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

As observed from the top of an 80 m tall lighthouse, the angles of depression of two ships on the same side of the lighthouse of the horizontal line with its base are 30° and 40° respectively. Find the distance between the two ships. Give your answer correct to the nearest meter.

Prove that `(tan^2 theta)/(sec theta - 1)^2 = (1 + cos theta)/(1 - cos theta)`

A Mathematics aptitude test of 50 students was recorded as follows:

Draw a histogram from the above data using a graph paper and locate the mode.

Prove that (cosec A – sin A)(sec A – cos A) sec² A = tan A.

Without using trigonometric tables evaluate

`(sin 35^@ cos 55^@ + cos 35^@ sin 55^@)/(cosec^2 10^@ - tan^2 80^@)`

Use ruler and compasses only for this question:

I. Construct  ABC, where AB = 3.5 cm, BC = 6 cm and ABC = 60o.
II. Construct the locus of points inside the triangle which are equidistant from BA and BC.
III. Construct the locus of points inside the triangle which are equidistant from B and C.
IV. Mark the point P which is equidistant from AB, BC and also equidistant from B and C. Measure and records the length of PB.

Use the Remainder Theorem to factorise the following expression:]

`2x^3 + x^2 - 13x + 6`

Prove the following trigonometric identities

`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`

Prove the following trigonometric identities.

`(1 + sin θ)/cos θ+ cos θ/(1 + sin θ) = 2 sec θ`

Prove the following trigonometric identities. `(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`

Prove the following trigonometric identities.

`1/(sec A - 1) + 1/(sec A + 1) = 2 cosec A cot A`

if `x/a cos theta + y/b sin theta = 1` and `x/a sin theta - y/b cos theta = 1` prove that `x^2/a^2 + y^2/b^2  = 2`

Find the remainder when x⁴ – 3x² + 2x + 1 is divided by x – 1.

Find the remainder when x³ + 3x² – 12x + 4 is divided by x – 2.

Find the remainder when x⁴ + 1 is divided by x + 1.

Use the Remainder Theorem to find which of the following is a factor of 2x³ + 3x² – 5x – 6. 

x + 1

Use the Remainder Theorem to find which of the following is a factor of 2x³ + 3x² – 5x – 6. 

2x – 1 

Use the Remainder Theorem to find which of the following is a factor of 2x³ + 3x² – 5x – 6. 

x + 2

Find the value of a, if the division of ax³ + 9x² + 4x – 10 by x + 3 leaves a remainder 5.

When x³ + 2x² – kx + 4 is divided by x – 2, the remainder is k. Find the value of constant k.