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

Describe completely the locus of a point in the following case:

Midpoint of radii of a circle. 

Describe completely the locus of a point in the following case:

Centre of a ball, rolling along a straight line on a level floor. 

Describe completely the locus of a point in the following case:

Point in a plane equidistant from a given line. 

Prove the following identity : 

`1/(cosA + sinA - 1) + 2/(cosA + sinA + 1) = cosecA + secA`

Describe completely the locus of a point in the following case:

Centre of a circle of varying radius and touching the two arms of ∠ ABC. 

Describe completely the locus of a point in the following case:

Centre of a circle of radius 2 cm and touching a fixed circle of radius 3 cm with centre O. 

Using only ruler and compasses, construct a triangle ABC 1 with AB = 5 cm, BC = 3.5 cm and AC= 4 cm. Mark a point P, which is equidistant from AB, BC and also from Band C. Measure the length of PB. 

Construct a triangle ABC, such that AB= 6 cm, BC= 7.3 cm and CA= 5.2 cm. Locate a point which is equidistant from A, B and C.

Prove the following identity : 

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

Prove the following identity : 

`(cosecθ)/(tanθ + cotθ) = cosθ`

Prove the following identity : 

`(1 + tan^2θ)sinθcosθ = tanθ`

Prove the following identity : 

`(1 + sinθ)/(cosecθ - cotθ) - (1 - sinθ)/(cosecθ + cotθ) = 2(1 + cotθ)`

Prove the following identity : 

`(1 + cotA + tanA)(sinA - cosA) = secA/(cosec^2A) - (cosecA)/sec^2A`

Draw and describe the locus in the following case:

The locus of a point in the rhombus ABCD which is equidistant from the point  A and C.

Prove the following identity : 

`2(sin^6θ + cos^6θ) - 3(sin^4θ + cos^4θ) + 1 = 0`

Prove the following identity : 

`sin^8θ - cos^8θ = (sin^2θ - cos^2θ)(1 - 2sin^2θcos^2θ)`

Prove the following identity :

`tan^2θ/(tan^2θ - 1) + (cosec^2θ)/(sec^2θ - cosec^2θ) = 1/(sin^2θ - cos^2θ)`

Prove the following identity :

`(sec^2θ - sin^2θ)/tan^2θ = cosec^2θ - cos^2θ`

Prove the following identity :

`(cos^3θ + sin^3θ)/(cosθ + sinθ) + (cos^3θ - sin^3θ)/(cosθ - sinθ) = 2`

Prove the following identity :

`(tanθ + sinθ)/(tanθ - sinθ) = (secθ + 1)/(secθ - 1)`