Calculus PYQ — Page 19

For the differential equation $x\frac{dy}{dx} + 2y = x^2\log_e x$ (A) Integrating factor is $2x$ (B) Integrating factor is $x^2$ (C) General Solution is $y = \frac{x^2}{16}(4\log_e|x| - 1) + Cx^{-2}$ Where C is an arbitrary constant. (D) General Solution is $y = \frac{x^4}{16}(4\log_e|x| - 1) + C$ Where C is an arbitrary constant. Choose the correct answer from the options given below:

$\int \frac{x}{(x-1)(x-2)} dx$ is equal to ( where $C$ is a constant of integration)

If $y = \sin^{-1} \sqrt\frac{x}{x+1} + \sec^{-1}\sqrt{\frac{x+1}{x}}$, then $\frac{dy}{dx}$ is

If $\int \frac{x^4}{x-2}dx = px + qx^2 + rx^3 + sx^4 + t\log |x - 2| + C$, where C is an arbitrary constant and p, q, r, s, t are real numbers, then the correct arrangement of p, q, r, s, t is:

For $|x| < 1$, if $x = \cos\left(\frac{1}{a}\log y\right)$, then

If $y = \log_e(\sec e^{x^2})$ then $\frac{dy}{dx} =$

The solution of the differential equation $log_e\left(\frac{dy}{dx}\right) = 3x + 4y$ is given by

The area of the region $\{(x, y): x^2 + y^2 \leq 1 \leq x + y\}$ is

The area (in sq. units) of the bigger portion of region enclosed by the curves $4x^2 + 9y^2 = 36$ and $2x + 3y = 6$ is

Consider a closed cylinder of radius $r$ with a fixed surface area. The volume of the cylinder is maximum when its height is

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 + 26x^2 + 15x + 400$. The marginal cost when 10 items are produced is:

The interval in which the function $g(x) = x^2 e^{-x}$ is increasing is:

If $y = \sqrt{ax + b}$ then $y\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 =$

If $f(x) = \sin x - \cos x$, $x \in [0, 2\pi]$ then (A) $f(x)$ is increasing in $(0, \frac{3\pi}{4})$ (B) $f(x)$ is decreasing in $(0, \frac{3\pi}{4})$ (C) $f(x)$ is decreasing in $(\frac{3\pi}{4}, \frac{7\pi}{4})$ (D) $f(x)$ is decreasing in $(\frac{7\pi}{4}, 2\pi)$ Choose the correct answer from the options given below:

$\int_0^1 tan^{-1}\left(\frac{2x-1}{1+x-x^2}\right)dx$ is equal to

Which of the following functions $f(x)$ are differentiable at $x = 0$? (A) $|x|$ (B) $|x - 1|$ (C) $|\sin x|$ (D) $|\cos x|$ (E) $x^2$ Choose the correct answer from the options given below:

The general solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = ax + by$ is

Differentiation of $\log[\log(\log x^5)]$ with respect to $x$ is

The function $f(x) = x^2e^{-2x}$ increases on

For the function $f(x) = e^{-2x}(2-x)^2$, the point of local maxima is:

If $x = a\cos\alpha + b\sin\alpha$ and $y = a\sin\alpha - b\cos\alpha$, then $\left(x\frac{dy}{dx} - y^2\frac{d^2y}{dx^2}\right)$ is equal to:

The area (in sq. units) of the region bounded by the line $y = x + 2$, $x = 0$, $x = 1$ and $y = 0$ is

If $x = \frac{1-t}{1+t}$ and $y = \frac{3t}{1+t}$, then $\frac{d^2y}{dx^2}$ is equal to

$\int_1^{\sqrt{3}} \frac{1}{1+x^2} dx$ is equal to: