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

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.

If the polynomials ax³ + 4x² + 3x - 4 and x³ - 4x + a leave the same remainder when divided by (x - 3), find the value of a.

In the following two polynomials. Find the value of ‘a’ if x + a is a factor of each of the two:

x³ + ax² − 2x + a + 4

In the following two polynomials. Find the value of ‘a’ if x + a is a factor of each of the two:
x⁴ - a²x² + 3x - a.

In the following two polynomials, find the value of ‘a’ if x – a is a factor of each of the two:
x⁶ - ax⁵ + x⁴ - ax³ + 3a + 2

In the following two polynomials, find the value of ‘a’ if x – a is a factor of each of the two:
x⁵ - a²x³ + 2x + a + 1.

If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x² - 3x + 5a

If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x³ + 2ax² + ax - 1