Calculus PYQ — Page 18

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Differential Equation** | **Degree** | | (A) $xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0$ | (I) 3 | | (B) $\frac{d^2y}{dx^2} + \log\left(\frac{dy}{dx}\right) = 0$ | (II) 1 | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + \frac{dy}{dx} + 1 = 0$ | (III) not defined | | (D) $2x^2\left(\frac{d^2y}{dx^2}\right)^3 - 5\left(\frac{dy}{dx}\right)^3 + y = 0$ | (IV) 2 | Choose the correct answer from the options given below:

The general solution of differential equation $\frac{dy}{dx} = e^{x+y}$ is

Match List-I with List-II | List-I | List-II | | --- | --- | | Function f(x) | Interval for increasing/decreasing of function f(x) | | --- | --- | | (A) $f(x) = x\vert x\vert $ | (I) Decreases on $(0, \infty)$ | | (B) $f(x) = x^2 + 2x - 5$ | (II) Increases on $(3, \infty)$ | | (C) $f(x) = x^2 - 6x + 9$ | (III) Decreases on $(-\infty, -1)$ | | (D) $f(x) = -x^2$ | (IV) Increases on $(-\infty, \infty)$ | Choose the correct answer from the options given below:

Value of $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \log(\tan x)dx$ is

Area (in sq. units) of the region bounded by the curve $y^2 = 4x$, $y$-axis and the line $y = 3$ is

Value of $\int_2^3 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{5-x}}dx$ is

The value of $\int_1^3 \frac{x^2}{x^3+1}dx$

Match List-I with List-II | List-I | List-II | |---|---| | Differential Equations | Order and degree | | (A) $ydx + x\log(y/x)dy - 2xdy = 0$ | (I) Order : 2, degree:1 | | (B) $\left(\frac{d^3y}{dx^3}\right)^2 + 3\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^4 = y^2$ | (II) Order :1, degree:1 | | (C) $\frac{dy}{dx} + \log\left(\frac{dy}{dx}\right) + x = y$ | (III) Order : 3, degree:2 | | (D) $\left(\frac{ds}{dt}\right)^4 + 2s\frac{d^2s}{dt^2} = 0$ | (IV) Order : 1, degree: Not defined | Choose the correct answer from the options given below:

Match List-I with List-II $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) The minimum value of } f(x) = (2x - 1)^2 + 3 & \text{(I) } 4 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) The maximum value of } f(x) = -|x + 1| + 4 & \text{(II) } 10 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) The minimum value of } f(x) = \sin(2x) + 6 & \text{(III) } 3 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) The maximum value of } f(x) = -(x - 1)^2 + 10 & \text{(IV) } 5 \\[1.2ex] \hline \end{array}$ Choose the correct answer from the options given below:

The rate of change of area of a circle with respect to its circumference when radius is 4cm, is

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

The sum of order and degree of the differential equation $(x^2\frac{d^2y}{dx^2})^{3/4} = 5(\frac{dy}{dx})^2 - 3$ is equal to

If $f(x) = a \log_e|x| + bx^2 + x$ has critical points at $x = -2$ and $x = 1$, then

If $y = \sin^{-1} x + \sin^{-1} \sqrt{1-x^2}, x \in (-1, 0)$, then $\frac{dy}{dx}$ is equal to

Consider the function $f(x) = \sin x$ in the interval $[\pi, 2\pi]$ then which of the following statements are correct? (A) $x = \frac{3\pi}{2}$ is its stationary point. (B) Its maximum value is 1 (C) Its minimum value is -1 (D) It attains its maximum value at $\pi$ and $2\pi$ Choose the **correct** answer from the options given below:

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

The equation of tangent line to $y = 2x^2 + 7$, which is parallel to the line $4x - y + 3 = 0$ is

The function $f(x) = x^3 + 3x^2 + 4x + 4$, $x \in \mathbb{R}$ (set of real numbers) :

The real valued function $f(x) = 12x^\frac{4}{3} - 6x^\frac{1}{3}, x \in [-8, 8]$ has absolute maximum value equal to

The function $f(x) = [x]$, where $[x]$ denotes the greatest integer function, is continuous at $x =$ (A) 2.9 (B) 5 (C) -3 (D) 6.5 Choose the correct answer from the options given below:

The Value of $\int_1^3 |2x - 1|dx$ equal to

Let $x = t^2, y = t^3$. Then $\frac{d^2y}{dx^2}$ is equal to

$\int \left(\frac{\cos x - \sin x}{1 + \sin 2x}\right) dx$ is equal to

The sides of an equilateral triangle are increasing at the rate of 5 cm/sec. The rate at which the area increases when the side is 20 cm, is