Period of Trigonometric Ratios

Q. The number of complex numbers $p$ such that $|p|=1$ and imaginary part of $p^4$ is $0$ , is

1. infinitely many
2. $4$
3. $2$
4. $8$

Q. The fundamental period of $f\left(x\right)=\sin 2x$ is

1. $\frac{\pi }{2}$
2. $\pi$
3. $2\pi$
4. $4\pi$

Q. The range of $f\left(x\right)=\sec \left(\frac{\pi }{4}{\cos }^{2}x\right)$ is

1. $[1,\sqrt{2}]$
2. $(1,\infty )$
3. $[-\sqrt{2},-1]\cup [1,\sqrt{2}]$
4. $(-\infty ,-1)\cup (1,\infty )$

Q. Which of the following options is (are) true regarding the function $f\left(x\right)=\left|\sin (x-\frac{\pi }{3})\right|$?

1. Period of $f\left(x\right)$ is $\frac{2\pi }{3}$
2. Period of $f\left(x\right)$ is $\pi$
3. Range of $f\left(x\right)$ is $[0,1]$
4. $f\left(x\right)=0$ has only one solution in $[0,\pi ]$

Q. The period of $\left|\sin \frac{x}{2}\right|+\left|\cos (\frac{x}{4}-\frac{\pi }{6})\right|$ is

1. $2\pi$
2. $\pi$
3. $4\pi$
4. $\frac{\pi }{2}$

Q. The value of
${\cos }^2\left(\frac{\pi }{16}\right)+{\cos }^2\left(\frac{3\pi }{16}\right)+{\cos }^2\left(\frac{5\pi }{16}\right)+{\cos }^2\left(\frac{7\pi }{16}\right)$ is

Q. The least positive value of $x$ satisfying $\tan x=x+1$ lies in the interval

1. $(\frac{\pi }{2},\pi )$
2. $(\frac{\pi }{4},\frac{\pi }{2})$
3. $(\pi ,\frac{3\pi }{2})$
4. $(\frac{3\pi }{2},\frac{5\pi }{2})$

Q. Period of $f\left(x\right)=\cos (7x-5)$ is

1. $\frac{2\pi -5}{7}$
2. $2\pi -5$
3. $\frac{2\pi }{7}$
4. $\frac{\pi }{7}$

Q. If $f\left(x\right)=\sin \frac{x}{3}+\cos \frac{3x}{10}$ and $f(n\pi +x)=f\left(x\right),$ then the least value of $n$ is

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{x(3e^{1/x}+4)}{2-e^{1/x}},& x\ne 0\\ 0,& x=0\end{array},$ then $f\left(x\right)$ is

1. Continuous as well as differentiable at $x=0$
2. Continuous but not differentiable at $x=0$
3. Neither Continuous nor differentiable at $x=0$
4. $R.H.D.$ at $(x=0)$ equals to $2$

Q. The domain of $f\left(x\right)=\sqrt{{\log }_x\left(\cos 2\pi x\right)},$ is

1. $(0,\frac{1}{4})\cup (\frac{3}{4},1)\cup N-\left\{1\right\}$
2. $(0,\frac{1}{4})\cup (\frac{3}{4},1)$
3. $(0,\frac{1}{4})\cup (\frac{3}{4},1)\cup N$
4. $(0,1)\cup N-\left\{1\right\}$

Q. If ${\cos }^{-1}\left(\cos \frac{x}{5}\right)=\frac{x-10\pi }{5}$ holds good for some $x\in R,$ then the number of integral values of $x$ satisfying it is

Q. The period of $3\sin (2x+\frac{\pi }{3})-4\cos (\pi x-\frac{\pi }{6})$ is

1. Not defined
2. $\pi$
3. $2+\pi$
4. $1$

Q. The period of the function $f\left(x\right)=\sin |100x|$ is

1. $\frac{\pi }{50}$
2. $\frac{\pi }{100}$
3. $\frac{\pi }{25}$
4. Not defined

Q. Fundamental period of the function $f\left(x\right)=\sin \sqrt{x}$ is

1. $\pi$
2. $2\pi$
3. $\sqrt{\pi }$
4. None of these

Q. Period of $5+3\cos (\pi x+\frac{\pi }{4})$ is

1. $2$
2. $5$
3. $6$
4. none of the above

Q. For the function $f\left(x\right)=\left[\frac{1}{\left[x\right]}\right]$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, which of the following statements are true?

1. The domain is $(-\infty ,\infty )$
2. The range is $\left\{0\right\}\cup \{-1\}\cup \left\{1\right\}$
   
3. The domain is $(-\infty ,0)\cup [1,\infty )$
   
4. The range is $\left\{0\right\}\cup \left\{1\right\}$

Q. The fundamental period of the function $f\left(x\right)=\left|\sin 2x\right|+\left|\cos 2x\right|$ is

1. $\frac{\pi }{2}$
2. $\pi$
3. $2\pi$
4. $\frac{\pi }{4}$

Q. The fundamental period of $f\left(x\right)=\sin 2x$ is

1. $\frac{\pi }{2}$
2. $\pi$
3. $2\pi$
4. $4\pi$

Q. Which of the following option(s) is (are) CORRECT ?

1. Fundamental period of the function $f\left(x\right)={\sin }^{2}2x+{\cos }^{4}2x+2$ is $\frac{\pi }{4}$
2. Number of integral values of $a$ for which $f\left(x\right)=\log \left({\log }_{1/3}\right({\log }_7(\sin x+a)\left)\right)$ is defined for all $x\in R$ is $3$
3. Number of integral values of $a$ for which $f\left(x\right)=\log \left({\log }_{1/3}\right({\log }_7(\sin x+a)\left)\right)$ is defined for all $x\in R$ is $7$
4. Fundamental period of the function $f\left(x\right)={\sin }^{2}2x+{\cos }^{4}2x+2$ is $\frac{\pi }{2}$

Q.

The value of $sin\left[2co{s}^{-1}\frac{\sqrt{5}}{3}\right]$ is

1. 
   

2. 
   

3. 
   

4. 
   

Q. The period of the function $f\left(x\right)=x\cos x$ is

1. $\frac{\pi }{2}$
2. $\pi$
3. $2\pi$
4. Not periodic.

Q. For the function given by the graph, which among the following option(s) is/are true

1. Range of the function is $[-3,1]$
2. Period of the function is $2\pi$
3. Period of the function is $4\pi$
4. Range of the function is $[-1,1]$

Q. Period of $5+3\cos (\pi x+\frac{\pi }{4})$ is

1. $5$
2. $6$
3. none of the above
4. $2$

Q. Correct graph of the function $y=3\cos (\frac{2x}{3}-\pi )-7$

1. 
2. 
3. 
4. 

Q. The period of the curve $y=\sin 4x$ is equal to

1. $e^{\ln \sqrt{\frac{{\pi }^2}{4}}}$
2. $e^{\ln \left|\frac{\pi }{4}\right|}$
3. $e^{\frac{1}{2}\ln \left(\frac{{\pi }^2}{16}\right)}$
4. Period of $y=\cos 4x$

Q. The period of the curve $y=\sin 4x$ is equal to

1. $e^{\ln \sqrt{\frac{{\pi }^2}{4}}}$
2. $e^{\ln \left|\frac{\pi }{4}\right|}$
3. $e^{\frac{1}{2}\ln \left(\frac{{\pi }^2}{16}\right)}$
4. Period of $y=\cos 4x$

Q. Which of the following options is (are) true regarding the function $f\left(x\right)=\left|\sin (x-\frac{\pi }{3})\right|$?

1. Period of $f\left(x\right)$ is $\frac{2\pi }{3}$
2. Period of $f\left(x\right)$ is $\pi$
3. Range of $f\left(x\right)$ is $[0,1]$
4. $f\left(x\right)=0$ has only one solution in $[0,\pi ]$

Q. The number of solutions of the equation $(|\sin x|-1)(5|\sin x|-1)=0$ in $[0,2\pi ]$ is

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

Q. The graph of $f\left(x\right)=a\cos (bx+c)$ with $b>0$ is shown below. Then choose the correct option(s) for the given function?

1. Period of $f\left(x\right)$ is $\frac{2\pi }{3}$
2. $a=-1$
3. $b=3$
4. $a=1$