Calculus PYQ — Page 30

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

If D is the distance between Sumit and Amit, then the value of $x$ for which D is minimum, is :

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

The area (in square units) bounded by the curve $y^2 = 4x$ and the line $x=1$ is :

The value of $\int_{-3}^{3} \log_e\left(\frac{4+x}{4-x}\right) dx$ 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

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

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 :

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

Sumit's position, when $x = 10$ is :

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

$\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx =$

The position of Sumit when Amit will hit the paper ball is :

Minimum value of D is :

Let $y = \log(x + \sqrt{x^2+1})$, and $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} = c$. Then identify the correct statements about the values of $a$, $b$ and $c$ : (A) $a = 1 + x^2$ (B) $b = 0$ (C) $c = 0$ (D) $b = x$ (E) $c = 2$ Choose the correct answer from the options given below :

$\int \frac{x^2+1}{(x+1)^2} e^x \, dx$ is equal to

Match List I with List II. | List I | List II | |---|---| | A. $\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$ | I. order 3, degree 1 | | B. $\left(\frac{d^2y}{dx^2}\right)^2 = 0$ | II. order 2, degree 2 | | C. $\frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y = 0$ | III. order 2, degree 1 | | D. $\sin\left(\frac{dy}{dx}\right) + 5y = 0$ | IV. order 1, degree is not defined | Choose the correct answer from the options given below:

Match List I with List II | List I | List II | |---|---| | A. If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then $\frac{g(x)}{f(x)}$ is | I. discontinuous at exactly three points. | | B. The function $f(x) = \frac{4-x^2}{4x-x^3}$ is | II. continuous everywhere | | C. The function $f(x) = \lvert x \rvert + \lvert x-1 \rvert$ is | III. discontinuous at $x = 0$. | | D. The function $f(x) = \lvert \sin x \rvert$ is | IV. continuous at $x = 0$ and $x = 1$ | Choose the correct answer from the options given below:

The function $f(x) = |x - 1|$ is

Match List I with List II | List I | List II | |---|---| | A. $\int \frac{dx}{x^2 - a^2}$ is equal to | I. $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | | B. $\int \frac{dx}{\sqrt{x^2 + a^2}}$ is equal to | II. $\frac{1}{2a}\log\left\lvert \frac{x-a}{x+a}\right\rvert + C$ | | C. $\int \sqrt{a^2 - x^2} \, dx$ is equal to | III. $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$ | | D. $\int \sqrt{a^2 + x^2} \, dx$ is equal to | IV. $\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$ | Choose the correct answer from the options given below:

The given function $f(x) = x^5 - 5x^4 + 5x^3 - 1$; has/have (a) local maxima at $x = 1$ (b) local maximum value is 0 (c) local minimum at $x = 3$ (d) local minimum value is $-28$ (e) The point of inflexion is $x = 1$ Choose the correct answer from the options given below

The integrating factor of the differential equation $\cos x \frac{dy}{dx} + y\sin x = 1$ is

The order and degree of the differential equation $\left[\left(\frac{d^2y}{dx^2}\right)^2 - 3\right]^{\frac{1}{3}} = 2\left(\frac{dy}{dx}\right)^{\frac{1}{4}}$ are