## Selina ICSE Solutions for Class 10 Maths – Quadratic Equations

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**Selina ICSE Solutions for Class 10 Maths Chapter 5 Quadratic Equations**

**Exercise 5(A)**

**Question 1.**

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

(i) 7x^{2} – 9x +2 =0

(ii) 6x^{2} – 13x +4 =0

(iii) 25x^{2} – 10x +1=0

(iv) x^{2} + 2√3x – 9=0

(v) x^{2} – ax – b^{2} =0

(vi) 2x^{2} +8x +9=0

**Solution:**

**Question 2(i).**

Find the value of p, if the following quadratic equation has equal roots : 4x^{2} – (p – 2)x + 1 = 0

**Solution:**

**Question 2(ii).**

Find the value of ‘p’, if the following quadratic equations have equal roots : x^{2} + (p – 3)x + p = 0

**Solution:**

x^{2} + (p – 3)x + p = 0

Here, a = 1, b = (p – 3), c = p

Since, the roots are equal,

⇒ b^{2}– 4ac = 0

⇒ (p – 3)^{2}– 4(1)(p) = 0

⇒p^{2} + 9 – 6p – 4p = 0

⇒ p^{2}– 10p + 9 = 0

⇒p^{2}-9p – p + 9 = 0

⇒p(p – 9) – 1(p – 9) = 0

⇒ (p -9)(p – 1) = 0

⇒ p – 9 = 0 or p – 1 = 0

⇒ p = 9 or p = 1

**Question 3.**

The equation 3x^{2} – 12x + (n – 5)=0 has equal roots. Find the value of n.

**Solution:**

**Question 4.**

Find the value of m, if the following equation has equal roots : (m – 2)x^{2} – (5+m)x +16 =0

**Solution:**

**Question 5.**

Find the value of p for which the equation 3x^{2}– 6x + k = 0 has distinct and real roots.

**Solution:**

**Exercise 5(B)**

**Question 1.**

Solve : x² – 10x – 24 = 0

**Solution:**

**Question 2.**

Solve : x² – 16 = 0

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

Solve : x(x – 5) = 24

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

Solve : (2x – 3)² = 49

**Solution:**

**Question 10.**

Solve : 2(x² – 6) = 3(x – 4)

**Solution:**

**Question 11.**

Solve : (x + 1)(2x + 8) = (x + 7)(x + 3)

**Solution:**

**Question 12.**

Solve : x² – (a + b)x + ab = 0

**Solution:**

**Question 13.**

(x + 3)² – 4(x + 3) – 5 = 0

**Solution:**

**Question 14.**

4(2x – 3)² – (2x – 3) – 14 = 0

**Solution:**

**Question 15.**

**Solution:**

**Question 16.**

2x^{2} – 9x + 10 = 0, When

(i) x∈ N

(ii) x∈ Q

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

**Solution:**

**Question 19.**

**Solution:**

**Question 20.**

**Solution:**

**Question 21.**

Find the quadratic equation, whose solution set is :

(i) {3, 5} (ii) {-2, 3}

**Solution:**

**Question 22(i).**

**Solution:**

**Question 22(ii).**

**Solution:**

**Question 23.**

Find the value of x, if a + 1=0 and x^{2} + ax – 6 =0.

**Solution:**

**Question 24.**

Find the value of x, if a + 7=0; b + 10=0 and 12x^{2} = ax – b.

**Solution:**

If a + 7 =0, then a = -7

and b + 10 =0, then b = – 10

Put these values of a and b in the given equation

**Question 25.**

Use the substitution y= 2x +3 to solve for x, if 4(2x+3)^{2} – (2x+3) – 14 =0.

**Solution:**

**Question 26.**

Without solving the quadratic equation 6x^{2} – x – 2=0, find whether x = 2/3 is a solution of this equation or not.

**Solution:**

**Question 27.**

Determine whether x = -1 is a root of the equation x^{2} – 3x +2=0 or not.

**Solution:**

x^{2} – 3x +2=0

Put x = -1 in L.H.S.

L.H.S. = (-1)^{2 }– 3(-1) +2

= 1 +3 +2=6 ≠ R.H.S

Then x = -1 is not the solution of the given equation.

**Question 28.**

If x = 2/3 is a solution of the quadratic equation 7x^{2}+mx – 3=0; Find the value of m.

**Solution:**

**Question 29.**

If x = -3 and x = 2/3 are solutions of quadratic equation mx^{2 }+ 7x + n = 0, find the values of m and n.

**Solution:**

**Question 30.**

If quadratic equation x^{2} – (m + 1) x + 6=0 has one root as x =3; find the value of m and the root of the equation.

**Solution:**

**Question 31.**

Given that 2 is a root of the equation 3x² – p(x + 1) = 0 and that the equation px² – qx + 9 = 0 has equal roots, find the values of p and q.

**Solution:**

**Question 32.**

**Solution:**

**Question 33.**

**Solution:**

**Question 34.**

**Solution:**

**Exercise 5(C)**

**Question 1.**

**Solution:**

**Question 2.**

Solve each of the following equations for x and give, in each case, your answer correct to one decimal place :

(i) x^{2} – 8x+5=0

(ii) 5x^{2} +10x – 3 =0

**Solution:**

**Question 3(i).**

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

(i) 2x^{2} – 10x +5=0

**Solution:**

**Question 3(ii).**

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

4x + 6/x + 13 = 0

**Solution:**

**Question 3(iii).**

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

x^{2} – 3x – 9 =0

**Solution:**

**Question 3(iv).**

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

x^{2} – 5x – 10 = 0

**Solution:**

**Question 4.**

Solve each of the following equations for x and give, in each case, your answer correct to 3 decimal places :

(i) 3x^{2} – 12x – 1 =0

(ii) x^{2} – 16 x +6= 0

(iii) 2x^{2} + 11x + 4= 0

**Solution:**

**Question 5.**

Solve:

(i) x^{4} – 2x^{2} – 3 =0

(ii) x^{4} – 10x^{2} +9 =0

**Solution:**

**Question 6.**

Solve :

(i) (x^{2} – x)^{2} + 5(x^{2} – x)+ 4=0

(ii) (x^{2} – 3x)^{2} – 16(x^{2} – 3x) – 36 =0

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

Solve the following equation and give your answer correct to 3 significant figures:

5x² – 3x – 4 = 0

**Solution:**

**Question 10.**

Solve for x using the quadratic formula. Write your answer correct to two significant figures.

(x – 1)^{2} – 3x + 4 = 0

**Solution:**

**Exercise 5(D)**

**Question 1.**

**Solution:**

**Question 2.**

Solve: (2x+3)^{2}=81

**Solution:**

**Question 3.**

Solve: a²x² – b² = 0

**Solution:**

**Question 4.**

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

Solve: 2x^{4} – 5x² + 3 = 0

**Solution:**

**Question 7.**

Solve: x^{4} – 2x² – 3 = 0.

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

**Solution:**

**Question 11.**

Solve : (x² + 5x + 4)(x² + 5x + 6) = 120

**Solution:**

**Question 12.**

Solve each of the following equations, giving answer upto two decimal places.

(i) x^{2} – 5x -10=0 (ii) 3x^{2} – x – 7 =0

**Solution:**

**Question 13.**

**Solution:**

**Question 14.**

Solve :

(i) x^{2} – 11x – 12 =0; when x ∈ N

(ii) x^{2} – 4x – 12 =0; when x ∈ I

(iii) 2x^{2} – 9x + 10 =0; when x ∈ Q

**Solution:**

**Question 15.**

Solve : (a + b)²x² – (a + b)x – 6 = 0; a + b ≠ 0

**Solution:**

**Question 16.**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

**Solution:**

**Question 19.**

**Solution:**

**Question 20.**

Without solving the following quadratic equation, find the value of ‘m’ for which the given equation has real and equal roots.

x² + 2(m – 1)x + (m + 5) = 0

**Solution:**

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