Properties of Periodic Function

Q.

Let A ={1, 2, 3}, B={4, 5, 6, 7} and let f={(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.

Q. Let $f$ be a function satisfying $f\left(x\right)+f(x+6)=f(x+3)+f(x+9).$ If fundamental period of $f\left(x\right)$ is $T,$ then $T$ equals

Q. Fundamental period of the function $f\left(x\right)=\frac{(1+\sin x)(1+\sec x)}{(1+\cos x)(1+\text{cosec}x)},x\in R-\left\{\right(2n+1)\pi ,\frac{(4m-1)\pi }{2},n,m\in Z\}$ is

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

Q.

The period of the function $f\left(x\right)=\sin \left\{\sin \left(\frac{x}{5}\right)\right\}$ is:

1. $2\mathrm{\pi }$

2. $\frac{2\mathrm{\pi }}{5}$

3. $10\mathrm{\pi }$

4. $5\mathrm{\pi }$

Q. Let $f,g:N\to N$ such that $f(n+1)=f\left(n\right)+f\left(1\right),\forall n\in N$ and $g$ be any arbitrary function. Which of the following statements is $\text{NOT}$ true $?$

1. $f$ is one -one
2. If $fog$ is one-one, then $g$ is one-one
3. If $g$ is onto, then $fog$ is one-one
4. If $f$ is onto, then $f\left(n\right)=n\forall n\in N$

Q. The graph of the function y = f(x) is symmetrical about the line x = 2, then

1. f ( x+2) = f (x - 2)
2. f (2 + x) =f (2 - x)
3. f(x) = f(-x)
4. f(x) = - f(-x)

Q.

What is the period of f(x)= 2tanx +3 ?

1. π

2. π/2

3. 2π

4. π+3

Q. Which of the following is TRUE ?

1. The fundamental period of $|\sin x|+|\cos x|$ is $\pi$
2. The fundamental period of $|\sin x\cdot \cos x|$ is $\pi$.
3. The fundamental period of $|\sin x|-|\cos x|$ is $\pi$
4. The fundamental period of $|\sin x|$ is $2\pi$

Q. What is the graph of equation x square is equal to K Y + p,

Q. If the function $f\left(x\right)=\lambda \left|\sin x\right|+{\lambda }^2\left|\cos x\right|+g\left(\lambda \right),\lambda \in R$ is periodic with fundamental period $\frac{\pi }{2},$ then

1. $\lambda =0,1$
2. $\lambda =1$
3. $\lambda =0$
4. $\lambda =-1$

Q.

If a function defined on f : R $\to$ R with 6 as the period of the f(x). If f(0) = 13 then find f(96) ?

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Q. Fundamental period of $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$ is

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

Q. If $f\left(g\right(x\left)\right)=|\sin x|\text{and}g\left(f\right(x\left)\right)={\sin }^2\sqrt{x},$ then

1. $f\left(x\right)$ is not periodic while $g\left(x\right)$ is a periodic function
2. $f\left(x\right)$ and $g\left(x\right)$ are both periodic functions.
3. $f\left(x\right)$ is periodic but $g\left(x\right)$ is not a periodic function
4. $f\left(x\right)$ and $g\left(x\right)$ are both non-periodic functions.

Q.

What is the period of f(x) = Sinx. Cos3x ?

1. 2π

2. 1

3. 2

4. π/2

Q. If the function $f\left(x\right)=\lambda |\sin x|+{\lambda }^2|\cos x|+g\left(\lambda \right),\lambda \in R,$ where $g$ is a function of $\lambda ,$ is periodic with fundamental period $\frac{\pi }{2},$ then

1. $\lambda =0,1$
2. $\lambda =1$
3. $\lambda =0$
4. $\lambda =-1$

Q. If the function $f\left(x\right)=\lambda |\sin x|+{\lambda }^2|\cos x|+g\left(\lambda \right),\lambda \in R,$ where $g$ is a function of $\lambda ,$ is periodic with fundamental period $\frac{\pi }{2},$ then

1. $\lambda =0,1$
2. $\lambda =1$
3. $\lambda =0$
4. $\lambda =-1$

Q. If $f\left(g\right(x\left)\right)=|\sin x|\text{and}g\left(f\right(x\left)\right)={\sin }^2\sqrt{x},$ then

1. $f\left(x\right)$ and $g\left(x\right)$ are both periodic functions.
2. $f\left(x\right)$ is periodic but $g\left(x\right)$ is not a periodic function
3. $f\left(x\right)$ is not periodic while $g\left(x\right)$ is a periodic function
4. $f\left(x\right)$ and $g\left(x\right)$ are both non-periodic functions.

Q. If the function $f\left(x\right)=\lambda |\sin x|+{\lambda }^2|\cos x|+g\left(\lambda \right),\lambda \in R,$ where $g$ is a function of $\lambda ,$ is periodic with fundamental period $\frac{\pi }{2},$ then

1. $\lambda =0,1$
2. $\lambda =1$
3. $\lambda =0$
4. $\lambda =-1$

Q.

What is the period of f(x)= 2tanx +3 ?

1. $2\pi$

2. $\pi$

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

4. $\pi +3$

Q.

What is the period of f(x)= 2tanx +3 ?

1. 2π

2. π

3. π/2

4. π+3

Q.

If a function defined on f : R $\to$ R with 6 as the period of the f(x). If f(0) = 13 then find f(96) ?

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Q. The graph of the function y = f(x) is symmetrical about the line x = 2, then

1. f ( x+2) = f (x - 2)
2. f (2 + x) =f (2 - x)
3. f(x) = f(-x)
4. f(x) = - f(-x)

Q. The graph of the function y = f(x) is symmetrical about the line x = 2, then

1. f ( x+2) = f (x - 2)
2. f (2 + x) =f (2 - x)
3. f(x) = f(-x)
4. f(x) = - f(-x)

Q. The value of $\Delta =\left|\begin{array}{ccc}0& \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0& \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$ is

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

Q. which of the following relations are odd?
I. y=2
II. y=x
III. x^2 + y^2 = 1
A) only II
B) only I and II
c) only I and III
D) only II and III
E) I, II and III

Q.

What is the period of f(x) = Sinx. Cos3x ?

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

2. $1$

3. $2\pi$

4. $2$

Q.

If a function defined on f : R $\to$ R with 6 as the period of the f(x). If f(0) = 13 then find f(96) ?

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Q.

If a function defined on f : R $\to$ R with 6 as the period of the f(x). If f(0) = 13 then find f(96) ?

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Q.

What is the period of f(x)= 2tanx +3 ?

1. $2\pi$

2. $\pi$

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

4. $\pi +3$

Q. x^2+x+1 is one-one function, why?