## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.4

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 5 Quadratic Equations in One Variable Ex 5.4

Question 1.

Find the discriminant of the following equations and hence find the nature of roots:

(i) 3x² – 5x – 2 = 0

(ii) 2x² – 3x + 5 = 0

(iii) 7x² + 8x + 2 = 0

(iv) 3x² + 2x – 1 = 0

(v) 16x² – 40x + 25 = 0

(vi) 2x² + 15x + 30 = 0.

Solution:

Question 2.

Discuss the nature of the roots of the following quadratic equations:

(i) x² – 4x – 1 = 0

(ii) 3x² – 2x + \(\\ \frac { 1 }{ 3 } \) = 0

(iii) 3x² – 4√3x + 4 = 0

(iv) x² – \(\\ \frac { 1 }{ 2 } x\) + 4 = 0

(v) – 2x² + x + 1 = 0

(vi) 2√3x² – 5x + √3 = 0

Solution:

Question 3.

Find the nature of the roots of the following quadratic equations:

(i) x² – \(\\ \frac { 1 }{ 2 } x\) – \(\\ \frac { 1 }{ 2 } \) = 0

(ii) x² – 2√3x – 1 = 0 If real roots exist, find them.

Solution:

Question 4.

Without solving the following quadratic equation, find the value of ‘p’ for which the given equations have real and equal roots:

(i) px² – 4x + 3 = 0

(ii) x² + (p – 2)x + p = 0.

Solution:

Question 5.

Find the value (s) of k for which each of the following quadratic equation has equal roots:

(i) kx² – 4x – 5 = 0

(ii) (k – 4) x² + 2(k – 4) x + 4 = 0

Solution:

Question 6.

Find the value(s) of m for which each of the following quadratic equation has real and equal roots:

(i) (3m + 1)x² + 2(m + 1)x + m = 0

(ii) x² + 2(m – 1) x + (m + 5) = 0

Solution:

Question 7.

Find the values of k for which each of the following quadratic equation has equal roots:

(i) 9x² + kx + 1 = 0

(ii) x² – 2kx + 7k – 12 = 0

Also, find the roots for those values of k in each case.

Solution:

Question 8.

Find the value(s) of p for which the quadratic equation (2p + 1)x² – (7p + 2)x + (7p – 3) = 0 has equal roots. Also find these roots.

Solution:

Question 9.

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

Solution:

Question 10.

Find the value(s) of p for which the equation 2x² + 3x + p = 0 has real roots.

Solution:

Question 11.

Find the least positive value of k for which the equation x² + kx + 4 = 0 has real roots.

Solution:

Question 12.

Find the values of p for which the equation 3x² – px + 5 = 0 has real roots.

Solution: