Prove the following identity:

(sin² θ − 1)(tan² θ + 1) + 1 = 0

(1 + sin A)(1 − sin A) is equal to: (a) cosec2 A (b) sin2 A (c) sec2 A (d) cos2 A

Prove that:

Factorize: sin³ θ + cos³ θ. Hence, prove the identity:

Statement (i): sin² θ + cos² θ = 1 Statement (ii): cosec² θ + cot² θ = 1 Which of the following is valid? (a) Only (i) (b) Only (ii) (c) Both (i) and (ii) (d) Neither (i) nor (ii)

Assertion (A): If sin² A + sin A = 1, then cos⁴ A + cos² A = 1. Reason (R): 1 − sin² A = cos² A. (a) A is true, R is false. (b) A is false, R is true. (c) Both A and R are true, and R is the correct reason for A. (d) Both A and R are true, and R is not the correct reason for A.

In the given figure ‘O’ is the centre of the circle. PQ is a tangent to the circle at B and AB = AC. If ∠CBQ = 40°, find the unknown angles x, y, z and w.

A flagpole is erected at the top of a building. The angle of elevation of the top and foot of the flagpole from a point 100 m away, on the same level as that of the foot of the building, are 33° and 31° respectively. Find the height of the flagpole. Give your answer correct to the nearest metre.

A solid wooden toy is prepared by joining a cone, a cylinder and a sphere, as shown in the given diagram. The radius of each of the three solids is 7 cm and heights of each of the cone and the cylinder is 24 cm. Find : (a) the total surface area of the given solid. (b) the cost of painting the total surface at the rate of ₹ 0.50 per cm2.

In the given diagram, DE ∥ BC and AD : DB = 2 : 3. (a) Prove that: ΔADE ~ ΔABC and hence find DE : BC. (b) Prove: ΔDFE ~ ΔCFB. (c) Given, area of ΔDFE = 16 square units, find the area of ΔCFB.

The given figure shows an eye-drop bottle consisting of a hemispherical part and a cylindrical part, with a conical cap. The height of the cylindrical part and the height of the conical cap are each equal to the diameter, 7 cm. (a) Find the minimum height of a cylindrical box in which the bottle can be packed. (b) Find the volume of the liquid medicine that the bottle can contain. Use π = 22/7.

Ms. Sushmita went to a fair and participated in a game. A box contained number cards numbered from 01 to 30. The prizes were awarded as follows: Prize Number on the card drawn at random is a Wall clock Perfect square Water bottle Even number which is also a multiple of 3 Purse Prime number Find the probability of winning: (a) a wall clock. (b) a water bottle. (c) a purse.

The difference of two natural numbers is 5 and sum of their reciprocals is . Find the two numbers.