Calculus PYQ — Page 21

The demand function for a commodity is $p = 35 - 2x - x^2$, then the consumer's surplus at equilibrium price $p_0 = 20$ is

The function $f(x) = x + \frac{a^2}{x}$, $a > 0$, $x \neq 0$ has a local maxima at

The area (in sq. units) of the region enclosed by the curve $9x^2 + 4y^2 = 36$ is

If $x = a\sin 2t(1 + \cos 2t)$ and $y = b\cos 2t(1 - \cos 2t)$, then $(\frac{dy}{dx})_{\text{at } x=\frac{\pi}{4}}$ is equal to

If $\int \frac{dx}{(x-1)^3/^4. (x+2)^5/^4} = a[1 - g(x)]^b + c$, where $c$ is a constant of integration, then which of the following are true? (A) $a = \frac{2}{3}$ (B) $\beta = \frac{3}{4}$ (C) $3\alpha + 4\beta = 5$ (D) $g(x) = \frac{3}{(x+2)}$ Choose the **correct** answer from the options given below:

The solution of the differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is (Here C is an arbitrary constant)

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

$\int_0^{\pi/2} \frac{\sin^8 x}{\sin^8 x + \cos^8 x} dx$ is equal to

Consider the region bounded by the lines $y - 1 = x, x = -2, x = 3$ and $x$ - axis. Then (A) The area of the bounded region is given by $\int_{-2}^{3}(x + 1)dx$ (B) The numerical value of the area is $\frac{15}{2}$ sq. units (C) The numerical value of the area is 8 sq. units (D) The numerical value of the area is $\frac{17}{2}$ sq. units Choose the **correct** answer from the options given below:

Let $f(x)=\begin{cases} |x|+3 & \text{if } x\le -3 \\ -2x & \text{if } -3<x<3 \\ 6x+2 & \text{if } x\ge 3 \end{cases}$ Then, which of the following is true?

If the area of an equilateral triangle is increasing at the rate of $4\sqrt{3}$ cm²/sec, then the rate of increase of its perimeter when the side is 4cm, is

The number of arbitrary constants in the general solution of a differential equation of order 4 and degree 1 is

The function $f(x) = \log_e(\sin x), x \in (0, \pi)$ is (A) strictly increasing on $\left(0, \frac{\pi}{2}\right)$ (B) strictly decreasing on $\left(0, \frac{\pi}{2}\right)$ (C) strictly increasing on $\left(\frac{\pi}{2}, \pi\right)$ (D) strictly decreasing on $\left(\frac{\pi}{2}, \pi\right)$ (E) strictly increasing on $(0, \pi)$ Choose the correct answer from the options given below:

Match **List-I** with **List-II** | List-I | List-II | | :--- | :--- | | **Function** | **Points of discontinuity** | | (A) $f(x) = \frac{x^2 + 1}{x}$ | (I) $x = 4$ | | (B) $f(x) = \frac{\vert x - 1 \vert}{x - 1}$ | (II) $x = 2$ | | (C) $f(x) = \begin{cases} x - 1, & x < 2 \\ x + 1, & x \ge 2 \end{cases}$ | (III) $x = 0$ | | (D) $f(x) = \frac{1 - x}{(x - 4)}$ | (IV) $x = 1$ | Choose the **correct** answer from the options given below:

If $y = \left(x + \sqrt{x^2+1}\right)^m$, then $\frac{dy}{dx}$ is

In which of the following intervals, the function $f(x) = -x^2 - 2x + 15$ is decreasing?

The integral I = $\int \frac{e^{5\log_e x} - e^{4\log_e x}}{e^{3\log_e x} - e^{2\log_e x}} dx$ is equal to

Particular solution of the differential equation $x(1 + y^2)dx - y(1 + x^2)dy = 0$, given $y = 0$ when $x = 1$, is

The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)

Value of $\int \left(\frac{1}{logx} - \frac{1}{(logx)^2}\right)dx$ is

The function $f(x) = \frac{x - 2}{x + 1}, x \neq -1$ is increasing when (Where $\mathbb{R}$ is a set of real numbers)

The volume of spherical balloon is increasing at the rate of $4 \text{ cm}^3/ \text{sec}$. The rate of increase of its surface area, when the radius is 3cm will be :-

If $y = \sin^{-1}x$, then $(1-x^2)\frac{d^2y}{dx^2}$ is equal to

$\int \frac{1}{x(x^5-1)} dx$ is equal to