16. Problem: Identify the correct trigonometric identity or inequality among the options. Solution: - Option a) $\tan^2\theta - \sec^2\theta = 1$ is incorrect because the identity is $1 + \tan^2\theta = \sec^2\theta$. - Option b) $\sin\theta + \cos\theta > 1$ is not always true; maximum value is $\sqrt{2} \approx 1.414$ but depends on $\theta$. - Option c) $\sin\theta + \cos\theta = 1$ is not generally true. - Option d) $\sin\theta + \cos\theta < 1$ is sometimes true but not always. Therefore, none are strictly identities; however, the known identity is option a) rewritten as $1 + \tan^2\theta = \sec^2\theta$. Answer: a) is closest to a known identity but stated incorrectly, so no correct option exactly. 17. Problem: Given $1 + \tan^2\theta = 4$, find $\theta$. Solution: From identity: $1 + \tan^2\theta = \sec^2\theta$. Here, $1 + \tan^2\theta = 4 \Rightarrow \sec^2\theta = 4$. So $\sec\theta = 2$ or $-2$. Recall $\sec\theta = \frac{1}{\cos\theta}$, so $$\frac{1}{\cos\theta} = 2 \Rightarrow \cos\theta = \frac{1}{2}.$$ Therefore, $\theta = 60^\circ$ or $300^\circ$ in $0^\circ-360^\circ$. Answer: (d) $60^\circ$ 18. Problem: If $\sin^2 A = \frac{1}{2}$, find $\cos 2A$. Solution: Using double angle formula: $$\cos 2A = 1 - 2\sin^2 A = 1 - 2(\frac{1}{2}) = 1 - 1 = 0.$$ Answer: (d) 0 19. Problem: Given ratios $a:b=3:4$ and $b:c=6:7$, find $a:b:c$. Solution: - From $a:b=3:4$, if we express in terms of $b$: $a = 3k$, $b=4k$. - From $b:c=6:7$, if $b=6m$, then $c=7m$. Equate $b$: $$4k = 6m \Rightarrow k = \frac{3m}{2}.$$ Then: $$a = 3k = 3 \times \frac{3m}{2} = \frac{9m}{2}.$$ So values: $$a:b:c = \frac{9m}{2} : 6m : 7m = 9:12:14$$ Dividing all by 3: $$3:4:\frac{14}{3},$$ not correct. Better to multiply all by 2: $$9:12:14$$ but to have integer ratio we keep as $9:12:14$. Answer: None of the listed options match exactly. But closest from options is (গ) $7:12:14$ which doesn't fit. So correct combined ratio is $9:12:14$. 20. Problem: In sugarcane juice, sugar to water ratio is $3:7$. What is the percentage of sugar? Solution: Total parts = $3 + 7 = 10$. Sugar percentage: $$\frac{3}{10} \times 100 = 30\%.$$ Answer: (d) 30% 21. Problem: How many zeros does a real number have? This question is ambiguous. Possibly referring to roots of an equation or zeros of a function. Assuming asking about zeros of a polynomial or function. Answer choices: - 2, 0, 1, infinite. Without more context, assuming 2 (most common for quadratic), but can't solve definitively. 22. Problem: A cube with surface area $4\sqrt{2}$ cm. Find its volume. Solution: Surface area of cube: $$S = 6a^2 = 4\sqrt{2}$$ Solve for $a$: $$a^2 = \frac{4\sqrt{2}}{6} = \frac{2\sqrt{2}}{3}$$ Volume: $$V = a^3 = a \times a^2 = a \cdot \frac{4\sqrt{2}}{6}$$ But better to find $a$ first: $$a = \sqrt{\frac{2\sqrt{2}}{3}} = \sqrt{\frac{2 \times 1.414}{3}} = \sqrt{\frac{2.828}{3}}= \sqrt{0.943} \approx 0.971$$ Then volume: $$V = a^3 = (0.971)^3 \approx 0.915$$ Choices are large numbers, likely error in question. Possibly surface area given in error; no exact match. 23. Problem: Maximum number of shadows (projections) of a solid volume? Typically 3 (projections on 3 perpendicular planes). Answer: (b) 3 24. Problem: For the given shape (rectangle with inscribed circle), find length of one diagonal with area equal to given area. Area of circle = 25 cm² Radius $r= \sqrt{25/\pi} = \sqrt{7.96} = 2.82$cm approx. Rectangle side DC = 7cm. Diagonal length calculation using Pythagoras: Diagonal $= \sqrt{7^2 + (something)^2}$. Options closest to 4.18 or 4.39. Exact calculation depends on shape, answer: (a) 4.18 Final answers: 16: a 17: d 18: d 19: $9:12:14$ (not in options) 20: d 21: can't decide 22: no match 23: b 24: a