Calculus PYQ — Page 34

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$, then the value of k is :

The curve passing through the point $(-1, 1)$, given that the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2x}{y^2}$ also passes through the point $\left( k, -\frac{1}{2} \right)$, then

$\int e^x \left(\frac{1}{x} - \frac{2}{x^3}\right) dx =$

The area of the region bounded by $f(x) = x|x|$ and x-axis from $x = 0$ to $x = 4$ is :

$y(x)$ is strictly increasing in the interval

The area bounded by the parabola $y^2 = 4ax$ and $x^2 = 4ay$ is :

The slope of normal to the curve $y = 3x^2 + 3 \sin 3x$, at $x = 0$ is:

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =$

The solution of differential equation $y(1 - x^2)\frac{dy}{dx} = x(1 + y^2)$ is :

The derivative of $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ w.r.t. $\tan^{-1}\left(\frac{2x}{1-x^2}\right)$ is

The solution of the differential equation $(x+1)\frac{dy}{dx} = 1 + y$ is

$\int_{1}^{5} |x - 2| \, dx =$

$\int \frac{1}{\cos^2 x (1 + \tan x)^3} \, dx =$

If $y = \frac{\log_e x}{x}$, then $\frac{d^2y}{dx^2} =$

The portion of the area enclosed by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, that lies in the first quadrant is

Derivative of $x^3 + 1$ with respect to $x^2 + 1$ is

The number of arbitrary constants in the general solution of a differential equation of fourth order is:

The function $f(x) = e^{|x|}$ is (a) continuous everywhere on $R$ (b) not continuous at $x = 0$ (c) Differentiable everywhere on $R$ (d) not differentiable at $x = 0$ (e) continuous and differentiable on $R$ Choose the most appropriate answer from the options given below :

The sum of order and degree of differential equation $2x^2 \cdot \left(\frac{d^2y}{dx^2}\right) - 3 \cdot \left(\frac{dy}{dx}\right)^3 + y = 0$ is