Contents
(Term II)
Preface
(v)
Latest Syllabus
(vii)
4. Quadratic Equations
...
2
5. Arithmetic Progressions
...
14
7. Coordinate Geometry
...
28
9. Some Applications of Trigonometry
...
41
10. Circles
...
53
11. Constructions
...
74
12. Areas Related to Circles
...
84
13. Surface Areas and Volumes
...
102
15. Probability
...
119
Value Based Questions
...
133
Practice Papers (I–III)
...
141
xv
Each Chapter Contains:
★ TIPS AND TRICKS
★ FORMATIVE ASSESSMENT
★ SUMMATIVE ASSESSMENT
• CBSE and Other Important Questions
• Objective Type Questions
• Higher Order Thinking Skills (HOTS) Questions
• NCERT Textbook Exercises
4
Quadratic Equations
CHAPTER
Tips and Tricks
• Standard form of a quadratic equation: The standard
form of a quadratic equation in variable x is
ax2 + bx + c, a ≠ 0
• Roots of a quadratic equation: A real number α is said
to be a root of the quadratic equation ax2 + bx + c = 0 if
aα2 + bα + c = 0.
Note: The roots of the quadratic equation ax2 + bx + c = 0
are the same as the zeroes of the quadratic polynomial
ax2 + bx + c.
• Solving a quadratic equation
(i) by factorisation: If we can factorise the quadratic
polynomial ax2 + bx + c into two real linear factors, then
the roots of the quadratic equation ax2 + bx + c = 0 can
be found by equating each factor to zero.
(ii) by completing the square: By adding and subtracting
a suitable constant, as required, we club the x2 and x terms
in the quadratic equation so that they become a complete
square and then solve for x.
(iii) by quadratic formula: The roots of the quadratic
equation ax2 + bx + c = 0 are given by
− b ± b − 4ac
2
x=
2a
, provided b2 – 4ac ≥ 0
Note: The expression b2 – 4ac is called the discriminant
of the quadratic equation.
• Nature of roots:
Value of ‘b2 – 4ac’
(i) > 0
(ii) = 0
(iii) < 0
ILLUSTRATIVE EXAMPLES
CBSE Exam.
1. Solve for x:
4
5
3
−3=
, x ≠ 0, −
x
2x + 3
2
Sol. We have,
4
5
−3=
x
2x + 3
⇒
4 − 3x
5
=
2x + 3
x
⇒
(4 – 3x) (2x + 3) = 5x
⇒
8x + 12 – 6x2 – 9x = 5x
⇒
6x2 + 6x – 12 = 0
⇒
x2 + x – 2 = 0
⇒
x2 + 2x – x – 2 = 0
⇒
x(x + 2) – 1(x + 2) = 0
⇒
(x + 2) (x – 1) = 0
⇒
x + 2 = 0 or x – 1 = 0
⇒
x=–2
⇒
x = – 2, 1.
or x = 1
2. Solve for x:
Nature of roots
two distinct real roots
two equal real roots
no real roots
x +1 x − 2
+
= 3; x ≠ 1, – 2
x −1 x + 2
Sol. We have,
x +1 x − 2
+
=3
x −1 x + 2
A-2
3
QUADRATIC EQUATIONS
⇒
( x + 1) ( x + 2) + ( x − 2) ( x − 1)
=3
( x − 1) ( x + 2)
⇒
( x − 10) + x
8
=
x ( x − 10)
75
⇒
x2 + 2x + x + 2 + x2 − x − 2x + 2
=3
x2 + 2x − x − 2
⇒
8
2 x − 10
=
2
75
x − 10 x
⇒
⇒
2x2 + 4
=3
x2 + x − 2
2x2 + 4 = 3(x2 + x – 2)
2 ( x − 5)
8
=
2
75
x − 10 x
⇒
2x2 + 4 = 3x2 + 3x – 6
⇒
x−5
4
=
x 2 − 10 x 75
⇒
⇒
x2 + 3x – 10 = 0
⇒
+ 5x – 2x – 10 = 0
⇒
⇒
x(x + 5) – 2(x + 5) = 0
4x2 – 115x + 375 = 0
⇒
⇒
(x + 5) (x – 2) = 0
⇒
4x2 – 100x – 15x + 375 = 0
⇒
⇒
x2
x + 5 = 0 or x – 2 = 0
⇒
x = – 5 or
⇒
x = – 5, 2.
x=2
3
hours. The
8
tap of larger diameter takes 10 hours less than the smaller
one to fill the tank separately. Find the time in which
each tap can separately fill the tank.
3. Two water taps together can fill a tank in 9
Sol. Let the tap of smaller diameter take x hours to fill the
tank alone.
Then, the tap of larger diameter will take (x – 10) hours
to fill the tank.
1
x
Tank filled by the tap of larger diameter in 1 hour
1
=
x − 10
∴ Tank filled by both the tanks together in 1 hour
Tank filled by the tap of smaller diameter in 1 hour =
1
1
= +
x x − 10
∴
⇒
(x – 25) (4x – 15) = 0
⇒
x – 25 = 0
or 4x – 15 = 0
⇒
x = 25 or x =
⇒
x = 25,
15
4
15
4
15
is inadmissible as then x – 10 is negative which is
4
not possible.
[Time cannot be –ve]
∴
x = 25
x=
⇒ x – 10 = 15
Hence, tap of smaller and larger diameter take respectively 25 hours and 15 hours to fill the tank alone.
4. Find value of p such that the quadratic equation
(p – 12)x2 – 2(p – 12)x + 2 = 0
has equal roots.
Sol. The given quadratic equation is
(p – 12)x2 – 2(p – 12)x + 2 = 0
Comparing it with
FG
IJ
H
K
75 F 1
1 I
+
=
G
J
8 H x x − 10 K
3 1
1
+
=9
8 x x − 10
According to the question,
⇒
4x2 – 40x = 75x – 375
⇒ 4x (x – 25) – 15(x – 25) = 0
3
Tank filled by both the tanks together in 9 hours
8
FG
H
4 (x2 – 10x) = 75(x – 5)
IJ
K
75 1
1
+
=1
8 x x − 10
1
1
8
+
=
x x − 10 75
Ax2
+ Bx + C = 0, we get
A = p – 12
B = – 2(p – 12)
C=2
For equal roots,
Discriminant = 0
⇒
⇒
B2 – 4AC = 0
B2 = 4AC
⇒
{– 2(p – 12)}2 = 4(p – 12) (2)
⇒
4(p – 12)2 = 4(p – 12) (2)
⇒
(p – 12)2 = 2 (p – 12)
...(1)
4
CCE MATHEMATICS–X
⇒
(p – 12)2 – 2 (p – 12) = 0
⇒
(p – 12) {(p – 12) – 2} = 0
⇒
(p – 12) (p – 14) = 0
⇒
p – 12 = 0
⇒
or p – 14 = 0
⇒
p = 12
⇒
p = 12, 14
1
1
11
−
=
; x ≠ – 4, 7
x + 4 x − 7 30
Sol. The given equation is
5. Solve:
= (3x + 2)x m2
According to the question,
(3x + 2)x = 120
⇒
3x2 + 2x = 120
3x2 + 2x – 120 = 0
11
1
1
=
−
x + 4 x − 7 30
⇒
3x2 + 20x – 18x – 120 = 0
( x − 7) − ( x + 4) 11
=
30
( x + 4) ( x − 7)
⇒
(3x + 20) (x – 6) = 0
⇒
3x + 20 = 0 or x – 6 = 0
⇒
⇒
x2 + 4x – 7x – 28 = – 30
⇒
x2 – 3x + 2 = 0
⇒
x2 – x – 2x + 2 = 0
⇒ x(3x + 20) – 6(3x + 20) = 0
⇒
⇒
(x – 1) (x – 2) = 0
⇒
x–1=0
20
or x = 6
3
20
x=−
,6
3
x= −
⇒
20
is inadmissible as breadth cannot be –ve.
3
x = 6 ∴ 3x + 2 = 3(6) + 2 = 20
x= −
∴
Hence, the length and breadth of the plot are 20 m and
6 m, respectively.
or x – 2 = 0
x = 1 or
3x = – 20 or x = 6
⇒
⇒ x (x – 1) – 2(x – 1) = 0
⇒
∴ Area of the plot = length × breadth
⇒
− 11
11
=
( x + 4) ( x − 7) 30
(x + 4) (x – 7) = – 30
⇒
6. The length of a rectangular plot is greater than thrice its
breadth by 2 m. The area of the plot is 120 sq. m. Find
the length and breadth of the plot.
Sol. Let the breadth of the plot be x m. Then, length of the
plot = (3x + 2) m
or p = 14
p = 12 is inadmissible as then from the given equation
2 = 0 which is impossible.
∴
p = 14
⇒
x = 1, 2
x=2
Formative Assessment
ORAL QUESTIONS (Conversation Type)
1. What is the standard form of a quadratic equation in
variable x?
2. What is the expression for the discriminant of the quadratic equation ax2 + bx + c = 0?
3. What is the maximum number of roots of a quadratic
equation?
4. In a quadratic equation ax2 + bx + c = 0, if a = 0, then
how will you call this equation?
5. What is the condition that a root of the quadratic equation ax2 + bx + c = 0 is 1?
TRUE OR FALSE
1. If the coefficient of x2 and the constant term have the
same sign and if the coefficient of x term is zero, then
the quadratic equation has no real roots.
2. If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic
equation has real roots.
3. Every quadratic equation has at most two roots.
4. Every quadratic equation has at least two roots.
5. Every quadratic equation has exactly one root.
5
QUADRATIC EQUATIONS
Assignments
Name: ....................................... Class: ........
Section: .......
CLASS ASSIGNMENT 1
1. Find the roots of the equation 4x2 – 25 = 0.
Roll No.: ...... Grade: ......
Teacher’s sign.: .....................
6. The sum of a number and its reciprocal is
10
. Form a
3
quadratic equation for this situation.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
2. Find the discriminant of the quadratic equation x2 + x – 5
= 0.
7. Does the equation ( x − 3 ) 2 = 0 have two equal roots?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
3. Is
2
3 a root of the equation x – 3 = 0?
____________________________________________
____________________________________________
____________________________________________
4. What must be added and subtracted to solve the quadratic equation x2 – 3x + 2 = 0 by the method of completing the square?
8. Is x3 – x2 = (x + 1)3 a quadratic equation?
____________________________________________
____________________________________________
____________________________________________
9. The sum of the squares of two consecutive natural numbers is 25. Form a quadratic equation for this situation.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
10. What is the sum of the roots of the quadratic equation
5. If –1 is a root of the quadratic equation x2 + x + k = 0,
then find the value of k.
2x2 −
3
x + 1 = 0?
2
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
6
CCE MATHEMATICS–X
Name: ....................................... Class: ........
Section: .......
CLASS ASSIGNMENT 2
(From CBSE Examination Paper)
1. If r = 3 is a root of the quadratic equation kr2 – kr – 3 = 0,
find the value of k.
Roll No.: ...... Grade: ......
Teacher’s sign.: .....................
6. Find the roots of the equation x2 + x – p(p + 1) = 0, where
p is a constant.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
7. Find the roots of the equation x2 – 3x – m (m + 3) = 0,
where m is a constant.
2. Find the nature of the roots of the equation 3x2 – 4x + 3
= 0.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
1
5
is a root of the equation x2 + kx + = 0, then find
2
4
the value of k.
____________________________________________
8. For what value of k, the quadratic equation
9x2 + 8kx + 16 = 0 has equal roots?
3. If
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
4. Prove that x = 3 is a root of the equation 2x2 – 5x – 3 = 0.
____________________________________________
9. For the quadratic equation x2 – 2x + 1 = 0, find the value
1
of x + .
x
____________________________________________
____________________________________________
____________________________________________
5. If one root of the equation 2x2 – 10x + p = 0 is 2, then
find the value of p.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
10. Find the nature of roots of the quadratic equation
3
1
2x2 −
x+
= 0.
2
2
____________________________________________
____________________________________________
____________________________________________
____________________________________________
7
QUADRATIC EQUATIONS
Name: ....................................... Class: ........
Section: .......
Roll No.: ...... Grade: ......
Teacher’s sign.: .....................
6. Is x2 + 2x + 1 = (4 – x)2 + 1 a quadratic equation?
HOME ASSIGNMENT 1
1. The area of a triangle is 25 cm2. The altitude to the base
is 2 cm less than its corresponding base. Form a quadratic equation for this situation.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
2. One diagonal of a rhombus is 2 cm less than the other.
The area of the rhombus is 36 cm2. Form a quadratic
equation for this situation.
7. Find the discriminant of the quadratic equation 5x2 – 3x
+ 1 = 0.
____________________________________________
____________________________________________
____________________________________________
8. What are the roots of the equation x2 – 4x = 0?
____________________________________________
____________________________________________
3. Can you say that 0.1 is a root of the equation x2 – 0.01
= 0?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
9. What is the product of the roots of the quadratic equation 2 x 2 − 5 x + 1 = 0?
4. What must be added and subtracted to solve the qua3
dratic equation 9 x 2 + x − 2 = 0 by the method of
4
completing the square?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
5. If 1 is a root of the quadratic equation
find the value of k.
x2
+ x + k = 0, then
10. For the quadratic equation ax2 + bx + c = 0, what is the
expression for the value of x using quadratic formula?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
8
CCE MATHEMATICS–X
Name: ....................................... Class: ........
Section: .......
HOME ASSIGNMENT 2
(From NCERT Exemplary Problems)
Roll No.: ...... Grade: ......
Teacher’s sign.: .....................
6. Find the roots of 6x2 – 2 x – 2 = 0 by the factorisation
of the corresponding quadratic polynomial.
1. Is 0.2 a root of the equation x2 – 0.4 = 0?
____________________________________________
____________________________________________
____________________________________________
____________________________________________
2. Does (x – 1)2 + 2(x + 1) = 0 have a real root?
____________________________________________
____________________________________________
7. The square of a natural number diminished by 84 is equal
to thrice of 8 more than the given number. Form a quadratic equation for this situation.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
3. Which constant should be added and subtracted to solve
the quadratic equation 4x2 – 3 x – 5 = 0 by the method
of completing the square?
8. A natural number, when increased by 12, equals 160 times
its reciprocal. Form a quadratic equation for this situation.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
4. How many real roots does the equation (x2 + 1)2 – x2 = 0
have?
F
H
9. Is – 2x2 = (5 – x) 2 x −
I
K
2
a quadratic equation?
3
____________________________________________
____________________________________________
____________________________________________
____________________________________________
5. Find the roots of the quadratic equation
= 0 by using the quadratic formula.
2x2
–
5x – 2
10. Find the value of k for which the quadratic equation 2x2
– kx + k = 0 has equal roots.
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____________________________________________
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