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

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta`

What must be subtracted from 16x³ – 8x² + 4x + 7 so that the resulting expression has 2x + 1 as a factor?

How much should a man invest in Rs. 50 shares selling at Rs. 60 to obtain an income of Rs. 450, if the rate of dividend declared is 10%. Also find his yield percent, to the nearest whole number.

The angles of depression of two ships A and B as observed from the top of a light house 60 m high are 60° and 45° respectively. If the two ships are on the opposite sides of the light house, find the distance between the two ships. Give your answer correct to the nearest whole number.

Using remainder theorem, find the value of k if on dividing 2x³ + 3x² – kx + 5 by x – 2, leaves a remainder 7.

If `x/a=y/b = z/c` show that `x^3/a^3 + y^3/b^3 + z^3/c^3 = (3xyz)/(abc)`.

The histogram below represents the scores obtained by 25 students in a mathematics mental test. Use the data to:

1. Frame a frequency distribution table.
2. To calculate mean.
3. To determine the Modal class.

Prove that `cosA/(1+sinA) + tan A =  secA`

Prove that `sqrt(sec^2 theta + cosec^2 theta) = tan theta + cot theta`

A man invests Rs. 22,500 in Rs. 50 shares available at 10% discount. If the dividend paid by the company is 12%, calculate:

1. The number of shares purchased.
2. The annual dividend received.
3. The rate of return he gets on his investment. Give your answer correct to the nearest whole number.

Prove that (1 + cot θ – cosec θ)(1+ tan θ + sec θ) = 2

Use Remainder theorem to factorize the following polynomial:

`2x^3 + 3x^2 - 9x - 10`

Find 'a' if the two polynomials ax³ + 3x² – 9 and 2x³ + 4x + a, leaves the same remainder when divided by x + 3.

Prove that `(sin theta)/(1-cottheta) + (cos theta)/(1 - tan theta) = cos theta + sin theta`

Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 105°

Hence:

1) Construct the locus of points equidistant from BA and BC

2) Construct the locus of points equidistant from B and C.

3) Mark the point which satisfies the above two loci as P. Measure and write the length of PC.

Using the Remainder and Factor Theorem, factorise the following polynomial:

`x^3 + 10x^2 - 37x + 26`

Prove the identity (sin θ + cos θ)(tan θ + cot θ) = sec θ + cosec θ.

Show that `sqrt((1-cos A)/(1 + cos A)) = sinA/(1 + cosA)`

Using the Remainder Theorem, factorise the following completely:

3x³ + 2x² – 19x + 6

Evaluate without using trigonometric tables:

`cos^2 26^@ + cos 64^@ sin 26^@ + (tan 36^@)/(cot 54^@)`