Period of a Function

Q. The period of the function f(x) = sin 3x is

1. 
2. 
3. 
4. 

Q.

If $f\left(x\right)$ is an odd periodic function with period $2$, then $f\left(4\right)$ is equal to.

1. $-4$

2. $4$

3. $2$

4. $0$

Q. If $[.]$ denotes the greatest integer function, then fundamental period of the function $f\left(x\right)=1-2(x-\left[x\right])$ is

1. $\frac{1}{2}$
2. $-\frac{1}{2}$
3. $1$
4. $f\left(x\right)$ is non-periodic

Q. The fundamental period of the function $f\left(x\right)=sin\left(\frac{\pi }{4}\right[x\left]\right)+tan\left(\frac{\pi }{2}\right[x\left]\right)+\left[x\right]+[x+\frac{1}{3}]$ $+[x+\frac{2}{3}]-3x$ is
$$\left[\begin{array}{c}where[.]\to \text{the greatest integer function}\end{array}\right]$$

1. 8
2. 2
3. $\frac{1}{3}$
4. Not periodic

Q. If f(x) is a function that is odd and even simultaneously, then f(3)- f(2) is equal to

1. 2
2. 1
3. -1
4. 0

Q.

Period of a function is always a positive integer.

1. True

2. False

Q. If f (x) +f (x+a) + f(x+2a) +....+ f (x+na) = constant; $\forall x$ $ϵR$ and $a>0$ and f(x) is periodic, then period of f(x), is

1. (n+1) a
2. $e^{n+1}a$
3. na
4. $e^{na}$

Q.

Period of a function is always a positive integer.

1. True

2. False

Q. Given the graph of $y=5\cos (2x+1);\forall x\in R.$ Select the correct options.

1. Range:$[-5,5]$
2. Period: $\pi$
3. Range: $[-1,1]$
4. Period: $4\pi$

Q.

Given f(x) defined on f: R $\to$ R is a periodic function with the fundamental period 2 then f(3) is not equal to -

1. f(2)

2. f(9)

3. f(5)

4. f(-1)

Q. If cos $(A-B)=sin(A+B)=\frac{1}{2}$, where A and B are positive, then smallest positive value of $A+B$ (in degrees) is

1. ${45}^0$
2. ${105}^0$
3. ${150}^0$
4. ${60}^0$

Q.

Period of a function is always a positive integer.

1. True

2. False

Q. If f(x)is an odd periodic function with period 2, then f(4) equals

1. 2
2. 4
3. -4
4. 0

Q. The fundamental period of the function $f\left(x\right)=2cos\frac{1}{3}(x-\pi )$ is

1. $6\pi$
2. $4\pi$
3. $2\pi$
4. $\pi$

Q.

If ∝, β are the roots of the equation $x^2$ - ax + b = 0, then ${\alpha }^4$+${\alpha }^3$β+${\alpha }^2$${\beta }^2$+α${\beta }^3$+${\beta }^4$

1. a⁴ + 3a²b + b²

2. a⁴ - 3a²b + b²

3. a³ + 3a²b + b²

4. None of these

Q. The period of the function $|sinx|+|cosx|$ is

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

Q. If f (x) +f (x+a) + f(x+2a) +....+ f (x+na) = constant; $\forall x$ $ϵR$ and $a>0$ and f(x) is periodic, then period of f(x), is

1. (n+1) a
2. $e^{n+1}a$
3. na
4. $e^{na}$

Q. The fundamental period of the function $f\left(x\right)=sin\left(\frac{\pi }{4}\right[x\left]\right)+tan\left(\frac{\pi }{2}\right[x\left]\right)+\left[x\right]+[x+\frac{1}{3}]$ $+[x+\frac{2}{3}]-3x$ is
$$\left[\begin{array}{c}where[.]\to \text{the greatest integer function}\end{array}\right]$$

1. 8
2. 2
3. $\frac{1}{3}$
4. Not periodic

Q.

Given f(x) defined on f: R $\to$ R is a periodic function with the fundamental period 2 then f(3) is not equal to -

1. f(2)

2. f(9)

3. f(5)

4. f(-1)

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

1. The fundamental period of the function $f\left(x\right)=\text{cosec}\left(\frac{x}{2}\right)+\sec \left(\frac{x}{3}\right)+\cot \left(\frac{x}{4}\right)+\tan \left(\frac{x}{5}\right)+\cos \left(\frac{x}{6}\right)+\sin \left(\frac{x}{7}\right)$ is $420\pi$.
2. If $f$ is a function such that $f\left(s\right)=f\left(t\right),$ then $s=t$.
3. A vertical line intersects the graph of a function at most once.
4. If $h\left(x\right)=\{\begin{array}{cc}{x}^2,& x\le 0\\ x,& x>0\end{array},$ then $h\left(h\right(x\left)\right)=h\left(x\right)\forall x\in R$.

Q. The fundamental period of the function $f\left(x\right)=sin\left(\frac{\pi }{4}\right[x\left]\right)+tan\left(\frac{\pi }{2}\right[x\left]\right)+\left[x\right]+[x+\frac{1}{3}]$ $+[x+\frac{2}{3}]-3x$ is
$$\left[\begin{array}{c}where[.]\to \text{the greatest integer function}\end{array}\right]$$

1. 8
2. 2
3. $\frac{1}{3}$
4. Not periodic

Q. Find the greatest & the least values of the following functions in the given interval, if they exist. $y=\sin 2x-x[-\frac{\pi }{2},\frac{\pi }{2}]$

1. $\frac{\pi }{2}$ and $-\frac{\pi }{2}$
2. None of the above
3. $\pi$ and $-\pi$
4. $\frac{3\pi }{2}$ and $-\frac{3\pi }{2}$

Q. The period of the function f(x) = sin 3x is

1. $2\pi$
2. $\frac{2\pi }{3}$
3. $6\pi$
4. $8\pi$

Q.

How do you evaluate $\sin \left(\frac{7\pi }{2}\right)$?

Q. Let $f\left(x\right)=\sin x+e^x+x,$ then which of the following(s) is(are) true about $f\left(x\right)$

1. increasing for all $x<0$
2. decreasing for all $x\in R-\left\{0\right\}$
3. increasing at $x=0$
4. neither increasing nor decreasing at $x=0$

Q. Which of the following is least ? (All angles have been measured in radians)

1. sin3
2. sin1
3. sin7
4. sin2