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

Use factor theorem to factorise the following polynomials completely: x³ – 19x – 30

If x³ – 2x² + px + q has a factor (x + 2) and leaves a remainder 9, when divided by (x + 1), find the values of p and q. With these values of p and q, factorize the given polynomial completely.

If (x + 3) and (x – 4) are factors of x³ + ax² – bx + 24, find the values of a and b: With these values of a and b, factorise the given expression.

f 2x³ + ax² – 11x + b leaves remainder 0 and 42 when divided by (x – 2) and (x – 3) respectively, find the values of a and b. With these values of a and b, factorize the given expression.

If (2x + 1) is a factor of both the expressions 2x² – 5x + p and 2x² + 5x + q, find the value of p and q. Hence find the other factors of both the polynomials.

If 3 is a root of the quadratic equation x² – px + 3 = 0, then p is equal to ______.

In the given figure ∠BAP = ∠DCP = 70°, PC = 6 cm and CA = 4 cm, then PD : DB is ______.

Solve the following quadratic equation:

x² + 4x – 8 = 0

Give your Solution correct to one decimal place.

(Use mathematical tables if necessary.)

Factorize completely using factor theorem:

2x³ – x² – 13x – 6

The roots of the quadratic equation px² – qx + r = 0 are real and equal if ______.

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.

Input tax credit shall only be allowed against ______.

Input tax credit means ______.

For a registered dealer, if input tax is ₹ x and output tax is ₹ y, then the input tax credit for the dealer is ______.