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

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

If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:
x⁵ - 3x⁴ - ax³ + 3ax² + 2ax + 4.

In the given figure, O is the centre of the circle and ∠PBA = 45°. Calculate the value of ∠PQB.

Show that x² - 9 is factor of x³ + 5x² - 9x - 45.

In the given figure, BAD = 65°, ABD = 70°, BDC = 45°.
(i) Prove that AC is a diameter of the circle.
(ii) Find ACB.

In Fig, Chord ED is parallel to the diameter AC of the circle. Given ∠CBE = 65°, Calculate ∠ DEC.

In the figure, ∠DBC = 58°. BD is diameter of the circle.

Calculate:

1. ∠BDC
2. ∠BEC
3. ∠BAC

In the figure given alongside, AD is the diameter of the circle. If ∠ BCD = 130°, Calculate: (i) ∠ DAB (ii) ∠ ADB.