Calculus PYQ — Page 14

If $\frac{d}{dx}[ax^3 + ax^2 + ax + 1] = 9x^2 + 6x + 3$, then $a$ is equal to

The particular solution of the differential equation $e^x\sqrt{1-y^2}dx + \frac{y}{x}dy = 0$, given that $y = 1$, when $x = 0$ is:

Which one of the following equations is a homogeneous differential equation?

Match List-I with List-II (Given that $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \frac{dx}{\sqrt{a^2 - x^2}} =$ | (I) $\log_e \vert x + \sqrt{x^2 - a^2}\vert + c$ | | (B) $\int \sqrt{a^2 - x^2} dx =$ | (II) $\sin^{-1} \frac{x}{a} + c$ | | (C) $\int \sqrt{x^2 - a^2} dx =$ | (III) $\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + c$ | | (D) $\int \frac{dx}{\sqrt{x^2 - a^2}} =$ | (IV) $\frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log_e \vert x + \sqrt{x^2 - a^2}\vert + c$ | Choose the correct answer from the options given below:

If the integral $I = \int \left[log_e(log_e x)^2 + \frac{a}{log_e x}\right] dx = x log_e(log_e x)^2 + C$, where C is constant of integration. Then the value of $a$ is:

The number of tangents to the curve $xy - 3y + 2 = 0$ having slope 2 is:

The area (in sq. units) of the region $\{(x,y): 3x^2 \leq y \leq |x|\}$ is equal to

If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

If the maximum value of the function $f(x) = \frac{\log_ex}{x}$, $x > 0$ occurs at $x = a$, then $a^2f''(a)$ is equal to

$\int_1^4 |x - 2|dx$ is equal to

Which of the following are linear first order differential equations? (A) $\frac{dy}{dx} + P(x)y = Q(x)$ (B) $\frac{dx}{dy} + P(y)x = Q(y)$ (C) $(x - y)\frac{dy}{dx} = x + 2y$ (D) $(1 + x^2)\frac{dy}{dx} + 2xy = 2$ Choose the correct answer from the options given below:

If $e^y = \log x$, then which of the following is true?

Match List-I with List-II $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = |x| & \text{(I) Not differentiable at } x=-2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x+2| & \text{(II) Not differentiable at } x=0 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = |x^2-4| & \text{(III) Not differentiable at } x=2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x-2| & \text{(IV) Not differentiable at } x=2,-2 \text{ only} \\[1.2ex] \hline \end{array}$ Choose the correct answer from the options given below:

The function $f(x) = tanx - x$

The function $f(x) = x^2 - 4x + 6$ is (A) Strictly decreasing on $(-\infty, 2) \cup (2, \infty)$ (B) Strictly increasing on $(2, \infty)$ (C) Strictly increasing on $(-\infty, \infty)$ (D) Strictly decreasing on $(-\infty, 2)$ Choose the correct answer from the options given below:

The equation of the tangent line to the curve $y = x^2 - 2x + 5$ which is parallel to the line $4x - y + 1 = 0$ is

$\int \frac{dx}{(1+5\sin^2 x)}$ is equal to

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Function** | **Increasing on the interval** | | (A) $f(x) = -x^2 - 2x + 1$ | (I) $(-\infty, -1)$ | | (B) $f(x) = x^2 + 1$ | (II) $(1, \infty)$ | | (C) $f(x) = x^2 - 2x + 3$ | (III) $(-\infty, 0)$ | | (D) $f(x) = -x^2$ | (IV) $(0, \infty)$ | Choose the correct answer from the options given below:

For what value of $\alpha$, the function $f$ defined by $f(x) = \begin{cases} \alpha(x^2 - 2x + 1), & \text{if } x \leq 0 \\ 2x + 1, & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$?

$\int_0^{\pi/2} \sqrt{1 - \sin 2x}\,dx$ is equal to:

The equation of the tangent to the curve $y = \frac{(x - 3)}{(x - 1)(x - 2)}$ at the point, where it cuts x-axis is:

The area (in sq. units) of the region bounded by the curve $y = \sqrt{16-x^2}$ and x-axis is

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $f(x) = x \sin x$ | (I) is not continuous at $x = -3$ | | (B) $f(x) = \frac{\vert x\vert }{x}, x \neq 0$ and $f(x) = 1 \text{ at } x = 0$ | (II) is continuous everywhere | | (C) $f(x) = x - [x]$, $[x]$ denotes greatest integer function | (III) is not differentiable at $x = 1$ | | (D) $f(x) = e^{\vert x - 1\vert }$ | (IV) is not continuous at $x = 0$ | Choose the correct answer from the options given below:

The value of $\int_{-a}^a f(x)dx$ where $f(x) = \frac{7^x}{1+7^x}$ is: