 RS Aggarwal Class 10 Solutions Chapter 10 - Quadratic Equations Ex 10B (10.2)

RS Aggarwal Class 10 Chapter 10 - Quadratic Equations Ex 10B (10.2) Solutions Free PDF

The RS Aggarwal Solutions for Class 10 Chapter 10 are considered extremely helpful in solving difficult questions which may be asked in the exam. These solutions help in simplify all your maths doubts and complete the maths syllabus of Class 10 in a proper time. By practicing the questions from the RS Aggarwal Maths textbook you can enhance your problem-solving skills. Students can refer to these solutions whenever they get stuck or have any doubt.

These solutions help you to revise different parts of the syllabus which increases your chances for scoring good marks in the exam. The RS Aggarwal Maths solutions is an important study material to practice before the exam so that you feel confident while writing the final exam.

Also, by using these solutions, you can revise different parts of the syllabus quite meticulously which all the more enhances your chance of scoring good marks in the exam.

Q1: 2x2 + ax – a2 = 0

We write, ax = 2ax – ax as 2x2 × (-a2) = -2a2x2 = 2ax × (-ax)

2x2 + ax – a2 = 0

2x2 + 2ax – ax – a2 = 0

2x(x + a) – a(x + a) =0

(x + a)(2x – a) = 0

x+a=0 or 2x-a=0

x = -a or x =$\frac{a}{2}$

Hence, -a and $\frac{a}{2}$ are the roots of the given equation.

Q2: 4x² + 4bx – (a² – b²) = 0

We write, 4bx = 2(a + b)x – 2(a – b)x as

4x2 ([- (a2 -b2)] = -4(a2 -b2) x2= 2(a + b)x ( [-2(a – b)x]

4x2 + 4bx – (a2 – b2) = 0

4x2 + 2(a + b)x – 2(a – b)x – (a – b)(a + b) = 0

2x[2x + (a + b)] – (a – b)[2x + (a + b)] = 0

[2x + (a + b)][2x – (a – b)] = 0

2x + (a + b) = 0 or 2x – (a – b) = 0

Hence, and are the roots of the given equation.

Q3: 4x² -4a²x + (a4 –b4) = 0

We write, -4a2x = -2(a2 b2)x – 2 (a2 – b2)x as

4x2 ( a4-b4)=4(a4-b4)x2 = [-2(a2 +b2)]x ( [-2(a2 -b2)]x

4x2 -4a2x + (a4 –b4) = 0

4x2 – 2 (a2 + b2)x – 2 (a2 – b2)x+ (a2 – b2) (a2 +b2) = 0

2x [2x – (a2+b2)] – (a2 – b2) [2x- (a2 + b2)] = 0

[2x – (a2 + b2)] [2x- (a2 – b2)] = 0

2x – (a2 +b2) = 0 or 2x – (a2 -b2) = 0

x=$\frac{a^{2} +b^{2}}{2}$ or x=$\frac{ a^{2} -b^{2}}{2}$ –*

Hence,* are the roots of the given equation.

Q4: x2 +5x – (a2 +a-6) =0

We write, 5x = (a + 3)x – (a – 2)x as

x2 ([- (a2 + a – 6)] = – (a2 + a – 6)x2 = (a + 3)x (-(a – 2)x])

.’. x2 +5x – (a2 +a-6) =0

x2 + (a + 3)x – (a – 2)x – (a + 3)(a – 2) = 0

x[x + (a + 3)] – (a – 2)[x + (a + 3)] = 0

[x + (a + 3)][x – (a – 2)] =0

x+(a+3)=0 or x – (a -2)=0

x=-(a+3) or x=a-2

Hence, -(a + 3) and (a – 2) are the roots of the given equation.

Q5: x² – 2ax – (4b² – a²) = 0

We write, -2ax = (2b – a)x – (2b+ a)x as

x2 (- (4b2 – a2)1 = – (4b2 a2) x2=(2b – a)x [- (2b + a)x]

.’. X2 – 2ax – (4b2 – a2) = 0

x2 + (2b – a)x – (2b+ a)x – (2b – a)(2b+ a) = 0

x[x + (2b – a)] – (2b + a)[x + (2b – a)] = 0

[x + (2b – a)][x – (2b + a)] =0

.x+(2b-a)=0 or x-(2b+a)=0

x=-(2b-a) or x=2b+a x=a-2b or x=a+2b

Hence, a – 2b and a + 2b are the roots of the given equation.

Q6: x2 – (2b – 1)x + (b² – b – 20) = 0

We write, (2b – 1)x = -(b- 5)x – (b + 4)x as

x2 (b2 – b – 20) = (b2 – b – 20) x2 = [-(b – 5)x] [-(b + 4)x]

x2 – (2b – 1)x + (b2 – b – 20) = 0

x2– (b – 5)x – (b+ 4)x + (b – 5)(b+ 4) =0

x[x – (b – 5)] – (b + 4)[x – (b – 5)] = 0

x – (b – 5)][x – (b+ 4)] = 0

x – (b – 5) = 0 or x – (b + 4) = 0

x=b-5 or x=b+4

Hence, b – 5 and b+ 4 are the roots of the given equation.

Q7: x2 + 6x – (a² + 2a – 8)= 0

We write, 6x = (a +4)x – (a – 2)x as

x2 [-(a2 + 2a – 8)] = -(a2 + 2a – 8)x2 =(a + 4)x [- (a – 2)x]

x2 + 6x – (a2 + 2a – 8)= 0

x2 + (a+ 4)x – (a – 2)x – (a + 4)(a – 2) = 0

x[x + (a+ 4)] – (a – 2)[x + (a +4)] = 0

[x+ (a + 4)][x – (a – 2)] = 0

x + (a + 4) =0 or x- (a- 2)= 0

x = -(a + 4) or x = a – 2

Hence, -(a + 4) and (a – 2) are the roots of the given equation.

Q8: abx2 + (b2 – ac)x – bc = 0

Given:

abx2 + (b2 – ac)x – bc = 0

abx2 + b2x – acx – bx = 0

bx(ax + b) – c(ax + b) = 0

(bx – c)(ax + b) = 0

bx – c = 0 or ax + b = 0

x= $\frac{c}{b}$ or $\frac{-b}{a}$

Q9: x2 – 4ax – b2 + 4a2 = 0

We write, -4ax = -(b + 2a)x+ (b – 2a)x as

x2 x (-b2 +4a2)= (-b2+4a2)x2= -(b+2a)x (b- 2a)x

x2 – 4ax – b2 + 4a2 = 0

x2 – (b + 2a)x + (b – 2a)x – (b – 2a)(b + 2a) = 0

x[x – (b + 2a)] + (b – 2a)[x – (b + 2a)] =0

[x-(b+2a)][x+(b-2a)] =0

x-(b+2a)=0 or x+(b-2a)=0

x=2a+b or x=-(b-2a)

x=2a+b or x=2a-b

Hence, (2a+ b) and (2a- b) are the roots of the given equation.

Q10: 4x2 – 2 (a2 ± b2 )x + a2b2 = 0

Given :

4x2 – 2 (a2 ± b2 )x + a2b2 = 0

4x2 – 2a2x – 2b2x + a2b2 = 0

2x (2x – a2) – b2 (2x – a2) = 0

(2x – b2) (2x – a2) = 0

2x – b2 = 0 or 2x – a2=0

x=$\frac{ b^{2}}{2}$ or x=$\frac{ a^{2}}{2}$

Hence, the roots of the equation.

Q11: 12abx2 – 9a2x + 8b2x – 6ab = 0

Given : 12abx2 – (9a2– 8b2)x – 6ab = 0

12abx2 – 9a2x + 8b2x – 6ab = 0

3ax(4bx – 3a) + 2b(4bx – 3a) = 0

(3ax + 2b)(4bx – 3a) = 0

3ax + 2b = 0 or 4bx – 3a = 0

X=$\frac{ -2b}{3a}$ or x= $\frac{ 3a}{4b}$

Hence, the roots of the equation

Q12: a2b2x2 + b2x – a2x – 1 = 0

Given :

a2b2x2 + b2x – a2x – 1 = 0

b2x(a2x + 1) – 1(a2x + 1) =0

(b2x – 1) (a2x + 1) = 0

(b2x – 1) = 0 or (a2x + 1) = 0

x=$\frac{ 1}{b^{2}}$ or x=$\frac{-1}{ a^{2}}$

Hence, 1 and are the roots of the given equation.

Q13: $\frac{x}{x – 1} + \frac{x – 1}{x} = 4 \frac{1}{4} , x \neq 0, 1$

$\frac{x}{x – 1} + \frac{x – 1}{x} = 4 \frac{1}{4} , x \neq 0, 1$

$\Rightarrow \frac{x^{2} + (x – 1)^{2}}{x(x – 1)} = \frac{17}{4}$

$\Rightarrow \frac{x^{2} + x^{2} – 2x + 1}{x^{2} – x} = \frac{17}{4}$

$\Rightarrow \frac{2x^{2} – 2x + 1}{x^{2} – x} = \frac{17}{4}$

$\Rightarrow$ 8x2 – 8x + 4 = 17x2 – 17x

$\Rightarrow$ 9x2 – 9x – 4 = 0

$\Rightarrow$ 9x2 – 12x + 3x – 4 = 0

$\Rightarrow$ 3x(3x – 4) + 1(3x – 4) = 0

$\Rightarrow$ (3x – 4)(3x + 1) = 0

$\Rightarrow$ 3x – 4 = 0 or 3x + 1 = 0

$\Rightarrow x = \frac{4}{3} or x = \frac{-1}{3}$

Hence, $\frac{4}{3} and \frac{-1}{3}$ are the roots of the given equation.

Q14: 9x2 — 9(a+ b)x+ (2a2 + 5ab+ 2b2) = 0

We write, —9(a+ b)x = —3(2a + b)x – 3(a+ 2b)x as

9x2 x (2a2 +5ab+ 2b2) = 9(2a2 + 5ab+2b2)x2 = [-3(2a + b)x] x [-3(a +2b)x]

Therefore, 9x2 — 9(a+ b)x+ (2a2 + 5ab+ 2b2) = 0

$\Rightarrow$ 9x2 — 3(2a + b)x — 3(a + 2b)x + (2a+ b)(a + 2b) = 0

$\Rightarrow$ 3x[3x — (2a+ b)] – (a +2b)[3x — (2a+ b)] = 0

$\Rightarrow$ [3x — (2a + b)] [3x — (a + 2b)] = 0

$\Rightarrow$ 3x — (2a+ b) = 0 or 3x — (a + 2b) = 0

$\Rightarrow$ X=$\frac{2a+b}{3}$ or x=$\frac{a+2b}{3}$

Hence, $\frac{2a+b}{3}$ and $\frac{a+2b}{3}$ are the roots of the given equation.

Q15: $\frac{16}{x}$-1= $\frac{15}{x+1}$,x $\neq$0, —1

$\Rightarrow$ $\frac{16}{x}$$\frac{15}{x+1}$=1

$\Rightarrow$ $\frac{16x+16-15x}{x(x+1)}$=1

$\Rightarrow$

$\frac{x+16}{x^{2}+x}$=1

$\Rightarrow$ x2 + x = x + 16 (Cross multiplication)

$\Rightarrow$ x2 — 16 = 0

$\Rightarrow$ (x + 4)(x — 4) = 0

$\Rightarrow$ x+4=0 or x-4=0

$\Rightarrow$ x = —4 or x = 4

Hence, -4 and 4 are the roots of the given equation.

Q16: $\frac{4}{x}$-3=$\frac{5}{2x+3}$,x$\neq$0,- $\frac{3}{2}$

$\frac{4}{x}$-3=$\frac{5}{2x+3}$,x$\neq$0,- $\frac{3}{2}$

$\Rightarrow$ $\frac{4}{x}$$\frac{5}{2x+3}$=3

$\Rightarrow$ $\frac{8x+12-5x}{x(2x+3)}$=3

$\Rightarrow$  $\frac{3x+12}{2x^{2}+3x}$=3

$\Rightarrow$  $\frac{x+4}{2x^{2}+3x}$=1

$\Rightarrow$ 2x2+3x = x+4 (Cross multiplication)

$\Rightarrow$ 2x2 + 2x — 4 = 0

$\Rightarrow$ x2+x —2=0

$\Rightarrow$ x2+2x-x-2=0

$\Rightarrow$ x(x + 2) — 1(x + 2) = 0

$\Rightarrow$ (x + 2)(x — 1)=0

$\Rightarrow$ x+2=0 or x-1=0

$\Rightarrow$ x=-2 or x=1

Hence, -2 and 1 are the roots of the given equation.

Q17: $\frac{3}{x+1}$$\frac{1}{2}$ = $\frac{2}{3x-1}$,x$\neq$-1, $\frac{1}{3}$

$\frac{3}{x+1}$$\frac{1}{2}$ = $\frac{2}{3x-1}$,x$\neq$-1, $\frac{1}{3}$

$\Rightarrow$ $\frac{3}{x+1}$$\frac{2}{3x-1}$= $\frac{1}{2}$

$\Rightarrow$ $\frac{9x-3-2x-2}{(x+1)(3x-1)}$ = $\frac{1}{2}$

$\Rightarrow$  $\frac{6}{x^{2}+4x-5}$= $\frac{6}{7}$

$\Rightarrow$ 3x2 + 2x — 1 = 14x — 10 (Cross multiplication)

$\Rightarrow$ 3x2 — 12x + 9 = 0

$\Rightarrow$ x2 —4x+ 3= 0

$\Rightarrow$ x2 — 3x—x+ 3 =0

$\Rightarrow$ x(x — 3) — 1(x — 3) = 0

$\Rightarrow$ (x — 3)(x — 1) = 0

$\Rightarrow$ x — 3 = 0 or x — 1 = 0

$\Rightarrow$ x = 3 or x = 1

Hence, 1 and 3 are the roots of the given equation.

Q18: $\frac{1}{x-1}$$\frac{1}{x+5}$= $\frac{6}{7}$,x$\neq$1, -5

$\frac{1}{x-1}$$\frac{1}{x+5}$= $\frac{6}{7}$,x$\neq$1, -5

$\Rightarrow$ $\frac{x+5-x+1}{(x-1)(x+5)}$ =$\frac{6}{7}$

$\Rightarrow$  $\frac{-(2a+b)}{4x^{2}+4ax+2bx}$= $\frac{2a+b}{2ab}$

$\Rightarrow$ x2+4x-5= 7

$\Rightarrow$ x2+4x-12=0

$\Rightarrow$ x2 +6x-2x-12=0

$\Rightarrow$ x(x + 6) — 2(x + 6) = 0

$\Rightarrow$ (X + 6)(X — 2)=0

$\Rightarrow$ x+6=0 or x-2=0

$\Rightarrow$ x=-6 or x=2

Hence, -6 and 2 are the roots of the given equation.

Q19: $\frac{1}{2a+b+2x}$= $\frac{1}{2a}$+ $\frac{1}{b}$+ $\frac{1}{2x}$

$\frac{1}{2a+b+2x}$= $\frac{1}{2a}$+ $\frac{1}{b}$+ $\frac{1}{2x}$

$\Rightarrow$ $\frac{1}{2a+b+2x}$$\frac{1}{2x}$= $\frac{1}{2a}$+ $\frac{1}{b}$

$\Rightarrow$ $\frac{2x-2a-b-2x}{2x(2a+b+2x}$= $\frac{2a+b}{2ab}$

$\Rightarrow$  $\frac{-(2a+b)}{4x^{2}+4ax+2bx}$= $\frac{2a+b}{2ab}$

$\Rightarrow$ 4x2 + 4ax + 2bx = —2ab

$\Rightarrow$ 4x2+4ax+2bx+2ab=0

$\Rightarrow$ 4x(x + a) + 2b(x + a)= 0

$\Rightarrow$ (x + a)(4x 2b) = 0

$\Rightarrow$ X + a=0 or 4x+2b=0

$\Rightarrow$ x= —a or –$\frac{b}{2}$

Hence, —a and –$\frac{b}{2}$ are the roots of the given equation.

Q20: $\frac{x+3}{x-2}$$\frac{1-x}{x}$= $\frac{17}{4}$

Given:

$\frac{x+3}{x-2}$$\frac{1-x}{x}$= $\frac{17}{4}$

$\Rightarrow$ $\frac{x(x+3)-(1-x)(x-2)}{(x-2)x}$= $\frac{17}{4}$

$\Rightarrow$  $\frac{x^{2}+3x-(x-2-x^{2}+2x)}{x^{2}-2x}$= $\frac{17}{4}$

$\Rightarrow$  $\frac{x^{2}+3x+x^{2}-3x+2}{x^{2}-2x}$= $\frac{17}{4}$

$\Rightarrow$  $\frac{2x^{2}+2}{x^{2}-2x}$= $\frac{17}{4}$

$\Rightarrow$ 8x2 + 8 = 17x2 — 34x [On cross multiplying]

$\Rightarrow$ — 9x2 + 34x + 8 = 0

$\Rightarrow$ 9x2 — 34x — 8 = 0

$\Rightarrow$ 9x2 — 36x + 2x — 8 = 0

$\Rightarrow$ 9x(x — 4) + 2(x — 4) =0

$\Rightarrow$ (x — 4)(9x + 2) = 0

$\Rightarrow$ x— 4 = 0 or 9x + 2 = 0

$\Rightarrow$ x = 4 or x = $\frac{-2}{9}$

Hence, the roots of the equation are 4 and $\frac{-2}{9}$

Q21.$\frac{3x-4}{7}$+ $\frac{7}{3x-4}$= $\frac{5}{2}$,x$\neq$ $\frac{4}{3}$

$\Rightarrow$  $\frac{(3x-4)^{2}+49}{7(3x-4)}$= $\frac{5}{2}$

$\Rightarrow$  $\frac{9x^{2}-24x+16+49}{21x-8}$= $\frac{5}{2}$

$\Rightarrow$  $\frac{9x^{2}-24x+65}{21x-28}$= $\frac{5}{2}$

$\Rightarrow$ 18x2 — 48x + 130 = 105x — 140

$\Rightarrow$ 18x2 — 153x + 270 = 0

$\Rightarrow$ 2x2 — 17x + 30 = 0

$\Rightarrow$ 2x2 — 12x — 5x + 30 = 0

$\Rightarrow$ 2x(x — 6) — 5(x — 6) = 0

$\Rightarrow$ (x — 6)(2x — 5) = 0

$\Rightarrow$ x-6= 0 or 2x— 5= 0

$\Rightarrow$ x= 6 or x=$\frac{5}{2}$

Hence, 6 and $\frac{5}{2}$ are the roots of the given equation.

Q22: $\frac{x}{x + 1} + \frac{x + 1}{x} = 2 \frac{4}{15}, x \neq 0 , -1$

$\frac{x}{x + 1} + \frac{x + 1}{x} = 2 \frac{4}{15}, x \neq 0 , -1$

$\Rightarrow \frac{x^{2} + (x + 1)^{2}}{x (x + 1)} = \frac{34}{15}$

$\Rightarrow \frac{x^{2} + x^{2} + 2x + 1}{x^{2} + x} = \frac{34}{15}$

$\Rightarrow \frac{2x^{2} + 2x + 1}{x^{2} + x} = \frac{34}{15}$

$\Rightarrow$ 30x2 + 30x + 15 = 34x2 + 34x

$\Rightarrow$ 4x2 + 4x – 15 = 0

$\Rightarrow 4x^{2}+ 10x – 6x – 15 = 0$

$\Rightarrow 2x(2x + 5) – 3(2x + 5) = 0$

$\Rightarrow$ (2x + 5)(2x – 3) = 0

$\Rightarrow$ (2x + 5) = 0 or 2x – 3 = 0

$\Rightarrow x = \frac{-5}{2} or x = \frac{3}{2}$

Hence, $\frac{-5}{2} and \frac{3}{2}$ are the roots of the given equation.

Key Features of RS Aggarwal Class 10 Solutions Chapter 10– Quadratic Equations Ex 10B (10.2)

• All the solutions are prepared by highly skilled subject experts.
• It shows different techniques to solve difficult and tricky questions.
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• Students should practice RS Aggarwal Class 10 solutions to score good marks in their exam.

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Which of the following were not used in India in ancient times?