Functions Questions (992)

Let $f(x) = ([a]^2 - 5[a] + a)x^3 - (8[a]^2 - 5[a] + 1)x - (\tan x)\operatorname{sgn}x$, be an even function for all $x \in \{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\}$, then sum of all possible values of $a$ is: (where $[.]$ and $\{.\}$ denotes greatest integer function and fractional part functions, respectively)

Let $x=3^{4/25}$. There is a unique value of $y$ such that $0<y<x$ and $x^x=y^y$. The value of $y$, expressed in the form $c^{a/b}$ where $a$ and $b$ are relatively prime positive integers and $c$ is a prime number, gives $a+b-c=$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x) = x + |x|\cos x$. Then which of the following statements is TRUE?

Let $x=3^{4/25}$. There is a unique value of $y$ such that $0<y<x$ and $x^x=y^y$. The value of $y$, expressed in the form $c^{a/b}$ where $a$ and $b$ are relatively prime positive integers and $c$ is a prime number, gives $a+b-c=$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x) = x + |x|\cos x$. Then which of the following statements is TRUE?

Let $F:(1,\infty)\to\mathbb{R}$ be the function defined by $F(x)=\displaystyle\int_x^{x^2}\dfrac{dt}{t\ln t}$. Then:

Let $f(x)=\begin{cases}0&x=0\\x\sin(1/x)&x\neq 0\end{cases}$, $g(x)=\begin{cases}0&\text{otherwise}\\\frac{1}{2}-|x-\frac{1}{2}|&0\leq x\leq 1\end{cases}$, $h(x)=af(x)+b(g(x)+g(\frac{1}{2}-x))+c(x-g(x))+dg(x)$. Match P)$a=0,b=1,c=0,d=0$; Q)$a=1,b=0,c=0,d=0$; R)$a=0,b=0,c=1,d=0$; S)$a=0,b=0,c=0,d=1$ with properties 1)$h$ one-one; 2)$h$ onto; 3)$h$ differentiable on $\mathbb{R}$; 4)range$=[0,1]$; 5)range$=\{0,1\}$

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^2\sin\!\left(\dfrac{\pi}{x^2}\right)$ for $x\neq 0$, $f(0)=0$. Then which of the following is TRUE?

Let $f(x)=\cos^{-1}\!\left(\dfrac{a\cos x+b}{a+b\cos x}\right)-2\tan^{-1}\!\left(\sqrt{\dfrac{a-b}{a+b}}\tan\frac{x}{2}\right)$ ($0<b\leq a$, $x\geq 0$). $g(x)=2\tan^{-1}x+\sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)$, $x\geq 1$. $h(x)=\begin{cases}g(x)&x\in[0,1)\\f(x)&x\in[1,\infty)\end{cases}$ Match List-I with List-II: P)$f(\pi/2)$; Q)$[h(3/2)]$ (GIF); R)$g(1)+g(3/2)g(2)$; S) integers in range of $h(x)$ List-II: 1)0, 2)1, 3)2, 4)3, 5)4

Match each entry in List-I to the correct entry in List-II. **List-I:** P) Number of points of non-derivability of $f(x)=\left[\frac{2}{\pi}x\right]\mathrm{sgn}\!\left(\!\left\{\frac{1}{x}\right\}\!\right)$ in $(-2,2)$ Q) $f:[0,\infty)\to\mathbb{R}$, $f(x)=\frac{2\sin x+x\sin(1/x)}{x}$ for $x>0$, continuous at $x=0$ with $f(0)=K$; find $7K$ R) Locus of orthocentre of $\triangle ABC$ where $A=(1,2)$ and $B$, $C$ on $y=x+\lambda$; $y$-intercept of locus S) Number of integral values of $x$ satisfying $\frac{(2x^2-4)(x-1)}{x(x-4)(x-9)}<0$ **List-II:** 1) 7; 2) 3; 3) 4; 4) 5

Let $f(x)=\begin{cases}0&x=0\\x\sin(1/x)&x\neq 0\end{cases}$, $g(x)=\begin{cases}0&\text{otherwise}\\\frac{1}{2}-|x-\frac{1}{2}|&0\leq x\leq 1\end{cases}$, $h(x)=af(x)+b(g(x)+g(\frac{1}{2}-x))+c(x-g(x))+dg(x)$. Match P)$a=0,b=1,c=0,d=0$; Q)$a=1,b=0,c=0,d=0$; R)$a=0,b=0,c=1,d=0$; S)$a=0,b=0,c=0,d=1$ with properties 1)$h$ one-one; 2)$h$ onto; 3)$h$ differentiable on $\mathbb{R}$; 4)range$=[0,1]$; 5)range$=\{0,1\}$

Let $f(x)=\cos^{-1}\!\left(\dfrac{a\cos x+b}{a+b\cos x}\right)-2\tan^{-1}\!\left(\sqrt{\dfrac{a-b}{a+b}}\tan\frac{x}{2}\right)$ ($0<b\leq a$, $x\geq 0$). $g(x)=2\tan^{-1}x+\sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)$, $x\geq 1$. $h(x)=\begin{cases}g(x)&x\in[0,1)\\f(x)&x\in[1,\infty)\end{cases}$ Match List-I with List-II: P)$f(\pi/2)$; Q)$[h(3/2)]$ (GIF); R)$g(1)+g(3/2)g(2)$; S) integers in range of $h(x)$ List-II: 1)0, 2)1, 3)2, 4)3, 5)4

Match each entry in List-I to the correct entry in List-II. **List-I:** P) Number of points of non-derivability of $f(x)=\left[\frac{2}{\pi}x\right]\mathrm{sgn}\!\left(\!\left\{\frac{1}{x}\right\}\!\right)$ in $(-2,2)$ Q) $f:[0,\infty)\to\mathbb{R}$, $f(x)=\frac{2\sin x+x\sin(1/x)}{x}$ for $x>0$, continuous at $x=0$ with $f(0)=K$; find $7K$ R) Locus of orthocentre of $\triangle ABC$ where $A=(1,2)$ and $B$, $C$ on $y=x+\lambda$; $y$-intercept of locus S) Number of integral values of $x$ satisfying $\frac{(2x^2-4)(x-1)}{x(x-4)(x-9)}<0$ **List-II:** 1) 7; 2) 3; 3) 4; 4) 5

Let $F:(1,\infty)\to\mathbb{R}$ be the function defined by $F(x)=\displaystyle\int_x^{x^2}\dfrac{dt}{t\ln t}$. Then:

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^2\sin\!\left(\dfrac{\pi}{x^2}\right)$ for $x\neq 0$, $f(0)=0$. Then which of the following is TRUE?

Let \(f:[1,\infty)\to[3,\infty)\), \(f(x)=(\log_2 x)^2+2\log_2 x+3\). Which are correct?

Let $f(x) = \sin^3 x - \sin x \cos x + \cos^3 x$, then range of $f(x)$ is:

The range of $f(x) = \tan x + \frac{1}{2}\sin^{-1} x$ is:

Let $f(x) = 1 + 2\cos x + 3\sin x$. If real numbers $a, b, c$ are such that $a f(x) + b f(-x) = 1$ holds for any $x \in \mathbb{R}$ then $\frac{b\cos c}{a} =$

The reflection about the line $x + y = 0$ of the inverse function $f^{-1}(x)$ of a function $f(x)$ is:

Let $f(x)$ be any function. The graphs of $y = f(x-1)$ and $y = f(-x+1)$ are symmetric about the line:

Let $f(x)$ be such that $f(x+2) = f(x)$ and $f(-x) = f(x)$ for any real number $x$. On the interval $[2,3]$, $f(x) = x$. Then the formula of $f(x)$ given on $[-2,0]$ is:

A function $f: \mathbb{R} \to \mathbb{R}$ has property $f(x+y) = f(x) \cdot e^{f(y)-1}$, for every $x, y \in \mathbb{R}$ then positive value of $f(4)$ is:

Let $f(x)$ be a function such that $f(x+y) = f(x) \cdot f(y)$ for all $x, y \in \mathbb{N}$. If $f(1) = 3$ and $\displaystyle\sum_{k=1}^{n} f(k) = 3279$, then the value of $n$ is

The number of positive integers $x$ that satisfy $3^x = x^3 + 3x^2 + 2x + 1$ is:

Suppose $f(x) = x^3 + \log_2\left(x + \sqrt{x^2 + 1}\right)$. For any $a, b \in \mathbb{R}$ to satisfy $f(a) + f(b) \geq 0$, the condition $a + b \geq 0$ is:

The graph of the function $f(x) = \frac{9x+7}{3x+12}$ is symmetric to the point:

The shaded region in the given figure is(The figure shows three overlapping circles A, B, C where the region inside A but outside B and C is shaded.)

Let $f(x)=x^7(x^3+2x^2-x-2)+(x^3-x)(x+3)+2(x^2-1)$. Which is NOT true?

Let $f(x) = 2x^n + \lambda$, $\lambda \in \mathbb{R}$, $n \in \mathbb{N}$, and $f(4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $(f(3)-f(2))$ is

Let $R = \{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equivalence relation, is:

If the domain of the function $f(x) = \log(1 - \log(x^2 - 9x + 18))$ is $(\alpha, \beta) \cup (\gamma, \delta)$, then $\alpha + \beta + \gamma + \delta$ is equal to

The domain of $f(x)=\dfrac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where $[x]$ denotes the greatest integer $\leq x$)

For the function f(x) = \frac{e^{x} + 1}{e^{x} - 1}, if n(d) denotes the number of integers which are not in its domain and n(r) denotes the number of integers which are not in its range, then n(d) + n(r) is equal to -

The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is

Let $A=\{1,3,4,6,9\}$ and $B=\{2,4,5,8,10\}$. Relation $R=\{((a_1,b_1),(a_2,b_2)):\ a_1\leq b_2\ \text{and}\ b_1\leq a_2\}$ has how many elements?

Among S = {(a, b) : a, b ∈R \ {0}, 2 + a/b > 0} and T = {(a, b) : a, b ∈R, a2 −b2 ∈Z}:

Find the range of $f(x) = \frac{2x}{1+x^2}$.

Let A = A1 ∪A2 ∪· · · ∪Ak where Ai ∩Aj = ∅for i ̸= j. Define R = {(x, y) : y ∈Ai ⇔ x ∈Ai}. Then R is:

The reflection about the line $x + y = 0$ of the inverse function $f^{-1}(x)$ of a function $f(x)$ is:

Let the domain of the function $f(x)=\log_3\log_5\!\left(7-\log_2(x^2-10x+85)\right)+\sin^{-1}\!\left(\left|\dfrac{3x-7}{17-x}\right|\right)$ be $(\alpha,\beta]$. Then $\alpha+\beta$ is equal to:

Let $f(g(x))=x+3-\sqrt{x}$ where $g(x)=\sqrt{x}+1$. Then $f(0)$ is equal to

The equation $x^2 - 4x + [x] + 3 = x[x]$, where $[x]$ denotes the greatest integer function, has:

If \( f(x) + 2f\left(\frac{1}{x}\right) = 3x, x \neq 0, \) and S = \( \{x \in \mathbb{R} : f(x) = f(-x)\} \); then S :

Let R be a relation on \mathbb{N} \times \mathbb{N} defined by (a, b) R (c, d) if and only if ad(b - c) = bc(a - d). Then R is

Let $f(x) = a\sin x + b\sqrt[3]{x+4}$. If $f\bigl(\log_{10}(\log_3 10)\bigr) = 5$ and $f\bigl(\log_{10}(\log_3 3)\bigr) = 3$, then $f\bigl(\log_{10}(\log_3 3)\bigr)$ is

For \( x \in \mathbb{R} - \{0, 1\} \), let f_1(x) = \frac{1}{x}, f_2(x) = 1 - x and f_3(x) = \frac{1}{1 - x} \) be three given functions. If a function, J(x) satisfies \( f_2 \circ f_1 \circ f(x) = f_3(x) \) then J(x) is equal to :-

If R = {(x, y) : x, y ∈Z, x2 + 3y2 ≤8}, then the domain of R−1 is:

Let R1 = {(a, b) ∈R2 : a2 + b2 ∈Q} and R2 = {(a, b) ∈R2 : a2 + b2 /∈Q}, where Q is the set of rationals. Then:

If $f(x) = \dfrac{x^3+x-2}{x^3-1}$ and $g(x) = \dfrac{x^2+x+2}{x^2+x+1}$, then ($D_f$ represents domain and $R_f$ represents range of $f(x)$)