English Medium कक्षा १० - CBSE Question Bank Solutions for Mathematics

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² + 3x + k = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² + kx + 3 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² − 5x − k = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

kx² + 6x + 1 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

x² - kx + 9 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² + kx + 2 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

3x² + 2x + k = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

4x² - 3kx + 1 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² + kx - 4 = 0

For what value of k, (4 – k)x² + (2k + 4)x + (8k + 1) = 0, is a perfect square.

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

Find the values of k for which the given quadratic equation has real and distinct roots:

kx² + 2x + 1 = 0

Find the values of k for which the given quadratic equation has real and distinct roots:

kx² + 6x + 1 = 0

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

If the roots of the equation (a² + b²)x² – 2(ac + bd)x + (c² + d²) = 0 are equal, prove that `a/b = c/d`.

If the roots of the equations ax² + 2bx + c = 0 and `bx^2 - 2sqrt(ac)x + b = 0` are simultaneously real, then prove that b² = ac.

If p, q are real and p ≠ q, then show that the roots of the equation (p – q) x² + 5(p + q) x – 2(p – q) = 0 are real and unequal.

Show that the equation 2(a² + b²)x² + 2(a + b)x + 1 = 0 has no real roots, when a ≠ b.

Prove that both the roots of the equation (x – a)(x – b) + (x – b)(x – c) + (x – c)(x – a) = 0 are real but they are equal only when a = b = c.

If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax² + bx + c = 0 and –ax² + bx + c = 0 has real roots.