Calculus PYQ — Page 33

$\int_0^{4\pi} \frac{x}{1 + |\cos x|} dx =$

The value of $\int_{\pi/2}^{\pi} \frac{1}{1 + \cot x} dx$ is equal to:

If $y = \frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{x+1} - \sqrt{x-1}}$, then $(x^2-1)^{3/2} \frac{d^2y}{dx^2} =$

If $f'(x) = 4x^5 - 6x$ and $f(0) = 3$, then $f(3)$ is equal to

$\int \frac{x}{(x-1)(x-2)} dx =$ (where c is an arbitrary constant)

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$, then the value of k is :

$\int \frac{dx}{\sin^2 x \cos^2 x} =$

If $x = at^2$ and $y = 2at$, then the value of $\frac{d^2y}{dx^2}$ is ( where t is a parameter )

Consider the function $f(x) = x^{\frac{1}{x}}$. Its

The maximum height (in meters) achieved in the first jump is

Let $\int \frac{dx}{\left(\sqrt{x} - \sqrt{x-1}\right)^2} = \alpha u(x) + \beta v(x) + C$, where $u(x) = x^2 - x + \left(x - \frac{1}{2}\right)\sqrt{x^2 - x}$ and $v(x) = \log\left|x - \frac{1}{2} + \sqrt{x^2 - x}\right|$. The value of $\alpha + \beta$ is :

$\int \frac{x^2 + 4}{(x^2 + 3)(x^2 + 5)} \, dx = u \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + v \tan^{-1}\left(\frac{x}{\sqrt{5}}\right) + c$, where c is arbitrary constant, then value of $\frac{1}{u^2} + \frac{1}{v^2}$ is equal to :

The equation of the normal to the curve $y = x - \frac{1}{x}$ at $(1,0)$ is:

The area of the smaller region bounded by the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and the line $\frac{x}{3} + \frac{y}{2} = 1$ is :

The order and the degree of the differential equation $\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ are respectively:

The area of the region bounded by $f(x) = x|x|$, x-axis and from $x = -1$ to $x = 1$ is :

The integrating factor of the differential equation $x\frac{dy}{dx} - y = 2x^2$ is :

$\int e^x \left(\frac{1}{x} - \frac{2}{x^3}\right) dx =$

The area of the region bounded by $f(x) = x|x|$ and x-axis from $x = 0$ to $x = 4$ is :

Choose the correct statements: A. The order and degree (if defined) of a differential equation are always positive integrals B. The order of a differential equation is the highest order derivative of the dependent variable with respect to the independent variable involved in a differential equation C. If $\frac{dy}{dx} + P(x)y = Q(x)$ then Integrating factor $= e^{\int P(x)dx}$ D. The sum of order and degree of differential equation $1 + (y'')^5 = (y''')^3$ is $8$ E. If the solution of a differential equation of order $n$, contains $n$ arbitrary constant, then it is called a general solution Choose the correct answer from the options given below:

$y(x)$ is strictly increasing in the interval

$\int \frac{x^2 - 4}{(x+2)(x-1)(x-3)} dx =$

The minimum value of $x^2 - 8x + 17$ on the set $\mathbb{R}$ of all real numbers is:

The maximum slope of the tangents to the curve $y(x) = -x^3 + 3x^2 + 9x - 30$ is