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.
(c) Both A and R are true, and R is the correct reason for A.
Explanation:
Given: sin² A + sin A = 1
⇒ sin A = 1 − sin² A
Using 1 − sin² A = cos² A:
sin A = cos² A
Squaring both sides:
sin² A = cos⁴ A … (1)
Now: cos⁴ A + cos² A
= sin² A + cos² A, using (1)
= 1
Thus, the assertion is true. The reason is also true and is used directly to prove the assertion.
More ICSE Moderate level questions
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)
Factorize: sin³ θ + cos³ θ. Hence, prove the identity:
Prove that:
(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
In the given diagram, chords AC and BC are equal. If ∠ACD = 120°, then ∠AEC is: (a) 30° (b) 60° (c) 90° (d) 120°
The factor common to the two polynomials x² − 4 and x³ − x² − 4x + 4 is: (a) (x + 1) (b) (x − 1) (c) (x − 2) (d) (x − 4)
The quadratic equation 3x² + √7x + 2 = 0 has: (a) two equal real roots. (b) two distinct real roots. (c) more than two real roots. (d) no real roots.
A man invested in a company paying 12% dividend on its shares. If the percentage return on his investment is 10%, then the shares are: (a) at par (b) below par (c) above par (d) cannot be determined
The solution set for 0 < − < 2, x ∈ Z is: (a) {−5, −4, −3, −2, −1} (b) {−6, −5, −4, −3, −2, −1} (c) {−5, −4, −3, −2, −1, 0} (d) {−6, −5, −4, −3, −2, −1, 0} bb
In the given diagram, △ABC ∼ △EFG. If ∠ABC = ∠EFG = 60°, then the length of side FG is: (a) 15 cm (b) 20 cm (c) 25 cm (d) 30 cm
In the given diagram, O is the centre of the circle. Chord SR produced meets the tangent XTP at P. (a) Prove that △PTR ∼ △PST. (b) Prove that PT² = PR × PS. (c) If PR = 4 cm and PS = 16 cm, find the length of tangent PT.
