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

In the given diagram, ΔABC ∼ ΔPQR. If AD and PS are the bisectors of ∠BAC and ∠QPR, respectively, then ______.

In the given diagram, ΔADB and ΔACB are two right-angled triangles with ∠ADB = ∠BCA = 90°. If AB = 10 cm, AD = 6 cm, BC = 2.4 cm and DP = 4.5 cm.

1. Prove that ΔAPD ∼ ΔBPC
2. Find the length of BD and PB
3. Hence, find the length of PA
4. Find area ΔAPD : area ΔBPC.

The polynomial 3x³ + 8x² – 15x + k has (x – 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

The roots of quadratic equation x² – 1 = 0 are ______.

While factorizing a given polynomial using the remainder and factor theorem, a student finds that (2x + 1) is a factor of 2x³ + 7x² + 2x – 3.

1. Is the student’s solution correct in stating that (2x + 1) is a factor of the given polynomial?
2. Give a valid reason for your answer.

Also, factorize the given polynomial completely.

In the given figure, AC is the diameter of the circle with center O.

CD is parallel to BE.

∠AOB = 80° and ∠ACE = 20°

Calculate:

1. ∠BEC
2. ∠BCD
3. ∠CED

The roots of equation (q – r)x² + (r – p)x + (p – q) = 0 are equal.

Prove that 2q = p + r; i.e., p, q, and r are in A.P.

If 4 is a root of equation x² + kx – 4 = 0; the value of k is ______.

The roots of the quadratic equation x² – 6x – 7 = 0 are ______.

The roots of quadratic equation x(x + 8) + 12 = 0 are ______.

If one root of equation (p – 3) x² + x + p = 0 is 2, the value of p is ______.

Equation 2x² – 3x + 1 = 0 has ______.

Which of the following equations has two real and distinct roots?

If the roots of x² – px + 4 = 0 are equal, the value (values) of p is ______.

Which of the following equations has imaginary roots?

One root of equation 3x² – mx + 4 = 0 is 1, the value of m is ______.

If the quadratic equation kx² + kx + 1 = 0 has real and distinct roots, the value of k is ______.

One factor of x³ – kx² + 11x – 6 is x – 1. The value of k is ______.

If (x – a) is a factor of x³ – ax² + x + 5; the value of a is ______.

(x – 2) is a factor of ______.