MATHS QUESTION XII – (CBSE) RELATIONS AND FUNCTIONS Each question carries 1/2 marks 6. If f(x) = x + 7 and g(x) = x – 7, x R, find (fog) (7). Let * be a binary operation, defined by a * b = 3a + 4b – 2, find 4 * 5. If the binary operation * on the set of integers Z, is defined by a * b = a + 3b2 , the find the value of 2 * 4. Let * be a binary operation on N given by a * b = HCF (a, b), a, b N. Write the value of 22 * 4. If the binary operation *, defined on Q is defined as a * b = 2a + b – ab, for all a, b Q, find the value of 3 * 4. If f : R R be defined by f(x) = (3 – x3 )1/3 , then find fof (x). 7. If f is an invertible function, defined as f (x) 8. If f : R R and g : R R are given by f (x) = sin x and g(x) = 5x2 , find gof (x). If f(x) = 27x3 and g (x) = 1/3 , find gof (x). State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not. If f : R R is defined by f (x) = 3x + 2, define f (f(x)). Write fog, if f : R R and g : R R are given by f (x) = | x | and g(x) = |5x – 2|. Write fog, if f : R R and g : R R are given by f (x) = 8x3 and g(x) = x1/3 . Let * be a binary operation on N given by a * b = lcm (a, b) for all a, b N. Find 5 * 7. The binary operation * : R × R R is defined as a * b = 2a + b. Find (2 * 3) * 4. If the bhinary operation * on the set Z of integers is defined by a * b = a + b – 5, the write the identify element for the operation * in Z. 1. 2. 3. 4. 5. 9. 10. 11. 12. 13. 14. 15. 16. 17. 3x 4 , write f 1 (x). 5 Each question carries 4/6 marks 1. 2. 3. 4. 5. Show that the relation R defined by (a, b) R (c, d) a + d = b + c on the set N × N is an equivalence relation. Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b N} is an equivalence relation. Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. n 1 , if n is odd 2 Let f : N N be defined by f (n) for all n N. Find whether the function f is bijective. n , if n iseven 2 Show that the relation R in the set of real numbers, defined as R = {(a, b) : a b2 } is neither reflexive, nor 6. symmetric, nor transitive. Let Z be the set of all integers and R be the relation on Z defined as R = {(a, b) : a, b by 5}. Prove that R is an equivalence relation. 7. Let * be a binary operation on Q, defined by a *b Z and (a – b) is divisible 3ab . Show that * is commutative as well as associative. Also 5 find its identify, if it exists. 8. Show that the relation S in the set R of real numbers, defined as S = {(a, b) : a, b R and a b3} is neither reflexive, nor symmetric nor transitive. Page 1 9. Show that the relation S in the set A {x Z:0 x 12} given by S {(a,b):a, b Z,| a b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. y 6 1 . 3 [ 5, ) given by f (x) 9x 2 6x 5. Show that f is invertible with f 1 (y) 10. Consider f : R 11. 13. Let A = N × N and * be a binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Show that * is commutative and associative. Also, find the identify element for * on A, if any. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write the operation table of the operation*. Let f : R R be defined as f(x) = 10x + 7. Find the function g:R R such that gof = fog = IR. 14. A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as : a * b = 12. a b, if a b 6 Show that zero a b 6, if a b 6 is the identify for this operation and each element a of the set is invertible with 6 – a, being the inverse of a. 15. f 1 (y) 16. [4, ) given by f (x) x 2 4. Show that f is invertible with the inverse f Consider f : R 1 of f given by y 4, where R+ is the set of all non-negative real numbers. Let A = R – {3} and B = R – {1}. Consider the function f : A B defined by f (x) x 2 . Show that f is onex 3 one and onto and hence find f 1. x 1, if x is odd , is bijective (both one-one and onto). x 1, if x is even 17. Show that f : N N, given by f (x) 18. Consider the binary operation * : R R R and o: R R R defined as a * b = |a – b| and a o b = a for all a, b R. Show that * is commutative but not associative, o is associative but not commutative. 19. If f (x) 4x 3 2 ,x , show that f o f (x) = x for all x 6x 4 3 2 . What is the inverse of f? 3 INVERSE TRIGONOMETRIC FUNCTIONS Each question carries 1/2 marks 1. Show that sin 1(2x 1 x2 ) 2sin 1 x. 2. Solve for x: tan 1 (x 1) tan 1 (x 1) tan 3. Prove that 2tan 4. Prove that sin 5. Find the principal value of sin 6. Write the principal value of sin 1 7. Write the principal value of cos 1 8. Write the principal value of sin 1 1 11 5 12 13 tan 11 cos 8 1 tan 4 5 1 sin 8 . 31 4 . 7 tan 1 1 63 16 1 . 3 . 5 3 . 2 3 . 2 1 2 cos 1 1 . 2 Page 2 9. Write the principal value of sec 1 ( 2). 10. Write the principal value of cot 1 ( 11. Write the principal value of sin 12. What is the domain of the function sin 1 x? 13. Evaluate sin 14. Write the principal value of cos 1 cos 7 . 6 15. Write the principal value of tan 1 tan 3 . 4 16. Find the principal value of tan 1( 1). 17. Using principal value evaluate the following : cos 18. Solve for x : tan 19. sin 3 1 1 3). sin 4 . 5 1 . 2 1 tan 1 x, x 0. 2 1 2sin Write the principal value of cos 1 2 1 cos 2 3 sin 1 sin 2 . 3 11 x 1 x 20. Find the principal value of tan 21. Find the value of cot (tan 1 1 1 . 2 1 3 sec 1 ( 2). cot 1 ). Each question carries 4/6 marks 1 4 5 sin 1 5 13 1. Prove that sin 2. Solve for x : tan 1 3x tan 1 2x sin 1 16 65 3. 4 Solve for x : 2tan 1 (cos x) tan 1 (2 cosec x) 4. Prove the following : tan 5. Solve the following for x : cos 6. Solve for x : tan 7. 1 1 4 tan 1 2 9 1 x2 1 x2 1 2 1 3 cos 1 2 5 tan 2x 1 x 2 1 2 3 x 1 x 1 tan 1 . x 2 x 2 4 1 1 x cos 1 , x (0,1) Prove the following : tan 1 x 2 1 x 1 1 12 13 Prove the following : cos 9. Prove that tan 1 (1) tan 1 (2) tan 1 (3) 10. Prove that tan 1 sin 1 8. 3 5 1 1 1 1 tan 1 tan 1 tan 1 3 5 7 8 sin 4 1 56 65 . Page 3 2x 1 x2 11. Prove the following : tan 1 x tan 12. Prove the following : cos [tan 1 {sin (cot 1 x)}] 13. Prove that tan 14. Prove the following : cot 15. Find the value of tan 16. Prove that tan 17. Prove the following : 2tan 18. Prove that tan 19. Prove the following : 20. Prove that sin 21. Prove the following : cos sin 22. Prove that : sin 23. Solve for x : 2tan 1 (sin x) tan 1 (2secx),x 1 sin x 1 sin x 1 x y 1 tan 1 x 1 x 1 1 2 1 8 17 1 1 9 8 63 65 1 5 tan sin 1 1 1 7 1 1 8 1 1 3 1 3 5 1 x ,x 2 1 cos 1 x, 2 4 tanh 9 sin 4 sin 1 sin x 1 sin x 1 2 1 1 3x x3 1 3x 2 1 0, 4 x y . x y 1 1 x 1 x tan tan 1 x2 2 x2 1 a 2b tan cos 1 . 4 2 b a 1 a cos 1 2 b 4 1 1 x 1 31 . 17 . 1 2 2 3 36 . 85 cos 1 3 3 cot 1 5 2 5 13 tan 4 9 sin 4 1 2 cos 6 5 13 3 5 1 2 MATRICES Each question carries 1/2 marks x 3 4 y 4 x y 5 4 , find x and y. 3 9 1. If 2. If matrix A = [1 2 3], write AA’, where A’ is the transpose of matrix A. 3. Find the value of x, if 4. If 5. y 2x 5 x 3 If A = (aij ) 3x y 2y x y 3 1 2 . 5 3 7 5 , find the value of y. 2 3 2 3 1 4 0 7 5 9 and B (bij ) 2 2 1 3 4 1 5 1 4 , then find a22 b21. 2 Page 4 1 2 3 1 3 4 2 5 7 11 , then write the value of k. k 23 6. If 7. If A 8. Write a square matrix of order 2, which is both symmetric and skew symmetric. 9. From the matrix equation, find the value of x : 10. If 11. If a matrix has 5 elements, write all the possible orders it can have. 12. x y z Write the values of x – y + z from the following equations : x z y z 13. 1 Write the order of the product matrix 2 [2 3 4] 3 14. If 15. Simplify : cos 16. Find the value of x + y from the following equations : 2 cos sin sin cos 3 4 x 2 x 1 2 3 5 7 T then for what value of 1 2 3 4 3 4 . 5 6 cos sin sin cos sin sin cos cos sin . x 5 7 y 3 3 1 4 2 7 6 15 14 1 2 1 , then find AT BT . 1 2 3 If A 18. If A is a square matrix such that A2 19. If x 1 1 9 5 7 4 6 , write the value of x. 9 x 3 4 1 4 and B 0 1 y x y 4 5 3y 19 , find the value of x. 15 17. 2 3 is A an identify matrix? A, then write the value of (I A)2 3A. 10 , write the value of x. 5 Each question carries 4/6 marks 1. Let A 3 2 5 4 1 3 . Express A as a sum of two matrices such that one is symmetric and the other is skew0 6 7 symmetric. 2. 3. If A 1 2 2 2 1 2 , verify that A2 4A 5I 0. 2 2 1 Using elementary transformations, find the inverse of the following matrix 1 2 2 2 5 4 3 7 . 5 Page 5 3 0 2 3 0 4 1 0 1 4. Obtain the inverse of the following matrix, using elementary operations : A 5. Using elementary row operations, find the inverse of the following matrix : 6. Express the following matrix as the sum of a symmetric and skew symmetric matrix, and verify your result: 3 3 1 2 2 1 2 5 1 3 4 5 2 1 4 ,B [ 1 2 1] 3 7. For the following matrices A and B, verify that (AB)' B'A'. A 8. Using elementary transformations, find the inverse of the following matrix : A 9. 10. Using elementary transformations, find the inverse of the matrix 1 3 3 0 2 1 Using elementary transformations, find the inverse of the matrix 1 1 2 1 2 3 . 3 1 1 6 5 . 5 4 2 1 0 DETERMINANTS Each question carries 1/2 marks a ib c id . c id a ib 1. Evaluate 2. 2 Find the cofactor of a12 in the following : 6 1 3. Evaluate 4. If 5. 2 3 4 8 . Write the value of the determinant 5 6 6x 9x 12x 6. Find the value of x, from the following : 7. If A sin30 sin60 2x 5 3 5x 2 9 3 0 5 5 4 . 7 cos30 . cos60 0, find x. x 4 2 2x 0 1 2 , then find the value of k if | 2A | = k | A |. 4 2 Page 6 8. 0 2 0 What is the value of the determinant 2 3 4 ? 4 5 6 9. 2 Find the minor of the element of second row and third column (a 23 ) in the following determinant : 6 1 10. If A is a square matrix of order 3 and | 3A | = k | A |, then write the value of k. 11. What positive value of x makes the following pair of determinants equal? 12. Write the adjoint of the following matrix : 2 4 13. What is the value of the following determinant? 14. For what value of x, the matrix 15. Write A 1 for A 16. Evaluate 17. If A 18. If 2 5 x x 1 x cos15 sin75 3 5 0 4 5 7 2x 3 16 3 , 5 x 5 2 1 . 3 4 a b c 4 b c a 4 c a b 5 x x 1 is singular? 2 4 2 5 . 1 3 sin15 . cos75 3 , write A 1 in terms of A. 2 3 4 , write the positive value of x. 1 2 5 3 8 2 0 1 , write the minor of the element a 23. 1 2 3 19. If 20. Let A be a square matrix of order 3 × 3. Write the value of |2A|, where |A| = 4. 21. 102 18 36 3 4 Write the value of the determinant 1 17 3 6 Each question carries 4/6 marks 1. a b c 2a 2a 2b b c a 2b Using properties of determinants, prove that : 2c 2c c a b 2. b c a b Using properties of determinants, show that : c a c a a b b c (a b c)3 . (a b c) (a c)2 Page 7 3. x y z 2 2 Prove the following, using properties of determinants : x y z2 y z z x x y (x y)(y z)(z x)(x y z). a2 4. bc c2 ac Using properties of determinants, prove the following : a 2 ab b2 ca 2 ab b bc c2 5. Using matrices, solve the following system of equations : x + 2y + z = 7; x + 3z = 11; 2x – 3y = 1. 6. 1 1 1 Using properties of determinants, prove that : a b c a3 b3 c3 7. Show that the matrix A 8. a b b c c a Prove using the properties of determinants : b c c a a b c a a b b c 9. a 2 2a 2a 1 1 Using properties of determinants, prove that : 2a 1 a 2 1 (a 1)3. 3 3 1 10. a b c 2 2 Using properties of determinants, prove that : a b c2 b c c a a b 11. x x 2 1 x3 If x y z and y y2 1 y3 z z2 1 z3 12. 3x 8 3 3 3x 8 3 Solve for x 3 3 3 3x 8 13. 1 a 2 bc a3 Using properties of determinants, prove the following : 1 b2 ca b3 1 c2 ab c3 14. 1 a 2 b2 2ab 2b 2 2 2ab 1 a b 2a Using properties of determinants, prove the following : 2b 2a 1 a 2 b2 15. Using matrices, solve the following system of equations : 2x 3y 5z 11; 3x 2y 4z (a b)(b c)(c a)(a b c) 3 1 satisfies the equation A2 5A 7I O. Hence find A 1. 1 2 a b c 2b c a. c a b (a b c)(a b)(b c)(c a) 0, then show that xyz = 1. 0. a 16. 4a 2b2c2 b (a b)(b c)(c a)(a 2 b2 c2 ). (1 a 2 b2 )3. 5; x y 2z 3 c Using properties of determinants, prove the following : a b b c c a b c c a a b a 3 b3 c3 3abc Page 8 17. a 2 1 ab ac 2 Using properties of determinants, show that ab b 1 bc 1 a 2 b2 c2 ca cb c2 1 18. 1 1 p 1 p q Using properties of determinants, prove the following : 2 3 2p 4 3p 2q 1. 3 6 3p 10 6p 3q 19. x y x x Using properties of determinants, prove that 5x 4y 4x 2x 10x 8y 8x 3x 20. 1 x x2 Using properties of determinants, prove that x2 1 x x x2 1 21. a bx c dx p qx Using properties of determinants, prove that ax b cx d px q u v w 22. (b c)2 ab ca 2 Using properties of determinants, prove that ab (a c) bc ac bc (a b)2 23. x x2 1 px3 y y2 1 py3 z z2 1 px3 x3. (1 x3 )2 . a c p (1 x ) b d q . u v w 2 2 abc (a b c)3. (1 pxyz)(x y)(y z)(z x), where p is any scalar. 24. Using matrices, solve the following system of equations : x 2y 3z 25. a b c If a, b, c are positive and unequal, show that the value of determinant b c a is negative. c a b 26. If A 2 3 1 3 2 1 3x 2y 4z 4; 2x 3y 2z 2; 3x 3y 4z 11 5 4 , find A 1. Using A 1 solve the following system of equations : 2x 3y 5z 16; 2 4; x y 2z 3 27. a bx2 c dx2 p qx2 Prove the following, using properties of determinants : ax2 b cx2 d px2 q u v w 28. x 4 2x 2x Prove that 2x x 4 2x 2x 2x x 4 (x 4 b d q 1) a c p u v w (5x 4)(4 x)2 . 2 29. Show that 2 4 2 2 2 2 Page 9 2 x 3 10 4 4; y z x 6 5 6 1; y z x 9 20 2. y z 30. Solve for x, y, z 31. x 2 2x 3 3x 4 Using properties of determinants, solve the following for x : x 4 2x 9 3x 16 x 8 2x 27 3x 64 32. x a x x Using properties of determinants, solve the following for x : x x a x x x x a 33. a b 2c a b Using properties of determinants, prove that c b c 2a b c a c a 2a 34. y k y y Prove using properties of determinants y y k y y y y k 35. Use product 1 0 3 1 2 2 2 3 4 2 0 9 2 6 1 1 3 2 0 0 2(a b c)3. k 2 (3y k) to solve the system of equations x y 2z 1; 2y 3z 1; 3x 2y 4z 2 36. 1 a 1 1 Prove without expanding that 1 1 b 1 1 1 1 c abc ab bc ca abc 1 1 a 1 1 1 or 1 b c a 1 1 is a b c factor of determinant. 37. 1 1 1 Using properties of determinants, prove that : a b c a3 b3 c3 38. b c c a a b Prove that q r r p p q y z z x x y 39. Using matrices solve the following system of linear equations : (a b)(b c)(c a)(a b c) a b c 2p q r. x y z x y 2z 7; 3x 4y 5z 5, 2x y 3z 12 40. b c a a c a b Using properties of determinants prove that b c c a b 41. a a b a b c Prove that : 2a 3a 2b 4a 3b 2c 3a 6a 3b 10a 6b 3c 42. a b c 2 2 b c2 Using properties of determinants prove that a b c c a a b 4abc. a3. (a b c)(a b)(b c)(c a) Page 10 43. If A 3 15 5 1 1 6 2 1 5 and B 2 1 1 0 2 3 2 2 0 , find (AB) 1. 1 CONTINUITY AND DIFFERENTIABILITY Each question carries 4/6 marks kx2 , x 1 is continuous at x = 1. 4, x 1 1. Find the value of k, if the function f (x) 2. Find the derivative of the following functions w.r.t. x : (sin x)tan x 3. If y = sin (log x), prove that x2 d2 y dy x y 0 2 dx dx x2 25 , if x 5 is continuous at x = 5 x 5 k, if x 5 4. Find k, so that the function f (x) 5. If y Aemx 6. If y log 7. For what value of k is the following function continuous at x = 2? f (x) Benx , prove that d2 y dy (m n) mny 0 2 dx dx 1 cos2x dy , show that 2 cosec 2x. 1 cos x dx 2x 1 ; if x 2 k ; if x 2 3x 1 ; if x 2 8. Find the derivative of the following function w.r.t. x : y tan 9. Find the derivative of the following function w.r.t. x : y 10. Find the derivative of the following function w.r.t. x : y tan 11. (cos x)secx . 1 1 x 1 x 1 x . 1 x 1 x x2 1 log 1 Determine the values of a, b and c for which the function f (x) 1 sin x 1 sin x 1 1 . x2 1 sin x . 1 sin x sin(a 1)x sin x , if x 0 x c , if x 0 may be x bx2 b x3 x , if x 0 continuous at x = 0. 12. Find derivative of the following function w.r.t. x : y sin 13. If x = a (cos 1 5x 12 1 x 2 13 dy logtan ) and y = a sin θ, find the value of at 2 dx 4 . Page 11 14. 1 sin3 x 3cos2 x , if x a , if x b(1 sin x) , if ( 2x)2 x Let f (x) 2 2 . If f(x) be a continuous function at x 2 , find a and b. 2 15. Find the derivative of the following function w.r.t. x : y x x 16. If y [log (x x2 1)]2 , show that (1 x 2 ) (sin x)x . d2 y dy x 2 0. 2 dx dx x4 2x3 x2 tan 1 x 0 17. Discuss the continuity of the following function at x = 0: f (x) 18. Find the derivative of following function w.r.t. x : y 19. Verify Lagrange’s mean value theorem for the function functions : f (x) x 2 20. Find the derivative of the following function w.r.t. x : (x 2 , x 0 . , x 0 secx 1 . secx 1 y2 ) 2 2x 3 in [4, 6] xy. APPLICATION OF DERIVATIVES Each question carries 4/6 marks 1. 2. 3. Prove that curve x a n y b n 2 touches the straight line x a y 2 at (a, b) for all values of n N at the b point (a, b). An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be the least when the depth of the tank is half of its width. Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle 30 is 4 3 h. 81 4. Find the intervals in which function f(x) is increasing or descreasing : f (x) x3 12x 2 36x 17. 5. A wire of length 28 metres is to be cut into two pieces. One of the pieces is to be made into a circle and the other into a square. What should be the length of the two pieces so that the combined area of the square and the circle is minimum? Of all the rectangle each of which has perimeter 40 metres, find one which has maximum area. Find the area also. Show that the right circular cyclinder of given volume open at the top has minimum total surface area, provided its height is equal to radius of its base. 6. 7. 8. Find the equation of tangent to the curve x = sin 3t, y = cos 2t, at t 9. Show that the height of the right circular cylinder of maximum volume that can be inscribed in a given right circular cone of height h is 10. . h . 3 Show that the semi-vertical angle of the right circular cone of maximum volume and given slant height, is tan 11. 4 1 2. Find the interval(s) in which following function f(x) is increasing or decreasing : f (x) 2x3 9x 2 12x 15. Page 12 12. At what points will the tangent to the curve y 2x3 15x2 36x 21 be parallel to the x-axis? Also find the 13. equations of the tangents to the curve at these points. A point on the hypotenuse of a right angled triangle is at distance a and b from the sides. Show that the length of the hypotenuse is at least (a 2/3 b2/3 )3/2 . 14. Show that the curves x y2 and xy = k cut at right angles, if 8k2 1. 15. Find the greatest area of isosceles triangle that can be inscribed in a given ellipse 16. coinciding with one extremity of the major axis. Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is sin 1 x2 a2 y2 1 with its vertex b2 1 . 3 1 ,x 0 x3 17. Find the interval(s) in which function f(x) is increasing or decreasing : f (x) x3 18. Find the equation of the tangent to the curve y 19. Find the volume of the largest cylinder that can be inscribed in a sphere of radius r. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and 20. 3x 2 which is parallel to the line 4x 2y 5 0. volume is 8m3. If building of tank costs Rs. 70 per sq metre for the base and Rs. 45 per sq metre for sides, what is the cost of least expensive tank? INTEGRALS Each question carries 1/2 marks 1. Evaluate x2 dx. 1 x3 2. Evaluate 2cosx dx. 3sin2 x 3. Evaluate 2x dx . 0 1 x2 4. Evaluate x 2 4x dx. x3 6x 2 5 5. Evaluate x cos6x dx. 3x2 sin6x 6. Evaluate sec2 x dx. 3 tan x 7. Evaluate sec2 (7 x)dx 8. If 9. Evaluate sin x dx x 10. Evaluate sec2 x dx x 11. Evaluate dx. 12. Evaluate 1 dx x xlog x 13. Evaluate eax e eax e dx 14. Evaluate 15. Evaluate log x dx x 16. Evaluate 17. Evaluate sec2 (7 4x)dx. 18. If (ax b)2 dx f (x) c, find f(x). 19. Evaluate 20. Evaluate sec x (sec x tan x) dx. 1 1 2 0 1 0 1 1 x 2 ax ax dx . 1 x2 1 0 (3x 2 2x k)dx 0, find the value of k. 1 dx 2x 3 1 0 /2 /2 sin5 x dx. Page 13 Each question carries 4/6 marks 1. Evaluate 3. Evaluate 5. Evaluate 7. Evaluate 9. Prove that 10. Prove that 12. Evaluate 14. Evaluate 16. Prove that 18. Evaluate 20. Evaluate 3 (2x 2 3x 5) dx. 2. Evaluate sin x dx (1 cosx)(2 cosx) 4. Evaluate cos(x a) dx sin(x b) 6. Evaluate 8. Evaluate 0 1/ 2 0 a 0 (sin 1 x) dx (1 x2 )3/2 f (x)dx /2 0 a 0 f (a x)dx. Using it, evaluate sin x dx sin x cosx 4 . x2 cot 1 x dx 1 0 cot 1 (1 x x 2 )dx. a 0 /2 0 sin 1 x a x dx a ( 2 2). log (sin x) dx. 2 0 1 0 (3x 2 2x 1)dx. x 4 x dx x2 1 x2 4 dx. x 4 16 0 x tan x dx. secxcosecx x 2 xdx. 11. Evaluate 13. Evaluate 15. Prove that 17. Evaluate 19. Evaluate 1 sin x dx. 1 sin x 1 dx 2 2 a sin x b2 cos2 x tan a a 1 0 1 a x dx a . a x log(1 x) dx. 1 x2 ex 5 4ex e2x dx. (x 4)ex dx. (x 2)3 APPLICATION OF INTEGRALS Each question carries 4/6 marks 1. Find the area of the region enclosed between the two circles : x 2 y2 1, (x 1)2 y2 1. 2. Using integration, find the area of the circle x 2 3. Find the area of the region included between the parabola y 4. Using integration, find the area of the region bounded by the parabola y2 5. Find the area bounded by the lines x 2y 2, y x 1 and 2x y 7. 6. Prove that the curves y2 y2 16 which is exterior to the parabola y2 6x. 3 2 x and the line 3x 2y 12 0. 4 4x and the circle 4x2 4y2 9. 4x and x 2 4y divide the area of the square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts. 7. Find the area of the region lying between the parabolas y2 4ax and x 2 8. Find the area of the region included between the parabola y2 9. Using integration, find the area of the region : {(x, y) :9x 2 10. Using integration find the area of the region : {(x, y) : 25x 2 9y2 11. Using integration find the area of the following region : (x, y) : 4ay, where a > 0. x and the line x + y = 2. y2 36 and 3x y 6}. x2 9 225 and 5x 3y 15}. y2 x 1 4 3 y 2 Page 14 5 x2 12. Using integration find the area of the following region : (x, y):| x 1| y 13. Compute, using integration, the area of the region bounded by the line y 4x 5, y 5 x and 4y x 5. 14. Find the area of the circle 4x 2 15. Using integration, find the area of the following region : {(x,y):| x 2| y 16. Sketch the graph of y = |x + 3| and evaluate the area under the curve y = |x + 3| above x-axis and between x = - 6 to x = 0. Using the method of integration find the area of the region bounded by the lines 2x y 4; 3x 2y 6 and 17. 4y2 9 which is interior of the parabola x 2 4y. 20 x2 }. x 3y 5 0 18. Using the method of integration, find the area of the region bounded by the lines 3x 2y 1 0, 2x 3y 21 0 and x 5y 9 0. 19. Find the area of the region {(x, y) : x 2 20. Using integration find the area of the region in the first quadrant enclosed by the x-axis, line x circle x 2 21. y2 y2 4,x y 2}. 3y and the 4. Using the method of integration, find the area of the ABC, coordinates of whose vertices are A(2, 0), B(4, 5) and C(6, 3). DIFFERENTIAL EQUATIONS Each question carries 1/2 marks 1. What is the degree of the following differential equation? 5x dy dx 2 d2 y 6y log x dx2 Each question carries 4/6 marks dy dx x . ex logx ex . xcos y 1. Solve the differential equation 2. Form the differential equation representing the family of curves y = A cos (x + B), where A and B are constants. 3. Solve the differential equation x2 4. 5. dy 2xy y2 . dx dy Solve the differential equation cosx y sin x, given that y = 2 when x = 0. dx Solve the differential equation (3xy y2 )dx (x 2 xy)dy 0. 6. Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis. 7. Solve the differential equation (1 x2 ) 8. Solve the differential equation 9. dy y tan 1 x. dx dy y cosx sin x dx dy y y cosec 0; y 0 when x = 1. Solve the differential equation dx x x dy dx y x tan y . x 10. Solve the differential equation x 11. Form the differential equation of the family of circles touching the y-axis at origin. Page 15 12. Form the differential equation representing the family of curves given by (x a)2 2y2 a 2 , where a is an arbitrary constant. 14. dy (x 2) (y 2), find the solution curve passing through the point (1, -1). dx Solve the differential equation (x3 y3 )dy x 2 y dx 0. 15. Solve the differential equation 16. Solve the differential equation xlogx 13. 17. 18. 19. 20. For the differential equation xy dy dx ytan x, given that y = 1 when x = 0. dy 2 y logx. dx x dy 2 Solve the differential equation (x2 1) 2xy 2 . dx x 1 dy Show that the differential equation (x y) x 2y, is homogeneous and solve it. dx dx Solve the differential equation (3x2 y) x, x 0, when x = 1, y = 1. dy Show that the following differential equation is homogeneous, and then solve it : y dx xlog y dy 2x dy 0. x VECTOR ALGEBRA Each question carries 1/2 marks 1. Find a unit vector in the direction of a 3i 2j 6k. 2. Find the angle between the vectors a i j k and b i 3. For what value of λ are the vectors a 2i 4. If P (1, 5, 4) and Q (4, 1, - 2), find the direction ratios of PQ . 5. If a i 2j k and b 3i 6. If | a | 7. If | a | 2, | b| 8. If a i 2j 3k and b 2i 4j 9k, find a unit vector parallel to a b. 9. If | a | 10. Find a vector in the direction of a i 2j whose magnitude is 7. 11. Find the projection of a on b if a . b 8 and b 2i 6j 3k. 12. Write a unit vector in the direction of a 2i 6j 3k. 13. Write the value of p for which a 3i 2j 9k and b i pj 3k are parallel vectors. 14. Find the angle between two vectors a and b with magnitudes 1 and 2 respectively and when | a b | 15. Find the value of p if (2i 6j 27k) (i 3j pk) 0. 16. Write the direction cosines of a line equally inclined to the three coordinate axes. 17. If p is a unit vector and (x p) (x p) 80, then find | x |. 18. Write a vector of magnitude 15 units in the direction of vector i 2j 2k . j k. j k and b i 2j 3k perpendicular to each other? j 5k, find a unit vector in the direction of a b. 3, | b| 2 and a . b 3, find the angle between a and b. 3 and a.b 3, find the angle between a and b. 3, | b| 2 and angle between a and b is 60º, find a.b. 3. Page 16 19. What is the cosine of the angle which the vector 2i j k makes with y-axis? 20. Write a vector of magnitude 9 units in the direction of vector 2i j 2k . ANSWERS 1. 6. 3 2 6 i j k 7 7 7 7. 6 1 3 2. 3. i 2j 2k 3 3 3 8. 3 12. 2 6 3 i j k 7 7 7 13. 17. 6i 3j 6k 18. 5 2 2 3 14. 3 19. 9. 3 4 4. 3, 4, 6 5. 3 10. 7(i 2j) 5 15. 5i 10j 10k 20. 2 i 21 1 j 21 11. 8 7 16. cos 4 k 21 1 2 3 Each question carries 4/6 marks 1. Find the projection of b c on a, where a 2i 2j k, b i 2j 2k and c 2i j 4k. 2. If a = i + j + k and b = j k, find a vector c such that a c b and a. b 3. 3. If a b c 0 and | a | 3, | b| 5 and | c | 7, show that the angle between a and b is 60º. 4. Show that the area of the parallelogram having diagonals (3i 5. If i j k, 2i 5j, 3i 2j 3k and i 6j k are the position vectors of the points A, B, C and D, find the angle j 2k) and (i 3j 4k) is 5 3 sq units. between AB and CD. Deduce that AB and CD are collinear. 6. The scalar product of the vector i j k with the unit vector along the sum of vectors 2i 4j 5k and i 2j 3k is equal to one. Find the value of λ. 7. Define the scalar and vector product of two vectors a and b. If for three non-zero vectors a , b and c ; a b = 8. a c and a b a c, then show that b c. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2a b) and (a 3b) respectively, externally in the ratio 1:2. Also, show that P is the mid-point of the line segment RQ. 9. If a i j k, b 4i 2j 3k and c i 2j k, find a vector of magnitude 6 units which is parallel to the vector 2a b 3c. 10. Let a i 4j 2k, b 3i 2j 7k and c 2i j 4k. Find a vector d which is perpendicular to both a and b and c. d 18. 11. The scalar product of the vector i 2j 3k and i 4j 5k is equal to one. Find the value of λ. 12. Find a unit vector perpendicular to each of the vectors a b and a b, where a 3i 2j 2k and b i 2j 2k. 13. If the vectors a and b are such that | a | = 2, | b | = 1 and a b 1, then find the value of (3a 5b) . (2a 7b). 14. Using vectors, find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5). 15. If Vectors a 2i 2j 3k, b i 2j k and c 3i j are such that a b is perpendicular to c , then find the value of λ. Page 17 16. If a, b, c are three vectors such that | a | 5, | b | 12 and | c| 13, and a b c 0, find the value of a b b c c a. 17. Let a i 4j 2k, b 3i 2j 7k and c 2i j 4k. Find a vector p which is perpendicular to both a and b and p c = 18. 18. If 3i 4j 5k and is perpendicular to 19. If 20. 2i j 3k then express sin 1 |a b| 2 2 where 1 2, 2 where 1 is parallel to and 2 in the form 1 1 is parallel to and 2 is . If a & b are unit vectors inclined at an angle (a) in the form . 3i j and perpendicular to 2i j 4k, then express (b) cos 2 then prove that. 1 |a b| 2 (c) tan | a b| 2 | a b| 21. | a b| 60 | a b| 40 | b| 46 then show that | a | 22 22. Show that the points A, B, C with position vectors 2i j j, i 3j 5k and 3i 4j 4k are the vertices of a right angled triangle. 23. Dot product of a vector with (i j 3k) , (i 3j 2k), (2i j 4k) are 0, 5, 8 respectively then show that this vector is (i 2j k) . 24. Show that the area of triangle whose vertices are A (3, -1, 2), B(1, -1, -3), C(4, -3, 1) is 25. Show that the area of parallelogram having diagonals (3i 165 2 j 2k) and (i 3.j 4k) is 5 3 ANSWERS 2 9. 2i 4j 4k 13. 0 18. 2i j 4k 2. 14. 5 2 2 i j k 3 3 3 c 1. 10. 3i 3 j 2k 2 3 4 i j k 5 5 3. 60 5 3 4. d 64i 2j 28k 15. 8 5. 11. 16. 13 9 i j 3k 5 5 169 6. 2 7. 12. 2a b 8. 2 2 1 i j k 3 3 3 (64i 2j 28k) 17. 19. 1 2i j 3k 3 1 i j 2 2 1 3 i j 3k 2 2 THREE DIMENSIONAL GEOMETRY Each question carries 1/2 marks 1. Find the direction cosines of the line passing through the following points : (-2, 4, -5), (1, 2, 3). 2. Write the vector equation of the following line : 3. Write the position vector of the midpoint of the vector joining the points P(2, 3, 4) and Q(4, 1, -2). 4. Write the Cartesian equation of the following line given in vector form : r 2i j 4k 5. Write the intercept cut off by the plane 2x y z 5 on x-axis. 6. What are the direction cosines of a line which makes equal angles with the coordinate axes? x 5 3 y 4 7 6 z 2 (i j k). Page 18 7. 8. 9. 10. If a line has direction ratios 2, -1, -2, then what are its direction cosines? Find the distance of the plane 3x 4y 12z 3 from the origin. Write the direction cosines of a line parallel to z-axis. The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, -2) is 4. Find its z-coordinate. ANSWERS 1. 4. 3 2 8 , , 77 77 77 x 2 y 1 z 4 1 1 1 2. r (5i 4j 6k) 5. 5 2 1 , 3 6. (3i 7j 2k) 1 , 3 7. 3 3i 2j k 3. 2 1 2 , , 3 3 3 3 13 8. 001 9. 10. -1 Each question carries 4/6 marks 1. Find the equation of the plane which is perpendicular to the plane 5x 3y 6z 8 0 and which contains the line of intersection of the planes x 2y 3z 4 0 and 2x y z 5 0. x 3 1 y 5 2 z 7 x 1 y 1 z 1 and 1 7 6 1 2. Find the shortest distance between the following lines : 3. Find the point on the line 4. Find the equation of the plane passing through the point (-1, -1, 2) and perpendicular to each of the planes 2x 3y 3z 2 and 5x 4y z 6. 5. Find the length and the foot of the perpendicular drawn from the point (2, -1, 5) to the line 6. From the point P(1, 2, 4) a perpendicular is drawn on the plane 2x y 2z 3 0. Find the equation, the length x 2 3 y 1 z 3 at a distance 3 2 from the point (1, 2, 3). 2 2 x 11 y 2 10 4 z 8 . 11 and the coordinates of the foot of the perpendicular. 7. Find the shortest distance between the line r i 2j 3k 8. Find the distance of the point (1, -2, 3) from the plane x – y + z = 5 measured parallel to the line 9. Find the value of λ so that the lines 1 x 3 7y 14 2 (2i 3j 4k) and r 2i 4j 5k 5z 10 7 7x and 11 3 (3i 4j 5k). x 2 y 3 z . 6 y 5 6 z are perpendicular to 1 5 each other. 10. Find the shortest r (2i j k) distance between the following lines : r (1 )i (2 )j ( 1)k and (2i j 2k). x 1 3y 5 3 z and the plane 10x 2y 11z 3. 2 9 6 y 1 z 5 x 1 y 2 z 5 and are coplanar. Also find the equation of the 1 5 1 2 5 11. Find the angle between the line 12. Show that the lines 13. plane containing the lines. Find the equation of the perpendicular drawn from the ppoint (1, -2, 3) to the plane 2x 3y 4z 9 0. Also 14. find the coordinates of the foot of the perpendicular. Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1, 2) and parallel to the line x 4 1 y 3 4 x 3 3 z 1 . 7 Page 19 15. Write the vector equations of the following lines and hence determine the distance between them : x 1 y 2 2 3 16. z 4 x 3 y 3 z 5 ; 4 6 12 6 x 2 y 1 z 3 Find the points on the line at a distance of 5 units from the point P(1, 3, 3). 3 2 2 17. Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point P(3, 2, 1) from the plane 2x y z 1 0. Find also the image of the point in the plane. 18. Find the equation of the plane passing through the point P(1, 1, 1) and containing the line r ( 3i j 5k) 19. (i 2j 5k). Find the shortest distance between the following pair of lines and hence write whether the lines are intersecting or not : 20. (3i j 5k). Also, show that the plane contains the line r ( i 2j 5k) x 1 y 1 x 1 y 2 z; ;z 2 2 3 5 1 Find the angle between the following pair of lines : x 2 2 y 1 z 3 x 2 and 7 3 1 2y 8 4 z 5 and check 4 whether the lines are parallel or perpendicular. 21. ANSWERS 1. 5. 11. 15. 19. 51x 15y 50z 173 0 14 sin 1 293 7 9 195 6. 8 21 x 1 y 2 2 1 2. 2 29 3. z 4 11 19 34 1 , , , , 2 9 9 9 3 12. x 2y z 0 16. ( 2, 1,3) or (4,3,7) 20. 90 lines are perpendicular 13. 56 43 111 , , 17 17 17 7. 1 6 x 1 y 2 2 3 17. 8. 1 4. 9x 17y 23z 20 0 9. 7 z 3 , ( 1,1, 1) 4 (1,3,0), ( 1,4, 1) 18. 3 2 10. 14. x 19y 11z 0 r. (i 2j k) 0 LINEAR PROGRAMMING Each question carries 4/6 marks 1. 2. 3. If a young man rides his motorcycle at the speed of 25 km/hour, he had to spend Rs. 2 per km on petrol. If he rides it at a faster speed of 40 km/hour, the petrol cost increases to Rs. 5 per km. He has at most Rs. 100 to spend on petrol and wishes to find the maximum distance that he can travel in one hour. Express this as a LPP and solve it graphically. A factory owner purchases two types of machines A and B for his factory. The requirements and the limitations for the machines are as follows : Machine Area occupied Labour force Daily output (in units) 2 A 1000 m 12 men 60 B 1200 m2 8 men 40 2 He has maximum area of 9000 m available and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximize the daily output? A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 5% phosphoric acid, and type B which contains 6% nitrogen and 10% phosphoric acid. After testing the soil conditions of the field, the Page 20 4. farmer finds that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid is required for producing a good crop. The fertilizer of type A costs Rs. 5 per kg and the type B costs Rs. 3 per kg. How many kg of each type of the fertilizer should be used to meet the requirement at the minimum possible cost? Using LPP solve the above problem graphically. A diet is the contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs. 4 per unit and F2 costs Rs. 6 per unit. One unit of food F1 contains 3 units of Vitamin A and 4 units of mninerals. One unit of food F2 contains 6 units of Vitamin A and 3 units of minerals. Formulate this as a 5. 6. 7. 8. 9. 10. 11. linear programming problem. Find graphically the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has space for at the most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. He expects to sell a fan at a profit of Rs. 22 and a sewing machine for a profit of Rs. 18. Assuming that he can sell all the items he buy, how should he invest money to maximize his profit? Solve it graphically. One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes. Formulate this problem as a linear programming problem and solve graphically. A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per da y is at most 24. It takes 1 hour to make a ring and 30 minutes to make chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as a LPP and solve it graphically. A factory makes two types of items A and B made of plywood. One piece of item A requires 5 minutes for cutting and 10 minutes for assembling. One piece of item B requires 8 minutes for cutting and 8 minutes for assembling. There are 3 hours and 20 minutes available for cutting and 4 hours for assembling. The profit on one piece of item A is Rs. 5 and that on item B is Rs. 6. How many pieces of each type should the factory make so as to maximize profit? Make it as LPP and solve it graphically. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman’s time. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as a LPP and solve graphically. A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakh and his profit on the desktop model is Rs. 4,500 and on the portable model is Rs. 5,000. Make a LPP and solve it graphically. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of grindling/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes one hour on the grindling/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The proft from the sale of a lamp is Rs. 5 and that from a shade is Rs. 3. Assuming that the manfucaturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit? Make a LPP and solve it graphically. Page 21 12. 13. 14. A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs. 17.50 per package on nuts and Rs. 7 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day? Formulate this problem as a linear programming problem and solve it graphically. A dietician wishes to mix two types of food in such a way that vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. If costs Rs. 5 per kg to purchase Food I and Rs. 7 per kg to purchase Food II. Determine the minimum cost of such a mixture. Formulate this problem as a LPP and solve it graphically. A company produces soft drinks that has a contract which requires a minimum of 80 units of chemical A and 60 units of chemical B go into each bottle of the drink. The chemical are available in prepared mix packets from two different suppliers. Supplier S has a packet of mix of 4 units of A and 2 units of B that costs Rs. 10 and the supplier T has a packet of mix of 1 unit of A and 1 unit of B that costs Rs. 4. How many packets of mixes from S and T should the company purchase to honour the contract requirement and yet minimize cost? Make a LPP and solve graphically. PROBABILITY Each question carries 4/6 marks 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Find mean μ, variance 2 , for the following probability distribution : X 0 1 2 3 P(X) 1/8 3/8 3/8 1/8 Find the binomial distribution for which mean is 4 and variance 3. Bag A contains 6 red and 5 blue balls and another bag B contains 5 red and 8 blue balls. A ball is drawn from bag A without seeing its colour and it is put into the bag B. Then a ball is drawn from bag B at random. Find the probability that the ball drawn is blue in colour. Out of 9 outstanding students of a school, there are 4 boys and 5 girls. A team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected. A bag X contains 2 white and 3 red balls and a bag Y contains 4 white and 5 red balls. One ball is drawn at random from one of the bag and is found to be red. Find the probability that it was drawn from bag Y. An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of getting (i) 2 red balls (ii) 2 blue balls (iii) 1 red and 1 blue ball. There are 2000 scooter drivers, 4000 car drivers and 6000 truck drivers all insured. The probabilities of an accident involving a scooter, a car, a truck are 0.01, 0.03, 0.15 respectively. One of the insured drivers meets with an accident. What is the probability that he is a scooter driver? 12 cards, numbered 1 to 12, are placed in a box, mixed up thoroughly and then one card is drawn at randomly from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number. In a bulb factory machines A, B and C manufacture 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A. Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the mean and standard deviation of the number of kings. In a factory which manufactures bolts, machines A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their outputs 5, 4 and 2 per cent are respectively defective bolts. A bolt is drawn at random from the production and is found to e defective. Find the probability that it is manufactured by the machine B. Page 22 12. 13. 14. 15. 16. 17. 18. 19. 20. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die. Three bags contain balls as shown in the table below : Bag Number of white balls Number of black balls Number of red balls I 1 2 3 II 2 1 1 III 4 3 2 A bag is chosen at random and two balls are drawn from it. They happen to be white and red. What is the probability that they came from the III bag? Two groups are competing for the positions on Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and corresponding probability is 0.3 if second group wins. Find the probability that the new product introduced was by second group. There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up tails 25% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows head, what is the probability that it was from the two headed coin? A man is known to speak the truth 3 out of 5 times. He throws a die and reports that it is a number greater than 4. Find the probability that it is actually a number greater than 4. From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the mean and variance of the number of defective bulbs. On a multiple choice examination with three possible answers (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answer just by guessing? A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn at random and are found to both clubs. Find the probability of the lost card being of club. A family has 2 children. Find the probability that both are boys, if it is known that (i) at least one of the children is a boy. (ii) the elder child is a boy Page 23 1. Consider the point A(0, 1) and B (2, 0) and P be a point on the line 4x 3 y 9 0 . Coordinates of P such that | PA PB | is maximum, are 84 13 6 17 (c) (d) (0, –3) , , 5 5 5 5 The vertices of a triangle are A( x1, x1 tan ) , B( x2 , x2 tan ) and C( x3 , x3 tan ) . If the circumecentre of a triangle ABC coincides with the origin and H(a, b) be orthocentre, then b cos cos cos sin sin sin sin sin sin tan tan tan (a) (b) (c) (d) cos .cos .cos sin .sin .sin cos cos cos tan .tan .tan 12 17 , 4 5 (a) 2. 3. (b) The joint equation of two altitudes of an equilateral triangle is ( 3x y 8 4 3) ( 3x y 12 4 3) 0 . The third altitude has the equation 3x 2 4 3 3x 2 4 3 (d) y 10 0 If the area of the rhombus enclosed by the lines lx my n 0 be 2 square units, then (a) 4. (a) l, m, n are in G.P. 5. (b) y 10 0 (c) (b) l, n, m are in G.P. (c) lm (d) l n n m If the point (1 cos ,sin ) lies between the region corresponding to acute angle between the lines x y 0 and 6 y x, then R n ,n I (a) 6. 2 xy by that the equation ax (a) a K2 b 2 (d) None of these K2 b (b) a can be found such 2K (x y 1) 0 represents a pair of straight lines is (c) K 2 The line lx my 1 intersects the circle x2 then a2 (l 2 a or K 2 a (d) K 2a or K 2b y2 a2 at points A, B, If AB subtends 45º at the origin m2 ) (a) 4 2 2 8. n ,n I (c) If a and b are positive numbers (a < b), then the range of value of K for which a real 2 7. (2n 1) , n I 2 (b) (b) 4 2 6 3 The equation x 2 6x y 11xy 2 (c) 2 6 3 6y (d) 4 6 0 represent three straight lines passing through the origin, the slopes of which form a/an (a) A.P. 9. (c) H.P. 2 If the slope of one of the lines represented by ax then 2hxy by (d) None of these 2 0 be the square of the other, a b 8h2 h ab (a) 4 10. (b) G.P. (b) 6 (c) 8 2 The locus of the centre of a circle touching the circle x (d) None of these y2 4 y 2x 2 3 1 internally and tangents on which from (1, 2) is making a 60º angle with each other, is (a) ( x 1)2 ( y 2)2 3 (b) ( x 2)2 ( y 1)2 1 2 3 Page 24 (c) x2 11. y2 1 (d) None of these The least distance between two points P and Q on the circles x2 y2 8x 10 y 37 0 and x2 y2 16x 55 0 (a) 5 units 12. (c) 5 5 units (b) 8 units Tangents are drawn from O (origin) to touch the circle x2 (d) None of these y2 2gx 2 fy c 0 at points P and Q. The equation of the circle circumscribing triangle OPQ is 13. (a) 2x2 2 y2 gx fy 0 (c) x2 y2 2gx 2 fy 0 (b) x2 y2 gx fy 0 (d) None of these The line 4x 3 y 4 0 divides the circumference of the circle centered at (5, 3), in the ratio 1 : 2. Then the equation of the circle is 14. (a) x2 y2 10x 6 y 66 0 (b) x2 y2 10x 6 y 100 0 (c) x2 y2 10x 6 y 66 0 (d) x2 y2 10x 6 y 100 0 If p and q be the longest and the shortest distance respectively of the point (–7, 2) from any point ( , ) on the curve whose equation is x2 (a) 2 11 15. (b) 2 5 (c) 13 PQ is any focal chord of the parabola y2 (a) 40 16. y2 10x 14 y 51 0 then G.M. of p and q is (d) 11 32x . The length of PQ can never be less than (b) 45 (c) 32 If the tangent at the point P( x1, y1) to the parabola y2 (d) 48 4ax meets the parabola y2 4a(x b) at Q and R, then the mid-point of QR is (a) ( x1 b, y1 b) 17. (b) ( x1 b, y1 b) (c) ( x1, y1 ) (d) ( x1 b, y1 b) If perpendiculars be drawn from any two fixed points on the axis of a parabola equidistant from the focus on any tangent to it, then the difference of their squares is [ l is the length of latus rectum and 2d is the distance between two points] (b) 2ld (a) ld 18. A focal chord of parabola y2 (c) 4ld (d) 4x is inclined at an angle of 4 l2 d2 with positive x-direction, then the slope of normal drawn at the ends of chord will satisfy the equation (a) m2 19. The parabola y2 (a) r 20. 2m 1 0 20 (b) m2 2m 1 0 (c) m2 1 0 (d) m2 2m 2 0 4x and the circle ( x 6)2 y2 r 2 will have no common tangent if ‘r’ is equal to (b) r 20 (c) r 18 (d) R ( 20, 28) The ends of line segement are P(1, 3) and Q(1, 1), R is a point on the line segement PQ such that PR : RQ 1: . If R is an interior point of parabola y2 4x , then Page 25 (0,1) (a) 21. 3 ,1 5 (b) (1, ) (d) Coordinates of the vertices B and C of a triangle ABC are (2, 0) and (8, 0) respectively. The vertex A is varying in such a way that 4 tan (a) 22. ( 1,0) (c) ( x 5)2 25 y2 1 16 (b) B C tan 1. Then locus of A is C 2 ( x 5)2 16 y2 1 25 (c) ( x 5)2 25 y2 1 9 (d) ( x 5)2 9 y2 1 25 In an ellipse, if the lines joining a focus to the extremities of the minor axis make an equilateral triangle with the minor axis, then the eccentricity of the ellipse is (a) 23. 3 2 Tangents are drawn to the ellipse (a) 27 24. 3 4 (b) (b) (d) (c) 27 4 (d) x2 a 27 55 y2 1 from two points on b2 a2 b2 from the centre is (b) 2b2 (c) a2 b2 (d) a2 The locus of the foot of perpendicular from the centre on any tangent to the ellipse (a) A circle 1 2 y2 1at ends of laterarecta. The area of quadrilateral so formed is 5 The sum of the squares of the perpendiculars on any tangent to the ellipse (a) 2a 2 26. x2 9 27 2 the minor axis each at a distance 25. 1 2 (c) (b) A pair of straight lines (c) Another ellipse If the normal to the rectangular hyperbola xy c2 at the point ct, x2 a2 b2 y2 1 is b2 (d) None of these c c meets the curve again at ct ', , t t' then 3 (a) t t ' 1 27. If | z i | 2 and z0 (a) 2 28. 3 (b) t t ' 31 31 2 (d) tt ' (c) 7 The complex number z1, z2 and z3 satisfying 1 (d) 5 1 i 3 are the vertices of a triangle, which is 2 z1 z3 z2 z3 (b) Right angled isosceles (c) Equilateral (d) Obtuse angled isosceles For all complex numbers z1, z2 satisfying | z1 | 12 and | z2 3 4i | 5 , the minimum value of | z1 (a) 0 30. (c) tt ' 1 5 3i , then the maximum value of | iz z0 | is (b) (a) Of area zero 29. 1 If | z (b) 2 2 2 1| | z | (c) 7 (d) 17 (c) Circle (d) None of these z2 | is 1, then locus of z is (a) Real axis (b) Imaginary axis TOP IC : 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. B D B B D D A C B D Page 26 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. B B A A C C A B B A 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. A A A A D B C C B B Page 27
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