Calculus PYQ — Page 2

If the curve $y = f(x)$ passes through the point $(1, e)$ and satisfies the differential equation $dy = y(2 + \log_e x)\,dx$, $x > 0$, then $f(e)$ is equal to :

Let $f(x)= \begin{cases}\frac{\mathrm{a} x^{2}+2 \mathrm{a} x+3}{4 x^{2}+4 x-3} &, x \neq-\frac{3}{2}, \frac{1}{2} \\ \mathrm{~b} &, x=-\frac{3}{2}, \frac{1}{2}\end{cases}$ be continuous at $x=-\frac{3}{2}$. If $f \circ f(x)=\frac{7}{5}$, then $x$ is equal to:

The number of critical points of the function $f(x) = \begin{cases} \left|\dfrac{\sin x}{x}\right|, & x \neq 0 \\ 1, & x = 0 \end{cases}$ in the interval $(-2\pi, 2\pi)$ is equal to :

Let $f(x) = \int \left(\dfrac{16x + 24}{x^2 + 2x - 15}\right) dx$. If $f(4) = 14 \log_e(3)$ and $f(7) = \log_e(2^{\alpha} \cdot 3^{\beta})$, $\alpha, \beta \in \mathbb{N}$, then $\alpha + \beta$ is equal to:

Let $[\cdot]$ denote the greatest integer function and $f(x)=\lim _{\mathrm{n} \rightarrow \infty} \frac{1}{\mathrm{n}^{3}} \sum_{\mathrm{k}=1}^{\mathrm{n}}\left[\frac{\mathrm{k}^{2}}{3^{x}}\right]$. Then $12 \sum_{\mathrm{j}=1}^{\infty} f(\mathrm{j})$ is equal to $\_\_\_\_$.

Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f\left(\frac{t x}{36}\right) d t=4 \alpha f(x)$. If $y=f(x)$ is a standard parabola passing through the points $(2,1)$ and $(-4, \beta)$, then $\beta^{\alpha}$ is equal to $\_\_\_\_$.

The value of the integral $\int_{\frac{\pi}{24}}^{\frac{5 \pi}{24}} \frac{\mathrm{~d} x}{1+\sqrt[3]{\tan 2 x}}$ is :

Let $A = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix}$ be a singular matrix. Let $f(x) = \int\limits_0^x (t^2 + 2t + 3)\,dt$, $x \in [1, \alpha]$. If $M$ and $m$ are respectively the maximum and the minimum values of $f$ in $[1, \alpha]$, then $3(M - m)$ is equal to :

If the function $f(x)=\frac{e^{x}\left(e^{\tan x-x}-1\right)+\log _{e}(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal to

Let $\lim\limits_{x \to 2} \dfrac{(\tan(x-2))(rx^2 + (p-2)x - 2p)}{(x-2)^2} = 5$ for some $r, p \in \mathbb{R}$. If the set of all possible values of $q$, such that the roots of the equation $rx^2 - px + q = 0$ lie in $(0, 2)$, be the interval $(\alpha, \beta]$, then $4(\alpha + \beta)$ equals :

Let $(2^{1-a} + 2^{1+a})$, $f(a)$, $(3^a + 3^{-a})$ be in A.P. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\int_{\log_e(\alpha-1)}^{\log_e(\alpha)} \dfrac{dx}{(e^{2x} - e^{-2x})}$ is :

Let $y=y(x)$ be the solution of the differential equation $x \frac{d y}{d x}-\sin 2 y=x^{3}\left(2-x^{3}\right) \cos ^{2} y, x \neq 0$. If $y(2)=0$, then $\tan (y(1))$ is equal to

Let $f(x)$ be a polynomial of degree $5$, and have extrema at $x = 1$ and $x = -1$. If $\displaystyle\lim_{x \to 0} \left(\dfrac{f(x)}{x^3}\right) = -5$, then $f(2) - f(-2)$ is equal to:

For the function $f(x) = e^{\sin|x|} - |x|$, $x \in \mathbb{R}$, consider the following statements: Statement I: $f$ is differentiable for all $x \in \mathbb{R}$. Statement II: $f$ is increasing in $\left(-\pi, -\dfrac{\pi}{2}\right)$. In the light of the above statements, choose the correct answer from the options given below:

Let $y=y(x)$ be the solution of the differential equation: $\dfrac{dy}{dx}+\left(\dfrac{6x^2+(3x^2+2x^3+4)e^{-2x}}{(x^3+2)(2+e^{-2x})}\right)y=2+e^{-2x}$, $x \in (-1,2)$, satisfying $y(0)=\dfrac{3}{2}$. If $y(1)=\alpha(2+e^{-2})$, then $\alpha$ is equal to:

The number of elements in the set $\mathrm{S}=\left\{x: x \in[0,100]\right.$ and $\left.\int_{0}^{x} t^{2} \sin (x-t) \mathrm{d} t=x^{2}\right\}$ is $\_\_\_\_$.

The area of the region $\{(x, y) : 0 \leq y \leq 6 - x, y^2 \geq 4x - 3, x \geq 0\}$ is:

The number of points in the interval $[2, 4]$, at which the function $f(x) = \left[x^2 - x - \dfrac{1}{2}\right]$, where $[\cdot]$ denotes the greatest integer function, is discontinuous, is _______.

Let $f: \mathbf{R} \rightarrow(0, \infty)$ be a twice differentiable function such that $f(3)=18, f^{\prime}(3)=0$ and $f^{\prime \prime}(3)=4$. Then $\lim _{x \rightarrow 1}\left(\log _{\mathrm{e}}\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^{2}}}\right)$ is equal to :

Let $f(x)=\begin{cases} e^{x-1}, & x<0 \\ x^2-5x+6, & x \geq 0 \end{cases}$ and $g(x)=f(|x|)+|f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha+\beta$ is equal to ______

Let $\alpha, \beta \in \mathbb{R}$ be such that the function $f(x)= \begin{cases}2 \alpha\left(x^{2}-2\right)+2 \beta x &, x<1 \\ (\alpha+3) x+(\alpha-\beta) &, x \geq 1\end{cases}$ be differentiable at all $x \in \mathbb{R}$. Then $34(\alpha+\beta)$ is equal to

Let $[\cdot]$ denote the greatest integer function, and let $f(x)=\min \left\{\sqrt{2} x, x^{2}\right\}$. Let $\mathrm{S}=\left\{x \in(-2,2)\right.$ : the function $\mathrm{g}(x)=|x|\left[x^{2}\right]$ is discontinuous at $\left.x\right\}$. Then $\sum_{x \in S} f(x)$ equals

$\max_{0 \leq x \leq \pi}\left(16\sin\left(\dfrac{x}{2}\right)\cos^3\left(\dfrac{x}{2}\right)\right)$ is equal to:

Let $(2 \alpha, \alpha)$ be the largest interval in which the function $f(t)=\frac{|t+1|}{t^{2}}, t<0$, is strictly decreasing. Then the local maximum value of the function $g(x)=2 \log _{\mathrm{e}}(x-2)+\alpha x^{2}+4 x-\alpha, x>2$, is $\_\_\_\_$