Calculus PYQ — Page 26

The area of the region bounded by the parabola $y^2 = 4ax$ and its latus rectum is:

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

The area enclosed by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is given by :

The area enclosed by the ellipse $\frac{x^2}{9^2} + \frac{y^2}{6^2} = 1$ is:

The area enclosed between the curves $y = x^2$ and $x = y^2$ is

The area enclosed between the curve $y = x^2 + 2$ and x-axis between $x = 0$ and $x = 3$ is :

The area enclosed between the curve $x^2 + y^2 = 16$ and the coordinate axes in the first quadrant is :

The area enclosed between $y^2 = 4x$, $x = 1$, $x = 4$ in first quadrant is :

The approximate volume of a cube of side a meters on increasing the side by 4% is:

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

The angle of intersection between the curves $y = 4 - x^2$ and $y = x^2$ is :

Solution of $\frac{dy}{dx} = (1+x^2)(1+y^2)$ is:

Solution of differential equation $x dy - y dx = 0$ respresents :

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =$

$\int e^x (\tan x + \log_e \sec x) \, dx =$

$\int \left(\frac{1+x+x^2}{1+x^2}\right) e^{\tan^{-1} x} dx =$

$\int_1^2 \frac{x \, dx}{(x+1)(x+2)} =$

Points of discontinuity of the greatest integer function $f(x) = [x]$, where $[x]$ denotes integer less than or equal to $x$, are

Particular solution of the differential equation $\log\left(\frac{dy}{dx}\right) = x + y$, given that when $x = 0, y = 0$ is:

Match List I with List II | LIST I | LIST II | |---|---| | A. Maximum value of $f(x) = -\lvert x+1 \rvert + 3$ | I. 6 | | B. Minimum value of $f(x) = (2x-1)^2 + 5$ | II. 5 | | C. Maximum value of $f(x) = 6 - x^2$ | III. no maximum value | | D. Maximum value of $f(x) = x^3 + 1$ | IV. 3 | Choose the correct answer from the options given below:

Match List I with List II | LIST I | LIST II | |---|---| | A. $\int \frac{\sin x}{1 + \cos x} \, dx$ | I. $e^{\tan^{-1} x} + C$ | | B. $\int \frac{1}{1 - \tan x} \, dx$ | II. $\log(\log x + 1) + C$ | | C. $\int \frac{e^{\tan^{-1} x}}{1 + x^2} \, dx$ | III. $-\log\lvert 1+\cos x \rvert + C$ | | D. $\int \frac{1}{x + x \log x} \, dx$ | IV. $\frac{x}{2} - \frac{1}{2}\log\lvert \cos x - \sin x \rvert + C$ | Choose the correct answer from the options given below:

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $x = 2at^2, y = at^4$ | (I) | Inverse trignometric function | | (B) | $f(x) = (2x + 3)^3$ | (II) | Implicit function | | (C) | $xy + y^2 = \tan(x + y)$ | (III) | Parametric function | | (D) | $y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right), -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}$ | (IV) | Composite function | Choose the **correct** answer from the options given below :

Match List - I with List - II. Match the integrating factors : | List - I (Differential Equation) | List - II (Integrating factor) | |---|---| | (A) $\frac{dy}{dx} + 3y = e^{-2x}$ | (I) $\frac{1}{x}$ | | (B) $x\frac{dy}{dx} + y = 3x^2$ | (II) $e^{-x}$ | | (C) $x\frac{dy}{dx} - y = 3x^2$ | (III) $x$ | | (D) $\frac{dy}{dx} - y = x$ | (IV) $e^{3x}$ | Choose the correct answer from the options given below :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $y = \log(\sin x)$ | (I) | $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$ | | (B) | $y = e^{(1 + \log x)}$ | (II) | $\frac{d^2y}{dx^2} = 2$ | | (C) | $y = \log\lvert x \rvert$ | (III) | $\frac{d^2y}{dx^2} = 0$ | | (D) | $y = x^2 + 4x - 1$ | (IV) | $\frac{d^2y}{dx^2} = -\csc^2 x$ | Choose the **correct** answer from the options given below :