Calculus PYQ — Page 3

Match List-I with List-II | List-I | List-II | |---|---| | **Differential equation** | **Degree** | | (A) $\frac{d^2y}{dx^2} + \sqrt{\frac{dy}{dx}} - y = 0$ | (I) 6 | | (B) $\sqrt{\frac{d^3y}{dx^3}} - \sqrt[12]{\frac{d^2y}{dx^2}} = 0$ | (II) Not defined | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} + e^{\frac{dx}{dx}} = x^2$ | (III) 3 | | (D) $\sqrt[3]{\frac{dy}{dx}} - \frac{d^2y}{dx^2} = e^x$ | (IV) 2 | Choose the correct answer from the options given below:

Which of the following are NOT correct regarding the equation of tangent and normal to the curve $y = \frac{x-11}{(x-2)(x-3)}$ at the point, where it cuts the $x$-axis? (A) The point of contact is (11, 0). (B) The equation of tangent is $x - 72y - 11 = 0$ (C) The equation of normal is $72x + y - 11 = 0$ (D) The slope of the tangent at the given point of contact is $\frac{1}{88}$ Choose the **correct** answer from the options given below:

If $y = \frac{1}{\sqrt[3]{1-x^3}}$ then $\frac{dy}{dx}$ is equal to

Let $[x]$ denote the greatest integer $\leq t$ and $a \mathbb{Z} = [ax: x \in \mathbb{Z}, a \in \mathbb{R}]$ (where $\mathbb{Z}$ is set of integer and $\mathbb{R}$ is set of real number). The set of points of discontinuity of the function $f(x) = [2x]$ is given by

$\int e^{-x}(\cot x + \cosec^2 x)dx =$

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{dy}{dx}$ at $ \theta = \frac{\pi}{3}$ is

The demand function for a certain product is represented by the equation: $p = 20 + 5x - 3x^2$, where $x$ is the number of units demanded and $p$ is the price per unit (in Rs.), then the marginal revenue when 2 units are sold is:

Curd is at 80° F, five minutes later it came down at 60°F. After another 5 minutes, its temperature became 50° F. Given that the rate of change of temperature is proportional to (T - S), where S is temperature of the surroundings and T is temperature of the curd at any time t. Then the temperature of the surroundings is :

The function $f(x) = kx^3 + 6kx^2 + 18x + 17$ is increasing on $\mathbb{R}$(set of real numbers) if:

If $y = ax^2 + bx$ has minima at $x = 2$ and the minimum value is -12, then which of the following are correct? (A) $a = 3$ (B) $a = -3$ (C) $b = 12$ (D) $b = -12$ Choose the correct answer from the options given below:

The value of the definite integral $I = \int_{1}^{2} \frac{1}{x(1 + x^2)}dx$ is:

The solution of the differential equation $\frac{dr}{dt} = -rt, r(0) = r_0$ is

The integrating factor of the differential equation $(x log_e x)\frac{dy}{dx} + y = 2log_e x$ is

The area (in sq. units) of the region bounded by $y = 2\sqrt{1 - x^2}$, $x \in [0, 1]$ and $x$-axis is equal to

$\int_{\pi/6}^{\pi/3} \frac{tan x}{tan x + cot x} dx$ is equal to

Match **List-I** with **List-II** The function $f(x) = 2x^3 - 15x^2 + 36x + 5$ for $x \in [2,5]$ has | List-I | List-II | |---|---| | (A) absolute maximum value | (I) 5 | | (B) absolute minimum value | (II) 60 | | (C) point of absolute maxima | (III) 3 | | (D) point of absolute minima | (IV) 32 | Choose the **correct** answer from the options given below:

The total cost $c(x)$ associated with the production of $x$ units of an item is given by $c(x) = 0.001x^3 + 0.06x^2 + 20x + 500$. The marginal cost when 10 units are produced is:

General solution of the differential equation $\frac{dy}{dx} = e^{\frac{x^2}{2}} + xy$ is

Match List-I with List-II | List-I | List-II | | --- | --- | | Function | Derivative | | --- | --- | | (A) $y = \sin^{-1} x + \sin^{-1} \sqrt{1 - x^2}; \vert x\vert < 1$ | (I) $\frac{dy}{dx} = \frac{1}{2y-1}$ | | (B) $y = \sqrt{x + y}, x+y > 0 \text{ and } y \neq \frac{1}{2}$ | (II) $\frac{dy}{dx} = 10^x \log_e 10$ | | (C) $y = \log_{10} x, x > 0$ | (III) $\frac{dy}{dx} = 0$ | | (D) $y = 10^x$ | (IV) $\frac{dy}{dx} = \frac{1}{x \log_e 10}$ | Choose the correct answer from the options given below:

Match **List-I** with **List-II**. Here [x] denotes the greatest integer function $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = [x] & \text{(I) is continuous everywhere but not differentiable at } x=-1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x-1| & \text{(II) is continuous everywhere except at all integral values} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = e^{|x|} & \text{(III) is continuous everywhere but not differentiable at } x=1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x+1| & \text{(IV) is continuous everywhere but not differentiable at } x=0 \\[1.2ex] \hline \end{array}$ Choose the **correct** answer from the options given below:

$\int_0^2 x(2-x)^n dx$ is equal to

If $f(x) = x^3\log_e x$, Then $f''(e^2)$ is equal to

$\int_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a-x}} dx$ is equal to

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{1}{\sin^2 x + \sin 2x} dx$ is equal to