(English Medium) ICSE Class 10 - CISCE Question Bank Solutions for Mathematics

Without actually determining the roots comment upon the nature of the roots of each of the following equations:
x² - 5x + 7 = 0

Without actually determining the roots comment upon the nature of the roots of each of the following equations:
x² - 4x + 1 = 0

Without actually determining the roots comment upon the nature of the roots of each of the following equations:
x² + 5x + 15 = 0.

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

Solve the following by reducing them to quadratic equations:
x⁴ - 26x2 + 25 = 0

Solve the following by reducing them to quadratic equations:
z⁴ - 10z² + 9 = 0.

Solve for x : `9^(x + 2) -6.3^(x + 1) + 1 = 0`.

Solve for x: (x² - 5x)² - 7(x² - 5x) + 6 = 0; x ∈ R.

Determine whether the given quadratic equations have equal roots and if so, find the roots:
x² + 5x + 5 = 0

Determine whether the given quadratic equations have equal roots and if so, find the roots:
x² + 2x + 4 = 0

Determine whether the given quadratic equations have equal roots and if so, find the roots:
`(4)/(3)x^2 - 2x + (3)/(4) = 0`

Determine whether the given quadratic equations have equal roots and if so, find the roots:
3x² - 6x + 5 = 0

Find the value of k so that sum of the roots of the quadratic equation is equal to the product of the roots:
kx² + 6x - 3k = 0, k ≠ 0

Find the value of k so that sum of the roots of the quadratic equation is equal to the product of the roots:
(k + 1)x² + (2k + 1)x - 9 = 0, k + 1 ≠ 0.

Find the values of k so that the sum of tire roots of the quadratic equation is equal to the product of the roots in each of the following:
kx² + 2x + 3k = 0

Find the values of k so that the sum of tire roots of the quadratic equation is equal to the product of the roots in each of the following:
2x² - (3k + 1)x - k + 7 = 0.

Given that one root of the quadratic equation ax² + bx + c = 0 is three times the other, show that 3b² – 16ac.

If one root of the quadratic equation ax² + bx + c = 0 is double the other, prove that 2b² = 9 ac.

If the ratio of the roots of the equation
lx² + nx + n = 0 is p: q, Prove that
`sqrt(p/q) + sqrt(q/p) + sqrt(n/l) = 0.`

Find the value of a and b so that the polynomial x³ - ax² - 13x + b has (x - 1) (x + 3) as factor.