ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2

More Exercises

Solve the following equations (1 to 24) by factorization

Question 1.
(i) 4x² = 3x
(ii) \(\frac { { x }^{ 2 }-5x }{ 2 } =0\)
Solution:
(i) 4x² = 3x
x(4x – 3) = 0
Either x = 0,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q1.1

Question 2.
(i) (x – 3) (2x + 5) = 0
(ii) x (2x + 1) = 6
Solution:
(i) (x – 3) (2x + 5) = 0
Either x – 3 = 0,
Then x = 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q2.1

Question 3.
(i) x² – 3x – 10 = 0
(ii) x(2x + 5) = 3
Solution:
(i) x² – 3x – 10 = 0
⇒ x² – 5x + 2x – 10 = 0
⇒ x(x – 5) + 2(x – 5) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q3.1

Question 4.
(i) 3x² – 5x – 12 = 0
(ii) 21x² – 8x – 4 = 0
Solution:
(i) 3x² – 5x – 12 = 0
⇒ 3x² – 9x + 4x – 12 = 0
⇒ 3x (x – 3) + 4(x – 3) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q4.1

Question 5.
(i) 3x² = x + 4
(ii) x(6x – 1) = 35
Solution:
(i) 3x² = x + 4
⇒ 3x² – x – 4 = 0
⇒ 3x² – 4x + 3x – 4 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q5.1

Question 6.
(i) 6p² + 11p – 10 = 0
(ii) \(\frac { 2 }{ 3 } { x }^{ 2 }-\frac { 1 }{ 3 } x=1 \)
Solution:
(i) 6p² + 11p – 10 = 0
⇒ 6p² + 15p – 4p – 10 = 0
⇒ 3p(2p + 5) – 2(2p + 5) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q6.1

Question 7.
(i) (x – 4)² + 5² = 13²
(ii) 3(x – 2)² = 147
Solution:
(i) (x – 4)² + 5² = 13²
x² – 8x + 16 + 25 = 169
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q7.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q7.2

Question 8.
(i) \(\\ \frac { 1 }{ 7 } \)(3x – 5)² = 28
(ii) 3(y² – 6) = y(y + 7) – 3
Solution:
(i) \(\\ \frac { 1 }{ 7 } \)(3x – 5)² = 28
(3x – 5)² = 28 × 7
⇒ 9x² – 30x + 25 = 196
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q8.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q8.2

Question 9.
x² – 4x – 12 = 0,when x∈N
Solution:
x² – 4x – 12 = 0
⇒ x² – 6x + 2x – 12 = 0
⇒ x (x – 6) + 2 (x – 6) = 0
⇒ (x – 6) (x + 2) = 0
Either x – 6 = 0, then x = 6
or x + 2 = 0, then x = -2
But -2 is not a natural number
∴ x = 6

Question 10.
2x² – 8x – 24 = 0 when x∈I
Solution:
2x² – 8x – 24 = 0
⇒ x² – 4x – 12 = 0 (Dividing by 2)
⇒ x² – 6x + 2x – 12 = 0
⇒ x (x – 6) + 2 (x – 6) = 0
⇒ (x – 6) (x + 2) = 0
Either x – 6 = 0, then, x = 6
or x + 2 = 0, then x = – 2
Hence x = 6, – 2

Question 11.
5x² – 8x – 4 = 0 when x∈Q
Solution:
5x² – 8x – 4 = 0
∵ 5 × ( – 4) = – 20
-20 = – 10 + 2
-8 = – 10 + 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q11.1

Question 12.
2x² – 9x + 10 = 0,when
(i)x∈N
(ii)x∈Q
Solution:
2x² – 9x + 10 = 0
⇒ 2x² – 4x – 5x + 10 = 0
⇒ 2x(x – 2) – 5(x – 2) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q12.1

Question 13.
(i) a²x² + 2ax + 1 = 0, a≠0
(ii) x² – (p + q)x + pq = 0
Solution:
(i) a²x² + 2ax + 1 = 0
⇒ a²x² + ax + ax + 1 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q13.1

Question 14.
a²x² + (a² + b²)x + b² = 0, a≠0
Solution:
a²x² + (a² + b²)x + b² = 0
⇒ a²x(x + 1) + b²(x + 1) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q14.1

Question 15.
(i) √3x² + 10x + 7√3 = 0
(ii) 4√3x² + 5x – 2√3 = 0
Solution:
(i) √3x² + 10x + 7√3 = 0
[ ∵ √3 x 7√3 = 7 x 3 = 21]
⇒ √3x(x + √3) + 7(x + √3) = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q15.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q15.3

Question 16.
(i) x² – (1 + √2)x + √2 = 0
(ii) \(x+ \frac { 1 }{ x } \) = \(2 \frac { 1 }{ 20 } \)
Solution:
(i) x² – (1 + √2)x + √2 = 0
⇒ x² – x – √2x + √2 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q16.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q16.2

Question 17.
(i) \(\frac { 2 }{ { x }^{ 2 } } -\frac { 5 }{ x } +2=0,x\neq 0 \)
(ii)\(\frac { { x }^{ 2 } }{ 15 } -\frac { x }{ 3 } -10=0 \)
Solution:
(i) \(\frac { 2 }{ { x }^{ 2 } } -\frac { 5 }{ x } +2=0,x\neq 0 \)
⇒ 2 – 5x + 2x² = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q17.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q17.2

Question 18.
(i) \(3x-\frac { 8 }{ x } =2 \)
(ii) \(\frac { x+2 }{ x+3 } =\frac { 2x-3 }{ 3x-7 } \)
Solution:
(i) \(3x-\frac { 8 }{ x } =2 \)
\(\frac { { 3x }^{ 2 }-8 }{ x } =2 \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q18.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q18.2

Question 19.
(i) \(\frac { 8 }{ x+3 } -\frac { 3 }{ 2-x } =2 \)
(ii) \(\frac { x }{ x-1 } +\frac { x-1 }{ x } =2\frac { 1 }{ 2 } \)
Solution:
(i) \(\frac { 8 }{ x+3 } -\frac { 3 }{ 2-x } =2 \)
\(\frac { 16-8x-3x-9 }{ (x+3)(2-x) } =2 \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q19.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q19.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q19.3

Question 20.
(i) \(\frac { x }{ x+1 } +\frac { x+1 }{ x } =\frac { 34 }{ 15 } \)
(ii) \(\frac { x+1 }{ x-1 } +\frac { x-2 }{ x+2 } =3 \)
Solution:
(i) \(\frac { x }{ x+1 } +\frac { x+1 }{ x } =\frac { 34 }{ 15 } \)
\(\frac { { x }^{ 2 }+{ x }^{ 2 }+2x+1 }{ x(x+1) } =\frac { 34 }{ 15 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q20.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q20.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q20.3

Question 21.
(i) \(\frac { 1 }{ x-3 } -\frac { 1 }{ x+5 } =\frac { 1 }{ 6 } \)
(ii) \(\frac { x-3 }{ x+3 } +\frac { x+3 }{ x-3 } =2\frac { 1 }{ 2 } \)
Solution:
(i) \(\frac { 1 }{ x-3 } -\frac { 1 }{ x+5 } =\frac { 1 }{ 6 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q21.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q21.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q21.3

Question 22.
(i) \(\frac { a }{ ax-1 } +\frac { b }{ bx-1 } =a+b,a+b\neq 0,ab\neq 0\)
(ii) \(\frac { 1 }{ 2a+b+2x } =\frac { 1 }{ 2a } +\frac { 1 }{ b } +\frac { 1 }{ 2x } \)
Solution:
(i) \(\frac { a }{ ax-1 } +\frac { b }{ bx-1 } =a+b\)
⇒ \(\left( \frac { a }{ ax-1 } -b \right) +\left( \frac { b }{ bx-1 } -a \right) =0\)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q22.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q22.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q22.3

Question 23.
\(\frac { 1 }{ x+6 } +\frac { 1 }{ x-10 } =\frac { 3 }{ x-4 } \)
Solution:
\(\frac { 1 }{ x+6 } +\frac { 1 }{ x-10 } =\frac { 3 }{ x-4 } \)
⇒ \(\frac { x-10+x+6 }{ (x+6)(x-10) } =\frac { 3 }{ x-4 } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q23.1

Question 24.
(i) \(\sqrt { 3x+4 } =x\)
(ii) \(\sqrt { x(x-7) } =3\sqrt { 2 } \)
Solution:
(i) \(\sqrt { 3x+4 } =x\)
Squaring on both sides
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q24.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q24.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q24.3

Question 25.
Use the substitution y = 3x + 1 to solve for x : 5(3x + 1 )² + 6(3x + 1) – 8 = 0
Solution:
y = 3x + 1
Now, 5(3x + 1)² + 6(3x + 1) – 8 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q25.1

Question 26.
Find the values of x if p + 1 =0 and x² + px – 6 = 0
Solution:
p + 1 = 0, then p = – 1
Substituting the value of p in the given quadratic equation
x² + ( – 1)x – 6 = 0
⇒ x² – x – 6 = 0
⇒ x² – 3x + 2x – 6 = 0
⇒ x (x – 3) + 2 (x – 3) = 0
⇒ (x – 3) (x + 2) = 0
Either x – 3 = 0, then x = 3
or x + 2 = 0, then x = – 2
Hence x = 3, -2

Question 27.
Find the values of x if p + 7 = 0, q – 12 = 0 and x² + px + q = 0,
Solution:
p + 7 = 0, then p = – 7
and q – 12 = 0, then q = 12
Substituting the values of p and q in the given quadratic equation,
x² – 7x + 12 = 0
⇒ x² – 3x – 4x + 12 = 0
⇒ x (x – 3) – 4 (x – 3) = 0
⇒ (x – 3) (x – 4) = 0
Either x – 3 = 0, then x = 3
or x – 4 = 0, then x = 4
Hence x = 3, 4

Question 28.
If x = p is a solution of the equation x(2x + 5) = 3, then find the value of p.
Solution:
Given, x = p and x(2x + 5) = 3
Substituting the value of p, we get
p(2p + 5) = 3
⇒ 2p² + 5p – 3 = 0
⇒ 2p² + 6p – p – 3 = 0
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.2 Q28.1

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