Calculus PYQ — Page 28

The integral $\int e^x \left(\frac{x-1}{2x^2}\right) dx$ is equal to:

$\int_{0}^{1.5} [x] dx$, where $[x]$ denotes the greatest integer function $\leq x$, is equal to :

The appropriate change in the volume V of a cube of side $x$ metres caused by increasing the side by 2% is :

The differential equation $y = xp + \sqrt{x^2 p^3 + 4}$ where $p = \frac{dy}{dx}$ is : (A) of order 1 (B) of degree 1 (C) of order 2 (D) of degree 3 Choose the **correct** answer from the options given below :

The solution of the differential equation $\frac{dy}{dx} = \frac{6}{x^2}$; $y(1) = 3$ is :

The area of the region bounded by the lines $x = 2y + 3, x = 0, y = 1$ and $y = -1$ is:

The integral $\int \frac{dx}{x^2(x^4+1)^{\frac{3}{4}}}$ equals __________.

If $f(x) = \begin{cases} \frac{x^2 - 9}{x - 3}, & x \neq 3 \\ 5, & x = 3 \end{cases}$ then $f(x)$ :

If $y = x^x$, then value of $\frac{dy}{dx}$ at $x = 2$ is :

The line $y = mx$ ($m > 0$) partitions the area of the circle $x^2 + y^2 = a^2$ ($a > 0$) in the ratio:

If $\int (1 + e^{-x} + e^{-2x} + ...)dx = \log\phi(x) + C$, then $\phi(x)$ is equal to :

The smaller area enclosed by the curve $y = |x|$ and the circle $(x-a)^2 + y^2 = a^2$ is:

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

$\int_{0}^{\pi} \frac{e^{\cos x}}{e^{\cos x} + e^{-\cos x}} dx =$

The solution of differential equation $\sqrt{x+1} - \sqrt{x-1}\frac{dy}{dx} = 0$ is

The integrating factor of differential equation $x\frac{dy}{dx} + 2y = x^2 \log x$ is

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

The value of k, for which the function $f(x) = \begin{cases} \frac{\sin kx}{x} + 3\cos x, & x \neq 0 \\ 7, & x = 0 \end{cases}$ is continuous at $x = 0$, is

Based on above information answer the following question : $x$ and $y$ will satisfy :

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

A function $f(x)$ is defined by : $f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ -x + 2, & \text{if } x > 0 \end{cases}$ Which of the following is true ?

If $\sqrt{y+x} + \sqrt{y-x} = a$, $a > 1$, $\frac{d^2y}{dx^2}$ is equal to :

The interval in which the function given by $f(x) = x^2 e^{-x}$ is strictly increasing is:

The slope of the tangent to the curve $y = 3x^2 + 2kx - 5$ at $x=1$ is $9$. The value of $k$ is :