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

Use ruler and compasses only for the following questions:
Construct triangle BCP, when CB = 5 cm, BP = 4 cm, ∠PBC = 45°.
Complete the rectangle ABCD such that :
(i) P is equidistant from AB and BC and
(ii) P is equidistant from C and D. Measure and write down the length of AB.

Prove that `((tan 20°)/(cosec 70°))^2 + ((cot 20°)/(sec 70°))^2  = 1`

Prove that `(sin 70°)/(cos 20°) + (cosec 20°)/(sec 70°) - 2 cos 70° xx cosec 20°` = 0.

If x = h + a cos θ, y = k + b sin θ. 
Prove that `((x - h)/a)^2 + ((y - k)/b)^2 = 1`.

Ruler and compass only may be used in this question. All construction lines and arcs must be clearly shown, and be of sufficient length and clarity to permit assessment.
(i) Construct Δ ABC, in which BC = 8 cm, AB = 5 cm, ∠ ABC = 60°.
(ii) Construct the locus of point 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 as P, the point which is equidistant from AB, BC and also equidistant from B and C.
(v) Measure and record the length of PB.

Ruler and compasses only may be used in this question. All construction lines and arcs must be clearly shown, and be of sufficient length and clarity to permit assessment.
(i) Construct a ΔABC, in which BC = 6 cm, AB = 9 cm and ∠ABC = 60°.
(ii) Construct the locus of the vertices of the triangles with BC as base, which are equal in area to ΔABC.
(iii) Mark the point Q, in your construction, which would make ΔQBC equal in area to ΔABC, and isosceles.
(iv) Measure and record the length of CQ.

Given ∠BAC (Fig), determine the locus of a point which lies in the interior of ∠BAC and equidistant from two lines AB and AC.

If A + B = 90°, show that sec² A + sec² B = sec² A. sec² B.

If A + B = 90°, show that `(sin B + cos A)/sin A = 2tan B + tan A.`

Prove the following identities: sec² θ + cosec² θ = sec² θ cosec² θ.

Prove the following identities: cot θ - tan θ = `(2 cos^2 θ - 1)/(sin θ cos θ)`.

Prove the following identities:
`1/(sin θ + cos θ) + 1/(sin θ - cos θ) = (2sin θ)/(1 - 2 cos^2 θ)`.

Prove the following identities:

`(1 - tan^2 θ)/(cot^2 θ - 1) = tan^2 θ`.

Prove that the following identities:
Sec A( 1 + sin A)( sec A - tan A) = 1.

Prove that: sin⁴ θ + cos⁴θ = 1 - 2sin²θ cos² θ.

Prove that: `cos^2 A + 1/(1 + cot^2 A) = 1`.

Prove that identity:
`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`

Prove that:  `1/(sec θ - tan θ) = sec θ + tan θ`.

Prove that: `(sin θ - 2sin^3 θ)/(2 cos^3 θ - cos θ) = tan θ`.

Prove that `(tan θ + sin θ)/(tan θ - sin θ) = (sec θ + 1)/(sec θ - 1)`