Calculus PYQ — Page 3

Let $f$ be a twice differentiable non-negative function such that $(f(x))^{2}=25+\int_{0}^{x}\left((f(\mathrm{t}))^{2}+\left(f^{\prime}(\mathrm{t})\right)^{2}\right) \mathrm{dt}$. Then the mean of $f\left(\log _{\mathrm{e}}(1)\right), f\left(\log _{\mathrm{e}}(2)\right), \ldots.., f\left(\log _{\mathrm{e}}(625)\right)$ 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 ________.

Let $y = y(x)$ be the solution of the differential equation $\dfrac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)$, $y(0) = \dfrac{1}{2}$. Then $(2y(1) - 1)$ is equal to:

If $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{(\mathrm{a}-1) x}+2 \cos \mathrm{~b} x+(\mathrm{c}-2) \mathrm{e}^{-x}}{x \cos x-\log _{\mathrm{e}}(1+x)}=2$, then $\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}$ is equal to :

Let $[\cdot]$ denote the greatest integer function. Then $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right) \mathrm{d} x$ is equal to :

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

Let $x = x(y)$ be the solution of the differential equation $2y^2 \dfrac{dx}{dy} - 2xy + x^2 = 0$, $y > 1$, $x(e) = e$. Then $x(e^2)$ is equal to:

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function $f(x)=\left\{\begin{array}{cl}b^{2} \sin \left(\frac{\pi}{2}\left[\frac{\pi}{2}(\cos x+\sin x) \cos x\right]\right), & x<0 \\ \frac{\sin x-\frac{1}{2} \sin 2 x}{x^{3}} &, x>0 \\ a &, x=0\end{array}\right.$ is continuous at $x=0$, then $a^{2}+b^{2}$ is equal to

The value of $\lim_{x \to 0}\left(\dfrac{x^2\sin^2 x}{x^2 - \sin^2 x}\right)$ is:

Let $y = y(x)$ be the solution curve of the differential equation $(1 + \sin x)\dfrac{dy}{dx} + (y+1)\cos x = 0$, $y(0) = 0$. If the curve $y = y(x)$ passes through the point $\left(\alpha, \dfrac{-1}{2}\right)$, then a value of $\alpha$ is :

Let $y = y(x)$ be the solution of the differential equation $(\tan x)^{1/2}\,dy = (\sec^3 x - (\tan x)^{3/2} y)\,dx$, $0 < x < \dfrac{\pi}{2}$, $y\left(\dfrac{\pi}{4}\right) = \dfrac{6\sqrt{2}}{5}$. If $y\left(\dfrac{\pi}{3}\right) = \dfrac{4}{5}\alpha$, then $\alpha^4$ equals _______.

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 $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that the quadratic equation $f(x) \mathrm{m}^{2}-2 f^{\prime}(x) \mathrm{m}+f^{\prime \prime}(x)=0$ in m, has two equal roots for every $x \in \mathbf{R}$. If $f(0)=1, f^{\prime}(0)=2$, and $(\alpha, \beta)$ is the largest interval in which the function $f\left(\log _{\mathrm{e}} x-x\right)$ is increasing, then $\alpha+\beta$ 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

The number of points, at which the function $f(x) = \max\{6x, 2 + 3x^2\} + |x - 1|\left|\cos\left|x^2 - \dfrac{1}{4}\right|\right|$, $x \in (-\pi, \pi)$, is not differentiable, is _____.

Let $y = y(x)$ be the solution of the differential equation $(x^2 - x\sqrt{x^2 - 1})dy + (y(x - \sqrt{x^2 - 1}) - x)dx = 0$, $x \geq 1$. If $y(1) = 1$, then the greatest integer less than $y(\sqrt{5})$ is _______.

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

Let $y=y(x)$ be the solution of the differential equation $\sec x \frac{\mathrm{~d} y}{\mathrm{~d} x}-2 y=2+3 \sin x, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, $y(0)=-\frac{7}{4}$. Then $y\left(\frac{\pi}{6}\right)$ is equal to :

Let $f(t)=\int\left(\frac{1-\sin \left(\log _{e} t\right)}{1-\cos \left(\log _{e} t\right)}\right) d t, t>1$. If $f\left(e^{\pi / 2}\right)=-e^{\pi / 2}$ and $f\left(e^{\pi / 4}\right)=\alpha e^{\pi / 4}$, then $\alpha$ equals

Let $y=y(x)$ be the solution of the differential equation $x^{4} \mathrm{~d} y+\left(4 x^{3} y+2 \sin x\right) \mathrm{d} x=0, x>0, y\left(\frac{\pi}{2}\right)=0$. Then $\pi^{4} y\left(\frac{\pi}{3}\right)$ 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 $y=y(x)$ be the solution of the differential equation $x \frac{\mathrm{~d} y}{\mathrm{~d} x}-y=x^{2} \cot x, x \in(0, \pi)$. If $y\left(\frac{\pi}{2}\right)=\frac{\pi}{2}$, then $6 y\left(\frac{\pi}{6}\right)-8 y\left(\frac{\pi}{4}\right)$ 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

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 $\_\_\_\_$.