Sets, Relations and Functions

The function $f:[0,3]\to[1,29]$, defined by $f(x)=2x^3-15x^2+36x+1$, is

Let $f:(-1,1)\to\mathbb R$ be such that $f(\cos4\theta)=\dfrac2{2-\sec^2\theta}$ for $\theta\in\left(0,\dfrac\pi4\right)\cup\left(\dfrac\pi4,\dfrac\pi2\right)$. Then the value(s) of $f\left(\dfrac13\right)$ is (are)

Let $f(x)=x^2$ and $g(x)=\sin x$ for all $x\in\mathbb R$. Then the set of all $x$ satisfying $(f\circ g\circ g\circ f)(x)=(g\circ g\circ f)(x)$, where $(f\circ g)(x)=f(g(x))$, is

Let $f:(0,1)\to\mathbb R$ be defined by
$$f(x)=\frac{b-x}{1-bx},$$
where $b$ is a constant such that $0<b<1$. Then

Match the statements given in Column I with the intervals/union of intervals given in Column II

For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined on the interval [−10, 10] by
$$f(x)=\begin{cases}x-[x]&\text{if }[x]\text{ is odd,}\\1+[x]-x&\text{if }[x]\text{ is even.}\end{cases}$$
Then the value of $\dfrac{\pi^2}{10}\displaystyle\int_{-10}^{10}f(x)\cos\pi x\,dx$ is

Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to

Let $f$ be a real-valued function defined on the interval $(-1,1)$ such that $e^{-x}f(x)=2+\displaystyle\int_0^x\sqrt{t^4+1}\,dt$, for all $x\in(-1,1)$, and let $f^{-1}$ be the inverse function of $f$. Then $(f^{-1})'(2)$ is equal to

If the function $f(x)=x^3+e^{\frac x2}$ and $g(x)=f^{-1}(x)$, then the value of $g'(1)$ is

$\displaystyle\int_{-1}^1g'(x)\,dx=$

Let the function $g:(-\infty,\infty)\to\left(-\frac\pi2,\frac\pi2\right)$ be given by $g(u)=2\tan^{-1}(e^u)-\frac\pi2$. Then, $g$ is

Let $f(x)=\dfrac{x}{\left(1+x^n\right)^{1/n}}$ for $n\ge2$ and $g(x)=\underbrace{(f\circ f\circ\cdots\circ f)}_{f\text{ occurs }n\text{ times}}(x)$. Then $\displaystyle\int x^{n-2}g(x)\,dx$ equals

Let $f(x)=\dfrac{x^2-6x+5}{x^2-5x+6}$.

Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.