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

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^@)`

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

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