ICSE Class 10

General instructions: Answer all questions. Marks are shown in brackets [ ].

Objective

1. 1.
   Prove the trigonometric identity by selecting the correct simplification of the expression: $(1 - \cos^2 A)\csc^2 A$

   1. A. $(1 - \cos^2 A) \times \frac{1}{\sin^2 A} = 1$
   2. B. $\sin^2 A \times \sec^2 A = 1$
   3. C. $(1 - \cos^2 A) \times \sin^2 A = 1$
   4. D. $\cos^2 A \times \csc^2 A = 1$
   [1]
2. 2.
   If $\sin \theta = \frac{a}{b}$, then $\cos \theta$ is equal to

   1. a. $\frac{b}{\sqrt{b^2 - a^2}}$
   2. b. $\frac{b}{a}$
   3. c. $\frac{\sqrt{b^2 - a^2}}{b}$
   4. d. $\frac{a}{\sqrt{b^2 - a^2}}$
   [1]
3. 3.
   (i) Using tables, find the value of: $\sin 83^\circ 12'$

   1. A. 0.9930
   2. B. 0.9390
   3. C. 0.9903
   4. D. 0.9929
   [1]
4. 4.
   In the adjoining figure, the length of $BC$ is

   1. a. $2\sqrt{3}$ cm
   2. b. $3\sqrt{3}$ cm
   3. c. $4\sqrt{3}$ cm
   4. d. $3$ cm
   [1]
5. 5.
   Using the table, find the value of: $\cos 23^\circ 6'$

   1. A. 0.9205
   2. B. 0.9182
   3. C. 0.9250
   4. D. 0.4067
   [1]
6. 6.
   (i) Using tables, find the acute angle θ, when sin θ = 0.36.

   1. A. 21° 6'
   2. B. 15° 42'
   3. C. 36°
   4. D. 21.6°
   [1]
7. 7.
   In $\Delta ABC$, which of the following correctly proves the identity $\tan \left(\frac{B + C}{2}\right) = \cot \frac{A}{2}$?

   1. A. $\tan \left(\frac{B + C}{2}\right) = \tan \left(90^\circ - \frac{A}{2}\right) = \cot \frac{A}{2}$
   2. B. $\tan \left(\frac{B + C}{2}\right) = \tan \left(180^\circ - \frac{A}{2}\right) = -\cot \frac{A}{2}$
   3. C. $\tan \left(\frac{B + C}{2}\right) = \cot \left(90^\circ - \frac{A}{2}\right) = \tan \frac{A}{2}$
   4. D. $\tan \left(\frac{B + C}{2}\right) = \sin \left(\frac{B + C}{2}\right) / \cos \left(\frac{A}{2}\right)$
   [1]
8. 8.
   Which of the following correctly shows that $\sin 42^\circ \sec 48^\circ + \cos 42^\circ \csc 48^\circ = 2$?

   1. A. $\sin 42^\circ \sec 48^\circ + \cos 42^\circ \csc 48^\circ = \sin 42^\circ \csc 42^\circ + \cos 42^\circ \sec 42^\circ = 1 + 1 = 2$
   2. B. $\sin 42^\circ \sec 48^\circ + \cos 42^\circ \csc 48^\circ = \sin 42^\circ \cos 48^\circ + \cos 42^\circ \sin 48^\circ = \sin 90^\circ = 1$
   3. C. $\sin 42^\circ \sec 48^\circ + \cos 42^\circ \csc 48^\circ = \sec 42^\circ \csc 42^\circ = 1/(\sin 42^\circ \cos 42^\circ) = 2$
   4. D. $\sin 42^\circ \sec 48^\circ + \cos 42^\circ \csc 48^\circ = \sin^2 42^\circ + \cos^2 42^\circ = 1$
   [1]
9. 9.
   Which of the following correctly proves the identity: $\csc A (1 + \cos A)(\csc A - \cot A) = 1$?

   1. A. $(1 + \cos A)(1 - \cos A) = 1 - \cos^2 A = \sin^2 A$, which simplifies to 1 when multiplied by $\csc A$.
   2. B. $\csc A (1 + \cos A)(\csc A - \cot A) = (1 + \cos A)(\csc^2 A - \csc A \cot A) = 1 + \cos A - \cos A = 1$.
   3. C. $\csc A (1 + \cos A)(\csc A - \cot A) = (1 + \cos A)(1 - \sin A) = 1 - \sin A + \cos A - \sin A \cos A$.
   4. D. $\csc A - \cot A = \sin A$, so the expression simplifies to $\csc A (1 + \cos A) \sin A = 1 + \cos A$.
   [1]
10. 10.
    If $f(x) = \cos^2 x + \sec^2 x$, then $f(x)$

    1. a. $\geq 1$
    2. b. $\leq 1$
    3. c. $\geq 2$
    4. d. $\leq 2$
    [1]
11. 11.
    Prove the following trigonometric identity by selecting the correct simplified form of the left-hand side (L.H.S.): $\frac{\sin A}{\sin (90^\circ - A)} + \frac{\cos A}{\cos (90^\circ - A)}$

    1. A. $\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}$
    2. B. $\sec A \csc A$
    3. C. $\tan A + \cot A$
    4. D. $\frac{1}{\sin A \cos A}$
    [1]
12. 12.
    Which of the following correctly proves the identity: $(\sec A - \cos A)(\sec A + \cos A) = \sin^2 A + \tan^2 A$?

    1. A. $(\sec A - \cos A)(\sec A + \cos A) = \sec^2 A - \cos^2 A = 1 + \tan^2 A - \cos^2 A = \sin^2 A + \tan^2 A$
    2. B. $(\sec A - \cos A)(\sec A + \cos A) = \sec^2 A - \sin^2 A = 1 + \tan^2 A - \sin^2 A = \cos^2 A + \tan^2 A$
    3. C. $(\sec A - \cos A)(\sec A + \cos A) = \cos^2 A - \sec^2 A = \cos^2 A - (1 + \tan^2 A) = -\sin^2 A - \tan^2 A$
    4. D. $(\sec A - \cos A)(\sec A + \cos A) = \sec A \cos A = 1 = \sin^2 A + \cos^2 A$
    [1]
13. 13.
    Which of the following correctly simplifies the expression $\sin A (1 + \tan A) + \cos A (1 + \cot A)$?

    1. A. $\sec A + \csc A$
    2. B. $\sin A + \cos A$
    3. C. $\frac{1 + \sin A + \cos A}{\cos A \sin A}$
    4. D. $\tan A + \cot A$
    [1]
14. 14.
    Prove the trigonometric identity: $(\sec^2 A - 1)\cot^2 A = 1$. Which of the following simplifications correctly demonstrates this identity?

    1. A. $(\sec^2 A - 1)\cot^2 A = \tan^2 A \cot^2 A = \tan^2 A \times \frac{1}{\tan^2 A} = 1$
    2. B. $(\sec^2 A - 1)\cot^2 A = \sin^2 A \cot^2 A = \sin^2 A \times \frac{\cos^2 A}{\sin^2 A} = \cos^2 A$
    3. C. $(\sec^2 A - 1)\cot^2 A = (1 - \sec^2 A)\cot^2 A = -\tan^2 A \cot^2 A = -1$
    4. D. $(\sec^2 A - 1)\cot^2 A = \sec^2 A (1 - \tan^2 A) = \sec^2 A - \sec^2 A \tan^2 A$
    [1]
15. 15.
    Prove the trigonometric identity by selecting the correct simplification of the left-hand side: $\frac{1}{1 + \sin A} + \frac{1}{1 - \sin A} = 2 \sec^2 A$. Which of the following steps correctly simplifies the left-hand side?

    1. A. $\frac{2}{1 - \sin^2 A} = \frac{2}{\cos^2 A} = 2 \sec^2 A$
    2. B. $\frac{2}{1 + \sin^2 A} = \frac{2}{\cos^2 A} = 2 \sec^2 A$
    3. C. $\frac{2}{1 - \sin A} = \frac{2}{\cos^2 A} = 2 \sec^2 A$
    4. D. $\frac{2}{1 + \sin A} = \frac{2}{\cos^2 A} = 2 \sec^2 A$
    [1]
16. 16.
    $$3 \cos 80^{\circ} \csc 10^{\circ} + 2 \cos 59^{\circ} \csc 31^{\circ}$$ equals which of the following?

    1. A. 3
    2. B. 5
    3. C. 6
    4. D. 1
    [1]
17. 17.
    Which of the following correctly proves the identity $2\sin^2 A + \cos^4 A = 1 + \sin^4 A$?

    1. A. $2\sin^2 A + (1 - \sin^2 A)^2 = 2\sin^2 A + 1 - 2\sin^2 A + \sin^4 A = 1 + \sin^4 A$
    2. B. $2\sin^2 A + \cos^4 A = 2\sin^2 A + \cos^2 A \cdot \cos^2 A = 2\sin^2 A + (1 - \sin^2 A) = 1 + \sin^2 A$
    3. C. $2\sin^2 A + \cos^4 A = 2(1 - \cos^2 A) + \cos^4 A = 2 - 2\cos^2 A + \cos^4 A = 2 - \cos^2 A$
    4. D. $2\sin^2 A + \cos^4 A = \sin^2 A + \sin^2 A + \cos^4 A = \sin^2 A + (\sin^2 A + \cos^4 A) = \sin^2 A + 1$
    [1]
18. 18.
    If $4\sin \theta = 3$, what is the value of $x$ in the equation $\sqrt{\frac{\operatorname{cosec}^2 \theta - \cot^2 \theta}{\sec^2 \theta - 1}} + 2 \cot \theta = \frac{\sqrt{7}}{x} + \cos \theta$?

    1. A. $\frac{3}{4}$
    2. B. $\frac{4}{3}$
    3. C. $\frac{3}{\sqrt{7}}$
    4. D. $\frac{\sqrt{7}}{3}$
    [1]
19. 19.
    Which of the following correctly proves the identity: $\frac{\sin A + \cos A}{\sin A - \cos A} + \frac{\sin A - \cos A}{\sin A + \cos A} = \frac{2}{2 \sin^2 A - 1}$?

    1. A. $\frac{(\sin A + \cos A)^2 + (\sin A - \cos A)^2}{\sin^2 A - \cos^2 A} = \frac{2}{\sin^2 A - \cos^2 A} = \frac{2}{2 \sin^2 A - 1}$
    2. B. $\frac{2(\sin^2 A + \cos^2 A)}{\sin^2 A - \cos^2 A} = \frac{2}{2 \cos^2 A - 1}$
    3. C. $\frac{(\sin A + \cos A) + (\sin A - \cos A)}{\sin A - \cos A} = \frac{2 \sin A}{\sin A - \cos A}$
    4. D. $\frac{2(\sin A + \cos A)^2}{\sin^2 A - \cos^2 A} = \frac{2}{1 - 2 \cos^2 A}$
    [1]
20. 20.
    Which of the following correctly proves the identity: $\frac{1}{\sec A + \tan A} = \sec A - \tan A$?

    1. A. Multiply numerator and denominator by $\sec A - \tan A$ and use $\sec^2 A - \tan^2 A = 1$.
    2. B. Multiply numerator and denominator by $\sec A + \tan A$ and simplify using $\sec^2 A + \tan^2 A = 1$.
    3. C. Rewrite $\sec A$ and $\tan A$ in terms of $\sin A$ and $\cos A$, then cross-multiply.
    4. D. Square both sides and use the identity $\sec^2 A - \tan^2 A = 1$ to equate them.
    [1]