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

Construct an isosceles triangle ABC such that AB = 6 cm, BC = AC = 4 cm. Bisect ∠C internally and mark a point P on this bisector such that CP = 5 cm. Find the points Q and R which are 5 cm from P and also 5 cm from the line AB. 

Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords AB and AC of the circle of lengths 6 cm and 5 cm respectively.
(i) Construct the locus of points, inside the circle, that are equidistant from A and C. prove your construction.
(ii) Construct the locus of points, inside the circle that are equidistant from AB and AC. 

Plot the points A(2, 9), B(–1, 3) and C(6, 3) on graph paper. On the same graph paper draw the locus of point A so that the area of ΔABC remains the same as A moves. 

Construct a triangle BCP given BC = 5 cm, BP = 4 cm and ∠PBC = 45°.

1. Complete the rectangle ABCD such that:
   1. P is equidistant from AB and BC.
   2. P is equidistant from C and D.
2. Measure and record the length of AB. 

Prove the following identities:

`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`

Prove the following identities:

`(1 + sin A)/(1 - sin A) = (cosec  A + 1)/(cosec  A - 1)`

Prove the following identities:

`1/(tan A + cot A) = cos A sin A`

Prove the following identities:

`tan A - cot A = (1 - 2cos^2A)/(sin A cos A)`

Prove the following identities:

(1 – tan A)² + (1 + tan A)² = 2 sec²A

Prove the following identities:

cosec⁴ A – cosec² A = cot⁴ A + cot² A

Prove the following identities:

cosec A(1 + cos A) (cosec A – cot A) = 1

Prove the following identities:

sec²A + cosec²A = sec²A . cosec²A

Prove the following identities:

`((1 + tan^2A)cotA)/(cosec^2A) = tan A`

Prove the following identities:

cot² A – cos² A = cos² A . cot² A

Prove the following identities:

(cosec A + sin A) (cosec A – sin A) = cot² A + cos² A

Prove the following identities:

(sec A – cos A) (sec A + cos A) = sin² A + tan² A 

Prove the following identities:

(cos A + sin A)² + (cos A – sin A)² = 2

Prove the following identities:

(cosec A – sin A) (sec A – cos A) (tan A + cot A) = 1

Prove the following identities:

`1/(secA + tanA) = secA - tanA`

Prove the following identities:

`cosecA + cotA = 1/(cosecA - cotA)`