Calculus PYQ — Page 7

Let for $f(x)=7 \tan ^8 x+7 \tan ^6 x-3 \tan ^4 x-3 \tan ^2 x, \quad \mathrm{I}_1=\int_0^{\pi / 4} f(x) \mathrm{d} x$ and $\mathrm{I}_2=\int_0^{\pi / 4} x f(x) \mathrm{d} x$. Then $7 \mathrm{I}_1+12 \mathrm{I}_2$ is equal to :

Let \(f:(0, \infty) \rightarrow \mathbf{R}\) be a twice differentiable function. If for some \(\mathrm{a} \neq 0, \int_0^1 f(\lambda x) \mathrm{d} \lambda=\mathrm{a} f(x), f(1)=1\) and \(f(16)=\frac{1}{8}\), then \(16-f^{\prime}\left(\frac{1}{16}\right)\) is equal to _______.

Let f be a differentiable function on $\mathbf{R}$ such that $\mathrm{f}(2) = 1$, $f^{\prime}(2)=4$. Let $\lim _{x \rightarrow 0}(f(2+x))^{3 / x}=e^\alpha$. Then the number of times the curve $y=4 x^3-4 x^2-4(\alpha-7) x-\alpha$ meets x -axis is :-

Let [.] denote the greatest integer function. If $\int_0^{e^3}\left[\frac{1}{\mathrm{e}^{\mathrm{x}-1}}\right] \mathrm{dx}=\alpha-\log _{\mathrm{e}} 2$, then $\alpha^3$ is equal to _______ .

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x)=[x]+|x-2|,-2 \lt x \lt 3$, is not continuous and not differentiable. Then $\mathrm{m}+\mathrm{n}$ is equal to :

Let $x=x(y)$ be the solution of the differential equation $y^2 \mathrm{~d} x+\left(x-\frac{1}{y}\right) \mathrm{d} y=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :

Let $y=f(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{x y}{x^2-1}=\frac{x^6+4 x}{\sqrt{1-x^2}},-1 \lt x \lt 1$ such that $f(0)=0$. If $6 \int_{-1 / 2}^{1 / 2} f(x) \mathrm{d} x=2 \pi-\alpha$ then $\alpha^2$ is equal to _______ .

Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{\mathrm{~d} y}{\mathrm{~d} x}=\sin 2 x-4 y \sin x, x \in\left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{3}\right)=0$, then $y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)$ is equal to $\qquad$ ________.

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y\gt0$ and $x(1)=\frac{\pi}{2}$. Then $\cos (x(2))$ is equal to :

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+1\right) y^{\prime}-2 x y=\left(x^4+2 x^2+1\right) \cos x$, $y(0)=1$. Then $\int_{-3}^3 y(x) d x$ is :

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+2 y \sec ^2 x=2 \sec ^2 x+3 \tan x \cdot \sec ^2 x$ such that $\mathrm{y}(0)=\frac{5}{4}$. Then $12\left(\mathrm{y}\left(\frac{\pi}{4}\right)-\mathrm{e}^{-2}\right)$ is equal to _______.

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(x y-5 x^2 \sqrt{1+x^2}\right) d x+\left(1+x^2\right) d y=0, y(0)=0$. Then $y(\sqrt{3})$ is equal to

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}+3\left(\tan ^2 x\right) y+3 y=\sec ^2 x$ $y(0)=\frac{1}{3}+e^3$. Then $y\left(\frac{\pi}{4}\right)$ is equal to

Let $y=y(x)$ be the solution curve of the differential equation $x\left(x^2+e^x\right) d y+\left(e^x(x-2) y-x^3\right) d x=0, x \gt 0$ passing through the point $(1,0)$. Then $y(2)$ is equal to :

Let $(a, b)$ be the point of intersection of the curve $x^2=2 y$ and the straight line $y-2 x-6=0$ in the second quadrant. Then the integral $I=\int_a^b \frac{9 x^2}{1+5^x} d x$ is equal to :

Let $(2,3)$ be the largest open interval in which the function $f(x)=2 \log _{\mathrm{e}}(x-2)-x^2+a x+1$ is strictly increasing and (b, c) be the largest open interval, in which the function $\mathrm{g}(x)=(x-1)^3(x+2-\mathrm{a})^2$ is strictly decreasing. Then $100(a+b-c)$ is equal to :

Let $f:[0, \infty) \rightarrow \mathbb{R}$ be differentiable function such that $f(\mathrm{x})=1-2 \mathrm{x}+\int_0^x e^{x-t} f(t) \mathrm{dt}$ for all $\mathrm{x} \in[0, \infty)$. Then the area of the region bounded by $\mathrm{y}=f(\mathrm{x})$ and the coordinate axes is

Let $f(\mathrm{x})= \begin{cases}(1+\mathrm{ax})^{1 / \mathrm{x}} & , \quad \mathrm{x} \lt 0 \\ 1+\mathrm{b} & , \quad \mathrm{x}=0 \\ \frac{(\mathrm{x}+4)^{1 / 2}-2}{(\mathrm{x}+\mathrm{c})^{1 / 3}-2} & ,\end{cases}$ be continuous at $x=0$. Then $e^a b c$ is equal to

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2 x+2 y)-f(2 x-2 y))=(\cos x$ $\sin \mathrm{y})(f(2 \mathrm{x}+2 \mathrm{y})+f(2 \mathrm{x}-2 \mathrm{y}))$, for all $\mathrm{x}, \mathrm{y} \in \mathbf{R}$. If $f^{\prime}(0)=\frac{1}{2}$, then the value of $24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)$ is:

Let $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(2)=1$. If $\mathrm{F}(x)=x f(x)$ for all $x \in \mathbf{R}$, $\int_0^2 x \mathrm{~F}^{\prime}(x) \mathrm{d} x=6$ and $\int_0^2 x^2 \mathrm{~F}^{\prime \prime}(x) \mathrm{d} x=40$, then $\mathrm{F}^{\prime}(2)+\int_0^2 \mathrm{~F}(x) \mathrm{d} x$ is equal to :

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbf{R}$. If $f^{\prime}(0)=4 \mathrm{a}$ and $f$ satisfies $f^{\prime \prime}(x)-3 \mathrm{a} f^{\prime}(x)-f(x)=0$, $\mathrm{a}\gt0$, then the area of the region $\mathrm{R}=\{(x, y) \mid 0 \leq y \leq f(\mathrm{a} x), 0 \leq x \leq 2\}$ is:

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a thrice differentiable odd function satisfying $f^{\prime}(\mathrm{x}) \geq 0, f^{\prime}(\mathrm{x})=f(\mathrm{x}), f(0)=0, f^{\prime}(0)=3$. Then $9 f\left(\log _{\mathrm{c}} 3\right)$ is equal to _______.

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int_0^x \mathrm{t} f(\mathrm{t}) \mathrm{dt}$. If $\mathrm{g}\left(x^3\right)=x^6+x^7$, then value of $\sum_{r=1}^{15} f\left(\mathrm{r}^3\right)$ is :

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in \mathbf{R}$. Then $\sum_{\mathrm{n}=1}^{100} \log _{\mathrm{e}} f(\mathrm{n})$ is equal to :