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

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.

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

Prove the following identity: (sin2 θ − 1)(tan2 θ + 1) + 1 = 0

Prove that:

The circumcentre of a triangle is a point which is: (a) at an equal distance from the three sides of the triangle. (b) at an equal distance from the three vertices of the triangle. (c) the point of intersection of the medians. (d) the point of intersection of the altitudes.

The following table shows the distance travelled by a train moving with uniform speed: Distance (m) 60 90 y Time (s) 2 x 5 The values of x and y are: (a) x = 4, y = 150 (b) x = 3, y = 100 (c) x = 4, y = 100 (d) x = 3, y = 150

The points A(x, y), B(3, −2) and C(4, −5) are collinear. Then y in terms of x is: (a) 3x − 11 (b) 11 − 3x (c) 3x − 7 (d) 7 − 3x

In the given diagram, △ABC ∼ △PQR. If AD and PS are bisectors of ∠BAC and ∠QPR respectively, then: (a) △ABC ∼ △PQS (b) △ABD ∼ △PQS (c) △ABD ∼ △PSR (d) △ABC ∼ △PSR

In the given diagram, PS and PT are tangents to the circle. SQ ∥ PT and ∠SPT = 80°. The value of ∠QST is: (a) 140° (b) 90° (c) 80° (d) 50°

A rectangular sheet of paper of size 11 cm × 7 cm is first rotated about the side 11 cm and then about the side 7 cm to form a cylinder, as shown in the diagram. The ratio of their curved surface areas is: (a) 1 : 1 (b) 7 : 11 (c) 11 : 7 (d)

For an Intra-state sale, CGST paid = ₹120. The marked price of the article is ₹2000. The rate of GST is: (a) 6% (b) 10% (c) 12% (d) 16.67%