Calculus PYQ — Page 4

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 :

Let $f$ be a real polynomial of degree $n$ such that $f(x) = f'(x) f''(x)$, for all $x \in \mathbb{R}$. If $f(0) = 0$, then $36\left(f'(2) + f''(2) + \int_0^2 f(x)\,dx\right)$ 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 $f$ be a polynomial function such that $f\left(x^{2}+1\right)=x^{4}+5 x^{2}+2$, for all $x \in \mathbb{R}$. Then $\int_{0}^{3} f(x) d x$ is equal to

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f\left(\dfrac{x+y}{3}\right) = \dfrac{f(x) + f(y)}{3}$ for all $x, y \in \mathbb{R}$, and $f'(0) = 3$. Then the minimum value of the function $g(x) = 3 + e^x f(x)$, is:

Let $f$ be a differentiable function satisfying $f(x)=1-2 x+\int_{0}^{x} \mathrm{e}^{(x-t)} f(t) \mathrm{dt}, x \in \mathbf{R}$ and let $\mathrm{g}(x)=\int_{0}^{x}(f(\mathrm{t})+2)^{15}(\mathrm{t}-4)^{6}(\mathrm{t}+12)^{17} \mathrm{dt}, x \in \mathbf{R}$. If p and q are respectively the points of local minima and local maxima of g, then the value of $|\mathrm{p}+\mathrm{q}|$ is equal to $\_\_\_\_$.

Let $y=y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1)=2$, and $\lim _{t \rightarrow x}\left(\frac{t^{2} y(x)-x^{2} y(t)}{x-t}\right)=3$ for each $x>0$. Then $2 y(2)$ is equal to

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function. If $6 \int_{1}^{x} f(t) d t=3 x f(x)+x^{3}-4$ for all $x \geq 1$, then the value of $f(2)-f(3)$ is

Let $f: [1, \infty) \rightarrow \mathbf{R}$ be a differentiable function defined as $f(x) = \int_1^x f(t)\,dt + (1-x)(\log_e x - 1) + e$. Then the value of $f(f(1))$ is :

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 $f(x)=x^{2025}-x^{2000}, x \in[0,1]$ and the minimum value of the function $f(x)$ in the interval $[0,1]$ be $(80)^{80}(n)^{-81}$. Then $n$ is equal to

Let $\mathrm{I}(x)=\int \frac{3 d x}{(4 x+6)\left(\sqrt{4 x^{2}+8 x+3}\right)}$ and $\mathrm{I}(0)=\frac{\sqrt{3}}{4}+20$. If $\mathrm{I}\left(\frac{1}{2}\right)=\frac{a \sqrt{2}}{b}+\mathrm{c}$, where $a, b, \mathrm{c} \in \mathrm{N}, \operatorname{gcd}(a, b)=1$, then $a+b+c$ 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 $P_{1}: y=4 x^{2}$ and $P_{2}: y=x^{2}+27$ be two parabolas. If the area of the bounded region enclosed between $P_{1}$ and $P_{2}$ is six times the area of the bounded region enclosed between the line $y=\alpha x, \alpha>0$ and $P_{1}$, then $\alpha$ is equal to :

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x) = g''(x)$ for all $x \in \mathbf{R}$, $f'(1) = 2g'(1) = 4$ and $g(2) = 3f(2) = 9$. Then $f(25) - g(25)$ 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 ∫₀¹ x·eˣ dx is:

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

$6 \int_{0}^{\pi}|(\sin 3 x+\sin 2 x+\sin x)| d x$ is equal to $\_\_\_\_$.

If $\int_{0}^{1} 4 \cot ^{-1}\left(1-2 x+4 x^{2}\right) \mathrm{d} x=\mathrm{atan}^{-1}(2)-\mathrm{blog}_{\mathrm{e}}(5)$, where $\mathrm{a}, \mathrm{b} \in \mathbf{N}$, then $(2 \mathrm{a}+\mathrm{b})$ is equal to $\_\_\_\_$.

If $\int\left(\frac{1-5 \cos ^{2} x}{\sin ^{5} x \cos ^{2} x}\right) d x=f(x)+\mathrm{C}$, where C is the constant of integration, then $f\left(\frac{\pi}{6}\right)-f\left(\frac{\pi}{4}\right)$ is equal to

If $\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}} d x=$ $-\frac{p_{1}}{q_{1}}(\cot x)^{\frac{9}{2}}-\frac{p_{2}}{q_{2}}(\cot x)^{\frac{5}{2}}-\frac{p_{3}}{q_{3}}(\cot x)^{\frac{1}{2}}+\frac{p_{4}}{q_{4}}(\cot x)^{\frac{-3}{2}}+\mathrm{C}$, where $p_{i}$ and $q_{i}$ are positive integers with $\operatorname{gcd}\left(p_{i}, q_{i}\right)=1$ for $i=1,2,3,4$ and C is the constant of integration, then $\frac{15 p_{1} p_{2} p_{3} p_{4}}{q_{1} q_{2} q_{3} q_{4}}$ is equal to

If $\displaystyle\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\dfrac{\pi}{3}\right)\cot\left(x+\dfrac{\pi}{3}\right)+1\right)dx = \alpha\log_e(\sqrt{3}-1)$, then $9\alpha^2$ is equal to ________.

If $\displaystyle\lim_{x \to 2} \dfrac{\sin(x^3 - 5x^2 + ax + b)}{(\sqrt{x-1} - 1)\log_e(x-1)} = m$, then $a + b + m$ is equal to :