Calculus PYQ — Page 29

$\int_0^1 \frac{dx}{x^2 + x + 1}$

If the order and degree of the differential equation $\sqrt{\frac{d^2y}{dx^2}} = \left(1 + \frac{dy}{dx}\right)^{\frac{1}{3}}$ are $a$ and $b$ respectively, then the value of $a^2 + b^2$ is

An energy DRONE is flying along the curve $y = x^2 + 7$. A soldier is placed at $(3, 7)$. The nearest distance of the DRONE from soldier's position is

The differential equation representing family of curves $y = ae^{mx} + be^{nx}$, where $a$ and $b$ are arbitrary constants, is

The absolute maximum value of $y = x^3 - 3x + 2$, $0 \leq x \leq 2$, is

If $g(x) = \int \frac{dx}{x^{1/2} + x^{1/6}}$, then $g(1) - g(0)$ is :

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

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:

Match list I with list II | List - I | List - II | |---|---| | A. Slope of tangent to the curve $y = x^3 - x$ at $x = 2$ | I. $-2$ | | B. Slope of tangent to the curve $y = 3x^3 - 4x$ at $x = 0$ | II. $11$ | | C. Slope of normal to the curve $y = \sin\theta$ at $\theta = \frac{\pi}{3}$ | III. $2$ | | D. Slope of normal to the curve $y = \cos\theta$ at $\theta = \frac{\pi}{6}$ | IV. $-4$ | Choose the correct answer from the option given below :

Let $y = m\sin rx + n\cos rx$. What is the value of $\frac{d^2y}{dx^2}$?

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

The maximum value of $x^{-x}$ is

If the order and the degree of the differential equation $\left(\frac{dy}{dx}\right)^{\frac{1}{2}} = \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{5}}$ are O and S respectively, then $S - O$ is equal to

The area bounded by the curve $x^2 = 4y$ and the line $x = 4y - 2$ is :

Integrating factor of the differential equation $\frac{dy}{dx} - \frac{1}{x} y = 1$ is:

Based on above information answer the following question : $\frac{dA(x)}{dx} =$

Order and degree of the differential equation $y\frac{dy}{dx} + \frac{4}{\frac{dy}{dx}} = 5$ are

$\int \frac{xe^x}{(x+1)^2} dx =$

The interval in which the function, $f(x) = 7 - 4x - x^2$ is strictly increasing is

The area (in square units) of minor segment of the circle $x^2 + y^2 = 25$ cut off by the line $x = \frac{5}{2}$ is

If $x = 2\sin\theta$ and $y = 2\cos\theta$, then the value of $\frac{d^2y}{dx^2}$ at $\theta = 0$ is

Consider the differential equation $\frac{dy}{dx} = \frac{y+1}{x+1}$, and $y=0$ when $x=2$. The value of $y$ at $x=3$ is :

$\int \frac{dx}{(e^x - 1)} =$

The tangent to the curve $y = e^{3x}$ at the point $(0, 1)$, meets the x-axis at :