Domain and Range of Trigonometric Ratios

Q.

If $\sin \left(\theta \right)=3\sin \left(\theta +2\alpha \right)$, then the value of $\tan \left(\theta +\alpha \right)+2\tan \left(\alpha \right)$ is

1. $3$

2. $2$

3. $-1$

4. $0$

5. $1$

Q.

If $\sin \alpha$and $\cos \alpha$ are the roots of the equation $p{x}^2+qx+r=0$ then

1. $p^2+q^2-2pr=0$

2. $p^2-q^2+2pr=0$

3. $p^2-q^2-2pr=0$

4. $p^2+q^2+2pr=0$

Q. If $\theta$ is an acute angle and $sin\theta =\frac{p-6}{8-p},$ then p must satisfy

1. < 7
2. < 8
3. < 7
4. 

Q.

If $\cos (\theta -\alpha )=a,\sin (\theta -\beta )=b$, then $a^2-2ab\sin (\alpha -\beta )+b^2$ is equal to

1. ${\sin }^2(\alpha –\beta )$

2. ${\cos }^2(\alpha –\beta )$

3. $1$

4. $0$

Q.

The number of points of intersection of the two curves y=2sin x and $y=5x^2+2x+3$ is

1. 0

2. 1

3. 

4. 2

Q.

sin105° + cos105° is equal to:

1. 

2. 

3. 

4. 1

Q.

Convert the given complex number in polar form: -3

Q. The maximum value of $cos{\alpha }_1.cos{\alpha }_2......cos{\alpha }_n,$
under the restrictions $0\le {\alpha }_1{\alpha }_2,.....,{\alpha }_n\le \frac{\pi }{2}andcot{\alpha }_1.cot{\alpha }_2......cot{\alpha }_n=1$ is

1. 
2. 
3. 
4. 

Q. If $cose{c}^2\theta =\frac{4xy}{(x+y{)}^2}$, then

1. 
2. 
3. 
4. 

Q.

If $e^{sinx}-e^{-sinx}=a$ has alteast one real solution, then

1. $|a|\in (\frac{e^2-1}{e},\infty )$
   

2. $|a|\in (e,\infty )$
   

3. $|a|\in [0,e]$

4. $|a|\in [0,\frac{e^2-1}{e}]$

Q.

If 2 cos x +sin x =1 then 7 cos x + 6 sin x=___

1. 6

2. 2

3. 1

4. 3

Q.

$Iff\left(x\right)=co{s}^{2}x+se{c}^{2}x,then$

1. f(x) = 1

2. 1 < f(x) < 2

3. f(x) <1

4. f(x) 2

Q. If 3 – 2 cos x – 4 sin x – cos 2x + sin 2x = 0, then x =( n $ϵ$ Z)

1. 
2. 
3. 
4. 

Q.

If $y=\sqrt{\frac{1-cos2x}{1+cos2x}},find\frac{dy}{dx}$.

Q.

If $e^{sinx}-e^{-sinx}=a$ has alteast one real solution, then

1. $|a|\in (\frac{e^2-1}{e},\infty )$

2. $|a|\in [0,\frac{e^2-1}{e}]$

3. $|a|\in (e,\infty )$

4. $|a|\in [0,e]$

Q.

In any $△ABC,$ Prove that $\frac{b^2-c^2}{a^2}$ $\text{sin}$ 2 A + $\frac{c^2-a^2}{b^2}$ sin 2 B + $\frac{a^2-b^2}{c^2}$ sin 2 C = 0

Or

Two ships leave a port at the same time. One goes 24 km/h in the direction N 45${}^{\circ }$ E and the other travels 32 km/h in the direction S 75${}^{\circ }$ E. Find the distance between the ships at the end of 3 h.

Q.

For 0< $\varphi$ < $\frac{\pi }{2},ifx={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi ,y={\sum }_{n=0}^{\infty }si{n}^{2n}\varphi ,z={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi ,then$

1. xyz = xz + y

2. xyz = xy + z

3. xyz = x + y + z

4. xyz = yz + x

Q. The quadratic equation $(\cos p-1)x^2+\left(\cos p\right)x+\sin p=0$
(where $x\in R$) has real roots if $p$ lies in

1. $(0,2\pi )$
2. $(-\pi ,0)$
3. $(-\frac{\pi }{2},\frac{\pi }{2})$
4. $(0,\pi ]$

Q. The number of points of intersection of two curves $y=2\sin x$ and $y=5x^2+2x+3$ is

1. $0$
2. $1$
3. $2$
4. infinitely many

Q.

If $e^{sinx}-e^{-sinx}=a$ has alteast one real solution, then

1. $|a|\in (\frac{e^2-1}{e},\infty )$
2. $|a|\in [0,\frac{e^2-1}{e}]$
3. $|a|\in (e,\infty )$
4. $|a|\in [0,e]$

Q. Reduce the following equations into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive $x$-axis.
(i)$x-\sqrt{3}y+8=0$
(ii)$y-2=0$
(iii)$x-y=4$

Q. If the equation ${\sin }^{4}x+{\cos }^{4}x+\sin 2x+k=0$ have real solutions, then range of $k$ is

1. $-\frac{1}{2}\le k\le \frac{1}{2}$
2. $-\frac{3}{2}\le k\le \frac{1}{2}$
3. $-\frac{1}{2}\le k\le \frac{3}{2}$
4. $-\frac{3}{2}\le k\le \frac{3}{2}$

Q.

If $|cos\theta +\{sin\theta +\sqrt{si{n}^2\theta +si{n}^2\alpha }\}|\le k$, then the value of k is

1. $\sqrt{1+co{s}^2\alpha }$

2. $\sqrt{1+si{n}^2\alpha }$

3. $\sqrt{2+si{n}^2\alpha }$

4. $\sqrt{2+co{s}^2\alpha }$

Q. The number of ordered solution pairs $(x,y)$ in $[0,2\pi ]$ for the system of the equations $x+y=\frac{2\pi }{3}$ and $\cos x+\cos y=\frac{3}{2}$ is

Q.

For 0< $\varphi$ < $\frac{\pi }{2},ifx={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi ,y={\sum }_{n=0}^{\infty }si{n}^{2n}\varphi ,z={\sum }_{n=0}^{\infty }co{s}^{2n}\varphi ,then$

1. xyz = xz + y

2. xyz = xy + z

3. xyz = x + y + z

4. xyz = yz + x

Q.

The value of $sin\theta +cos\theta$ will be greatest when

1. $\theta ={30}^{\circ }$

2. $\theta ={45}^{\circ }$

3. $\theta ={60}^{\circ }$

4. $\theta ={90}^{\circ }$