЂ�rвЂ™ is given by пЃ± п‚ґ пЂЁ circumference of the circle пЂ© 360п‚° пЃ± 1 = Г— 2пЃ°r 360В° 180 пЃ± l пЂЅ п‚ґ пЃ°r 180п‚° l пЂЅ 6. If the arc subtends an angle пЃ±, then area of пЃ± п‚ґ пЃ°r 2 . the corresponding sector is 360п‚° 117 r r пЃ± l r [X вЂ“ Maths] 7. Angle described by minute hand in 60 minutes = 360В°. Angle described пѓ¦ 360п‚° пѓ¶ by minute hand in 1 minute пЂЅ пѓ§ пЂЅ 6 п‚°. пѓЁ 60 пѓ·пѓё 8. Total Surface area of cube of side a units = 6a2 units. 9. Volume of cube of side a units = a3 cubic units. 10. Total surface area of cuboid of dimensions l, b and h = 2(l Г— b + b Г— h + h Г— l) square units. 11. Volume of cuboid of dimensions l, b and h = l Г— b Г— h cubic units. 12. Curved surface area of cylinder of radius r and height h = 2пЃ°rh square units. 13. Total surface area of cylinder of radius r and height h = 2пЃ°r (r + h) square units. 14. Volume of cylinder of radius r and height h = пЃ°r2h cubic units. 15. Curved surface area of cone of radius r, height h and slant height l = пЃ°rl square units where l пЂЅ r 2 пЂ« h2 . 16. Total surface area of cone = пЃ°r (l + r) sq. units. 17. Volume of cone пЂЅ 18. Total surface area of sphere of radius r units = 4пЃ°r2 sq. units. 19. Curved surface area of hemisphere of radius r units 2пЃ°r2 sq. units. 20. Total surface area of a solid hemisphere of radius r units = 3пЃ°r2 sq. units. 21. Volume of sphere of radius r units пЂЅ 22. Volume of hemisphere of radius r units пЂЅ 23. Curved surface area of frustum = пЃ°l(r + R) sq. units, where l slant height of frustum and radii of circular ends are r and R. 24. Total surface area of frustum = пЃ°l (r + R) + пЃ°(r2 + R2) sq. units. 1 2 пЃ°r h cubic units. 3 118 4 3 пЃ°r cubic units. 3 2 3 пЃ°r cubic units. 3 [X вЂ“ Maths] 25. Volume of Frustum пЂЅ 1 пЃ°h пЂЁ r 2 пЂ« R2 пЂ« rR пЂ© cubic units. 3 MULTIPLE CHOICE QUESTIONS 1. 2. 3. 4. 5. 6. Find the area of circle whose diameter is вЂ�dвЂ™ (a) 2пЃ°d (b) пЃ°d 2 4 (c) пЃ°d (d) пЃ°d2 If the circumference and area of a circle are numerically equal then the radius of the circle is equal to (a) r = 1 (b) r = 7 (c) r = 2 (d) r = 0 The radius of a circle is 7 cm. What is the perimeter of the semi circle? (a) 36 cm (b) 14 cm (c) 7пЃ° cm (d) 14пЃ° cm The radius of two circles are 13 cm and 6 cm respectively. What is the radius of the circle which has circumference equal to the sum of the circumference of two circles? (a) 19пЃ° cm (b) 19 cm (c) 25 cm (d) 32 cm The circumference of two circles are in the ratio 4 : 5 what is the ratio of the areas of these circles. (a) 4 : 5 (b) 16 : 25 (c) 64 : 125 (d) 8 : 10 The area of an equilateral triangle is пѓ–3 m2, its one side is (a) 4 m (b) 3пѓ–3 m (c) 3 3 m 4 (d) 2 m 119 [X вЂ“ Maths] 7. 8. 9. 10. 11. 12. 13. The volume of a cuboid is 440 cm3. The area of its base is 66 cm2. What is its height? (a) 40 cm 3 (b) 20 cm 3 (c) 440 cm (d) 66 cm Volume of two cubes is in the ratio of 8 : 125. The ratio of their surface areas is (a) 8 : 125 (b) 2 : 5 (c) 4 : 25 (d) 16 : 25 If the length of arc of a circle is вЂ�lвЂ™ and radius is вЂ�rвЂ™ then the area of the sector is (a) l . r (b) l . r2 (c) lr 2 (d) l2. r An arc of a circle is of length 5пЃ° cm and the section it bounds has an area of 10пЃ° cm2. Then the radius of circle is : (a) 2 cm (b) 4 cm (c) 2пѓ–2 cm (d) 8 cm Three cubes each of side вЂ�aвЂ™ are joined from end to end to form a cuboid. The volume of the new cuboid : (a) a2 (b) 3a3 (c) a3 (d) 6a3 A wire is in the form of a circle of radius 7 cm. It is bent into a square the area of the square is : (a) 11 cm2 (b) 121 cm2 (c) 154 cm2 (d) 44 cm2 The area of a sector with radius 3 cm and angle of sector 40В° is (a) 2пЃ° cm2 (b) 120 пЃ° cm2 [X вЂ“ Maths] (c) 14. 15. 16. 3пЃ° cm2 (d) 4пЃ° cm2 Ratio of the volumes of the cylinder and cone having same height and same radius of the bases, is (a) 1 : 1 (b) 1 : 2 (c) 3 : 1 (d) 1 : 3 The radii of the circular ends of a frustum of height 16 cm are 8 cm and 20 cm. The slant height of the frustum is (a) 28 cm (b) 10 cm (c) 44 cm (d) 20 cm If вЂ�CвЂ™ and вЂ�AвЂ™ respectively be the circumference and area of the circle of radius вЂ�rвЂ™, then (a) C пЂЅ 1 Ar 2 (b) A пЂЅ 1 Cr 2 (c) C = 2 Ar (d) A = 2 Cr SHORT ANSWER TYPE QUESTIONS 17. What is the perimeter of a sector of angle 45В° of a circle with radius 7 cm. 18. From each vertex of trapezium a sector of radius 7 cm has been cut off. Write the total area cut off. 19. Write the ratio of the areas of two sectors of a circle having angles 120В° and 90В°. 20. How many cubes of side 4 cm can be cut from a cuboid measuring (16 Г— 12 Г— 8). 21. The diameter and height of a cylinder and a cone are equal. What is the ratio of their volume. 22. A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ratio in their volumes? 23. A bicycle wheel makes 5000 revolutions in moving 10 km. Write the perimeter of wheel. 121 [X вЂ“ Maths] 24. The sum of the radius of the base and the height of a solid cylinder is 15 cm. If total surface area is 660 cm2. Write the radius of the base of cylinder. 25. Find the height of largest right circular cone that can be cut out of a cube whose volume is 729 cm3. 26. What is the ratio of the areas of a circle and an equilateral triangle whose diameter and a side of triangle are equal. 27. The numerical difference between circumference and diameter is 30 cm. What is the radius of the circle? 28. The length of an arc of a circle of radius 12 cm is 10пЃ° cm. Write the angle measure of this arc. 29. The cost of fencing a circular field at the rate of Rs. 10 per meter is Rs. 440. What is the radius of the circular field? 30. Find the perimeter of the protactor if its diameter is 14 cm. 31. A path of 5 m is build round the circular park of radius 15m. Find the area of the path. 32. The radii of two circles are 4 cm and 3 cm respectively. Find the radius of a circle having area equal to the sum of the areas of the circles. 33. In the figure find length of arc AB if вЂ�OвЂ™ is the centre of the circle and 22 пѓ¶ пѓ¦ radius is 14 cm. пѓ§пѓЁ take пЃ° пЂЅ пѓ· 7 пѓё 14 cm o 90В° A B 34. ABC is an equilateral triangle of side 30m. A Cow is tied at vertex A by means of 10m long rope. What is the area the Cow can graze in? 35. Find the area of the four blades of same size of radius 20 cm and central angle 45В° of a circular fan. 122 [X вЂ“ Maths] 45В° 36. Find the perimeter of the shaded region. 4 cm A B 6 cm D 37. C Two concentric circles with centre вЂ�OвЂ™ have radii 7 cm and 14 cm. If пѓђAOC = 120В° what is the area of shaded region? o B 120В° A D C 123 [X вЂ“ Maths] 38. Find the perimeter of the shaded portion. A 39. C 14 14 B D 14 Find the circumference of the circle with centre вЂ�OвЂ™. P o 24 cm 7 cm R Q 40. The radius of two circles are in the ratio 3 : 4 and sum of the areas of two circles is equal to the area of third circle. What is the radius of third circle. If the radius of first is 6 cm. 41. What is the area of the largest triangle that can be inscribed in a semicircle of radius r cm. 42. A piece of wire 20 cm long is bent into an arc of a circle subtending an angle of 60В° at the centre then what is the radius of the circle? 43. The minute hand of a clock is пѓ–12 cm long. What is the area described by the minute hand between 8.00 a.m to 8.05 a.m.? 44. Find the area of shaded portion. 20 cm 20 cm 20 cm 20 cm 124 [X вЂ“ Maths] 45. Find the area of shaded portion. 2 cm 5 cm 2 cm 5 cm 46. In the figure find the area of sector. 60В° 3 cm 3 cm 47. ABCD is a square kite of side 4 cm. What is the are of the shaded portion. A D 4 cm 4 cm B 4 cm C 48. The volume of cube is 8a3. Find its surface area. 49. The length of a diagonal of a cube is 17.32 cm. Find the volume of cube (use пѓ–3 = 1.732). 125 [X вЂ“ Maths] 50. Three cubes of the same metal, whose edges are 6, 8, 10 cm are melted and formed into a single cube. Find the diagonal of the single cube. LONG ANSWER TYPE QUESTIONS 51. The height of frustum is 4 cm and the radii of two bases are 3 cm and 6 cm respectively. Find the slant height of the frustum. 52. Volume of right circular cylinder is 448пЃ° cm3 height of cylinder is 7cm. Find the radius. 53. If lateral surface area of a cube is 64 cm2. What is its edge? 54. The area of a rhombus is 24 cm2 and one of its diagonal is 8 cm. What is other diagonal of the rhombus? 55. What is the length of the largest rod that can be put in a box of inner dimensions 30cm, 24 cm and 18 cm? 56. Curved surface area of a cylinder is 16пЃ° cm2, radius is 4cm, then find its height. 57. 50 circular plates each of equal radius of 7 cm are placed one over the other to form a cylinder. Find the height and volume of the cylinder if 1 cm. thickness of plate is 2 58. A well of diameter 2m is dug 14 m deep. Find the volume of the earth dug out. 59. A largest sphere is carved out of a cube of side 7 cm. Find the radius. 60. If the semi vertical angle of a cone of height 3 cm is 60В°. Find its volume. 61. Find the edge of cube if volume of the cube is equal to the volume of cuboid of dimensions (8 Г— 4 Г— 2) cm. 62. Find the volume of cone of height 2h and radius r. 63. Is it possible to have a right circular cylinder closed at both ends, whose flat area is equal to its total curve surface. 64. In a shower, there is 5 cm rain falls. Find in cubic meter the volume of water that falls on 2 hectares of ground. (1 hectare = 10000 m2). 126 [X вЂ“ Maths] 65. Find the total surface area of a solid hemisphere of Radius вЂ�RвЂ™. 66. In figure, пЃ„ABC is equilateral triangle. The radius of the circle is 4 cm. Find the area of shaded portion. 4 cm A o m 4c 4 cm B 67. C Find the area of Shaded portion. 6c m 60В° 12 cm 12 cm 12 cm 68. Four Cows are tied with a rope of 7 cm at four corners of a quadrilateral field of unequal sides. Find the total area grazed. 69. A solid consists of a right circular cylinder with a right circular cone at the top. The height of cone is вЂ�hвЂ™ cm. The total volume of the solid is 3 times the volume of the cone. Find the height of the cylinder. 127 [X вЂ“ Maths] 70. A cylindrical vessel of 36 cm height and 18 cm radius of the base is filled with sand. The sand is emptied on the ground and a conical heap of sand is formed. The height of conical heap is 27 cm. Find the radius of base of sand. 71. The radii of circular ends of bucket are 5.5 cm and 15.5 cm height is 24 cm. Find the total surface area of bucket. 72. Water flows out through a circular pipe whose internal diameter is 2 cm at the rate of 6m/sec. into a cylindrical tank. If radius of base of the tank is 60 cm. How much will the level of the water rise in half an hour? 73. In the figure along side. Find the area of the Shaded portion. and its 5 cm 12 cm 74. Find the shaded area in the given figure. 28 cm 28 cm 128 [X вЂ“ Maths] 75. Find the shaded area, in the figure. 14 cm 14 cm 76. AB and CD are two perpendicular diameters and CD = 8 cm find the area of Shaded portion. A D o C B 77. In the adjoining figure ABC is a right angled triangle, right angled at A. Semi circles are drawn on AB, AC as diameters. Find the area of shaded portion. 129 [X вЂ“ Maths] A 4 cm 3 cm C B 78. A toy is in the form of a conemounted on a cone frustum. If the radius of the top and bottom are 14 cm and 7 cm and the height of cone and toy are 5.5 cm and 10.5 cm respectively. Find the volume of toy adj. fig. 14 cm 10.5 cm 5.5 cm 7 cm 79. In the adjoining figure, ABC is a right angled triangle at A. Find the area of Shaded region if AB = 6 cm, BC = 10 cm and O is the centre of the incircle of пЃ„ABC (take пЃ° = 3.14). 130 [X вЂ“ Maths] A 6 cm o C 10 cm B 80. The volume and surface area of a sphere are numerically equal. Find the radius of the sphere. 81. Find the perimeter of the figure in which a semicircle is drawn on BC as diameter, пѓђBAC = 90В°. A 5 cm 12 cm B 82. C Find the area of shaded region in the figure. 14 cm 9 cm 9 cm 14 cm 131 [X вЂ“ Maths] ANSWERS 1. b 2. c 3. c 4. b 5. b 6. d 7. b 8. c 9. c 10. b 11. b 12. b 13. b 14. c 15. d 16. b 17. 19.5 cm 18. 154 cm2 19. 4 : 3 20. 24 21. 3 : 1 22. 3 : 1 : 2 23. 20 m 24. 7 cm 25. 9 cm 26. пЃ° : пѓ–3 27. 14 cm 28. 150В° 29. 7 m 30. 36 cm 31. 550 m2 32. 5 cm 33. 22 cm 34. 35. 200пЃ°пЂ cm2 36. (16 + пЃ°) cm 37. 154 cm2 38. 42пЃ° 39. 25пЃ°пЂ cm 40. 10 cm 41. r2 42. 43. пЃ° cm2 44. 86 cm2 45. (25 вЂ“ 4пЃ°) cm2 46. 3пЃ° cm2 47. (16 вЂ“ 4пЃ°) cm2 48. 24 a2 49. 1000 cm3 50. 12пѓ–3 cm 132 50 пЃ°m2 3 60 cm пЃ° [X вЂ“ Maths] 51. 5 cm 52. 8 cm 53. 4 cm 54. 6 cm 55. 30пѓ–2 cm 56. 2 cm 57. 25 cm; 3850 cm3 58. 44 m3 59. 3.5 cm 60. 27пЃ° 61. 4 cm 62. 63. Yes, when r = h 64. 1000 m3 65. 3пЃ°R2 66. 29.46 cm2 67. пѓ¦ 660 пѓ¶ пЂ« 36 3 пѓ· cm2 пѓ§пѓЁ пѓё 7 68. 154 cm2 69. 2 h. 3 70. 36 cm 71. 1716 cm2 72. 3 m 73. 1019 cm2 14 74. 154 m2 75. 77 cm2 76. 77. 6 cm2 78. 2926 cm3 79. 11.44 cm2. 2 пЃ° . r2. h 3 108 cm2 7 [Hint : Join 0 to A, B and C. area of пЃ„ABC = area of пЃ„OAB + area of пЃ„OBC + area of пЃ„OAC пЂЅ пѓћ 1 1 1 AB п‚ґ r пЂ« п‚ґ BC п‚ґ r пЂ« AC п‚ґ r 2 2 2 (r = 2 cm)] 80. 3 units 82. 49 cm2 81. 133 37 3 cm 7 [X вЂ“ Maths] CHAPTER 8 PROBABILITY 1. The Theoretical probability of an event E written as (E) is P пЂЁE пЂ© пЂЅ Number of outcomes favourable to E Number of all possible outcomes of the experiment. 2. The sum of the probability of all the elementary events of an experiment is 1. 3. The probability of a sure event is 1 and probability of an impossible event is 0. 4. If E is an event, in general, it is true that P(E) + P (E ) = 1. 5. From the definition of the probability, the numerator is always less than or equal to the denominator therefore O п‚Ј P(E) п‚Ј 1. MULTIPLE CHOICE QUESTIONS 1. 2. 3. If E is an event then P(E) + P пЂЁ E пЂ© = ........ ? (a) 0 (b) 1 (c) 2 (d) вЂ“1 The probability of an event that is certain to happen is : (a) 0 (b) 2 (c) 1 (d) вЂ“1 Which of the following can not be the probability of an event : (a) 2 3 (b) вЂ“3 2 (c) 15% (d) 0.7 134 [X вЂ“ Maths] 4. 5. 6. 7. 8. 9. If P(E) is .65 what is P (Not E)? (a) .35 (b) .25 (c) 1 (d) 0 If P(E) is 38% of an event what is the probability of failure of this event? (a) 12% (b) 62% (c) 1 (d) 0 A bag contains 9 Red and 7 blue marbles. A marble is taken out randomly, what is the P (red marble)? (a) 7 16 (b) 9 16 (c) 18 16 (d) 14 16 In a Survey it is found that every fifth person possess a vehicle what is the probability of a person not possessing the vehicle? (a) 1 5 (b) 4 5 (c) 3 5 (d) 1 Anand and Sumit are friends what is the probability that they both have birthday on 11th Nov. (ignoring leap year). (a) 1 12 (b) 1 7 (c) 1 365 (d) 1 366 The number of face cards in a well shuffled pack of cards are : (a) 12 (b) 16 (c) 4 (d) 52 135 [X вЂ“ Maths] 10. 11. 12. 13. A die is thrown once. What is the probability of getting an even prime number? (a) 3 6 (b) 1 6 (c) 1 2 (d) 1 3 The probability of an impossible event is : (a) 0 (b) 1 (c) вЂ“1 (d) п‚Ґ From the letters of the word вЂњMobileвЂќ, a letter is selected. The probability that the letter is a vowel, is (a) 1 3 (b) 3 7 (c) 1 6 (d) 1 2 An arrow pointer is spined which is placed on a fixed circular number plate numbered from 1 to 12 at equal distance. The pointer is equally likely to rest at any number. What is the probability that it will rest at (a) number 10 (b) an odd number (c) a number multiple of 3 (d) an even number SHORT ANSWER TYPE QUESTIONS 14. Two dice are rolled once what is the probability of getting a doublet? 15. A die is rolled once. What is the probability of getting a prime number? 16. A bank A.T.M. has notes of denomination 100, 500 and 1000 in equal numbers. What is the probability of getting a note of Rs. 1000. 17. What is the probability of getting a number greater than 6 in a single throw of a die. 136 [X вЂ“ Maths] 18. A selection committee interviewed 50 people for the post of sales manager. Out of which 35 are males and 15 are females. What is the probability of a female candidate being selected. 19. A bag contains cards numbering from 5 to 25. One card is drawn from the bag. Find the probability that the card has numbers from 10 to 15. 20. In 1000 lottery tickets there are 5 prize winning tickets. Find the probability of winning a prize if a person buys one tickets. 21. It is known that in a box of 600 screws, 42 screws are defective. One screw is taken out at random from this box. Find the probability that it is not defective. 22. Write all the possible outcomes when a coin is tossed twice. 23. Two dice are rolled simultaneously. Find the probability that the sum is more than and equal to 10. 24. From the well shuffled pack of 52 cards. Two Black kings and Two Red Aces are removed. What is the probability of getting a face card. 25. In a leap year what is the probability of 53 Sundays. 26. A box contains cards numbered from 2 to 101. One card is drawn at random. What is the probability of getting a number which is a perfect square. 27. Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. What is the probability that the ticket has a number which is a multiple of 3 or 7? 28. A bag contains 5 red balls and вЂ�nвЂ™ green balls. If the P(green ball) = 3 Г— P (red ball) then what is the value of n. 29. If from the well shuffled pack of cards all the aces are removed, find the probability of getting red card. 30. What is the probability of getting a total of less than 12 in the throws of two dice? 31. From the data (1, 4, 9, 16, 25, 29). If 29 is removed what is the probability of getting a prime number. 32. A card is drawn from an ordinary pack of playing cards and a person bets that it is a spade or an ace. What are the odds against his winning the bet. 137 [X вЂ“ Maths] LONG ANSWER TYPE QUESTIONS 33. A coin is tossed thrice then find the probability of (i) 2 heads (ii) 2 tails (iii) 3 heads. 34. The king, queen and jack of clubs are removed from a deck of 52 playing cards and the remaining cards are shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of (i) heart; (ii) queen; (iii) Clubs. 35. A box contains 5 Red balls, 8 white balls and 4 Green balls. One ball is taken out of the box at random. What is the probability that ball is (i) red; (ii) white; (iii) Not green. 36. 12 defective pens are mixed with 120 good ones. One pen is taken out at random from this lot. Determine the probability that the pen taken out is not defective. 37. A number x is selected from the numbers 1, 2, 3 and then a second number y is randomly selected from the numbers 1, 4, 9. What is the probability that the product of two numbers will be less than 9? 38. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two digit number (ii) a perfect square number (ii) a number divisible by 5. 39. A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Anand wins if all the tosses give the same result i.e., three heads or three tails and loses otherwise. Calculate the probability that Anand will lose the game. 40. A die is thrown twice. What is the probability of getting : (i) The Sum of 7; (ii) The sum of greater than 10; (iii) 5 will not come up either time. 41. A card is drawn at randown from a well shuffled deck of playing cards. Find the probability that the card drawn is (i) (iv) a card of spade or an ace (ii) a red king either a king or a queen (iii) neither a king nor a queen 138 [X вЂ“ Maths] 42. A jar contains 24 balls, some are green and other are blue. If a ball is 2 drawn at random from the jar, the probability that it is green is . Find 3 the number of blue balls in the jar. ANSWERS 1. b 2. c 3. b 4. a 5. b 6. b 7. b 8. c 9. a 10. b 11. a 12. d 13. (i) 15. 14. 1 6 1 2 16. 1 3 17. 0 18. 3 10 19. 2 7 20. 1 200 21. 93 100 22. S = [HH, TT, HT, TH] 23. 1 6 24. 5 24 25. 2 7 26. 9 100 1 1 1 1 ; (ii) ; (iii) ; (iv) 12 2 3 2 139 [X вЂ“ Maths] 27. 2 5 28. 15 29. 1 2 30. 35 36 31. zero 32. 9 13 33. (i) 3 ; 8 34. (i) 13 ; 49 (ii) 3 ; 49 (iii) 35. (i) 5 ; 17 (ii) 8 ; 17 (iii) 13 17 36. 9 10 37. 5 9 38. (i) 9 ; 10 (ii) 1 ; 10 (iii) 1 5 39. 3 4 40. (i) 1 ; 6 41. (i) 4 1 11 2 ; (ii) ; (iii) ; (iv) 13 26 13 13 42. 8. (ii) (ii) 3 ; 8 1 ; 12 (iii) (iii) 1 8 10 49 25 36 140 [X вЂ“ Maths] SAMPLE QUESTION PAPER - I MATHEMATICS (SA - 1I) Time allowed : 3 hours Maximum Marks : 90 General Instructions 1. All questions are compulsory. 2. The question paper consists of 34 questions divided into four sections A, B, C and D. Section A comprises of 8 questions of 1 mark each. Section B comprises of 6 questions of 2 marks each. Section C comprises of 10 questions of 3 marks each and Section D comprises of 10 questions of 4 marks each. 3. Question numbers 1 to 8 in Section A are multiple choice questions where you are to select one correct option out of the given four. 4. There is no overall choice. 5. Use of calculator is not permitted. SECTION A 1. 2. The discriminant of quadratic equation x2 вЂ“ 4x + 1 = 0 is : (a) вЂ“20 (b) 20 (c) 12 (d) пѓ–12 In figure, PQ and PR are tangents to the circle with O, such that пѓђQPR = 50В°, пѓђOQR is equal to : Q O P R 141 [X вЂ“ Maths] 3. 4. 5. 6. (a) 25В° (b) 30В° (c) 40В° (d) 50В° How many tangent(s) to a circle can be constructed from an external point? (a) Three (b) One (c) Two (d) No The length of the shadow of a tower is пѓ–3 times that of its length. The angle of elevation of the sun is : (a) 90В° (b) 30В° (c) 45В° (d) 60В° Which of the following number cannot be the probability of an event ? (a) 1 3 (b) 0.1 (c) 3% (d) 17 16 If the probability of happening of an event is 3 , then the probability of 7 non-happening of this event is : 7. 8. (a) 0 (b) 1 (c) 2 7 (d) 4 7 The mid-point of segment AB is the point (0, 4). If the co-ordinates of B are (вЂ“2, 3), then the co-ordinates of A are : (a) (2, 5) (b) (вЂ“2, вЂ“5) (c) (2, 9) (d) (вЂ“2, 11) If the area of a circle is numerically equal to twice its circumference, then the diameter of the circle is : (a) 4 units (b) пЃ° units (c) 8 units (d) 2 units 142 [X вЂ“ Maths] SECTION B 9. If the equation kx2 вЂ“ 2kx + 6 = 0 has equal roots, then find the value of k. 10. Which term of the arithmetic progression 3, 10, 17, .... will be 84 more than its 13th term ? 11. Prove that in two concentric circles, the chord of the larger circle, which touches the smaller circle is bisected at the point of contact. 12. In the given figure, BOA is a diameter of a circle and the tangent at a point P meets BA produced at T. If пѓђPBO = 30В°, what is the measure of пѓђPTA? P B 13. 14. 30В° O A T An integer is chosen between 0 and 100. Find the probability that the number is dividible by 3 and 5 both. 22 пѓ¶ пѓ¦ If the perimeter of a protractor is 72 cm, calculate its area. пѓ§пѓЁ Take пЃ° пЂЅ пѓ· 7 пѓё SECTION C 15. Solve for x : 1 1 1 1 пЂЅ пЂ« пЂ« ; where a п‚№ 0, b п‚№ 0, x п‚№ 0 and a + b + x п‚№ 0. aпЂ«b пЂ«x a b x 16. The first and the last terms of an A.P. are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum? 17. Draw a triangle ABC in which BC = 6.5 cm, AB = 4.5 cm and пѓђABC = 3 60В°. Also construct a triangle similar to this triangle whose sides are 4 of the corresponding sides of the triangle ABC. 18. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eye of the girl at any instant is 60В°. After sometimes the 143 [X вЂ“ Maths] angle of elevation reduces at 30В°. Find the distance travelled by balloon during the interval (Take пѓ–3 = 1.7). 19. Find a point on the y-axis which is equidistant from the points A(6, 5) and B(вЂ“4, 3). 20. In what ratio does the point (ВЅ, 6) divide the line segment joint the point (3, 5) and (вЂ“7, 9). 21. A chord 10 cm long is drawn in a circle whose radius is пѓ–50 cm. Find the area of segments. 22. Find the area of the shaded region in the given figure, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle. Q O R P 23. The largest sphere is curved out of a cube of side 7 cm. Find the volume of sphere. 24. A glass cylinder with diameter 20 cm has water to a height of 9 cm. A metal cube of 8 cm edge is immersed in it completely. Calculate the 22 пѓ¶ пѓ¦ height by which water will rise in the cylinder. пѓ§пѓЁ Use пЃ° пЂЅ пѓ·. 7 пѓё SECTION D 25. 6500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got 30 less. Find the original number of persons. 26. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees that each section of each class will plant, will be the same as the class in which they are studying, e.g., a section of class I will plant one tree, a section of class II will plant two trees and so on till class XII. There are three sections of each class. Find : (a) How many trees will be planted by the students? 144 [X вЂ“ Maths] (b) Which mathematical concept is used in above problem? (c) Which value is depicted in this problem? 27. The sum of 5th and 9th terms of A.P. is 72, and the sum of 7th and 12th terms is 97. Find the A.P. 28. Prove that the length of tangents drawn from an external point to a circle are equal. 29. Prove that the intercept of a tangent between a pair of parallel tangents to a circle subtend a right angle at the centre of the circle. (Figure is given). O 30. There is a small island in the middle of a 100 meter wide river and a tall tree stands on the island. P and Q are points directly opposite to each other on two banks and in line with the tree. If the angles of elevation of the top of the tree from P and Q are respectively 30В° and 45В°, find the height of the tree. 31. All the three face cards of spades are removed from a well shuffled pack of 52 cards. A card is drawn at random from the remaining pack. Find the probability of getting : (a) a black-faced card (b) a queen. (c) a black card (d) a spade 32. Find the value of K, if the points A (2, 3), B (4, K) and C (6, вЂ“3) are collinear. 33. A cylinderical bucket, 32 cm high and with radius of base 18cm, is filled with sand. This bucket is emptied on the gound and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap. 34. A cap is shaped like the frustum of a cone. If its radius on the open side is 10 cm, and radius at the upper base is 4 cm and its slant height is 15 cm, find the area of cloth used for making it. 145 [X вЂ“ Maths] SAMPLE QUESTION PAPER - II MATHEMATICS (SA - 1I) Time allowed : 3 hours Maximum Marks : 90 General Instructions 1. All questions are compulsory. 2. The question paper consists of 34 questions divided into four sections A, B, C and D. Section A comprises of 8 questions of 1 mark each. Section B comprises of 6 questions of 2 marks each. Section C comprises of 10 questions of 3 marks each and Section D comprises of 10 questions of 4 marks each. 3. Question numbers 1 to 8 in Section A are multiple choice questions where you are to select one correct option out of the given four. 4. There is no overall choice. 5. Use of calculator is not permitted. SECTION A 1. 2. If one root of 2x2 + kx + 1 = 0 is вЂ“ 1 , then the value of k is 2 (a) 3 (b) вЂ“3 (c) 5 (d) вЂ“5 In the figure, if TP and TQ are two tangents to a circle with centre O then x + y + z is P x O Z y T Q 146 [X вЂ“ Maths] 3. 4. 5. 6. 7. 8. (a) 120В° (b) 150В° (c) 180В° (d) 360В° If two circles touch each other externally, then the number of common tangents are (a) 4 (b) 3 (c) 2 (d) 1 The length of the shadow of a man is equal to the height of man, the angle of elevation is (a) 90В° (b) 60В° (c) 45В° (d) 30В° If E be an event associated with a random experiment and 0 п‚Ј P(E) п‚Ј x, then the value of x is : (a) 0 (b) 1 (c) 2 (d) None of these From the letter of the word вЂ�вЂ�EducationвЂ™вЂ™ a letter is selected, the probability that the letter is a vowel is : (a) 7 9 (b) 2 9 (c) 4 9 (d) 5 9 The distance between two points A( 7 sin 43В°, 0) and B(0, 7 sin 47В°) is : (a) 7 (b) 49 (c) 1 (d) 0 The circumference of two circles are in the ratio 2 : 3, the ratio of their areas will be : (a) 2 : 3 (b) 4 : 9 (c) 9 : 4 (d) 8 : 27 147 [X вЂ“ Maths] SECTION B 9. Find the value of m so that the quadratic equation x2 вЂ“ 2x (1 + 3m) + 7(3 + 2m) = 0 has equal roots. 10. How many two digit numbers are there in between 6 and 102 which are divisible by 6 ? 11. A circle is inscribed in a пЃ„PQR having sides PQ = 10 cm, QR = 8 cm and PR = 12 cm. R U P T S Q Find PS and QT. 12. In the given figure, radii of two concentric circles are 5 cm and 8 cm. The length of tangent from P to bigger circle is 15 cm. Find the length of tangent to smaller circle. Q O P R 13. A card is drawn at random from a pack of 52 playing cards. Find the probability that the card drawn is neither an ace nor a king. 14. Find the diameter of a circle whose area is equal to the sum of area of the two circles of radii 24 cm and 7 cm. 148 [X вЂ“ Maths] SECTION C 15. Solve the equation 2x2 вЂ“ 5x + 3 = 0 by the method of completing the square. 16. Find the sum of first 15 terms of an A.P. whose nth term is 9 вЂ“ 5n. 17. Draw a triangle ABC with side BC = 7 cm, пѓђB = 45В°, пѓђA = 105В°, Then, 4 construct a triangle whose sides are times the corresponding sides of 3 пЃ„ABC. 18. As observed from the top of a 75 m high light house from the sea-level, the angles of depression of two ships are 30В° and 45В°. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. 19. Find a point on y-axis which is equidistant from the point (вЂ“2, 5) and (2, вЂ“3). 20. Find the ratio in which the y-axis divides the line segment joining the points (5, вЂ“6) and (вЂ“1, 4). 21. Find the area of shaded region in the following figure : A 7cm 7cm C 7cm 7cm 7cm B 22. The numerical difference between circumference and diameter is 30 cm. 22 пѓ¶ пѓ¦ What is the radius of the circle? пѓ§пѓЁ Take пЃ° пЂЅ пѓ· 7пѓё 23. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10km/h. How much area in hectares will it irrigate in 30 minute, if 8 cm of standing water is needed? (1 hectare = 10000 m2) 24. The total surface area of a solid right circular cylinder is 231 cm2, its 2 rd of the total surface area. Find the radius of the curved surface is 3 base and height. 149 [X вЂ“ Maths] SECTION D 25. Usha surveyed a class on World Food Day and observed that the number of students who like Junk food is 4 more than the number of students who like home made food. The sum of the squares of the number of two types of students is 400. Find the number of students who like Junk food and who like home made food? 26. If the sum of three consecutive terms of an increasing A.P. is 51 and the product of first and third terms is 273, find the third term. 27. An old lady Krishna Devi deposited Rs. 12000 in a bank at 8% simple interest p.a. She uses the annual interest to give five scholarships to the students of a school for their overall performances each year. The amount of each scholarship is 300 less than the preceding scholarship. Answer the following (i) Find amount of each scholarship? (ii) What values of the lady are reflected? 28. Prove that the tangents drawn at the ends of a diameter of a circle are parallel. 29. A quadrilateral ABCD is drawn to circumscribe a circle, Prove that AB + CD = AD + BC. 30. An eagle is sitting on the top of a tower, which is 80m high. The angle of elevation of the eagle from a point on the ground is 45В°. Th eagle flies away from the point of observation horizontally and remains at a constant height. After 2 seconds, the angle of elevation of the eagle from the point of observation become 30В°. Find the speed of flying the eagle. 31. A bag contains some cards which are numbered between 31 and 96 are placed. If one card is drawn from the bag, find the probability that the number on card is : (i) a multiple of 3 (ii) a perfect square (iii) a number not more than 40. 150 [X вЂ“ Maths] 32. The slant height of a frustum of a cone is 4 cm and the perimeter of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum. 33. If A(вЂ“5, 7), B(вЂ“4, вЂ“5), C(вЂ“1, вЂ“6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD. 34. A cone of radius 10 cm is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base, compare the volume of the two parts. ANSWERS 1. a 2. c 3. d 4. c 5. b 6. d 7. a 8. b 9. вЂ“ 10 ,2 9 10. 15 11. PS = 7 cm, QT = 3 cm 12. 13. 11 13 14. 50 cm 15. 3 ,1 2 16. вЂ“465 18. 75 пЂЁ 3 вЂ“ 1пЂ© m 19. (0, 1) 20. 5 : 1 21. 24.5 пЃ° cm2 22. 7 cm 23. 56.25 hectares 24. 3.5 cm, 7 cm 25. 12, 16 151 2 66 cm [X вЂ“ Maths] 26. 21 27. (i) 2520, 2220, 1920, 1620, 1320 (ii) Love for children, charity. 29. 29.28 m/sec. 31. (i) 32. 48 cm2 33. 72 square units 5 12 (ii) 1 16 (iii) 9 64 34. 1 : 7 152 [X вЂ“ Maths]

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