Calculus PYQ — Page 27

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 :

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

The two curves $x^3 - 3xy^2 + 15 = 0$ and $3x^2 y - y^3 + 17 = 0$ :

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

If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then which of the following can be a discontinuous function ?

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to :

The differential equation whose solution is $Ax^2 + By^2 = 1$ where A and B are arbitrary constant is of : (A) first order and first degree (B) second order and first degree (C) second order and second degree (D) second order Choose the correct answer from the options given below :

The condition on a and b, such that for $y = \frac{a}{x} - \frac{b}{x^2}$, $\frac{dy}{dx} = 0$ at $x=1$ is :

The derivative of $\sin(\tan^{-1} e^{2x})$ with respect to $x$ is:

The maximum value of $2x^3 - 24x + 107$ in the interval $[1, 3]$ is :

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

Integrating factor of the differential equation $(1 - y^2)\frac{dx}{dy} + xy = ay$ is :

If $x = a\left(t - \frac{1}{t}\right)$, $y = b\left(t + \frac{1}{t}\right)$, then $\frac{dy}{dx} =$

If $\cos y = x\cos(a+y)$, then $\frac{dy}{dx} = $

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:

The integral $\int_{0}^{1} x(1-x)^n dx$ is equal to :

Let $y = \log_e \left(\frac{a + b \sin x}{a - b \sin x}\right)$, then value of $\frac{dy}{dx}$ is :

The intervals for which $f(x) = x^4 - 2x^2$ is increasing are :

The solution of $y' - y'' = 2x$ is: A. $y = x^2 + 2x + 2$ B. $y = x^2 + 2x + 1$ C. $y = x + 2$ D. $y = x^2 - 2x + 1$ Choose the correct answer from the options given below:

Area of the region bounded by $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is :

The degree of the differential equation $\left[1 + \left(\frac{dy}{dx}\right)\right]^3 = \left(\frac{d^2 y}{dx^2}\right)^2$ is :

The value of C which satisfies Rolle's Theorem for $f(x) = \sin^4 x + \cos^4 x$ in $\left[0, \frac{\pi}{2}\right]$. Then C is :

The value of C, in Rolle's theorem for the function $f(x) = e^x \sin x$, when $x \in [0, \pi]$ is :

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