Principal Solution of Trigonometric Equation

Q. For $0<\theta <\frac{\pi }{2},$ the solution (s) of
${\sum }_{m=1}^{6}cosec(\theta +\frac{(m-1)\pi }{4})cosec(\theta +\frac{m\pi }{4})=4\sqrt{2}$ is/are

1. $\frac{\pi }{4}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{12}$
4. $\frac{5\pi }{12}$

Q. If $0\le x<2\pi$ , then the number of real values of x, which satisfy the equation $\cos x+\cos 2x+\cos 3x+\cos 4x=0$, is

1. $3$
2. $5$
3. $7$
4. $9$

Q. If $x-y=\frac{1}{3}$ and ${\cos }^2\left(\pi x\right)-{\sin }^2\left(\pi y\right)=\frac{1}{2}$, then $(x,y)$ can be

1. $(\frac{2}{3},\frac{1}{3})$
2. $(\frac{13}{6},\frac{11}{6})$
3. $(\frac{7}{6},\frac{5}{6})$
4. $(\frac{19}{6},\frac{17}{6})$

Q. The general solution of the equation $\sin x+\cos x=\frac{3}{2}$ is

1. $n\pi +\frac{\pi }{6},n\in Z$
2. $2n\pi +\frac{\pi }{3},n\in Z$
3. $n\pi +\frac{\pi }{3},n\in Z$
4. no solution

Q. The number of solution(s) of the equation $\tan x\tan 4x=1$ for $0<x<\pi$ is

1. $1$
2. $2$
3. $4$
4. $5$

Q.

$$\begin{array}{cc}\text{Trigonometric Equation}& \text{Principal Solutions}\\ 1.\sin x=\frac{\sqrt{3}}{2}& A.\frac{5\pi }{6}\\ 2.\tan x=\frac{-1}{\sqrt{3}}& B.\frac{5\pi }{12}\\ 3.\sec 2x=\frac{-2}{\sqrt{3}}& C.\frac{7\pi }{12}\\ 4.\cos 3x=\left(\frac{-1}{2}\right)& D.\frac{\pi }{3}\\ & E.\frac{2\pi }{9}\\ & F.\frac{4\pi }{9}\\ & G.\frac{11\pi }{6}\\ & H.\frac{2\pi }{3}\end{array}$$

1. 1 - D, 2 - A, 3 - B, 4 - E

2. 1 - D, 2 - G, 3 - B, 4 - F

3. 1 - H, 2 - A, 3 - C, 4 - F

4. 1 - H, 2 - G, 3 - C, 4 - E

Q. Number of solution(s) of the equation ${\cos }^{2}x+\cos x={\sin }^{2}x$ where $x\in [0,\pi ]$ is

Q. The number of values of $x$ in $[0,2\pi ]$ which satisfy $\tan x+\tan 4x+\tan 7x=\tan x\tan 4x\tan 7x$ is

Q. The sum of roots of ${\sin }^{2}x-5\sin x\cos x+2=0$, where $x\in [0,2\pi ]$ is

1. $\frac{3\pi }{2}$
2. $\frac{5\pi }{2}$
3. $\frac{5\pi }{2}+{\tan }^{-1}\frac{2}{3}$
4. $\frac{5\pi }{2}+{\tan }^{-1}\left(\frac{12}{5}\right)$

Q. Number of solution(s) of the equation $\ln (2-{\sin }^{2}x)=0$ where $x\in [0,10\pi ]$ is

Q. If ${\log }_3{\left(\frac{2\sin 2x}{3}\right)}^2+1=0,x\in [0,2\pi ],$
then which among the following value(s) of $x$ satisfying the above equation

1. $x=\frac{\pi }{6}$
2. $x=\frac{4\pi }{3}$
3. $x=\frac{\pi }{3}$
4. $x=\frac{7\pi }{6}$

Q. One of the principle solution of the equation $\tan \theta =1$ is $\frac{9\pi }{4}$.

1. False
2. True

Q.

The set of positive real values of the parameter 'a' for which the equation |sin2x|-|x|-a=0 does not have any real solution is

1. $(\frac{3\sqrt{3}+\pi }{6},\infty )$
   

2. $(\frac{3\sqrt{3}-\pi }{6},\infty )$
   

3. $(\frac{3\sqrt{3}-\pi }{12},\infty )$
   

4. $(\frac{3\sqrt{3}+\pi }{12},\infty )$

Q. If $x\in [0,3\pi ]$ and $r\in R$, then the number of pairs of $(r,x)$ satisfying $2\sin x=r^4-2r^2+3$ is

1. $2$
2. $3$
3. $4$
4. $6$

Q. If $\alpha ,\beta$ are the solutions of $\cot x=-\sqrt{3}$ in $[0,2\pi ]$ and $\alpha ,\gamma$ are the solutions of $\text{cosec}x=-2$ in $[0,2\pi ],$ then

1. $\alpha -\beta =\pi$
2. $\alpha +\gamma =3\pi$
3. $\beta +\gamma =2\pi$
4. $\alpha -\gamma =\pi$

Q. If $\alpha ,\beta$ are the solutions of $\sin x=-\frac{1}{2}$ in $[0,2\pi ]$ and $\alpha ,\gamma$ are the solutions of $\cos x=-\frac{\sqrt{3}}{2}$ in $[0,2\pi ],$ then

1. $\alpha -\beta =\pi$
2. $\beta -\gamma =\pi$
3. $\alpha -\gamma =\pi$
4. $\alpha +\beta =3\pi$

Q. If $0\le x\le 3\pi ,0\le y\le 3\pi$ and $\cos x\sin y=1$, then the possible number of values of the ordered pair $(x,y)$ is

1. $4$
2. $5$
3. $6$
4. $7$

Q. Number of solution(s) of the equation ${\cos }^{2}x+\cos x={\sin }^{2}x$ where $x\in [0,\pi ]$ is