Calculus PYQ — Page 4

Let $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x=0, x=1, y^{2}=x$ and $y=|\alpha x-5|-|1-\alpha x|+\alpha x^{2}$. Then $(f(0)+f(1))$ 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 :

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x$, where $[\cdot]$ denotes the greatest integer function, is

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 $f: \mathbf{R} \rightarrow \mathbf{R}$ 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

$6 \int_{0}^{\pi}|(\sin 3 x+\sin 2 x+\sin x)| d x$ 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 _______.

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 $y=y(x)$ 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 $y=y(x)$ 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 the solution curve $y=f(x)$ of the differential equation $\left(x^{2}-4\right) y^{\prime}-2 x y+2 x\left(4-x^{2}\right)^{2}=0, x>2$, passes through the point $(3,15)$, then the local maximum value of $f$ is $\_\_\_\_$.

Let $f(x)=\int \frac{\mathrm{d} x}{x^{\left(\frac{2}{3}\right)}+2 x^{\left(\frac{1}{2}\right)}}$ be such that $f(0)=-26+24 \log _{\mathrm{e}}(2)$. If $f(1)=\mathrm{a}+\mathrm{b} \log _{\mathrm{e}}(3)$, where $\mathrm{a}, \mathrm{b} \in \mathbf{Z}$, then $\mathrm{a}+\mathrm{b}$ is equal to :

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 :

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 $\_\_\_\_$.

Let $f$ be a twice differentiable function such that $f(x)=\int_{0}^{x}\tan(t-x)dt-\int_{0}^{x}f(t)\tan t\,dt$, $x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$. Then $f''\left(\dfrac{\pi}{6}\right)+12f'\left(-\dfrac{\pi}{6}\right)+f\left(\dfrac{\pi}{6}\right)$ is equal to ______

Let $y=y(x)$ be the solution of the differential equation $x\sqrt{1-x^2}\,dy + \left(y\sqrt{1-x^2} - x\cos^{-1}x\right)dx = 0$, $x \in (0, 1)$, $\displaystyle\lim_{x\to 1^-} y(x) = 1$. Then $y\left(\dfrac{1}{2}\right)$ equals:

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 :