SECTION C — Short Answer Type Questions [3 Marks Each]
- 3 Marks Q1. In $\triangle PQR$, right-angled at $Q$, $PR + QR = 25\text{ cm}$ and $PQ = 5\text{ cm}$. Determine the values of $\sin P$, $\cos P$, and $\tan P$.
- 3 Marks Q2. If $\cot \theta = \frac{15}{8}$, evaluate $\frac{(2 + 2\sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(2 - 2\cos \theta)}$.
- 3 Marks Q3. Given $\sec \theta = \frac{13}{12}$, calculate all other trigonometric ratios and verify that $\frac{1 - \tan^2\theta}{1 + \tan^2\theta} = \cos^2\theta - \sin^2\theta$.
- 3 Marks Q4. If $\angle A$ and $\angle B$ are acute angles such that $\sin A = \sin B$, then prove that $\angle A = \angle B$ without using angle values.
- 3 Marks Q5. In a right triangle $ABC$, right-angled at $B$, if $\tan A = \sqrt{3}$, find the value of $\sin A \cos C + \cos A \sin C$.
- 3 Marks Q6. If $4 \tan \theta = 3$, evaluate $\frac{4\sin \theta - \cos \theta + 1}{4\sin \theta + \cos \theta - 1}$
- 3 Marks Q7. Let $\triangle ABC$ be a right-angled triangle at $C$. If $A = 30^\circ$ and $AB = 40\text{ units}$, find the remaining angles and sides of the triangle.
- 3 Marks Q8. If $\tan(A - B) = \frac{1}{\sqrt{3}}$ and $\sin A = \frac{\sqrt{3}}{2}$, where $A$ and $B$ are acute angles ($A > B$), find the values of $A$ and $B$.
- 3 Marks Q9. In $\triangle XYZ$, right-angled at $Y$, $XZ = 13\text{ cm}$ and $XY = 5\text{ cm}$. Find $\tan X - \cot Z$.
- 3 Marks Q10. If $\sin(A + B) = 1$ and $\cos(A - B) = \frac{\sqrt{3}}{2}$, where $0^\circ < A + B \le 90^\circ$ and $A > B$, find the acute angles $A$ and $B$.
- 3 Marks Q11. Evaluate the expression: $\frac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$.
- 3 Marks Q12. Find the value of: $\frac{\tan^2 60^\circ + 4\cos^2 45^\circ + 3\sec^2 30^\circ + 5\cos^2 90^\circ}{\csc 30^\circ + \sec 60^\circ - \cot^2 30^\circ}$.
- 3 Marks Q13. If $\theta = 30^\circ$, verify that $\sin 2\theta = \frac{2\tan \theta}{1 + \tan^2 \theta}$.
- 3 Marks Q14. Verify the identity $\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$ by taking $\theta = 30^\circ$.
- 3 Marks Q15. If $\tan(A + B) = \sqrt{3}$ and $\tan(A - B) = \frac{1}{\sqrt{3}}$, use this to find whether $\sin(A+B) = \sin A \cos B + \cos A \sin B$ holds true for these values of $A$ and $B$.
- 3 Marks Q16. Find the acute angle $\theta$ satisfying $\cos^2 \theta - (1 + \sqrt{3})\cos \theta + \sqrt{3} = 0$.
- 3 Marks Q17. Evaluate without using direct tables: $\frac{\cos 0^\circ + \sin 30^\circ + \sin 45^\circ}{\sin 90^\circ + \cos 60^\circ - \cos 45^\circ}$
- 3 Marks Q18. If $x = \sin 60^\circ \cos 30^\circ - \cos 60^\circ \sin 30^\circ$, find the value of $\csc^2 x$ (in degrees interpretation or standard evaluation).
- 3 Marks Q19. Given that $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$, find the value of $\sin 75^\circ$ using standard angles $\alpha = 45^\circ$ and $\beta = 30^\circ$.
- 3 Marks Q20. Show that $\left(1 - \cos 30^\circ\right)\left(1 + \cos 30^\circ\right) = \sin^2 60^\circ$ and evaluate its numerical value
- 3 Marks Q21. Prove that $\sin 70^\circ \sec 20^\circ + \cos 70^\circ \csc 20^\circ = 2$.
- 3 Marks Q22. Evaluate: $\frac{2 \cos 58^\circ}{\sin 32^\circ} - \frac{\sqrt{3} \cos 38^\circ \csc 52^\circ}{\tan 15^\circ \tan 60^\circ \tan 75^\circ}$.
- 3 Marks Q23. If $\sin 3A = \cos(A - 26^\circ)$, where $3A$ is an acute angle, find the exact value of $A$.
- 3 Marks Q24. Prove that $\tan 1^\circ \tan 2^\circ \tan 3^\circ \dots \tan 89^\circ = 1$.
- 3 Marks Q25. Express $\sin 81^\circ + \tan 68^\circ$ in terms of trigonometric ratios of angles between $0^\circ$ and $45^\circ$.
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English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
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