Calculus PYQ — Page 2

Let $f(x) = \begin{cases} \dfrac{1}{3}, & x \leq \pi/2 \\ \dfrac{b(1-\sin x)}{(\pi-2x)^2}, & x > \pi/2 \end{cases}$. If $f$ is continuous at $x=\pi/2$, then the value of $\displaystyle\int_{0}^{3b-6} |x^2+2x-3|\,dx$ is:

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 $A = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix}$ 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 :

The area of the region $R = \{(x, y): xy \leq 27, 1 \leq y \leq x^2\}$ is equal to:

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:

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 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 :

If the area of the region bounded by $16x^2 - 9y^2 = 144$ and $8x - 3y = 24$ is A, then $3(A + 6 \log_e(3))$ 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 $\_\_\_\_$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be such that $f(xy) = f(x)f(y)$, for all $x, y \in \mathbb{R}$ and $f(0) \neq 0$. Let $g: [1, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $x^2 g(x) = \int\limits_1^x (t^2 f(t) - tg(t))\,dt$. Then $g(2)$ is equal to :

Let $f(x) = \begin{cases} x^3 + 8 ; & x < 0 \\ x^2 - 4 ; & x \geq 0 \end{cases}$ and $g(x) = \begin{cases} (x-8)^{1/3} ; & x < 0 \\ (x+4)^{1/2} ; & x \geq 0 \end{cases}$. Then the number of points, where the function $g \circ f$ is discontinuous, is __________.

Let $y = y(x)$ be the solution of the differential equation $x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx$, $y(1) = \dfrac{\pi}{2}$ and let $\alpha = \cos\left(\dfrac{y(e^{12})}{e^{12}}\right)$. Then the number of integral values of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is __________.

Let $f(x)=\int \frac{\left(2-x^{2}\right) \cdot \mathrm{e}^{x}}{(\sqrt{1+x})(1-x)^{3 / 2}} \mathrm{~d} x$. If $f(0)=0$, then $f\left(\frac{1}{2}\right)$ is equal to:

The value of $\lim _{x \rightarrow 0} \frac{\log _{e}\left(\sec (e x) \cdot \sec \left(e^{2} x\right) \cdot \ldots \cdot \sec \left(e^{10} x\right)\right)}{e^{2}-e^{2 \cos x}}$ is equal to

If $f(x)$ satisfies the relation $f(x)=e^{x}+\int_{0}^{1}\left(y+x e^{x}\right) f(y) d y$, then $e+f(0)$ is equal to $\_\_\_\_$.

The area of the region $\{(x, y) : x^2 - 8x \leq y \leq -x\}$ is :

The area of the region $\{(x, y): y \leq \pi - |x|, y \leq |x \sin x|, y \geq 0\}$ is:

The area of the region $\mathrm{R}=\left\{(x, y): x y \leq 8,1 \leq y \leq x^{2}, x \geq 0\right\}$ is

The area of the region $\mathrm{A}=\left\{(x, y): 4 x^{2}+y^{2} \leqslant 8\right.$ and $\left.y^{2} \leqslant 4 x\right\}$ is: