5.

16 Chapter 5

r­ emaining eight students are playing in the ground. 36. The roots of the equation 2x2 + 3x + c = 0 (where
What is the total number of students of the class? c < 0) could be ______.
29. If 2a and 3b are the roots of the equation x2 + ax
+ b = 0, then find the equation whose roots are 37. The roots of the equation 30x2 − 7 3x + 1 = 0 are
a, b. 38. If a and b are the roots of the quadratic equation
30. If a, b are the roots of the quadratic equation +
lx2 x2 + 3x − 4 = 0, then a−1 + b−1 = _____.
mx + n = 0, then evaluate the following expressions. x − 3 2x
39. The roots of the equation + = 1 , where
(a) a2 + b 2
x−2 x+3
x ≠ 2, −3, are
α β
(b)
+ 40. If the roots of the quadratic equation 4x2 − 16x +
β α
p = 0 are real and unequal, then find the value/s
1 1 of p.
(c)
+
α3 β3 41. If one root of the quadratic equation ax2 + bx
31. If the price of sugar is reduced by `1 per kg, + c = 0 is15 + 2 56 and a, b and c are rational,
5 ­kilograms more can be purchased for `1200. then find the quadratic equation.
What was the original price of sugar per kilogram?
42. If the roots of the equation ax2 + bx + 4c = 0 are in
32. The zeroes of the quadratic polynomial x2 − 24x + the ratio of 3 : 4, then find the relation between a,
143 are b and c.
33. Find the quadratic equation in x whose roots are 43. For which value of p among the following, does
−7 8 the quadratic equation 3x2 + px + 1 = 0 have real
and . roots?
2 3
34. If k1 and k2 are the roots of x2 − 5x − 24 = 0, then 44. If the product of the roots of ax2 + bx + 2 = 0
find the quadratic equation whose roots are −k1 is equal to the product of the roots of px2 + qx
and −k2. −1 = 0, then a + 2p = _____.
35. The product of the roots of the equations 45. Find the roots of quadratic equation ax2 + (a − b
1 1 1 + c) x − b + c = 0.
+ =
PRACTICE QUESTIONS

is _____.
x +1 x − 2 x + 2 46. If (2x − 9) is a factor of 2x2 + px − 9, then
p = _____.

Essay Type Questions

Solve the given equations:
47. (x + 3) (x + 4) (x + 6) (x + 7) = 1120. 49. x − 3 + 3x + 4 = 5.

48. (x2 + 3x)2 − 16 (x2 + 3x) − 36 = 0. 50. 3x4 − 10x3 − 3x2 + 10x + 3 = 0.

CONCEPT APPLICATION
Level 1
Direction for questions 1 to 20: Select the correct 2. The discriminant of the equation x2 − 7x + 2 = 0
alternative from the given choices. is
1. The solution of the equation x2 + x + 1 = 1 is are (a) 47
(b) 40
(a) x = 0
(c) 41
(d) −41
(b) x = −1
(c) Both (a) and (b) 3. Find the maximum value of the quadratic
(d) Cannot be determined expression −3x2 + 7x + 4.

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 Quadratic Expressions and Equations 5.17

1 1 (a) 9x2 − 2x + 7 = 0

(a) 8
(b) 8
6 12 (b) 9x2 − 2x − 7 = 0

1 (c) 9x2 + 2x − 7 = 0

(c) 8 (d) 12
4 (d) 9x2 + 2x + 7 = 0

4. If a and b are the roots of the equation x2 + 3x
− 2 = 0, then a2b + ab 2 = ? 12. The number of real roots of the quadratic equation
(x − 4)2 + (x − 5)2 + (x − 6)2 = 0 is
(a) −6
(b) −3
(a) 1
(b) 2
(c) 6
(d) 3
(c) 3
(d) None of these
5. If one of the roots of an equation, x2 − 2x + c
= 0 is thrice the other, then c = ? 13. The number of distinct real solutions of
2
1 4 x − 5 x + 6 = 0 is
(a) (b)
2 3 (a) 4
(b) 3
1 3 (c) 2
(d) 1
(c) − (d)
2 4 2
14. The number of real solutions of x − 5 x + 6
6. The number of real roots of the quadratic equa- = 0 is
tion 3x2 + 4 = 0 is
(a) 1
(b) 2
(a) 0
(b) 1
(c) 3
(d) 4
(c) 2
(d) 4
15. In writing a quadratic equation of the form x2
7. If a and b are the roots of the equation x2 + px + px + q = 0, a student makes a mistake in writing
+ q = 0 then (a - b)2 = _____. the coefficient of x and gets the roots as 8 and 12.
(a) q2 − 4p
(b) 4q2 − p Another student makes a mistake in writing the
constant term and gets the roots as 7 and 3. Find
(c) p2 − 4q
(d) p2 + 4q the correct quadratic equation.
8. Which of the following equations does not have (a) x2 − 10x + 96 = 0

real roots?
(b) x2 − 20x + 21 = 0

PRACTICE QUESTIONS

(a)
x2 + 4x + 4 = 0 (b) x2 + 9x + 16 = 0
(c) x2 − 21x + 20 = 0

(c) x2 + x + 1 = 0
(d) x2 + 3x + 1 = 0 (d) x2 − 96x + 10 = 0

9. The sum of the roots of the equation, + bx ax2 16. The roots of the equation 6x 2 − 8 2x + 4 = 0 are
+ c = 0 where a, b and c are rational and whose one
of the roots is 4 − 5, is 1 2
(a)
, 2 (b) ,1
3 3
(a) 8
(b) −2 5
2 3
(c) 2 5
(d) 11 (c)
, 2 (d) , 2
3 2
10. For the quadratic equation x2 + 3x − 4 = 0 which
17. Which of the following equations has roots as a, b
of the following is a solution?
and c?
(A) x = −4
(B) x = 3 (a) x3 + x2(a + b + c) + x(ab + bc + ca) + abc = 0

(C) x = 1
(b) x3 + x2(a + b + c) + x(ab + bc + ca) − abc = 0

(a) A and B
(b) B and C (c) x3 − x2(a + b + c) + x(ab + bc + ca) − abc = 0

(c) A and C
(d) Only A (d) x3 − x2(a + b + c) − x(ab + bc + ca) − abc = 0

11. Find the quadratic equation whose roots are 18. If the roots of the equation 2ax2 + (3b − 9) x + 1 =
reciprocals of the roots of the equation 7x2 − 2x + 0 are −2 and 3, then the values of a and b respec-
9 = 0. tively are

M05_TRISHNA_78982.indd 17 2/7/2018 12:43:51 PM

 5.18 Chapter 5

1 5 −1 −53 25. The age of a father is 25 years more than his son’s
(a)
, (b) , age. The product of their ages is 84 in years. What
12 18 12 18
will be son’s age in years, after 10 years?
−1 −5 −1 55
(c)
, (d) , (a) 3
(b) 28
12 8 12 18
(c) 13
(d) 18
19. The roots of the equation x2 + 5x + 1 = 0 are
26. If the roots of the equation ax2 − bx + 5c = 0 are in
5 + 21 5 − 21
(a)
, the ratio of 4 : 5, then
2 2
(a) ab = 18c2

−5 − 21 5 + 21
(b)
, (b) 81b2 = 4ac

2 2
(c) bc = a2

−5 + 21 −5 − 21
(c)
, (d) 4b2 = 81ac

2 2
27. The speed of Uday is 5 km/h more than that
−5 + 29 −5 − 29
(d) , of Subash. Subash reaches his home from office
2 2 2 hours earlier than Uday. If Subash and Uday stay
20. If a and b are the roots of the equation 3x2 − 2x 12 km and 48 km from their respective offices,
− 8 = 0, then a2 − ab + b 2 = _____. find the speed of Uday.
76 25 (a) 10 km/h

(a) (b)
9 3
(b) 4 km/h

16 32
(c) (d) (c) 9 km/h

3 3
(d) 8 km/h

21. If 2x + (2p − 13) x + 2 = 0 is exactly divisible by
2

x − 3, then the value of p is 28. If the roots of the quadratic equation c(a − b)x2 +
−16 19 a(b − c)x + b(c − a) = 0 are equal, then
(a) (b)
6 6 (a) 2b−1 = a−1 + c−1

16 −19 (b) 2c−1 = a−1 + b−1
PRACTICE QUESTIONS

(c)
(d)
6 6
(c) 2a−1 = b−1 + c−1

22. If x − ax − 6 = 0 and x + ax − 2 = 0 have one
2 2

common root, then a can be _____. (d) None of these

(a) −1
(b) 2 29. If the roots of the quadratic equation x2 − 3x − 304
(c) −3
(d) 0 = 0 are a and b, then the quadratic equation with
roots 3a and 3b is
2 1
23. The root of the equation + + (a) x2 + 9x − 2736 = 0

x −1 x + 2
3x( x + 1) (b) x2 − 9x − 2736 = 0

= 0 among the following is _____.
( x − 1)( x + 2) (c) x2 − 9x + 2736 = 0

(a) 2
(b) 3
(d) x2 + 9x + 2736 = 0

(c) −1
(d) 0
30. If x2 + a1x + b1 = 0 and x2 + a2x + b2 = 0 have a
24. If a and b are the roots of the equation, 2x2 − 5x common root (x − k), then find k.
+ 2 = 0, then (a − 1)b−1 = _____. where (a > b).
1 1 α 2 − α1 α 2 − α1
(a) k =
(b) k =
(a) (b) − β 2 − β1 β 2 − β1
2 2
1 α 2 − α1 α 2 − α1
(c)
(d) 1 (c) k =
(d) k =
2 β 2 − β1 β 2 − β1

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 Quadratic Expressions and Equations 5.19

Level 2

31. If one of the roots of x2 + (1 + k)x + 2k = 0 is twice 38. If (x + 2) is a common factor of the expressions x2
a2 + b2 + ax − 6, x2 + bx + 2 and kx2 − ax − (a + b), then
the other, then ______ . k = ______.
ab
(a) 2 (b) 1 (a) 2
(b) 3
(c) 4 (d) 7 (c) 1
(d) −2
32. If α and β are the roots of 2x2 − x − 2 = 0, then 39. The roots of a pure quadratic equation exists only
(α−3 + β −3 + 2α −1β −1) is equal to if ______.
17 23 (a) a > 0, c < 0
(b) c > 0, a < 0
(a) −
(b)
8 6 (c) a > 0, c ≤ 0
(d) Both (a) and (b)
37 29 x −1 x − 3 1
(c) (d) − 40. The roots of the equation + =2 ,
9 8 x −2 x −4 3
33. In a right angled triangle, one of the perpendicular (where x ≠ 2, 4) are
sides is 4 cm greater than the other and 4 cm lesser
than the hypotenuse. Find the area of triangle (a) 6 + 10, 6 − 10

in cm2.
(b) 6 + 2 10, 6 − 10

(a) 72 (b) 48
(c) 36 (d) 96 (c) 6 + 6 10, 6 − 6 10

34. In a fraction, the denominator is 1 less than
the numerator. The sum of the fraction and its (d) 2 + 2 10, 6 − 2 10

1 41. If x + 3 is the common factor of the expres-
reciprocal is 2 . Find the fraction.
56 sions ax2 + bx + 1 and px2 + qx −3, then
3 13 (9a + 3 p )
(a)
(b) − = ______ .
2 12 3b + q
(a) −2
(b) 2

PRACTICE QUESTIONS
18 8
(c)
(d)
17 7 (c) 3
(d) −1
35. The length of the rectangular surface of a table is 42. If the sum of the roots of an equation x2 + px + 1
10 m more than its breadth. If the area of the sur- = 0 (p > 0) is twice the difference between them,
face is 96 m2, its perimeter is (in m) _______. then p = ______.
(a) 64
(b) 44 1 3
(a) −
(b)
(c) 52
(d) 48 4 4
36. If α and β are the roots of the equation x2 + 9x
4 3
+ 18 = 0, then the quadratic equation having the (d)
(d)
roots α + β and α − β is ______, where (α > β). 3 2

(a) x2 + 6x − 27 = 0  (b) x2 − 9x + 27 = 0
5 5
43. The equation x + =3+ has
(c) x2 − 9x + 7 = 0      (d) x2 + 6x + 27 = 0
3−x 3−x
37. Find the minimum value of the quadratic (a) no real root.

expression 4x2 − 3x + 4. (b) one real root.

−55 55 (c) two equal roots.

(a) (b)
16 16 (d) infinite roots.

16 161 1 1
(c)
(d) 44. The roots of the equation − = 8 are
15 22 2x − 3 2x + 5

M05_TRISHNA_78982.indd 19 2/7/2018 12:43:57 PM

 5.20 Chapter 5

1 1 47. The roots of x2 − 2x − 1 = 0 are ______.

(a) 2 −
, 2−
2 2 (a)
2 + 1, 2 − 1
−1 + 17 −1 − 1 17 (b) 1, 2

(b)
,
2 2
(c) 1 + 2 , 1 − 2

1 1
(c) 2 +
, 2− (d) 2, 1

2 2
48. If 2x2 + 4x − k = 0 is same as (x − 5)  x +  = 0,
k
1 + 17 1 − 17  10 
(d) ,
2 2 then find the value of k.
45. If the quadratic expression x2 + (a − 4)x + (a + 4) (a) 100
(b) 90
is a perfect square, then a = ______.
(c) 70
(d) 35
(a) 0 and −4 (b) 0 and 6
49. If the numerically smaller root of x2 + mx = 2
(c) 0 and 12 (d) 6 and 12 is 3 more than the other one, find the value
46. The minimum value of 2x2 − 3x + 2 is ______. of m.
7 4 (a) −1 (b) 1
(a) (b) (c) 4 (d) −3
8 7 (c) −2 (d) 2

Level 3

50. Two persons A and B solved a quadratic equation the number are in the office and the remaining six
of the form x2 + bx + c = 0. A made a mistake in employees are on leave. What is the number of
noting down the coefficient of x and obtained the employees in the group?
roots as 18 and 2, where as B obtained the roots as (a) 49
(b) 64
−9 and −3 by misreading the constant term. The
(c) 36
(d) 100
correct roots of the equation are
(a) −6, −3
(b) −6, 6 55. Find the quadratic equation whose roots are 2
PRACTICE QUESTIONS

times the roots of x2 − 12x − 13 = 0.

(c) −6, −5
(d) −6, −6
(a) x2 − 24x − 52 = 0
51. If a and b are the roots of x2 − x + 2 = 0, then find (b) x2 − 24x − 26 = 0
the value of (a−6 + b −6 + 2a −3b−3)a6b6. (c) x2 − 14x − 15 = 0
(a) 16
(b) 25 (d) None of these
(c) 30 (d) 36
56. If one of the roots of ax2 + bx + c = 0 is thrice that
52. If b1, b2, b3, …, bn are positive, then the least value of the other root, then b can be
1 1 1 4ac 16ac
of (b1 + b2 + b3 + ∙∙∙ + bn)  + +  +  is (a) (b)
 b1 b2 bn  3 9
(a) b1b2 ∙∙∙ bn
(b) n2 + 1 ac 4ac
(c) 4
(d)
(c) n(n + 1)
(d) n2 3 3

53. The equation x + 1 − 4x − 1 = x − 1 has 57. If α, β are the roots of px2 + qx + r = 0, then
α3 + β 3 = _____.
(a) no solution.
(b) one solution.
3qpr − q3 3 pqr − 3q
(c) two solutions.
(d) more than two solutions. (a)
(b)
p3 p3
54. Out of the group of employees, twice the square
root of the number of the employees are on a trip pqr − 3q 3 pqr − q
(c)
(d)
to attend a conference held by the company, half p3 p3

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 Quadratic Expressions and Equations 5.21

1 2 (a) ax2 + (ab − ac)x − c = 0

58. If α and β are the roots of x2 − (a + 1) x + (a +
2 (b) ax2 + (b − c)x − bc = 0

a + 1) = 0 then α2 + β 2 = ______. (c) a2x2 + (b − c)x − ac = 0

(a) a (b) a2
(d) a2x2 + (ab − ac)x − bc = 0

(c) 2a (d) 1 66. Ramu swims a distance of 3 km each upstream and
59. The number of roots of the equation downstream. The total time taken is one hour. If
2
2 x −7 x +6 = 0 the speed of the stream is 4 km/h, then find the
speed of Ramu in still water.
(a) 4
(b) 3
(a) 12 km/h
(b) 9 km/h
(c) 2
(d) 1
60. In a quadratic equation ax2 − bx + c = 0, a, b, c (c) 8 km/h
(d) 6 km/h
are distinct primes and the product of the sum of 67. In solving a quadratic equation x2 + px + q = 0 a
91 student made a mistake in copying the coefficient
the roots and product of the roots is . Find the
9 of x and obtained the roots as 4, −3 but one of the
difference between the sum of the roots and the actual roots is 2 what is the difference between the
product of the roots. actual and wrong values of the coefficients of x?
(a) 2
(b) 3 (a) 5
(b) 4
(c) 4
(d) Cannot be determined (c) 7
(d) +6
2 + 12x − 3x 2 68. The roots of − bx + 2c = 0 are in the ratio of
ax2
61. Maximum value of is _____. 2 : 3, then _____.
2x 2 − 8x + 9
(a) a2 = bc
(b) 3b2 = 25ac
(a) 14
(b) 17
(c) 2b2 = 75c
(d) 5b2 = ac
(c) 11
(d) Cannot be determined
69. If the roots of 9x2 − 2x + 7 = 0 are 2 more than the
62. If x2 + ax + b and x2 + bx + c have a common factor
roots of ax2 + bx + c = 0, then 4a − 2b + c can be
(x − k), then k = _______.
(a) −2
(b) 7

PRACTICE QUESTIONS
a−b b−c
(a)
(b)
b−c c −a (c) 9
(d) 10

c−b c−b 70. If the roots of ax2 + bx + c = 0 are 2 more than

(c)
(d) the roots of px2 + qx + r = 0, then the value of c in
b−a a −b
terms of p, q and r is
y y 2 − 4xz (a) p + q + r
(b) 4p − 2q + r
63. If 9x − 3y + z = 0, then the value of + (c) 3p − q + 2r
(d) 2p + q − r
2x 4x 2
(where x, y, z are constants). 71. If the roots of 2x2 + 7x + 5 = 0 are the reciprocal
roots of ax2 + bx + c = 0, then a − c = _____.
(a) 9
(b) 2
(a) 3 (b) −3
(c) 3
(d) 6 (c) −2 (d) −5
64. If the roots of 3x2 − 12x + k = 0 are complex, then 1
72. If the roots of the equation ax2 + bx + c = 0 is
find the range of k. k
times the roots of px2 + qx + r = 0, then which of
(a) k < 22
(b) k < −10 the following is true?
(c) k > 11
(d) k > 12 a p
(a) a = pk
(b) =
65. If a, b are the roots of ax2 + bx + c = 0, then find b q
the quadratic equation whose roots are a + b, ab. (c) aq = pbk
(d) ab = pqk2

M05_TRISHNA_78982.indd 21 2/7/2018 12:44:01 PM

 5.22 Chapter 5

TEST YOUR CONCEPTS

Very Short Answer Type Questions

1. non-zero real numbers 11. Yes

2. root 12. −1
3. 3
13. real and distinct
4. quadratic
14. 4ac
3
5. 15. Zero
2
6. x2 + x(a + b) + ab = 0 16. zero
7. (x − 2) 17. ±6
8. real numbers 18. positive
9. 2, −3
19. a + c
−a ± a 2 − 4b
10.
2

Short Answer Type Questions

20. (a) (x + 2)(x + 3) 32. 11, 13
(b) (x − 9)(x + 4)
33. 6x2 + 5x − 59 = −3
(c) (x − 2)(2x + 9)
34. x2 + 5x − 24 = 0
21. (a) Complex conjugates 35. zero
(b) Real and distinct
36. rational or irrational, but unequal
(c) Real and equal
1 1
37. ,
22. −4 and −33 2 3 5 3
23. ± 2 3
38.
−2 4
24.
3 −1
25. 6 39. 3,
2
26. qx2 − (p2 − 2q)x + q = 0
ANSWER KEYS

40. p < 16
27. 8 km/h 41. x2 − 30x + 1 = 0
28. 12 or 24 42. 3b2 = 49ac
29. x2 − (6ab − 2a − 3b)x − (2a + 3b)(6ab) = 0 43. 4
m 2 − 2ln m 2 − 2ln 3lmn − m3 44. 0
30. (a)   (b)    (c) b−c
l2 ln n3 45. −1,
a
31. `16 46. −7

Essay Type Questions

47. x = 1, x = −11 49. 4

1 ± 37 3 ± 13
48. −1, −2, 3 and −6 50. ,
6 2

M05_TRISHNA_78982.indd 22 2/7/2018 12:44:02 PM

 Quadratic Expressions and Equations 5.23

CONCEPT APPLICATION
Level 1
1. (c) 2. (c) 3. (b) 4. (c) 5. (d) 6. (a) 7. (c) 8. (c) 9. (a) 10. (c)
11. (a) 12. (d) 13. (c) 14. (d) 15. (a) 16. (c) 17. (c) 18. (d) 19. (c) 20. (a)
21. (b) 22. (a) 23. (c) 24. (d) 25. (c) 26. (d) 27. (d) 28. (c) 29. (b) 30. (a)

Level 2
31. (d) 32. (d) 33. (d) 34. (d) 35. (b) 36. (a) 37. (b) 38. (c) 39. (d) 40. (a)
41. (d) 42. (c) 43. (a) 44. (b) 45. (c) 46. (a) 47. (c) 48. (c) 49. (b)

Level 3
50. (d) 51. (b) 52. (d) 53. (a) 54. (c) 55. (a) 56. (c) 57. (a) 58. (a) 59. (a)
60. (a) 61. (a) 62. (d) 63. (c) 64. (d) 65. (d) 66. (c) 67. (a) 68. (b) 69. (b)
70. (b) 71. (a) 72. (c)

ANSWER KEYS

M05_TRISHNA_78982.indd 23 2/7/2018 12:44:02 PM

 5.24 Chapter 5

CONCEPT APPLICATION

Level 1

1. Simplify and factorize. 17. Let (x − a)(x − b)(x − c) = 0 and expand.

2. Use the formula to find the discriminant. 18. Substitute x = −2 and x = 3 to get to equation in
a and b and then solve for a and b.
3. Use the formula to find the maximum value.
4. Find the sum and product of the roots. Let a2b + −b ± b 2 − 4ac
19. Apply formula x = .
ab 2 = ab(a + b). 2a
5. Let the roots be a and 3a. 20. Find ab and (a + b ) and use, a2 − ab + b2 =
(a + b )2 − 3ab.
6. Solve for x.
21. Substitute x = 3 in the given equation and simplify.
7. (a − b)2 = (a + b)2 − 4ab.
22. Find x in terms of a and substitute x = a in either
8. Find the value of the discriminant for each of the
of the equations.
equations.
23. Simplify and solve for x.
9. If one root is 4 − 5, then the other root is
4 + 5, because the coefficients of the xn terms 24. Factorize LHS of the given equation to find a
are rational. and b.

10. Solve for x. 25. (i) Form a quadratic equation by assuming the
age of son as r years. The roots of quadratic
11. The quadratic equation with reciprocals of the c
equation are 1 and only if sum of all the
roots of the equation f (x) = 0 is f   = 0.
1 a
H i n t s a n d E x p l a n at i o n

x coefficient s = 0.
12. (i) Use the concept of perfect square of a number. (ii) Assume the ages of father and son as x and
(ii) If a2 + b2 + c2 = 0 is true only when a = b = (25 − x) years.
c = 0. (iii) Write the relation in terms of x according to

13. (i) Solve the equation to find the number of real the data and then solve the equation.
roots.
26. (i) Use the concept of sum and product of the
(ii) Replace |x| by y and solve for y. roots of a quadratic equation.
(iii) Now, x = ±y.
(ii) Take the roots as 4a, 5a.
14. (i) Use the concept |x| and find the roots. (iii) Using the sum of the roots and product of the

(ii) Replace |x| by y and solve for y. roots eliminate a.

(iii) Now, x = ±y.

27. (i) Frame the quadratic equation from the given
data.
15. (i) Use the concept of sum and product of the
(ii) Assume the speed of Subhash as x and Subhash
roots of a quadratic equation.
reaches his home in t hours and Uday reaches
(ii) The product of the roots obtained by the first his home in (t + 2) hours.
student is product of the roots of the required
quadratic equation. distance
(iii) Now, use time =
, then solve the equa-
speed
(iii) The sum of the roots obtained by the second
tion for x.
student is sum of the roots of the required qua-
dratic equation. c
28. (i) If sum of all coefficients is zero, then 1, are
a
16. Take 2 as common and then factorize. the roots of ax2 + bx + c = 0.

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 Quadratic Expressions and Equations 5.25

(ii) If the sum of the coefficients is 0, then 1 and (ii) The quadratic equation whose roots are
c m times of the roots of the equation f (x)
are the roots of the equation.
a −1 55
= 0 is , = 0.
(iii) Use product of the roots concept. 12 18
30. (i) Use the concept of common root of given
29. (i) The quadratic equation with thrice the roots equations.
1
of f (x) = 0 as roots is f   = 0. (ii) Put x = k in the given equations and solve
x for k.

Level 2

31. (i) Use the concept of sum and product of the (iii) Find the value of ‘l’, using l(l − 10) = 96.

roots of a quadratic equation.
(iv) Calculate the perimeter of the rectangle using
(ii) Assume the roots as a and 2a. 2(l + l − 10).
(iii) Find the sum of the roots and product of the
36. Find a + b, ab and using these values find a − b.
roots.
37. Use the formula to find the minimum value.
(iv) From the above equation eliminate ‘a’.
38. Substitute x = −2 in the first two expressions,
k2 + 1 equated to zero.
(v) Then obtain the value of .
k
32. (i) Simplify the required expression and find 39. (i) A pure quadratic equation is ax2 + c = 0.
a + b and ab. (ii) Pure quadratic equation is ax2 + c = 0.
(ii) Find the sum of the roots and product of the

H i n t s a n d E x p l a n at i o n
40. (i) Simplify the equation 1.
roots.
(ii) Take the LCM of the equation.
(iii) Use relation, a3 + b3 = (a + b)3 − 3ab(a + b).

33. (i) Use Pythagorean theorem to find the sides of (iii) Convert it into quadratic equation.

the triangle. (iv) Solve the equation for x.

(ii) Assume the sides as x, x − 4 and hypotenuse as 41. (i) If x + k is the common root, then x = −k
x + 4. ­satisfies both the equations.
(iii) Find the value of x using the relation (ii) If x + a is factor of f (x), then f (−a) = 0.
(hypotenuse)2 = sum of the squares of the
other two sides. (iii) Write p in terms of q and b in terms of a.

1 (iv) Now substitute these values in the given
(iv) The area of triangle = × base × height.
2 expression and simplify.
34. (i) Form the quadratic equation and solve for x.
42. Find the sum and product of the roots and form
x
(ii) Assume the fraction as . the equation as per the condition given in the
x −1 problem.
x x − 1 21 43. (i) Simplify the equation.
(iii)
+ = .
x −1 x 16 f (x )
(ii) A rational function is defined only
(iv) Solve the above equation. g( x )
when g(x) > 0.
35. (i) Use the formula to find the area of the
rectangle. 44. (i) Simplify the equation.
(ii) Assume the length and breadth as lm and (ii) Take the LCM.
(l − 10) m. (iii) Convert it into quadratic equation.

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 5.26 Chapter 5

(iv) Solve the equation by using formula x 48. 2x2 + 4x − k = 0 (1)

−b ± b2 − 4ac \ ⇒ (x − 5) is a factor of Eq. (1).

= .
2a
⇒ x − 5 = 0 ⇒ x = 5
45. (i) If the equation is a perfect square, then it has
equal roots. \ 2(5)2 + 4(5) − k = 0

(ii) Quadratic equation is a perfect square, if b2 − 50 + 20 − k = 0

4ac = 0.
\ k = 70.

(iii) Substitute the value of b and c in the above
equation and obtained the value of a. 49. The difference of the roots of ax2 + bx + c = 0 is
46. The minimum value of ax2 + bx + c is b 2 − 4ac
for a > 0.
4ac − b 2 a
. (a > 0 )
4a m2 + 8 = 3
\ For x2 + mx − 2 = 0. It is

The minimum value of 2x2 − 3x + 2 =
⇒ m = ±1.
4 × 2 × 2 − ( −3)2 7
= . That is, the equation could be x2 + x − 2 = 0 or

4×2 8
x2 − x − 2 = 0.
47. Given x2 − 2x − 1 = 0
That is, (x + 2)(x − 1) = 0 or (x − 2)(x + 1) = 0

2 ± 4 − 4 × 1 × ( −1)
x= The roots are −2, 1 or −1, 2.

2×1
2± 8 2±2 2 As the numerically smaller root is greater, the roots

x = = =1± 2 are −2, 1 and m = 1.
2 2
x = 1 + 2 or 1 − 2 .
H i n t s a n d E x p l a n at i o n

Level 3

50. (i) Use the concept of sum of the roots and (ii) Square the given expression twice and then
product of the roots of a quadratic equation. solve for x.
(ii) The product of the roots obtained by A and 54. (i) Form the quadratic equation and solve
sum of the roots obtained by B is equal to the for x.
product and sum of the roots of the required (ii) Assume number of employees in the group
equation respectively. as x. Then write the quadratic equation in x
51. (i) Simplify the expression and find (a + b) and according to the data and solve it.
ab.
55. f(x) = x2 − 12x − 13 = 0
(ii) First find the sum of the roots and product of
If the roots of g(x) are 2 times the roots of f(x), then
the roots.
x
2 (α 3 + β 3 ) g( x ) = f   = 0.
(iii) α −6 + β −6 +
= 2
α β
3 3 α 6β 6
2
52. (i) Take b1 = b2 = b3 … bn = k and find the value. x x
⇒   − 12   − 13 = 0
2 2
(ii) AM (a1, a2, …, an) ≥ HM (a1, a2, …, an). 2
x 12x
⇒ − − 13 = 0
4 2
(iii) a1 + a2 +  + an ≥
n
.
n 1 1 1 ⇒ x 2 − 24x − 52 = 0.
+ +
a1 a2 a2 56. Let the roots be k and 3k.
53. (i) Simplify the equation. Sum of the roots = k + 3k

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 Quadratic Expressions and Equations 5.27

−b −b 2y − 3 = 0 or y − 2 = 0

⇒ 4k = ⇒k= .
a 4a 3
Product of the roots = k × 3k
y=
or y = 2
2
c 3
⇒ 3k 2 = |x| = or |x| = 2

a 2
2
 −b  c x=±
3
or x = ±2.
⇒ 3  =
 4a  a 2
3b 2 c \ x has 4 real solutions.

⇒ =
16a 2 a
⇒ 3b 2 = 16ac 60. Product of the sum of the roots and product of the
91
ac roots is , i.e.,
⇒ b = ±4 . 9
3
 b c  91
−q  × =
57. α + β = a a 9
p
r bc 91
α ⋅β= =
p a2 9

α 3 + β 3 = (α + β )3 − 3αβ (α + β ) bc 13 × 7
=
3 a×a 3×3
 −q   r   −q 
=   − 3   b 13 c 7 b 7 c 13
 p   p  p  ⇒ = , = or = , =
a 3 a 3 a 3 a 3
−q3 + 3 pqr

H i n t s a n d E x p l a n at i o n
= 13 7
p3 The required difference is
− = 2.
3 3
3 pqr − q3
∴α 3 + β 3 = . 2 + 12x − 3x 2
p3 61. For the maximum value of ,
2x 2 − 8x + 9
1 2 2 + 12x − 3x2 is maximum and 2x2 − 8x + 9 is
58. x2 − (a + 1)x + (a + a + 1) = 0 minimum.
2
a + b = a + 1 The maximum value of 2 + 12x − 3x2 and mini-

1 2 −b
ab =
(a + a + 1) mum value of 2x2 − 8x + 9 occurs at x = ,
2 2a
i.e., 2.
a2 + b2 = (a + b)2 − 2ab

When x = 2,

1 
= (a + − 2  (a 2 + a + 1)
1)2 2 + 12x − 3x 2 2 + 24 − 12
2  = = 14.
2x − 8x + 9 8 − 16 + 9
2
= a2 + 2a + 1 − a2 − a − 1 = a.
62. x2 + ax + b and x2 + bx + c have a common factor
59. 2|x|2 − 7|x| + 6 = 0 (x − k)
Let |x| = y
⇒ k2 + ak + b = 0 and k2 + bk + c = 0
2y2 − 7y + 6 = 0
⇒ k2 + ak + b = k2 + bk + c
2y2
− 3y − 4y + 6 = 0 ak + b = bk + c

y(2y − 3) − 2(2y − 3) = 0
c −b
k= .
(2y − 3)(y − 2) = 0
a −b

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 5.28 Chapter 5

63. 9x − 3y + z = 0 consider xa2 − ya + z = 0, a quadratic (x − 8) (x + 2) = 0

equation in a. Where x, y, z are constants. x = 8 km/h ( speed cannot be −2 km/h).

Let a = 3 ⇒ (3)2 x − 3y + z = 0 is a quadratic equa-
67. Quadratic equation with 4, −3 as roots is x2
tion in 3. − 1x − 12 = 0, quadratic equation whose product
of the roots is −12.(1)
−( −y ) + y 2 − 4 ⋅ x ⋅ z
3= As one of the actual roots is 2, the other root is −6.
2.x  (from (1))
y y 2 − 4xz The quadratic equation is x − (−6 + 2) x − 12 = 0
2
⇒3= + .
2x 4x 2 x2 + 4x − 12 = 0.
\ The difference between the coefficients of x =
64. Given the roots of the given equation are complex
4 − (−1) = 5.
⇒ b2 − 4ac < 0 68. Let a, b be the roots of ax2 − bx + 2c = 0
⇒ (−12)2 − 4(3) k < 0 α 2
Given =
=
144 − 12 k < 0
β 3
−12k < −144
2β
12 k > 144
⇒ α =
3
k > 12.
Product of the roots

65. a, b are the roots of ax2 + bx + c = 0 2c
α ⋅β =
−b a
⇒α + β = 2β 2c
a ×β =
c 3 a
α ⋅β = .
H i n t s a n d E x p l a n at i o n

3c
a β 2 = (1)
a
Quadratic equation whose roots are a + b, and ab
Sum of the roots = a + b
−b c −b c 2β
is x2 −  +  x +
b
× = 0. ⇒ +β =
 c a a a 3 a

b −c  bc 5β b
x2 + x − 2 =0 ⇒ =
 a  a 3 a

a2 x2 + (ab − ac)x − bc = 0.
3b
β =
66. Let the speed of Ramu = x km/h 5a
Total time taken is = 1 hour
9b 2
β 2 = (2)
That is,
25a 2

3 3 3c 9b 2
+ =1 From Eqs. (1) and (2),
= 3b2 = 25ac.
x − 4 x +4 a 25a 2
3x + 12 + 3x − 12 69. f(x) = ax2 + bx + c = 0
=1
x 2 − 16 f(x − 2) = 9x2 − 2x + 7 = 0

6x = x2 − 16
a(x − 2)2 + b(x − 2) + c = 9(x)2 − 2x + 7
\

x2 − 6x − 16 = 0
a(x2 − 4x + 4) + bx − 2b + c = 9x2 − 2x + 7

x2 − 8x + 2x − 16 = 0
ax2 − (4a − b)x + 4a − 2b + c = 9x2 − 2x + 7

x(x − 8) + 2(x − 8) = 0
⇒ 4a − 2b + c = 7.

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 Quadratic Expressions and Equations 5.29

70. Let f(x) ≡ px2 + qx + r = 0 2

That is, 2   + 7   + 5 = 0
1 1

Given the quadratic equation whose roots are 2
x x
more than the roots of f(x) as ax2 + bx + c = 0.
⇒ 2 + 7x + 5x2 = 0 ⇒ 5x2 + 7x + 2 = 0
⇒ f(x − 2) = ax2 + bx + c
a = 5, b = 7, c = 2

⇒ p(x − 2)2 + q(x − 2) + r = ax2 + bx + c
a − c = 5 − 2 = 3.

⇒ px2 − 4px + 4p + qx − 2q + r = ax2 + bx + c
72. ax2 + bx + c
⇒ px2 + (q − 4p)x + (4p − 2q + r) = ax2 + bx + c
= p(kx)2 + q(kx) + r = 0
⇒ c = 4p − 2q + r.
⇒ a = pk2, b = qk, c = r
71. If the roots of2x2 + 7x + 5 = 0 (1)
are the reciprocal roots of ax + bx + c = 0, then ax2
2 a pk 2
=
1 b qk
+ bx + c = 0 is obtained by substituting in Eq. (1).
x ⇒ aq = pbk.

H i n t s a n d E x p l a n at i o n

M05_TRISHNA_78982.indd 29 2/7/2018 12:44:10 PM

 Chapter
Chapter

12
6 Sets and
Kinematics
Relations

REMEMBER
Before beginning this chapter, you should be able to:

• Understand the basic definitions of sets

• Apply basic operations on sets
• Understand basic concept of Venn diagrams

KEY IDEAS
After completing this chapter, you should be able to:

• learn about sets, representation of sets and some

definitions related to sets
• Apply operations on sets
• Understand the Venn diagrams
• Obtain cartesian products of sets
• Know about relation of sets, its representation, types,
properties and to find its domain and range
• Understand the functions of sets
Figure 1.1

M06_TRISHNA_78982.indd 1 2/7/2018 12:51:09 PM

 6.2 Chapter 6

INTRODUCTION
In everyday life we come across different collections of objects. For example, A herd of sheep, a
cluster of stars, a posse of policemen, etc. In mathematics, we call such collections as sets. The
objects are referred to as the elements of the sets.

SET
A set is a well-defined collection of objects.
Let us understand what we mean by a well-defined collection of objects.
We say that a collection of objects is well-defined if there is some reason or rule by which we
can say whether a given object of the universe belongs to or does not belong to the collection.

Elements of a Set
The objects in a set are called elements or members of the set. We usually denote the sets by
capital letters A, B, C or X, Y, Z, etc.
If a is an element of a set A, then we say that a belongs to A and we write, a ∈ A.
If a is not an element of A, then we say that a does not belong to A and we write, a ∉ A.
To understand the concept of a set, let us look at some examples.
Examples:
1. L
 et us consider the collection of odd natural numbers less than or equal to 15.
In this example, we can definitely say what the collection is. The collection comprises the
numbers 1, 3, 5, 7, 9, 11, 13 and 15.
2. L
 et us consider the collection of students in a class who are good at painting. In this
example, we cannot say precisely which students of the class belong to our collection. So,
this collection is not well-defined.
Hence, the first collection is a set where as the second collection is not a set. In the first
example given, the set of odd natural numbers less than or equal to 15 can be represented as set
A = {1, 3, 5, 7, 9, 11, 13, 15}.

Some Sets of Numbers and Their Notations

N = Set of all natural numbers = {1, 2, 3, 4, 5, …}.
W = Set of all whole numbers = {0, 1, 2, 3, 4, 5, …}.
Z or I = Set of all integers = {0, ±1, ±2, ±3, …}.
p 
Q = Set of all rational numbers =  where p, q ∈ Z and q ≠ 0  .
q 

Cardinal Number of a Set

The number of elements in a set A is called its cardinal number. It is denoted by n(A). A set which
has finite number of elements is a finite set and a set which has infinite number of elements is an
infinite set.
Examples:
1. Set of English alphabets is a finite set.
2. Set of number of days in a month is a finite set.

M06_TRISHNA_78982.indd 2 2/7/2018 12:51:09 PM

 Sets and Relations 6.3

3. The set of all even natural numbers is an infinite set.

4. Set of all the lines passing through a point is an infinite set.
5. T
 he cardinal number of the set X = {a, c, c, a, b, a} is n(X) = 3 as in sets only distinct
elements are counted.

Representation of Sets
We represent sets by the following methods:

Roster or List Method

In this method, a set is described by listing out all the elements in the set.
Examples:
1. Let W be the set of all letters in the word JANUARY. Then we represent W as,
W = {A, J, N, R, U, Y}.
2. L
 et M be the set of all multiples of 3 less than 20. Then we represent the set M as,
M = {3, 6, 9, 12, 15, 18}.

Set Builder Method

In this method, a set is described by using a representative and stating the property or properties
which the elements of the set satisfy, through the representative.
Examples:
1. Let D be the set of all days in a week. Then we represent D as,
D = {x: x is a day in a week}.
2. Let N be the set of all natural numbers between 10 and 20, then we represent the set N as,
N = {x: 10 < x < 20 and x ∈ N}.

Some Simple Definitions of Sets

Empty Set or Null Set or Void Set
A set with no elements in it is called an empty set (or) void set (or) null set. It is denoted by { }
or f. (read as ‘phi’)
Examples:
1. Set of all positive integers less than 1 is an empty set.
2. Set of all mango trees with apples is an empty set.

Singleton Set
A set consisting of only one element is called a singleton set.
Examples:
1. T
 he set of all vowels in the word MARCH is a singleton, as A is the only vowel in the
word.
2. T
 he set of whole numbers which are not natural numbers is a singleton, as 0 is the only
whole number which is not a natural number.

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 6.4 Chapter 6

Equivalent Sets
Two sets A and B are said to be equivalent if their cardinal numbers are equal. We write this
symbolically as A ~ B or A ↔ B.
Examples:
1. Sets, X = {2, 4, 6, 8} and
Y = {a, b, c, d} are equivalent as n(X) = n(Y) = 3.
2. Sets, X = {Dog, Cat, Rat} and
Y = { , , } are equivalent.
3. Sets, X = {-1, -7, -5} and
Y = {Delhi, Hyderabad} are not equivalent as n(X) ≠ n(Y ).

Note If the sets A and B are equivalent, we can establish a one-to-one correspondence
between the two sets. That is, we can pair up elements in A and B such that every element
of set A is paired with a distinct element of set B and every element of B is paired with a
distinct element of A.

Equal Sets
Two sets A and B are said to be equal if they have the same elements.
Examples:
1. Set, A = {a, e, i, o, u} and B = {x: x is a vowel in the English alphabet} are equal sets.
2. Set, A = {1, 2, 3} and B = {x, y, z} are not equal sets.
3. Set, A = {1, 2, 3, 4, …} and B = {x: x is a natural number} are equal sets.

Note If A and B are equal sets, then they are equivalent but the converse need not
be true.

Disjoint Sets
Two sets A and B are said to be disjoint, if they have no elements in common.
Examples:
1. Sets X = {3, 6, 9, 12} and Y = {5, 10, 15, 20} are disjoint as they have no elements in
common.
2. Sets A = {a, e, i, o, u} and B = {e, i, j} are not disjoint as they have elements e and i in
common.

Subset
Let A and B be two sets. If every elements of set A is also an element of set B, then A is said to be a
subset of B or B is said to be a superset of A. If A is a subset of B, then we write A ⊆ B or B ⊇ A.
Examples:
1. Set A = {2, 4, 6, 8} is a subset of set B = {1, 2, 3, 4, 5, 6, 7, 8}.
2. Set of all primes except 2 is a subset of the set of all odd natural numbers.
3. Set A = {1, 2, 3, 4, 5, 6, 7, 8} is a superset of set B = {1, 3, 5, 7}.

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 Sets and Relations 6.5

Notes
1. Empty set is a subset of every set.
2. Every set is a subset of itself.
3. If a set A has n elements, then the number of subsets of A is 2n.
4. If a set A has n elements, then the number of non-empty subsets of A is 2n - 1.

Proper Subset and Superset

If A ⊆ B and A ≠ B, then A is called a proper subset of B and is denoted by A ⊂ B. If A ⊂ B,
then B is called a superset of A and is denoted as B ⊃ A.

Power Set
The set of all subsets of a set A is called power set of A. It is of A denoted by P(A).
Example: Let A = {x, y, z}. Then the subsets of A are f, {x}, {y}, {z}, {x, y}, {x, z}, {y, z},
{x, y, z}.
So, P(A) = {f, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.
We observe that the cardinality of P(A) is 8 = 23.

Notes
1. I f a set A has n elements, then the number of subsets of A is 2n, i.e., the cardinality of the
power set is 2n.
2. If a set A has n elements, then the number of proper subsets of A is 2n - 1.
3. If a set A has n elements, then the number of non-empty proper subsets is 2n - 2.

Universal Set
A set which consists of all the sets under consideration or discussion is called the universal set. It
is usually denoted by ∪ or µ.
Example:
Let A = {a, b, c}, B = {c, d, e} and C = {a, e, f, g, h}. Then, the set {a, b, c, d, e, f, g, h} can
be taken as the universal set here.
∴ µ = {a, b, c, d, e, f, g, h}.

Complement of a Set
Let µ be the universal set and A ⊆ µ. Then, the set of all those elements of µ which are not in
set A is called the complement of the set A. It is denoted by A′ or Ac.
A′ = {x : x ∈ µ and x ∉ A}.
Examples:
1. Let µ = {3, 6, 9, 12, 15, 18, 21, 24} and A = {6, 12, 18, 24}. Then, A′ = {3, 9, 15, 21}.
2. Let µ = {x: x is a student and x ∈ class X}.
And B = {x: x is a boy and x ∈ class X}.
Then, B′ = {x: x is a girl and x ∈ class X}.

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 6.6 Chapter 6

Notes
1. A and A′ are disjoint sets.
2. µ′ = f and f′ = µ.

Operations on Sets
Union of Sets
Let A and B be two sets. Then, the union of A and B, denoted by A ∪ B, is the set of all those
elements which are either in A or in B or in both A and B.
That is, A ∪ B = {x: x ∈ A or x ∈ B}.
Examples:
1. Let A = {-1, -3, -5, 0} and B = {-1, 0, 3, 5} then, A ∪ B = {-5, -3, -1, 0, 3, 5}.
2. Let A = {x: 5 ≤ 5x < 25 and x ∈ N} and B = {x: 5 ≤ (10x) ≤ 20 and x ∈ N}, then, A ∪ B
= {x: 5 ≤ 5x ≤ 20 and x ∈ N}.

Notes
1. If A ⊆ B, then A ∪ B = B.
2. A ∪ µ = µ and A ∪ f = A.
3. A ∪ A′ = µ.

Intersection of Sets
Let A and B be two sets. Then the intersection of A and B, denoted by A ∩ B, is the set of all
those elements which are common to both A and B.
That is, A ∩ B = {x: x ∈ A and x ∈ B}.
Examples:
1. Let A = {1, 2, 3, 5, 6, 7, 8} and B = {1, 3, 5, 7}. A ∩ B = {1, 3, 5, 7}.
2. Let A be the set of all English alphabet and B be the set of all consonants. A ∩ B is the set
of all consonants in the English alphabet.
3. Let E be the set of all even natural numbers and O be the set of all odd natural numbers.
E ∩ O = { } or f.

Notes
1. If A and B are disjoint sets, then A ∩ B = f and n(A ∩ B) = 0.
2. If A ⊆ B, then A ∩ B = A.
3. 3A ∩ µ = A and A ∩ f = f.
4. A ∩ A′ = f

Difference of Sets
Let A and B be two sets. Then the difference A - B is the set of all those elements which are in
A but not in B. That is, A - B = {x: x ∈ A and x ∉ B}.
Example: Let A = {3, 6, 9, 12, 15, 18} and B = {2, 6, 8, 10, 14, 18}.
A - B = {3, 9, 12, 15} and B - A = {2, 8, 10, 14}.

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