KC Sinha Mathematics Solution Class 10 Chapter 7 Quadratic Equations Exercise 7.4

Exercise 7.1
Exercise 7.2
Exercise 7.3
Exercise 7.4
Exercise 7.5

Exercise 7.4

Question 1 A

Write the discriminate of each of the following quadratic equation:

x² + 4x + 3 =0
Sol :
d = b² – 4ac
d = (4)² – 4 (1) (3)
d = 16 – 12
d = 4

Question 1 B

Write the discriminate of each of the following quadratic equation:

4x² + 5x + 7 = 0
Sol :
d = b² – 4ac
d = (5)² – 4 (4) (7)
d = 25 – 112
d = –87

Question 1 C

Write the discriminate of each of the following quadratic equation:

2x² + 4x + 5 = 0
Sol :
d = b² – 4ac
d = (4)² – 4 (2) (5)
d = 16 – 40
d = –24

Question 1 D

Write the discriminate of each of the following quadratic equation:

3x² + 5x + 6 = 0
Sol :
d = b² – 4ac
d = (5)² – 4 (3) (6)
d = 25 – 72
d = –47

Question 1 E

Write the discriminate of each of the following quadratic equation:

√3 x² — 2√2 – 2√3 = 0
Sol :
d = b²– 4ac
d = (–2√2)² – 4 (√3) (–2√3)
d = 8 + 24
d = 32

Question 2 A

Examine whether the following quadratic equations have real roots or not:

7x² + 8x — 1 = 0
Sol :
TO CHECK REAL ROOTS, ‘d’ SHOULD BE GREATER THAN 0.
d = b² – 4ac
d = (8)²2 – 4 (7) (–1)
d = 64 – 28
d = 92
Yes, roots are real.

Question 2 B

Examine whether the following quadratic equations have real roots or not:

2x² + 3x + 4 = 0
Sol :
TO CHECK REAL ROOTS, ‘d' SHOULD BE GREATER THAN 0.
d = b²– 4ac
d = (3)² – 4(2)(4)
d = 9 – 32
d = – 23
No, roots aren’t real.

Question 2 C

Examine whether the following quadratic equations have real roots or not:

x² — 12x — 16 = 0
Sol :
TO CHECK REAL ROOTS, ‘d' SHOULD BE GREATER THAN 0.
d = b² – 4ac
d = (–12)² – 4 (1) (–16)
d = 144 – 64
d = 208
Yes, roots are real .

Question 2 D

Examine whether the following quadratic equations have real roots or not:

x² + x – 1= 0
Sol :
TO CHECK REAL ROOTS, ‘d' SHOULD BE GREATER THAN 0.
d = b² – 4ac
d = (1)² – 4 (1) (–1)
d = 1 + 4
d = 5
Yes, roots are real.

Question 2 E

Examine whether the following quadratic equations have real roots or not:

x² — 10x + 2 = 0
Sol :
TO CHECK REAL ROOTS, ‘d' SHOULD BE GREATER THAN 0.
d = b² – 4ac
d = (–10)² – 4 (1) (2)
d = 100 – 8
d = 92
Yes, roots are real.

Question 3 A

Find whether the following quadratic equations have a repeated root :

9x²— 12x + 4 = 0
Sol :
REPEATED ROOTS MEAN d = 0.
d = b² – 4ac
d = (–12)² – 4(9)(4)
d = 144 – 144
d = 0
Yes, roots are repeated.

Question 3 B

Find whether the following quadratic equations have a repeated root :

y² — 6y + 6 = 0
Sol :
REPEATED ROOTS MEAN d = 0.
d = b² – 4ac
d = (–6)² – 4(1)(6)
d = 36 – 24
d = 12
∴ roots are not repeated.

Question 3 C

Find whether the following quadratic equations have a repeated root :

9x² + 4x + 6 = 0
Sol :
REPEATED ROOTS MEAN d = 0.
d = b² – 4ac
d = (4)² – 4 (9) (6)
d = 16 – 216
d = –200
∴ roots are not repeated.

Question 3 D

Find whether the following quadratic equations have a repeated root :

16y²— 40y + 25 = 0
Sol :
REPEATED ROOTS MEAN d = 0 .
d = b² – 4ac
d = (–40)² – 4 (16) (25)
d = 1600 – 1600
d = 0
∴ roots are repeated.

Question 3 E

Find whether the following quadratic equations have a repeated root :

x² + 6x + 9 = 0
Sol :

REPEATED ROOTS MEAN d = 0 .
d = b² – 4ac
d = (6)² – 4(1)(9)
d = 36 – 36
d = 0
∴ roots are repeated.

Question 4 A

Comment upon the nature of roots of the following equations:

4x² + 7x + 2 = 0
Sol :
d = b² – 4ac
d = (7)² – 4 (4) (2)
d = 49 – 32
d = 17
Since, d>0, roots are unique and real.

Question 4 B

Comment upon the nature of roots of the following equations:

x² + 10x + 39 = 0
Sol :
d = b² – 4ac
d = (10)² – 4 (1) (39)
d = 100 – 156
d = –56
Since, d < 0, no real roots exists.

Question 5 A

Without solving, determine whether the following equations have real roots or not. If yes, find them:

2x²— 4x + 3 = 0
Sol :
d = b² – 4ac
d = (–4)² – 4 (2) (3)
d = 16 – 24
d = –8
Since, d < 0, no real roots exist for the given equation.

Question 5 B

Without solving, determine whether the following equations have real roots or not. If yes, find them:

$\mathrm{y}^{2}-\frac{2}{3} \mathrm{y}+\frac{1}{9}=0$

Sol :
d = b² – 4ac
$\mathrm{d}=\left(\frac{-2}{3}\right)^{2}-4(1)\left(\frac{1}{9}\right)$
$d=\frac{4}{9}-\frac{4}{9}$
d = 0
Since, d = 0, roots are real and equal for the given equation.
$x=\frac{-b \pm \sqrt{d}}{2 a}$
$\mathrm{x}=\frac{-\left(-\frac{2}{3}\right) \pm \sqrt{0}}{2 \times 1}$
$x=\frac{2}{3} \times \frac{1}{2}$
$x=\frac{1}{3}$

Question 6 A

Without finding the roots, comment upon the nature of roots of each of the following quadratic equations:

2x²— 6x + 3 = 0
Sol :
d = b² – 4ac
d = (–6)² – 4 (2) (3)
d = 36 – 24
d = 12
∴ Roots are real and unique.

Question 6 B

Without finding the roots, comment upon the nature of roots of each of the following quadratic equations:

2x² — 5x — 3 = 0
Sol :
d = b² – 4ac
d = (–5)² – 4 (2) (–3)
d = 25 + 24
d = 49
∴ Roots are real and unique.

Question 7 A

Find the value of k for which the quadratic equation

4x² — 2 (k + 1) x + (k + 4) = 0 has equal roots.
Sol :
Since roots are equal
∴ d=0 ….(1)
4x² — 2 (k + 1) x + (k + 4) = 0
d = b² – 4ac
d = (–2 (k + 1)² – 4 (4) (k + 4)
d = (– 2k – 2)² – 16k – 64
d = 4k² + 4 + 8k – 16k – 64
(∵ (a – b)² = a² + b² – 2ab)
d = 4k² – 8k – 60
From (1), d = 0
∴ Equation will be:
4k² – 8k – 60 = 0
Dividing by 4
k² – 2k – 15 = 0
4k² – 5k + 3k – 15 = 0
k (k – 5) + 3(k – 5) = 0
(k – 5) (k + 3) = 0
K – 5 = 0 k + 3 = 0
K = 5 k = –3

Question 7 B

Find the value of k, so that the quadratic equation

(k + 1) x² – 2 (k — 1) x + 1 = 0 has equal roots.
Sol :
Since roots are equal
∴ d=0 ….(1)
(k + 1)x² — 2 (k – 1) x + 1 = 0
d = b² – 4ac
d = (–2(k–1))²– 4(k+1)(1)
d = (–2k+2)² – 4k – 4
d=4k² + 4 – 8k – 4k – 4
(∵ (a + b)² = a² + b² + 2ab)
d = 4k² – 12k
From (1), d = 0
∴ Equation will be:
0 = 4k² – 12k
4k² = 12k
$\mathrm{k}^{2}=\frac{12}{4} \mathrm{k}$
k² = 3k
k² – 3k = 0
k(k – 3) = 0
k = 0 or k – 3 = 0
k = 3
∴ Values of k are 0, 3.

Question 8 A

For what values of k, does the following quadratic equation has equal roots.

9x² + 8kx + 16 = 0
Sol :
Since roots are equal
∴ d=0 (1)
9x² + 8kx + 16 = 0
d = b² – 4ac
d = (8k)² – 4 (9) (16)
From (1), d = 0
∴ Equation will be:
0 = 64k² – 576
$\mathrm{k}^{2}=\frac{576}{64}$
k² = 9
k = ±√9
k = 3, –3
∴ values of k are –3, 3 .

Question 8 B

(k + 4)x² + (k + 1)x + 1 = 0

Sol :
Since roots are equal
∴ d=0 (1)
(k + 4)x² + (k + 1)x + 1 = 0
d=b²–4ac
d = (k – 1)²– 4 (k + 4) (1)
d = (–2k + 2)² – 4k – 4
d = k² + 1+ 2k – 4k – 16
From (1), d = 0
∴ Equation will be:
0 = k² + 1 + 2k – 4k – 16
k² – 2k – 15 = 0
k² – 5k + 3k – 15 = 0
k(k – 5) + 3 (k – 5) = 0
(k – 5) (k + 3) = 0
K – 5 = 0 k + 3 = 0
k = 5 k = –3
∴ Values of k are –3, 5.

Question 8 C

k²x² — 2(2k — 1)x + 4 = 0

Sol :
Since roots are equal
∴ d=0 (1)
k²x² — 2(2k — 1)x + 4 = 0
d = b² – 4ac
d = (–2(k–1))² – 4 (k²) (4)
d = (–2k+2)² – 4k – 4
d = (–4k+2)² – 16k²
(∵ (a – b)² = a² + b² – 2ab)
d = 16k² – 16k + 4 – 16k²
d = –16k + 4
From (1), d = 0
∴ Equation will be:
0 = –16k + 4
16k = 4

∴ Values of k are is

Question 9

If the roots of the equation (a — b)x² + (b — c) x + (c — a) = 0 are equal, prove that 2a = b + c.

Sol :
Since roots are equal
∴ d=0 (1)
(a — b)x² + (b — c) x + (c — a) = 0
d = b² – 4ac
d = (b–c)² – 4 (a–b) (c–a)
d = b² + c² – 2bc –4 [a (c – a) – b (c – a)]
d = b² + c² – 2bc – 4 [ac – a² – bc + ba]
From (1), d = 0
∴ Equation will be:
0 = b² + c² – 2bc – 4ac + 4a² + 4bc – 4ba
b² + c² – (2a)² 2bc + 2c (–2a) + 2(–2a)b = 0
(b + c – 2a)² = 0
(b + c – 2a) = 0
b + c = 2a
Hence proved.

Question 10

If — 5 is a root of the quadratic equation 2x² + 2px — 15 = 0 and the quadratic equation p (x² + x) + k = 0 has equal roots, find the value of k.

Sol :
2x² + 2px — 15 = 0
Put x = –5
2(–5)² + 2p(–5) — 15 = 0
50 – 10p – 15 = 0
35 = 10p
p = 3.5
Equation p (x² + x) + k = 0 has equal roots i.e. d = 0
p (x² + x) + k = 0
p = 3.5
3.5 (x² + x) + k = 0
3.5x² + 3.5x + k = 0
d = b² – 4ac
d = (3.5)² – 4(3.5)(k)
d = 12.25 – 14k
Putting d = 0
∴ Equation will be:
0 = 12.25 – 14k
14k = 12.25
$\mathrm{k}=\frac{12.25}{14}$
k = 0.875

Question 11 A

Find the values of k, for which the given equation has real roots:

2x² — 10x + k = 0

Sol :
Roots are equal
∴ d = 0
d = b² – 4ac
d = (–10)² – 4 (2) (k)
d = 100 – 8k
Put d = 0
0 = 100 – 8k
8k = 100

$\mathrm{k}=\frac{100}{8}$

$\mathrm{k}=\frac{25}{2}$

Question 11 B

Find the values of k, for which the given equation has real roots:

kx² — 6x — 2 = 0
Sol :
Roots are equal
∴ d = 0
d = b² – 4ac
d = (–6)² – 4 (–2) (k)
d = 36 + 8k
Put d = 0
0 = 36 + 8k
–8k = 36

$\mathrm{k}=\frac{-36}{8}$

$\mathrm{k}=\frac{-9}{2}$

Question 11 C

Find the values of k, for which the given equation has real roots:

kx²+ 4x + 1 = 0
Sol :
Roots are equal
∴ d = 0
d = b² – 4ac
d = (4)² – 4 (1) (k)
d = 16 – 4k
Put d = 0
0 = 16 – 4k
4k = 16
k = 4

Question 11 D

Find the values of k, for which the given equation has real roots:

kx² – 2√5x + 4 = 0
Sol :
Roots are equal
∴ d = 0
d = b² – 4ac
d = (–2√5)² – 4 (k) (4)
d = 20 – 16k
Put d = 0
0 = 20 – 16k
16k = 20

$\mathrm{k}=\frac{20}{16}$

$\mathrm{k}=\frac{5}{4}$

Question 11 E

Find the values of k, for which the given equation has real roots:

x² + k(4x + k — 1) + 2 = 0
Sol :
Roots are equal
∴ d = 0
d = b² – 4ac
d = (4k)² – 4 (k² – k + 2) (1)
d = 16k² – 4k² + 4k – 8
Put d = 0
0 = 16k² – 4k² + 4k – 8
12k² + 4k² + 4k – 8 = 0 (divide by 4)
3k² + k – 2 = 0
3k² + 3k – 2k – 2 = 0
3k (k + 1) – 2k (k + 1) = 0
(3k–2k)(k+1) = 0
(3k–2k)=0 or (k+1) = 0
k = 0 k = –1

Question 12

Prove that the equation x²(a² + b²) + 2x(ac + bd) + (c² + d²)= 0 has no real root, if ad ≠ bc.

Sol :
x²(a² + b²) + 2x(ac + bd) + (c² + d²)= 0
d = b² – 4ac
d = (2ac + 2bd)² – 4 (a² + b²) (c² + d²)
d = 4a²c² + 4b²d² + 8abcd – 4 [a² (c² + d²) + b² (c²+d²)]
d = 4a²c² + 4b²d² + 8abcd – 4 [a²c² + a²d² + b²c² + b²d²]
d = 4a²c² + 4b²d² + 8abcd – 4a²c² – 4a²d² – 4b²c² – 4b² d²
d = 8abcd – 4a²d² – 4b²c²
d = 8abcd – 4(a²d² + b² c²)
d = –4 (a² d² + b²c² – 2abcd)
d = –4 [(ad + bc)²]
For ad ≠ bc
d= –4 × [value of (ad + bc)²]
∴ d is always negative
So, d < 0
The given equation has no real roots.

S.no	Chapters	Links
1	Real numbers	Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4
2	Polynomials	Exercise 2.1 Exercise 2.2 Exercise 2.3
3	Pairs of Linear Equations in Two Variables	Exercise 3.1 Exercise 3.2 Exercise 3.3 Exercise 3.4 Exercise 3.5
4	Trigonometric Ratios and Identities	Exercise 4.1 Exercise 4.2 Exercise 4.3 Exercise 4.4
5	Triangles	Exercise 5.1 Exercise 5.2 Exercise 5.3 Exercise 5.4 Exercise 5.5
6	Statistics	Exercise 6.1 Exercise 6.2 Exercise 6.3 Exercise 6.4
7	Quadratic Equations	Exercise 7.1 Exercise 7.2 Exercise 7.3 Exercise 7.4 Exercise 7.5
8	Arithmetic Progressions (AP)	Exercise 8.1 Exercise 8.2 Exercise 8.3 Exercise 8.4
9	Some Applications of Trigonometry: Height and Distances	Exercise 9.1
10	Coordinates Geometry	Exercise 10.1 Exercise 10.2 Exercise 10.3 Exercise 10.4
11	Circles	Exercise 11.1 Exercise 11.2
12	Constructions	Exercise 12.1
13	Area related to Circles	Exercise 13.1
14	Surface Area and Volumes	Exercise 14.1 Exercise 14.2 Exercise 14.3 Exercise 14.4
15	Probability	Exercise 15.1