Functions

If $f(x)+f(1-x)=2$, then the value of $f\left(\dfrac{1}{2001}\right)+f\left(\dfrac{2}{2001}\right)+\cdots+f\left(\dfrac{2000}{2001}\right)$ is

If $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ are continuous functions, then evaluate $\displaystyle \int_{-\pi/2}^{\pi/2}[f(x)+f(-x)][g(x)-g(-x)],dx$

The number of functions $f$ from $A={0,1,2}$ into $B={0,1,2,3,4,5,6,7}$ such that $f(i) \le f(j)$ for $i
1 $8C_3$
2 $8C_3 + 2(8C_2)$
3 $10C_3$
4 None of these
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Solution

Number of non-decreasing functions = combinations with repetition $= \binom{8+3-1}{3}=\binom{10}{3}$

If $y = f(x)$ is odd and differentiable on $(-\infty,\infty)$ such that $f'(3) = -2$, then $f'(-3)$ equals:

If the function $f:[1,\infty)\to[1,\infty)$ is defined by $f(x)=2^{x(x-1)}$, then $f^{-1}(x)$ is:

The number of one-to-one functions from {1, 2, 3} to {1, 2, 3, 4, 5} is

A function $f : (0,\pi) \to R$ defined by $f(x) = 2 sin x + cos 2x$ has

If   and , then f(A)=

Let $f : \mathbb{R} \to \mathbb{R}$ be defined by $f(x)=\begin{cases} x \sin\left(\frac{1}{x}\right), & x>0,\\ 0, & x \le 0. \end{cases}$ 

Let S be the set $\{a\in Z^+:a\leq100\}$.If the equation $[tan^2 x]-tan x - a = 0$ has real roots (where [ . ] is the greatest integer function), then the number of elements is S is

If  is a continuous function at x = 0, then the value of k is

If $f(x)=\begin{cases}{{x}^2} & {,\leq0} \\ {2\sin x} & {,0}\end{cases}$, then x = 0 is

Test the continuity of the function at x = 2 

If $f\colon R\rightarrow R$ is defined by $f(x)=\begin{cases}{\frac{x+2}{{x}^2+3x+2}} & {,\, if\, x\, \in R-\{-1,-2\}} \\ {-1} & {,if\, x=-2} \\ {0} & {,if\, x=-1}\end{cases}$ , then f(x) is continuous on the set

The function $f(x)=\frac{x}{1+x\tan x}$ , $0\leq x\leq\frac{\pi}{2}$ is maximum when

If $\prod ^n_{i=1}\tan ({{\alpha}}_i)=1\, \forall{{\alpha}}_i\, \in\Bigg{[}0,\, \frac{\pi}{2}\Bigg{]}$ where i=1,2,3,...,n. Then maximum value of $\prod ^n_{i=1}\sin ({{\alpha}}_i)$.

The sum of infinite terms of decreasing GP is equal to the greatest value of the function $f(x) = x^3 + 3x – 9$ in the interval [–2, 3] and difference between the first two terms is f '(0). Then the common ratio of the GP is

In a reality show, two judges independently provided marks based on the performance of the participants. If the marks provided by the second judge are given by y= 1+ x, where x is the marks provided by the first judge. Then for a participant

A real valued function f is defined as $f(x)=\begin{cases}{-1} & {-2\leq x\leq0} \\ {x-1} & {0\leq x\leq2}\end{cases}$.  Which of the following statement is FALSE?

If n₁ and n₂ are the number of real valued solutions x = | sin^–1 x | & x = sin (x) respectively, then the value of n₂– n₁ is

Number of point of which f(x) is not differentiable $f(x)=|cosx|+3$ in $[-\pi, \pi]$

The maximum value of $f(x) = (x – 1)^2 (x + 1)^3$ is equal to $\frac{2^p3^q}{3125}$  then the ordered pair of (p, q) will be

If f(x) is a polynomial of degree 4, f(n) = n + 1 & f(0) = 25, then find f(5) = ?

Between any two real roots of the equation $e^x sin x = 1$, the equation $e^x cos x = –1$ has

Largest value of $cos^2\theta -6sin\theta cos\theta+3sin^2\theta+2 $ is

If the equation $|x^2 – 6x + 8| = a$ has four real solution then find the value of $a$?

The graph of function $f(x)=\log _e({x}^3+\sqrt[]{{x}^6+1})$ is symmetric about:

$\int f(x)\mathrm{d}x=g(x)$, then $\int {x}^5f({x}^3)\mathrm{d}x$

Consider the function $$f(x)=\begin{cases}{-{x}^3+3{x}^2+1,} & {if\, x\leq2} \\ {\cos x,} & {if\, 2{\lt}x\leq4} \\ {{e}^{-x},} & {if\, x{\gt}4}\end{cases}$$  Which of the following statements about f(x) is true:

Consider the function $f(x)={x}^{2/3}{(6-x)}^{1/3}$. Which of the following statement is false?

Let $f(x)=\begin{cases}{{x}^2\sin \frac{1}{x}} & {,\, x\ne0} \\ {0} & {,x=0}\end{cases}$

The value of $f(1)$ for $f\Bigg{(}\frac{1-x}{1+x}\Bigg{)}=x+2$ is

The number of one - one functions f: {1,2,3} → {a,b,c,d,e} is

Let $g:\mathbb{R}\rightarrow \mathbb{R}$ and $h:\mathbb{R}\rightarrow \mathbb{R}$, be two functions such that $h(x) = sgn(g(x))$. Then select which of the following is not true?( $\mathbb{R}$ denotes the set of all real numbers, sgn stands for signum function)

Let A = {1,2,3, ... , 20}. Let $R\subseteq A\times A$ such that R = {(x,y): y = 2x - 7}. Then the number of elements in R, is equal to