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The length of the minutes hand of a wall clock is 6 cm. Find the distance covered by the tip of the minutes hand in 25 minutes.
(1) $\frac{270}{7}\text{\hspace{0.17em}cm}$
(2) 110 cm
(3) $\frac{88}{7}\text{cm}$
(4) $\frac{110}{7}\text{cm}$

The value of tan 10° – tan 20° × tan 45° × tan 70° × tan 80° =
(1) 1
(2) $\frac{1}{\sqrt{3}}$
(3) 0
(4) 8

The value of sin θ in terms of tan θ is _______.
(1) $\frac{\tan \theta }{\sqrt{1-{\tan }^2\theta }}$
(2) $\frac{{\tan }^2\theta }{\sqrt{1+{\tan }^2\theta }}$
(3) $\frac{{\tan }^2\theta }{\sqrt{1-{\tan }^2\theta }}$
(4) $\frac{\tan \theta }{\sqrt{1+{\tan }^2\theta }}$

The value  of sin²60° + cos²30° – sin²45° is _________.
(1) 1
(2) sin 90°
(3) $\frac{1}{2}$
(4) Both (1) and (2)

The simplified value of ${\text{cosec}}^2\alpha (1+\frac{1}{\sec \alpha })(1-\frac{1}{\sec \alpha })$ is _______.
(1) 1
(2) 0
(3) 2
(4) –1

A wheel makes 240 revolutions in one minute. Find the measure of the angle it describes at the centre in 15 seconds.
(1) 60 π^c
(2) 120 π^c
(3) 8 π^c
(4) None of these

If 3 tan A = 4, then find the value of $\frac{2\sin A-7\cos A}{3\cos A+4}$.
(1) $\frac{-13}{29}$
(2) $\frac{13}{11}$
(3) ∞
(4) $\frac{29}{13}$

If tan α = 3 and $\beta =\frac{1}{2}$, then which of the following is true?
(1) $\alpha +\beta =\frac{{\pi }^c}{4}$
(2) $\alpha \beta =\frac{{\pi }^c}{4}$
(3) $\alpha -\beta =\frac{{\pi }^c}{4}$
(4) None of these

If $\tan A=\frac{1}{2}$ and $\tan B=\frac{1}{3}$, then which of the following is true?
(1) $A+B=\frac{{\pi }^c}{4}$
(2) $A-B=\frac{{\pi }^c}{4}$
(3) $2A+B=\frac{{\pi }^c}{4}$
(4) $A+2=\frac{{\pi }^c}{4}$

If $\sin A=\frac{1}{2}$ and 90° < A < 180°, then the value of A in circular measure is_________.
(1) $\frac{\pi }{6}$
(2) $\frac{\pi }{3}$
(3) $\frac{5\pi }{6}$
(4) $\frac{7\pi }{6}$

The value of $\frac{3{\pi }^c}{5}$ in sexagesimal measure is _______.
(1) 216°
(2) 144°
(3) 128°
(4) 108°

The value of log_secθ (1 – sin²θ) is _____.
(1) 2
(2) –2
(3) 0
(4) 1

If $\sin A=\frac{\sqrt{3}}{2}$ and A is an acute angle, then find the value of $\frac{\tan A-\cot A}{\sqrt{3}+\text{cosec}A}$.
(1) $\frac{-2}{5}$
(2) $\frac{2}{5}$
(3) $\frac{2}{3+2\sqrt{3}}$
(4) –2

The length of an arc, which subtends an angle of 30° at the centre of the circle of radius 42 cm is_______.
(1) 22 cm
(2) 44 cm
(3) 11 cm
(4) $\frac{22}{7}$ cm

If ${\sin }^4\theta +{\cos }^4\theta =\frac{1}{2}$, then find $\sin \theta \cos \theta$.
(1) $\pm \frac{1}{8}$
(2) $\pm \frac{1}{4}$
(3) $\pm 1$
(4) $\pm \frac{1}{2}$

If $\sin \theta =\frac{a}{b}$, then cos θ and tan θ in terms of a and b are
(1) $\frac{\sqrt{b^2-a^2}}{b}\text{\hspace{0.17em}and\hspace{0.17em}}\frac{b}{\sqrt{b^2-a^2}}$
(2) $\frac{b}{\sqrt{b^2-a^2}}\text{\hspace{0.17em}and\hspace{0.17em}}\frac{a}{\sqrt{b^2-a^2}}$
(3) $\frac{\sqrt{a^2-b^2}}{a}\text{\hspace{0.17em}and\hspace{0.17em}}\frac{b}{\sqrt{a^2-b^2}}$
(4) $\frac{\sqrt{b^2-a^2}}{b} \mathrm{and}\frac{a}{\sqrt{b^2-a^2}}$

The distance covered by the tip of a minute hand in 35 minutes is 33 cm. What is the length of the minute hand?
(1) 6 cm
(2) 9 cm
(3) 10 cm
(4) 12 cm

If $\sin \alpha =\frac{4}{5}$, where $\left(0^{\circ }\le \alpha \le {90}^{\circ }\right)$, then find sin 2a.
(1) $\frac{12}{25}$
(2) $-\frac{24}{25}$
(3) $\frac{25}{24}$
(4) $\frac{24}{25}$

In a $\Delta ABC,\cos \left(\frac{A+B}{2}\right)$ = _______.
(1) $\cos \frac{C}{2}$
(2) $-\sin \frac{C}{2}$
(3) $\cos \left(\frac{A-B}{2}\right)$
(4) $\sin \frac{C}{2}$

${\sin }^4\theta -{\cos }^4\theta =$
(1) –1
(2) cos2θ
(3) 2sin²θ – 1
(4) sin 2θ

If A = sin θ + cos θ and B = sinθ – cosθ, then which of the following is true?
(1) $A^2+B^2=1$
(2) $A^2-B^2=2$
(3) $A^2+B^2=2$
(4) $2A^2+B^2=4$

If cot (A – B) = 1 and $\cos (A+B)=\frac{1}{2}$, then find B.
(1) $42\frac{1}{2}°$
(2) $7\frac{1}{2}°$
(3) $15\frac{1}{2}°$
(4) 60°

Find the measure of angle A, if (2sin A + 1) (2sin A – 1) = 0.
(1) 75°
(2) 60°
(3) 45°
(4) 30°

The value of cos2θ in terms of cotθ is______.
(1) $\frac{{\cot }^2\theta +1}{{\cot }^2\theta -1}$
(2) $-\frac{1+{\cot }^2\theta }{{\cot }^2\theta -1}$
(3) $\frac{{\cot }^2\theta -1}{{\cot }^2\theta +1}$
(4) $\frac{1-{\cot }^2\theta }{1+{\cot }^2\theta }$

The simplified value of ${\sin }^4\alpha +{\cos }^4\alpha +\frac{1}{2}{\sin }^{2}2\alpha$ is
(1) –1
(2) $\sin \alpha +\cos \alpha$
(3) 0
(4) 1

$\sqrt{\frac{(1+\sin 2\theta )}{1-{\cos }^2\theta }}[\text{where\hspace{0.17em}}\theta \in [0,\frac{\pi }{4}]]$ =
(1) ${\text{cosec}}^2\theta$
(2) 1
(3) $1+\cot \theta$
(4) $1+\tan \theta$

If A and B are complementary angles, then the value of $\frac{{\sin }^{2}A+{\sin }^{2}B}{{\text{cosec}}^{2}A-{\tan }^{2}B}$ is______.
(1) 0
(2) 1
(3) –1
(4) 2

Find the value of $4({\sin }^4{30}^{\circ }+{\cos }^4{30}^{\circ })-3({\cos }^2{45}^{\circ }+{\sin }^2{90}^{\circ })$.
(1) $-\frac{1}{2}$
(2) –2
(3) 2
(4) $\frac{1}{2}$

If $\sec \theta +\tan \theta =\frac{4}{3}$, then $\sec \theta \tan \theta$ =
(1) $\frac{175}{24}$
(2) $\frac{25}{576}$
(3) $\frac{27}{576}$
(4) $\frac{175}{576}$

If $\sin \alpha +\cos a=n$, then ${\sin }^6\alpha +{\cos }^6\alpha$ in terms of n is_______.
(1) 4 + 3(n – 1)²
(2) $\frac{4+3(n^2-1)}{4}$
(3) $\frac{4-3{(n^2-1)}^2}{4}$
(4) None of these

Find the value of $\frac{\sin {25}^{\circ }}{\cos {35}^{\circ }}-\frac{\cos {25}^{\circ }}{\sin {35}^{\circ }}$.
(1) cosec 70°
(2) sin 70°
(3) –sin 70°
(4) –cosec 70°

The value of $\frac{3\tan {30}^{\circ }-{\tan }^3{30}^{\circ }}{1-3{\tan }^2{30}^{\circ }}$ is _______.
(1) tan 90°
(2) tan 60°
(3) tan 45°
(4) tan 30°

If ${\cot }^{4}x-{\cot }^{2}x=1$, then the value of ${\cos }^{4}x+{\cos }^{2}x$ is_____.
(1) –1
(2) 0
(3) 2
(4) 1

If $\frac{\cos (A-B)}{\cos (A+B)}=\frac{8}{3}$, then tanA ⋅ tanB is ______.
(1) $\frac{5}{11}$
(2) $\frac{7}{13}$
(3) $\frac{8}{5}$
(4) $\frac{11}{5}$

If A + B + C = 45°, then the value of $\sum (\tan A+\tan A\tan B)$ is______.
(1) $1-\pi \tan A$
(2) 1
(3) $1+\pi \tan A$
(4) 1 + ΣtanA

If $\cot \theta +\tan \theta =2$, then the value of ${\tan }^2\theta -{\cot }^2\theta$ is______.
(1) 1
(2) 0
(3) –1
(1) 2

The value of $\cot 5^{\circ }\cdot \cot {15}^{\circ }\cdot \cot {25}^{\circ }\cdot \cot {35}^{\circ }\cdot \cot {45}^{\circ }\cdot \cot {55}^{\circ }\cdot \cot {65}^{\circ }\cdot \cot {75}^{\circ }\cdot \cot {85}^{\circ }$ is______.
(1) 0
(2) –1
(3) –2
(4) 1

The simplified form of $\sqrt{1+\sin \left(\frac{\pi }{8}\right)}$ is________.
(1) $\sin \left(\frac{x}{8}\right)+\cos \left(\frac{x}{8}\right)$
(2) $\sin \left(\frac{x}{16}\right)+\cos \left(\frac{x}{16}\right)$
(3) $\sin \left(\frac{x}{4}\right)+\cos \left(\frac{x}{4}\right)$
(4) None of these

${\sin }^4\theta +{\cos }^4\theta$ in terms of sin q is _____.
(1) $2{\sin }^4\theta -2{\sin }^2\theta -1$
(2) $2{\sin }^4\theta -2{\sin }^2\theta +1$
(3) $2{\sin }^4\theta +2{\sin }^2\theta -1$
(4) $2{\sin }^4\theta -2{\sin }^2\theta$

If tan (A – B) = 1 and $(A+B)=\frac{\sqrt{3}}{2}$, then find B.
(1) $42\frac{1°}{2}$
(2) $7\frac{1^{\circ }}{2}$
(3) $15\frac{1^{\circ }}{2}$
(4) 60°

If $\sin \beta +\cos \beta =\frac{5}{4}$, then find the value of sin β ⋅ cos β.
(1) $\frac{1}{4}$
(2) $\frac{9}{32}$
(3) $\frac{5}{16}$
(4) $\frac{11}{32}$

If ${\sin }^2\alpha +\sin \alpha =1$, then the value of ${\cos }^4\alpha +{\cos }^2\alpha$ is______.
(1) 0
(2) –1
(3) 1
(4) 2

If $\tan P+\cot P=2$, then the value of ${\tan }^{n}P+{\cot }^{n}P$ is______.
(1) 2
(2) 2ⁿ
(3) $2^{n-1}$
(4) $2^{n+1}$

The value of *$\sqrt{3}\tan {10}^{\circ }+\sqrt{3}\tan {20}^{\circ }+\tan {10}^{\circ }\tan {20}^{\circ }$ is_______.
(1) –1
(2) 0
(3) 1
(4) 2

If tan (A + B) = 1 and $(A-B)=\frac{1}{\sqrt{2}}$, then find A and B.
The following are the steps involved in solving the above problem. Arrange them in sequential order.
(A) $\tan (A+B)=1\Rightarrow \tan (A+B)=\tan {45}^{\circ }$ and $\sin (A-B)=\frac{1}{\sqrt{2}}\Rightarrow \sin (A-B)=\sin {45}^{\circ }$.
(B) $2A={90}^{\circ }\Rightarrow A={45}^{\circ }$.
(C) $A+B={45}^{\circ }$ and $A-B={45}^{\circ }$.
(D) $\therefore A={45}^{\circ }$ and $B=0^{\circ }$.
(1) DBCA
(2) CABD
(3) ACDB
(4) ACBD

If $\cos (A-B)=\frac{1}{2}$ and $\sin (A+B)=\frac{\sqrt{3}}{2}$, then find A and B.
The following are the steps involved in solving the following problem. Arrange them in sequential order.
(A) $2A={120}^{\circ }\Rightarrow A={60}^{\circ }$
(B) $\therefore A={60}^{\circ },\text{\hspace{0.17em}}B=0^{\circ }.$
(C) $\cos (A-B)=\frac{1}{2}\Rightarrow \cos (A-B)=\cos {60}^{\circ }$ and $\sin (A+B)=\frac{\sqrt{3}}{2}\Rightarrow \sin (A+B)=\sin {60}^{\circ }$.
(D) $A+B={60}^{\circ }$ and $A-B={60}^{\circ }$.
(1) DCAB
(2) CADB
(3) DCBA
(4) CDAB

If $\sin \alpha +\sin \beta +\sin \gamma =3$, then ${\sin }^3\alpha +{\sin }^3\beta +{\sin }^3\gamma$ = ______.
(1) 0
(2) 2
(3) 3
(4) 1

${\sec }^4\theta -{\sec }^2\theta$ =_______.
(1) ${\tan }^2\theta {\sec }^2\theta$
(2) $\frac{{\tan }^2\theta }{{\sec }^2\theta }$
(3) ${\text{cosec}}^2\theta {\cot }^2\theta$
(4) $\frac{{\cot }^2\theta }{{\text{cosec}}^2\theta }$

$\sin \theta +\cos \theta =\sqrt{2}$, then ${\sin }^{16}\theta$ =______.
(1) $\frac{{\cos }^{16}\theta }{2^{16}}$
(2) $\frac{{\sec }^{16}\theta }{2^8}$
(3) $\frac{1}{2{\sec }^{16}\theta }$
(4) $\frac{1}{2^{16}{\cos }^{16}\theta }$

If $\frac{\sin (x-y)}{\sin (x+y)}=\frac{3}{5}$, then $\tan x\cdot \cot y$ is_______.
(1) 1
(2) 2
(3) 3
(4) 4

$\sqrt{\sqrt{16{\sin }^4\theta +{\text{cosec}}^4\theta +}8-4}=$
(1) $2\sin \theta -\text{cosec}\theta$
(2) $2\sin \theta +\text{cosec}\theta$
(3) $2\text{cosec}\theta \text{+cos}\theta$
(4) $2\text{cosec}\theta +\sin \theta$

If sin θ and cos θ are the roots of the quadratic Ration $p{x}^2+qx+r=0(p\ne 0)$, then which of the following relation holds good?
(1) $q^2-p^2=2pr$
(2) $p^2-q^2=2pr$
(3) $p^2+q^2+2pr=0$
(4) ${(p-q)}^2=2pr$

If $\sin \alpha -\cos \alpha =m$, then the value of ${\sin }^6\alpha +{\cos }^6\alpha$ in terms of m is_______.
(1) $1+\frac{3}{4}{(1+m^2)}^2$
(2) $1-\frac{4}{3}{(m^2-1)}^2$
(3) $1-\frac{3}{4}{(1-m^2)}^2$
(4) $1-\frac{3}{4}{(1+m^2)}^2$

The value of $\frac{8{\sec }^4\theta -8{\tan }^4\theta }{4+8{\tan }^2\theta }-\frac{2{\cos }^6\theta +2{\sin }^6\theta }{1-3{\sin }^2\theta {\cos }^2\theta }$ is_______.
(1) 0
(2) 1
(3) –1
(4) 3

If $\frac{1+\tan \theta }{1-\tan \theta }=\sqrt{3}$, then find the value of θ.
(1) 30°
(2) 25°
(3) 15°
(4) 45°

If $A\times B=1$, $A+B=\text{cosec}\theta \cdot \sec \theta$ then $\frac{A}{B}$ can be_______.
(1) ${\tan }^2\theta$
(2) ${\sec }^2\theta$
(3) ${\sin }^2\theta {\cos }^2\theta$
(4) ${\text{cosec}}^2\theta {\sec }^2\theta$

If $7{\sin }^2\theta +3{\cos }^2\theta =4$, then find tan θ.
(1) $\frac{1}{\sqrt{3}}$
(2) $\frac{2}{\sqrt{3}}$
(3) $\sqrt{3}$
(4) 1

If $\sin (A+B)=\frac{\sqrt{3}+1}{2\sqrt{2}}$ and sec A = 2, then the value of B in circular measure is______.
(1) $\frac{\pi }{12}$
(2) $\frac{3\pi }{5}$
(3) $\frac{7\pi }{5}$
(4) $\frac{5\pi }{12}$

If $\tan \theta -\cot \theta =7$, then the value of ${\tan }^3\theta -{\cot }^3\theta$ is
(1) 250
(2) 354
(3) 343
(4) 364

If $x^n=a^m{\cos }^4\theta$ and $y^n=b^m{\sin }^4\theta$, then
(1) $\frac{x^{n/2}}{a^{m/2}}+\frac{y^{n/2}}{b^{m/2}}=1$
(2) $\frac{x^n}{a^m}+\frac{y^n}{b^m}=1$
(3) $\frac{x^{n/2}}{y^{n/2}}+\frac{a^{m/2}}{b^{m/2}}=1$
(4) None of these

If ${\sin }^2\theta +2{\cos }^2\theta +3{\sin }^2\theta +4{\cos }^2\theta +\mathrm{...}+40\text{\hspace{0.17em}terms\hspace{0.17em}}=405$ where θ is acute, then find the value of tan θ.
(1) $\frac{1}{\sqrt{3}}$
(2) $\sqrt{3}$
(3) 1
(4) ∞

If $\sin \alpha +\sin \beta =2$, then find the value of ${\cos }^2\alpha +{\cos }^2\beta$.
(1) 0
(2) 1
(3) 2
(4) 3

If $x=a^2\cos 3\theta$ and $y=b^2\sin 3\theta$, then
(1) $\frac{x^2}{a}+\frac{y^2}{b}=1$
(2) ${\left(\frac{x}{a^2}\right)}^{1/3}+{\left(\frac{Y}{b^2}\right)}^{1/3}=1$
(3) ${\left(\frac{x^2}{a^2}\right)}^{2/3}+{\left(\frac{Y^2}{b^2}\right)}^{2/3}=1$
(4) ${\left(\frac{x}{a^2}\right)}^{2/3}+{\left(\frac{Y^2}{b^2}\right)}^{2/3}=1$

If ${\cos }^2\theta +2{\sin }^2\theta +3{\cos }^2\theta +4{\sin }^2\theta +\mathrm{...}\left(200\text{\hspace{0.17em}terms}\right)=10025$, where θ is acute, then the value of sin θ – cos θ is
(1) $\frac{1-\sqrt{3}}{2}$
(2) $\frac{1+\sqrt{3}}{2}$
(3) $\frac{\sqrt{3}-1}{2}$
(4) 0