Constant Function

Q. Find the domain and range of the real function $f$. defined by
$f\left(x\right)=|x-1|$

Q. Which of the following statements is (are) CORRECT for a constant function?

1. It is a function whose range consists of only one element
2. It is a type of polynomial function having degree $0$
3. Its graph is a line which is always parallel to $x-$axis.
4. Domain of constant function is $R$

Q. The graph of the function $f\left(x\right)=\frac{x}{1+x^2}$ is given as:

Then which of the following is/are true regarding $f\left(x\right)?$

1. $f\left(x\right)$ is one-one function
2. $f\left(x\right)$ is many-one function
3. Range of $f\left(x\right)$ is $[\frac{-1}{2},\frac{1}{2}]$
4. $f\left(x\right)$ is one-one function in $x\in [1,\infty )$

Q. If $f\left(x\right)$ is a non-zero polynomial of degree 4 , having local extreme points at $x=-1,0,1$; then the set
$S=\{x\in R:f(x)=f(0\left)\right\}$ contains exactly

1. two irrational and two rational numbers.
2. two irrational and one rational number.
3. four rational numbers.
4. four irrational numbers.

Q. If $f(x+y+1)={(\sqrt{f\left(x\right)}+\sqrt{f\left(y\right)})}^2\forall x,y\in R$ and $f\left(0\right)=1,$ then the value of $f\left(\frac{1}{2}\right)+f\left(1\right)+f\left(2\right)$ is

1. $\frac{61}{4}$
2. $\frac{29}{2}$
3. $\frac{31}{4}$
4. $\frac{21}{2}$

Q. If $f\left(x\right)=x,-1\le x\le 1$, then function f(x) is
​

1. Increasing
2. Decreasing
3. Stationary
4. Discontinuous

Q. If $f\left(x\right)=3x-2$ and $(gof{)}^{-1}=x-2$, then $\int g\left(x\right)dx$ is (where $C$ is constant of integration)

1. $\frac{3x^2}{2}-2x+C$
2. $-\frac{x^2}{6}-\frac{8}{3}x+C$
3. $\frac{x^2}{6}+\frac{8}{3}x+C$
4. $\frac{x^2}{2}+x+C$

Q.

show that the function defined by f(x) = |cos x | is a continuous function.

Q. Let $\left\{y\right\}$ and $\left[y\right]$ denote fractional part function and greatest integer function respectively.
If $f\left(x\right)={\sin }^{-1}\{[3x+2]-\{3x+(x-\left\{2x\right\}\left)\right\}\}$ for $x\in (0,\frac{\pi }{12})$ and $(g\circ f)\left(x\right)=x$ for all $x\in (0,\frac{\pi }{12}),$ then $g^{\prime }\left(\frac{\pi }{6}\right)$ is equal to

1. $\frac{\sqrt{3}}{8}$
2. $\frac{-1}{4}$
3. $\frac{1}{8}$
4. $\frac{-\sqrt{3}}{4}$

Q. Let f:R $\to$ R be defined by f(x)=|x-2n| for x $ϵ$ [2n-1, 2n+1], n $ϵ$ Z. Then f is periodic with period ___

Q.

When the first-order derivative is positive then the function is

Q. Let $2f\left(x\right)=f\left(xy\right)+f\left(\frac{x}{y}\right),x,y\in R^{+}.$ If $f\left(1\right)=0,$ then

1. $f\left(x^n\right)=(n-1)f\left(x\right)$
2. $f\left(x^n\right)=nf\left(x\right)$
3. $f\left(x^n\right)=(n+1)f\left(x\right)$
4. $f\left(x^n\right)=(n+2)f\left(x\right)$

Q. If the equation $x^5-10a^3{x}^2+b^{4}x+c^5=0$ has 3 equals roots, then

1. $2b^2-10a^3{b}^2+c^5=0$
2. $6a^5+c^5=0$
3. $2c^5-10a^3{b}^2+b^4{c}^5=0$
4. $b^4=15a^4$

Q. Let $F\left(x\right)$ be the primitive of $\frac{3x+2}{\sqrt{x-9}}$ with respect to $x$. If $F\left(10\right)=60,$ then the value of $F\left(13\right)$ is

Q.

Which situations can be modeled with a periodic function?

1. Flag on windmill

2. Height of baseball after being hit

3. Ball suspended from string

Q. The graph of the function $y=f\left(x\right)$ is given below

Then which of the following is/are true

1. $f^{\prime }\left(x\right)<0$ if $0<x<2$
2. $f^{\prime }\left(x\right)>0$ if $0<x<2$
3. $f^{\prime \prime }\left(x\right)>0$ if $0<x<1$
4. $f^{\prime \prime }\left(x\right)<0$ if $1<x<2$

Q. Which of the following statements is (are) CORRECT for a constant function?

1. It is a function whose range consists of only one element
2. It is a type of polynomial function having degree $0$
3. Its graph is a line which is always parallel to $x-$axis.
4. Domain of constant function is $R$

Q. Let $f\left(x\right)=\frac{\text{sin}\left(\pi [x-\pi ]\right)}{1+{\left[x\right]}^2},$ where $[.]$ denotes the greatest integer function. Then $f\left(x\right)$ is

1. discontinuous at integral points
2. continuous everywhere but not differentiable
3. differentiable once but $f^{\prime \prime }\left(x\right),f^{\prime \prime \prime }\left(x\right)\cdots$ doesn't exists
4. differentiable for all$x$

Q. If $g\left(x\right)=x^2+x-1$ and $(g\circ f)\left(x\right)=4x^2-10x+5,$ then $f\left(\frac{5}{4}\right)$ is equal to

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

Q. The solution of $\frac{dv}{dt}+\frac{k}{m}v=-g,$ where $m,g$ and $k$ are constants and $c$ is the constant of integration, is

1. $|v+\frac{mg}{k}|=e^{-\frac{kt}{m}+c}$
2. $|v+\frac{mg}{k}|=e^k-\frac{t}{m}+c$
3. $|v+\frac{mg}{k}|=e^{\frac{kt}{m}+c}$
4. $|v+\frac{mg}{k}|=e^{\frac{kt}{m}}+c$

Q.

$\sum _{n=1}^{\infty }\frac{1}{(n+1)(n+2)(n+3)....(n+k)}$ is equal to

1. 

2. 

3. 

4. 

Q. Which of the following functions is/are constant ?

1. $f\left(x\right)=x^2+2$
2. $f\left(x\right)=x+\frac{1}{x}$
3. $f\left(x\right)=7$
4. $f\left(x\right)=6+x$

Q. If $f\left(x\right)=2\left[x\right]+\cos ([-\pi ]x)$, where $[.]$ represents greatest integer function, then

1. $f\left(\frac{\pi }{2}\right)=3$
2. $f\left(\pi \right)=0$
3. $f\left(\frac{\pi }{4}\right)=-1$
4. $f(-\pi )=-7$

Q. The equation $12x^3+8x^2-x-1=0$ has

1. $3$ real and distinct roots
2. $3$ real roots of which two are equal
3. $2$ positive and $1$ negative roots
4. $2$ negative and $1$ positive roots

Q. Find the value(s) of $x$ for which $y=\left[x\right(x-2{)}^2]$ is an increasing function ?

Q. Let $A=\left(\begin{array}{ccc}3& 0& 3\\ 0& 3& 0\\ 3& 0& 3\end{array}\right).$ Then the roots of the equation $det(A-\lambda I_3)=0$ (where $I_3$ is the identity matrix of order $3$) are

1. $3,0,3$
2. $0,3,6$
3. $1,0,–6$
4. $3,3,6$

Q. The graph of a constant function $f\left(x\right)=k$ is?

1. A straight line parallel to $X$-axis
2. A straight line parallel to $Y$-axis
3. A straight line passing through orgin
4. None

Q. Which of the following graphs represent a constant function?

1. Only $\left(1\right)$ and $\left(4\right)$
2. Only $\left(1\right),\left(2\right)$ and $\left(4\right)$
3. Only $\left(1\right)$ and $\left(2\right)$
4. Only $\left(1\right)$

Q. If $\{(7,11),(5,a)\}$ represents a constant function, then the value of '$a$' is :

1. $11$
2. $9$
3. $5$
4. $7$

Q. Let a, b, c be three real numbers such that a < b < c. Let f(x) be continuous $\forall x\in [a,c]$ and differentiable $\forall x\in (a,c)$.If $f"\left(x\right)>0\forall x\in (a,c)$ then

1. (c – b) f(a) + (b – a) f(c) > (c – a) f(b)
2. (c – b) f(a) + (a – c) f(b) < (a – b) f(c)
3. f(a) < f(b) < f(c)
4. none of these