1. In triangle ABC , what is the value of sin⁡\(\cfrac{(B+C)}{2} \) ?

(a) sin⁡\(\cfrac{A}{2}\) (b) sinA (c) cosA (d) cos⁡ \(\cfrac{A}{2}\)

2. If \(\sin\theta + \cos\theta = \sqrt{2}\), then the value of \(\theta\) will be—

(a) \(\cfrac{\pi^c}{2}\) (b) \(\cfrac{\pi^c}{3}\) (c) \(\pi^c\) (d) \(\cfrac{\pi^c}{4}\)

3. If \( \sin^2 \theta + 2x \cos^2 \theta = 1 \), then the value of \( x \) will be _____

4. Asit and Dipak invested ₹22,500 and ₹35,000 respectively in a joint business. If the total profit is ₹13,800, then Dipak's share of the profit will be —

(a) ₹ 8400 (b) ₹ 8600 (c) ₹ 8200 (d) ₹ 8000

5. "If ???? sin ⁡ ???? = 7 2 and ???? cos ⁡ ???? = 7 3 2 , then what is the value of ???? ?

(a) \(49\) (b) \(7\) (c) \(\sqrt7\) (d) \(-7\)

6. What will be the minimum value of 4 cos ⁡ ???? − 3 sin ⁡ ???? ?

(a) 1 (b) 2 (c) 0 (d) None of the above

7. The value of sin12° − cos78° will be __.

8. For which value(s) of \(θ\) (where \(0^\circ ≤ θ ≤ 90^\circ\)) will \(2\sinθ\cosθ = \cosθ\) be true?

9. If \(\alpha\) and \(\beta\) are complementary angles, find the value of the expression: \[ (1 - \sin^2 \alpha)(1 - \cos^2 \alpha)(1 + \cot^2 \beta)(1 + \tan^2 \beta) \]

10. If \(\tan \theta \cos 60° = \cfrac{\sqrt{3}}{2}\), then the value of \(\sin (\theta - 15°)\) will be _____ .

11. If \(\alpha + \beta = 90^\circ\) and \(\alpha : \beta = 2 : 1\), then find the value of \(\sin \alpha : \sin \beta\).

(a) \(3:1\) (b) \(1:3\) (c) \(\sqrt3:1\) (d) \(1:\sqrt 3\)