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$$\begin{gathered} \mathrm{If} \sin \mathrm{x}° = \sin \mathrm{\alpha x}, \mathrm{then} \mathrm{\alpha } \mathrm{is} \\ 1. \frac{180}{\mathrm{\pi }} \\ 2. \frac{\mathrm{\pi }}{270} \\ 3. \frac{270}{\mathrm{\pi }} \\ 4. \frac{\mathrm{\pi }}{180} \end{gathered}$$

If in a triangle ABC , and A and B are complementary the tanC is
$$\begin{gathered} 1. \infty \\ 2. 0 \\ 3. 1 \\ 4. \sqrt{3} \end{gathered}$$

$$\begin{gathered} \mathrm{If} \mathrm{\alpha } = \frac{4}{5} \mathrm{and} \mathrm{\beta } = \frac{4}{5}, \mathrm{then} \mathrm{which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{true}? \\ 1. \mathrm{\alpha }< \mathrm{\beta } \\ 2. \mathrm{\alpha } > \mathrm{\beta } \\ 3. \mathrm{\alpha } = \mathrm{\beta } \\ 4. \mathrm{None} \mathrm{of} \mathrm{these} \end{gathered}$$

$$\begin{gathered} {\sin }^2 20 + {\sin }^{2}70 \mathrm{is} \mathrm{equal} \mathrm{to} \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. 1 \\ 2. -1 \\ 3. 0 \\ 4. 2 \end{gathered}$$

$$\begin{gathered} \cos 50°50\text{'} \cos 9°10\text{'} - \sin 50°50\text{'} \sin 9°10\text{'} = \\ 1. 0 \\ 2. \frac{1}{2} \\ 3. 1 \\ 4. \frac{\sqrt{3}}{2} \end{gathered}$$

$$\begin{gathered} \sin \mathrm{\theta } \cos (90° - \mathrm{\theta }) + \cos \mathrm{\theta } \sin (90° - \mathrm{\theta }) = \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. -1 \\ 2. 2 \\ 3. 0 \\ 4. 1 \end{gathered}$$

A wheel makes 20 revolutions per hour. The radians it turns through 25 minutes is __________ .
$$\begin{gathered} 1. \frac{50{\mathrm{\pi }}^{\mathrm{c}}}{7} \\ 2. \frac{250{\mathrm{\pi }}^{\mathrm{c}}}{3} \\ 3. \frac{150{\mathrm{\pi }}^{\mathrm{c}}}{7} \\ 4. \frac{50{\mathrm{\pi }}^{\mathrm{c}}}{3} \end{gathered}$$
 

$$\begin{gathered} \frac{{\sin }^4 \mathrm{\theta } - {\cos }^4 \mathrm{\theta }}{{\sin }^2 \mathrm{\theta } - {\cos }^2 \mathrm{\theta }} = \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. -1 \\ 2. 2 \\ 3. 0 \\ 4. 1 \end{gathered}$$

The simplified expression of ($\sec \mathrm{\theta } + \tan \mathrm{\theta }$) $(1 - \sin \mathrm{\theta }) \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_} .$
$$\begin{gathered} 1. {\sin }^2\mathrm{\theta } \\ 2. {\cos }^2\mathrm{\theta } \\ 3. {\tan }^2\mathrm{\theta } \\ 4. \cos \mathrm{\theta } \end{gathered}$$

$$\begin{gathered} \mathrm{If} \mathrm{a} = \sec \mathrm{\theta } - \tan \mathrm{\theta } \mathrm{and} \mathrm{b} = \sec \mathrm{\theta } + \tan \mathrm{\theta }, \mathrm{then} \\ 1. \mathrm{a} = \mathrm{b} \\ 2. \frac{1}{\mathrm{a}} = \frac{-1}{\mathrm{b}} \\ 3. \mathrm{a} = \frac{1}{\mathrm{b}} \\ 4. \mathrm{a} - \mathrm{b} = 1 \end{gathered}$$

$$\begin{gathered} \mathrm{If} \sec \mathrm{\alpha } + \tan \mathrm{\alpha } = \mathrm{m}, \mathrm{then} {\sec }^4 \mathrm{\alpha } - {\tan }^4 \mathrm{\alpha } - 2 \sec \mathrm{\alpha } \tan \mathrm{\alpha } \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_} . \\ 1. {\mathrm{m}}^2 \\ 2. -{\mathrm{m}}^2 \\ 3. \frac{1}{{\mathrm{m}}^2} \\ 4. \frac{-1}{{\mathrm{m}}^2} \end{gathered}$$

$$\begin{gathered} \mathrm{If} {\sin }^4 \mathrm{A} - {\cos }^4 \mathrm{A} = 1, \mathrm{then} (\mathrm{A}/2) \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_} . (0 < \mathrm{A} \le 90°). \\ 1. 45° \\ 2. 60° \\ 3. 30° \\ 4. 40° \end{gathered}$$

$$\begin{gathered} \mathrm{The} \mathrm{value} \mathrm{of} \tan 15° \tan 20° \tan 70° \tan 75° \mathrm{is} \\ 1. -1 \\ 2. 2 \\ 3. 0 \\ 4. 1 \end{gathered}$$

$$\begin{gathered} \mathrm{In} \mathrm{a} △ \mathrm{ABC}, \tan \left(\frac{\mathrm{A} + \mathrm{C}}{2}\right) = \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. \tan \frac{\mathrm{B}}{2} \\ 2. \cot \frac{\mathrm{B}}{2} \\ 3. -\tan \mathrm{B} \\ 4. \cot \mathrm{B} \end{gathered}$$

$$\begin{gathered} \mathrm{If} \tan (\mathrm{A} - 30°) = 2 - \sqrt{3} , \mathrm{then} \mathrm{find} \mathrm{A}. \\ 1. \frac{{\mathrm{\pi }}^{\mathrm{c}}}{6} \\ 2. \frac{{\mathrm{\pi }}^{\mathrm{c}}}{4} \\ 3. \frac{{\mathrm{\pi }}^{\mathrm{c}}}{6} \\ 4. \frac{{\mathrm{\pi }}^{\mathrm{c}}}{3} \end{gathered}$$

$$\begin{gathered} \mathrm{If} {\sin }^4 \mathrm{\theta } - {\cos }^4 \mathrm{\theta } = {\mathrm{K}}^4, \mathrm{then} {\sin }^2 \mathrm{\theta } - {\cos }^2 \mathrm{\theta } \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_\_\_} . \\ 1. {\mathrm{K}}^4 \\ 2. {\mathrm{K}}^3 \\ 3. {\mathrm{K}}^2 \\ 4. \mathrm{K} \end{gathered}$$

$$\begin{gathered} \frac{{\tan }^3 \mathrm{\theta } - 1}{\tan \mathrm{\theta } - 1} = \mathrm{\_\_\_\_\_\_\_\_\_\_\_} . \\ 1. {\sec }^2 \mathrm{\theta } + \tan \mathrm{\theta } \\ 2. {\sec }^2 \mathrm{\theta } - \tan \mathrm{\theta } \\ 3. 0 \\ 4. \tan \mathrm{\theta } - {\sec }^{2 \mathrm{\theta }} \end{gathered}$$

$$\begin{gathered} \mathrm{For} \mathrm{all} \mathrm{values} \mathrm{of} \mathrm{\theta }, 1 + \cos \mathrm{\theta } \mathrm{can} \mathrm{be} \mathrm{\_\_\_\_\_\_\_\_\_} . \\ 1. \mathrm{Positive} \\ 2. \mathrm{Negative} \\ 3. \mathrm{Non}-\mathrm{positive} \\ 4. \mathrm{non}-\mathrm{negative} \end{gathered}$$

$$\begin{gathered} \mathrm{If} \sin 3\mathrm{\theta } = \cos (\mathrm{\theta } - 6°), \mathrm{where} 3\mathrm{\theta } \mathrm{and} (\mathrm{\theta } - 6°) \mathrm{are} \mathrm{acute} \mathrm{angles} \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{\theta } \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_} . \\ 1. 42° \\ 2. 24° \\ 3. 12° \\ 4. 26° \end{gathered}$$

$$\begin{gathered} (\mathrm{cosec} \mathrm{A} - \sin \mathrm{A})(\sec \mathrm{A} - \cos \mathrm{A})(\tan \mathrm{A} + \cot \mathrm{A}) = \mathrm{\_\_\_\_\_\_\_\_\_\_\_} . \\ 1. -1 \\ 2. 2 \\ 3. 0 \\ 4. 1 \end{gathered}$$

$$\begin{gathered} \mathrm{If} \mathrm{x} = \mathrm{a} (\mathrm{cosec} \mathrm{\theta } + \cot \mathrm{\theta }) \mathrm{and} \mathrm{y} = \mathrm{b} (\cot \mathrm{\theta } - \mathrm{cosec} \mathrm{\theta }) , \mathrm{then} \\ 1. \mathrm{xy} - \mathrm{ab} = 0 \\ 2. \mathrm{xy} + \mathrm{ab} = 0 \\ 3. \frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} = 1 \\ 4.{\mathrm{x}}^2 {\mathrm{y}}^2 = \mathrm{ab} \end{gathered}$$

$$\begin{gathered} \mathrm{the} \mathrm{value} \mathrm{of} \frac{{\cos }^4 \mathrm{x} + {\cos }^2 \mathrm{x} {\sin }^2\mathrm{x} + {\sin }^2 \mathrm{x}}{{\cos }^4 \mathrm{x} + {\sin }^2 \mathrm{x} {\cos }^2 \mathrm{x} + {\sin }^4 \mathrm{x}} \mathrm{is} \\ 1. 2 \\ 2. 1 \\ 3. 3 \\ 4. 0 \end{gathered}$$

$$\begin{gathered} \frac{1}{ 1+ \sin \mathrm{\theta }} + \frac{1}{1 - \sin \mathrm{\theta }} \mathrm{is} \mathrm{equal} \mathrm{to} \mathrm{\_\_\_\_\_\_\_\_} . \\ 1. 2{\sec }^2\mathrm{\theta } \\ 2. 2{\cos }^2\mathrm{\theta } \\ 3. 0 \\ 4. 1 \end{gathered}$$

$$\begin{gathered} \mathrm{If} \tan (\mathrm{\alpha } + \mathrm{\beta }) = \frac{1}{2} \mathrm{and} \tan \mathrm{\alpha } = \frac{1}{2}, \mathrm{then} \tan \mathrm{\beta } = \mathrm{\_\_\_\_\_\_\_\_\_\_\_} . \\ 1. \frac{1}{6} \\ 2. \frac{1}{7} \\ 3. 1 \\ 4. \frac{7}{6} \end{gathered}$$

$$\begin{gathered} \mathrm{The} \mathrm{value} \mathrm{of} \log \sin 0° + \log \sin 1° + \log \sin 2° + ... + \log \sin 90° \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_\_\_\_} . \\ 1. 0 \\ 2. 1 \\ 3. -1 \\ 4. \mathrm{undefined} \end{gathered}$$

Which of the following is not possible?
$$\begin{gathered} 1. \sin \mathrm{\theta } = \frac{3}{5} \\ 2. \sec \mathrm{\theta } = 100 \\ 3. \mathrm{cosec} \mathrm{\theta } =0.14 \\ 4. \mathrm{None} \mathrm{of} \mathrm{these} \end{gathered}$$
 

$$\begin{gathered} {\sin }^2 20° + {\cos }^2 160° + - {\tan }^2 45° = \_\_\_\_\_\_\_\_\_\_ . \\ 1. 2 \\ 2. 0 \\ 3. 1 \\ 4. -2 \end{gathered}$$

$$\begin{gathered} \frac{\sin \mathrm{\theta } + \cos \mathrm{\theta }}{\sin \mathrm{\theta } - \cos \mathrm{\theta }} + \frac{ \sin \mathrm{\theta } - \cos \mathrm{\theta }}{\sin \mathrm{\theta } + \cos \mathrm{\theta }} = \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. \frac{2}{1 - 2{\cos }^2 \mathrm{\theta }} \\ 2. \frac{2}{2{\sin }^2 \mathrm{\theta } - 1} \\ 3. \mathrm{Both} (1.) \mathrm{and} (2.) \\ 4. \mathrm{none} \mathrm{of} \mathrm{these} \end{gathered}$$

The length of the side (in cm) of an equilateral triangle inscribed in a circle of radius 8 cm is ___________.
$$\begin{gathered} 1. 16\sqrt{3} \\ 2. 12\sqrt{3} \\ 3. 8\sqrt{3} \\ 4. 10\sqrt{3} \end{gathered}$$

$$\begin{gathered} \mathrm{which} \mathrm{among} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{true} ? \\ 1. \sin 1° > \sin 1^{\mathrm{c}} \\ 2. \sin 1° < \sin 1^{\mathrm{c}} \\ 3. \sin 1° = \sin 1^{\mathrm{c}} \\ 4. \mathrm{None} \mathrm{of} \mathrm{these} \end{gathered}$$

$$\begin{gathered} \mathrm{If} 2 \sin \mathrm{\alpha } + 3 \cos \mathrm{\alpha } = 2, \mathrm{then} 3 \sin \mathrm{\alpha } - 2\cos \mathrm{\alpha } = \mathrm{\_\_\_\_\_\_\_} . \\ 1. \pm 3 \\ 2. \pm 1 \\ 3. 0 \\ 4. \pm 2 \end{gathered}$$

$$\begin{gathered} \mathrm{If} \frac{{\sin }^2 \mathrm{\alpha } - 3\sin \mathrm{\alpha } + 2}{{\cos }^2 \mathrm{\alpha }} = 1, \mathrm{then} \mathrm{\alpha } \mathrm{can} \mathrm{be} \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. 60° \\ 2. 45° \\ 3. 0° \\ 4. 30° \end{gathered}$$

$$\begin{gathered} \mathrm{If} \cot \mathrm{A} = \frac{\mathrm{A}}{12} \mathrm{and} \mathrm{A} \mathrm{is} \mathrm{not} \mathrm{the} \mathrm{first} \mathrm{quadrant}, \mathrm{then} \frac{\sin \mathrm{A} - \cos \mathrm{A}}{1 + \cot \mathrm{A}} \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. \frac{-74}{25} \\ 2. \frac{-84}{221} \\ 3. \frac{-87}{223} \\ 4. \mathrm{None} \mathrm{of} \mathrm{these} \end{gathered}$$

$$\begin{gathered} \mathrm{If} \frac{1 + \sin \mathrm{\alpha }}{1 - \sin \mathrm{\alpha }} = \frac{{\mathrm{m}}^2}{{\mathrm{n}}^2}, \mathrm{then} \sin \mathrm{\alpha } \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_} . \\ 1. \frac{{\mathrm{m}}^2 + {\mathrm{n}}^2}{{\mathrm{m}}^2 - {\mathrm{n}}^2} \\ 2. \frac{{\mathrm{m}}^2 - {\mathrm{n}}^2}{{\mathrm{m}}^2 + {\mathrm{n}}^2} \\ 3. \frac{{\mathrm{m}}^2 + {\mathrm{n}}^2}{{\mathrm{n}}^{2 } - {\mathrm{m}}^2} \\ 4. \frac{{\mathrm{n}}^{2 } - {\mathrm{m}}^2}{{\mathrm{m}}^2 + {\mathrm{n}}^2} \end{gathered}$$

$$\begin{gathered} \mathrm{If} \sin \mathrm{\theta } - \cos \mathrm{\theta } = \frac{3}{5}, \mathrm{then} \sin \mathrm{\theta } \cos \mathrm{\theta } = \mathrm{\_\_\_\_\_\_\_\_} . \\ 1. \frac{16}{25} \\ 2. \frac{9}{16} \\ 3. \frac{9}{25} \\ 4. \frac{8}{25} \end{gathered}$$

$$\begin{gathered} \mathrm{If} \mathrm{ABCD} \mathrm{is} \mathrm{a} \mathrm{cyclic} \mathrm{quadrilateral}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} {\cos }^2 \mathrm{A} - {\cos }^2 \mathrm{B} - {\cos }^2 \mathrm{C} + {\cos }^2 \mathrm{D} \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_\_\_} . \\ 1. 0 \\ 2. 1 \\ 3. -1 \\ 4. 2 \end{gathered}$$

The length of the minute hand of a wall clock is 12cm. find the distance covered by the tip of the minute hand in 25 minutes.
$$\begin{gathered} 1. \frac{220}{7}\mathrm{cm} \\ 2. \frac{110}{7}\mathrm{cm} \\ 3. \frac{120}{7}\mathrm{cm} \\ 4. \frac{240}{7}\mathrm{cm} \end{gathered}$$

$$\begin{gathered} {\sin }^2 2° + {\sin }^{2}4° + {\sin }^{2}6° + ... + {\sin }^{2}90° = \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. 22 \\ 2. 23 \\ 3. 44 \\ 4. 45 \end{gathered}$$

a straight leads to the foot of a tower of height 50m. From the top of the tower, the angle of depression of two cars standing on the highway are $30° \mathrm{and} 60°$ . What is the distance between the two cars? (in meters)
$$\begin{gathered} 1. \frac{100}{\sqrt{30}} \\ 2. 50\sqrt{3} \\ 3. \frac{50}{\sqrt{3}} \\ 4. 100\sqrt{3} \end{gathered}$$

The angle of elevation of the top of a hill from the foot of a tower is and the angle of elevation of the top of the tower from the foot of the hill is 30. If the tower is 50m high, then what is the height of the hill?
1.  180m
2.  150m
3.  100m
4.  120m 

$$\begin{gathered} \tan 38° - \cot 22° = \mathrm{\_\_\_\_\_\_\_\_} . \\ 1. \frac{1}{2}\mathrm{cosec} 38° \sec 22° \\ 2. 2 \sin 22° \cos 38° \\ 3. -\frac{1}{2} \mathrm{cosec} 22° \sec 38° \\ 4. \mathrm{None} \mathrm{of} \mathrm{these} \end{gathered}$$

$$\begin{gathered} \frac{-1 \cos \mathrm{\theta }}{\sin \mathrm{\theta }} \hspace{0.17em}+ \frac{\sin \mathrm{\theta }}{1 - \cos \mathrm{\theta }} = \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. 2 \sin \mathrm{\theta } \\ 2. 2 \cos \mathrm{\theta } \\ 3. 2 \mathrm{cosec} \mathrm{\theta } \\ 4. 2 \sec \mathrm{\theta } \end{gathered}$$

$$\begin{gathered} \sqrt{- 4 +\sqrt{8 + 16 \cos e{c}^4\alpha {\sin }^4\alpha }} = \_\_\_\_\_\_\_. \\ 1. \cos ec \alpha - \sin \alpha \\ 2. 2\cos ec \alpha + \sin \alpha \\ 3. 2\cos ec \alpha - \sin \alpha \\ 4. \cos ec \alpha - 2\sin \alpha \end{gathered}$$

The angles of depression of the top and the bottom of a 7 m tall building from the top of a tower are $45° \mathrm{and} 60°$ respectively. Find the height of the tower in meters.
$1. 7(3 + \sqrt{3})$
$2. \frac{7}{2}(3 - \sqrt{3})$
$3. \frac{7}{2}(3 + \sqrt{3})$
$4. 7 (3 - \sqrt{3})$

$$\begin{gathered} \mathrm{If} \tan 86° = \mathrm{m}, \mathrm{then} \frac{\tan 176° + \cot 4°}{\mathrm{m} + \tan 4°} \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. \frac{{\mathrm{m}}^2 - 1}{{\mathrm{m}}^2 + 1} \\ 2. \frac{{\mathrm{m}}^2 + 1}{1- {\mathrm{m}}^2 } \\ 3. \frac{1 - {\mathrm{m}}^2 }{1 + {\mathrm{m}}^2 } \\ 4. \frac{{\mathrm{m}}^2 + 1}{{\mathrm{m}}^2 - 1} \end{gathered}$$

$$\begin{gathered} \mathrm{The} \mathrm{following} \mathrm{sentences} \mathrm{are} \mathrm{the} \mathrm{steps} \mathrm{involved} \mathrm{in} \mathrm{providing} \mathrm{the} \mathrm{result} \frac{\cos \mathrm{x}}{1 - \tan \mathrm{x}} + \frac{\sin \mathrm{x}}{1 - \cot \mathrm{x}} = \cos \mathrm{x} + \sin \mathrm{x}. \\ \mathrm{Arrange} \mathrm{them} \mathrm{in} \mathrm{sequential} \mathrm{order} \mathrm{from} \mathrm{first} \mathrm{to} \mathrm{last}. \\ \left(\mathrm{A}\right) \frac{{\cos }^2 \mathrm{x}}{\cos \mathrm{x} - \sin \mathrm{x}} + \frac{{\sin }^2 \mathrm{x}}{\sin \mathrm{x} - \cos \mathrm{x} } \\ \left(\mathrm{B})\right) \frac{{\cos }^2 \mathrm{x} - {\sin }^2 \mathrm{x}}{\cos \mathrm{x} - \sin \mathrm{x}} \\ \left(\mathrm{C}\right) \frac{\cos \mathrm{x}}{1 - {\displaystyle \frac{\sin \mathrm{x}}{\cos \mathrm{x}}}} + \frac{\sin \mathrm{x}}{1 - {\displaystyle \frac{\cos \mathrm{x}}{\sin \mathrm{x}}}} \\ 1. \left(\mathrm{A}\right),\left(\mathrm{B}\right) \mathrm{and} \left(\mathrm{C}\right) \\ 2. \left(\mathrm{C}\right), \left(\mathrm{A}\right) \mathrm{and} \left(\mathrm{B}\right) \\ 3. \left(\mathrm{C}\right),\left(\mathrm{B}\right) \mathrm{and} \left(\mathrm{A}\right) \\ 4. \mathrm{None} \mathrm{of} \mathrm{these} \end{gathered}$$

$$\begin{gathered} \mathrm{The} \mathrm{following} \mathrm{sentences} \mathrm{are} \mathrm{the} \mathrm{steps} \mathrm{involved} \mathrm{in} \mathrm{eliminating} \mathrm{\theta } \mathrm{from} \mathrm{the} \mathrm{equation} \mathrm{x} = \mathrm{y} \tan \mathrm{\theta } \mathrm{and} \mathrm{a} = \mathrm{b} \sec \mathrm{\theta } . \\ \mathrm{Arrange} \mathrm{them} \mathrm{in} \mathrm{sequential} \mathrm{order} \mathrm{from} \mathrm{first} \mathrm{to} \mathrm{last}. \\ \left(\mathrm{A}\right) \mathrm{substract} {\left(\frac{\mathrm{x}}{\mathrm{y}}\right)}^2 \mathrm{from} {\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^2 \\ \left(\mathrm{B}\right) {\left(\frac{\mathrm{x}}{\mathrm{y}}\right)}^2 - {\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^2 = 1 \\ \left(\mathrm{C}\right) \mathrm{Taking} \mathrm{squares} \mathrm{on} \mathrm{both} \mathrm{the} \mathrm{sides} \\ \left(\mathrm{D}\right) \mathrm{Find} \frac{\mathrm{x}}{\mathrm{y}} \mathrm{and} \frac{\mathrm{a}}{\mathrm{b}} \\ 1. \left(\mathrm{D}\right), \left(\mathrm{A}\right), \left(\mathrm{C}\right) \mathrm{and} \left(\mathrm{B}\right) \\ 2. \left(\mathrm{D}\right), \left(\mathrm{C}\right), \left(\mathrm{B}\right) \mathrm{and} \left(\mathrm{A}\right) \\ 3. \left(\mathrm{D}\right), \left(\mathrm{B}\right), \left(\mathrm{A}\right) \mathrm{and} \left(\mathrm{C}\right) \\ 4. \left(\mathrm{D}\right), \left(\mathrm{C}\right), \left(\mathrm{A}\right) \mathrm{and} \left(\mathrm{B}\right) \end{gathered}$$

There is a small island in the river which is 100m wide and a tall tree stands on the island. P and Q are points directly opposite to each other on the two banks and in line with the tree. if the angles of elevation of the top of the tree from P and Q are respectively are $30° \mathrm{and} 45°$ , find the height of the tree (in meters). 
$$\begin{gathered} 1. 50\left(\sqrt{3} - 1\right) \\ 2. 50\left(\sqrt{3} + 1\right) \\ 3. 100\left(\sqrt{3} + 1\right) \\ 4. 100\left(\sqrt{3} - 1\right) \end{gathered}$$

A balloon is connected to a meteorological ground station by a cable of length 215m inclined at $60°$ to the horizontal. Determine the height of the balloon from the ground. Assume that there is no slack in the cable.
$$\begin{gathered} 1. 107.5\sqrt{3} \mathrm{m} \\ 2. 100\sqrt{3} \mathrm{m} \\ 3. 215\sqrt{3} \mathrm{m} \\ 4. 215/\sqrt{3} \mathrm{m} \end{gathered}$$

$\mathrm{If} {\sin }^2 \mathrm{A} = 2{\sin }^2 \mathrm{A} \cos \mathrm{A} \mathrm{and} \sin 20° = \mathrm{K}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \cos 20° \cos 40° \cos 80° \cos 160° = \mathrm{\_\_\_\_\_\_\_\_\_\_} .$
$$\begin{gathered} 1. \mathrm{K} \\ 2. -\sqrt{1 - {\mathrm{K}}^2} \\ 3. \frac{\sqrt{1 - {\mathrm{K}}^2}}{8} \\ 4. -\frac{\sqrt{1 - {\mathrm{K}}^2}}{8} \end{gathered}$$

$$\begin{gathered} \mathrm{If} \sqrt{2}\cos \mathrm{\theta } - \sqrt{6} \sin \mathrm{\theta } = 2\sqrt{2}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{\theta } \mathrm{can} \mathrm{be} \mathrm{\_\_\_\_\_\_\_\_\_\_} . \\ 1. 0° \\ 2. -45° \\ 3. 30° \\ 4. -60° \end{gathered}$$

A circus artist climbs the ground along a rope that is stretched from the top of a verticle pole and tired at the ground at a certain distance from the foot of the pole. The height of the pole is 12m and the angle made by the rope with the ground is $30°$. Calculate the distance covered by the artist in reaching the top of the pole.
1.  24m
2.  6m
3.  12m
4.  None of these

$$\begin{gathered} \mathrm{Find} \mathrm{the} \mathrm{value} \mathrm{of} {\sin }^2 5° + {\sin }^{2}10° + {\sin }^{2}15° + ... + {\sin }^{2}90° \\ 1. 8 \\ 2. 9 \\ 3. \frac{17}{2} \\ 4. \frac{19}{2} \end{gathered}$$

$\mathrm{If} \sec \mathrm{\theta } + \tan \mathrm{\theta } = 2, \mathrm{then} \mathrm{find} \mathrm{the} \mathrm{vlaue} \mathrm{of} \sin \mathrm{\theta }.$
$$
$1. \frac{3}{5}$
$2. \frac{2}{5}$
$3. -\frac{3}{5}$
$4. -\frac{2}{5}$

$$\begin{gathered} \mathrm{If} \cos \mathrm{\theta } + \left(\frac{1}{\sqrt{3}}\right) \sin \mathrm{\theta } = \frac{1}{\sqrt{3}}, \mathrm{then} \mathrm{find} \mathrm{\theta } \mathrm{in} \mathrm{circular} \mathrm{measure}. \\ 1. \frac{{\mathrm{\pi }}^{\mathrm{c}}}{10} \\ 2. \frac{{\mathrm{\pi }}^{\mathrm{c}}}{9} \\ 3. \frac{{\mathrm{\pi }}^{\mathrm{c}}}{6} \\ 4. \frac{{\mathrm{\pi }}^{\mathrm{c}}}{3} \end{gathered}$$

$$\begin{gathered} \sqrt{\frac{1 + \sin \mathrm{\theta }}{1 - \sin \mathrm{\theta }}} = \mathrm{\_\_\_\_\_\_\_\_\_\_\_\_} . \\ 1. \sec \mathrm{\theta } + \tan \mathrm{\theta } \\ 2. \sec \mathrm{\theta } - \cot \mathrm{\theta } \\ 3. \mathrm{cosec} \mathrm{\theta } + \tan \mathrm{\theta } \\ 4. \mathrm{cosec} - \tan \mathrm{\theta } \end{gathered}$$

$$\begin{gathered} \mathrm{If} \frac{{\sin }^2 \mathrm{\theta } - 5 \sin \mathrm{\theta } + 3}{{\cos }^2 \mathrm{\theta }} = 1, \mathrm{then} \mathrm{\theta } \mathrm{can} \mathrm{be} \mathrm{\_\_\_\_\_\_\_\_\_\_\_} . \\ 1. 30° \\ 2. 45° \\ 3. 60° \\ 4. 0° \end{gathered}$$

$\mathrm{If} \cot \mathrm{\theta } = \frac{24}{7} \mathrm{and} \mathrm{\theta } \mathrm{is} \mathrm{not} \mathrm{in} \mathrm{the} \mathrm{first} \mathrm{quadrant}, \mathrm{then} \mathrm{find} \mathrm{the} \mathrm{value} \mathrm{of} \tan \mathrm{\theta } - \sec \mathrm{\theta }.$
$$
$1. 1$
$2. \frac{4}{3}$
$3. \frac{3}{2}$
$4. \frac{5}{4}$
$$

$\mathrm{If} \sin 20° = \mathrm{p}, \mathrm{then} \mathrm{find} \mathrm{the} \mathrm{value} \mathrm{of} \left(\frac{\sin 380° - \sin 340°}{\cos 380° + \cos 340°}\right).$
$$
$1. \sqrt{1 - {\mathrm{p}}^2}$
$2. \sqrt{\frac{1 - {\mathrm{p}}^2}{\mathrm{p}}}$
$3. \frac{\mathrm{p}}{\sqrt{1 - {\mathrm{p}}^2}}$
$4. \mathrm{None} \mathrm{of} \mathrm{these}$

Find the value of tan $\left(22 \frac{1°}{2}\right)$
$$\begin{gathered} 1. \sqrt{2} - 1 \\ 2. 1 + \sqrt{2} \\ 3. 2 + \sqrt{3} \\ 4. 2 - \sqrt{3} \end{gathered}$$

If the sun ray's inclination increases from $45°$ to $60°$ , the length of shadows of a tower decreases by 50 m. Find the height of the tower (in m). 
$1. 50\left(\sqrt{3} - 1\right)$
$2. 75\left(3 - \sqrt{3} \right)$
$3. 100\left(\sqrt{3} + 1\right)$
$4. 25\left(3 + \sqrt{3} \right)$

The angles of depression of two points from the top of the tower are $30° \mathrm{and} 60°$. If the height of the tower is 30 m, then find the maximum possible distance between the two points.
1. $40\sqrt{3}$
2. $30\sqrt{3}$
3. $20\sqrt{3}$
4. $10\sqrt{3}$

FRom a point on the ground, the angle of elevation of an aeroplane flying at an altitude of 500 m changes from $45° \mathrm{to} 30°$ in 5 seconds. Find the speed of the aeroplane (in kmph).
1. $720(\sqrt{3}-1)$
2. $720(\sqrt{3}+1)$
3. $360(\sqrt{3}-1)$
4. $360(\sqrt{3}+1)$

From the top of a building, the angles of elevation and depression of the top $60°$ and $30°$ bottom of a tower are and repectively. If the height of the building is 5 m, then find the height of the tower.
$1. 10\sqrt{3}\mathrm{m}$
$2. 15\mathrm{m}$
$3. 15\sqrt{3}\mathrm{m}$
$4. 20\mathrm{m}$

$\mathrm{In} \mathrm{the} \mathrm{figure} \mathrm{given} \mathrm{below} \left(\mathrm{not} \mathrm{t} \mathrm{scale}\right), \mathrm{ABCD}, \mathrm{BEFC} \mathrm{and} \mathrm{EGHF} \mathrm{are} \mathrm{three} \mathrm{squares}. \mathrm{Find} \angle \mathrm{FAE} + \angle \mathrm{HAG}.$
         
$1. 30°$
$2. 45°$
$3. 60°$
$4. 90°$