Calculus PYQ — Page 20

Area of the region bounded by the curve $y = \sin x$ and x-axis between $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$ is

If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

The particular solution of the differential equation $\frac{dy}{dx} = e^{x^2/2} + xy$, when $x = 0$, $y = 1$, is

Consider the function $f(x) = x^3 - 3x$. Then Match List-I with List-II | List-I | List-II | |---|---| | (A) Point of local Maxima | (I) 1 | | (B) Point of local Minima | (II) -1 | | (C) Local maximum value | (III) 2 | | (D) Local minimum value | (IV) -2 | Choose the correct answer from the options given below:

For $y \neq 0$, the particular solution of the differential equation $2ye^{x/y}dx + (y - 2xe^{x/y})dy = 0$ at the point (1, 1) is

The area (in sq. units) of the region bounded by the curve $x^2 = 250y$, $y = 0$ and $x = 50$ is

The function $f(x) = \frac{x}{3} + \frac{3}{x}$ is increasing in the interval:

Match List-I with List-II Consider the function f(x) = 2x³ - 21x² + 36x + 80, x∈[0, 6]. Then | List-I | List-II | |---|---| | (A) one of its critical points is at x = | (I) -28 | | (B) Its absolute maximum value is | (II) -42 | | (C) Its absolute minimum value is | (III) 97 | | (D) Its second derivative at x = 0 is | (IV) 6 | Choose the correct answer from the options given below:

$\int_0^1 \frac{dx}{\sqrt{1+x} - \sqrt{x}}$ is equal to

The general solution of the differential equation $\frac{dy}{dx} = xy + x + y + 1$ is

The integral $\int \frac{2dx}{e^{2x}-1}$ is equal to:

The area of the region bounded by the line $y = 2x$ and the x-axis between $x = -2$ and $x = 2$ is

$\int \frac{f'(x)}{f(x) \log_e[f(x)]} dx$ is equal to

If the function $f(x) = 2x^2 - kx + 7$, is increasing on $[1,2]$, then $k$ lies in the interval

$\int_0^8 (x^{\frac{2}{3}} + 1) dx$ is equal to

If the function defined by $f(x) = \begin{cases} \ kx^2 + 1, & \text{if } x \le 1 \\ 2 , & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then k is equal to

If $x = \frac{a}{1 + t}$ and $y = \frac{a}{(1 + t)^2}$ where $a > 0$ , then $\frac{d^2y}{dx^2}$ at $t = 1$ is

The area (in square units) bounded by the curve $y = \cos x$ between $x = 0$ and $x = 2\pi$ in first quadrant is equal to:

For the function $f(x) = 2x^3 - 3x^2 - 12x + 5$, the difference of maximum and minimum value of $f(x)$ is

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) The maximum value of $f(x) = \sin(3x) + 6$ | (I) 2 | | (B) The maximum value of $f(x) = -\vert x + 2\vert + 4$ | (II) 5 | | (C) The minimum value of $f(x) = (3x + 1)^2 + 5$ | (III) 7 | | (D) The minimum value of $f(x) = 2 \cos x + 4$ | (IV) 4 | Choose the correct answer from the options given below:

If $x = 4t$ and $y = \frac{4}{t}$, then $\frac{d^2y}{dx^2}$ is

$\int_{-1}^{1}(x^7 + x^5 + x^3 + x + 1)dx$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | **Differential Equation** | **Order and Degree** | | (A) $\left(\frac{d^2y}{dx^2}\right)^2 = e^x\left(\frac{dy}{dx}\right)^4 + 1 = 0$ | (I) order = 1 and degree = 2 | | (B) $\left(\frac{dy}{dx}\right)^2 + xy = 0$ | (II) order = 2 and degree = 1 | | (C) $\left(1 + \frac{dy}{dx}\right)^{3/2} = 4\left(\frac{d^2y}{dx^2}\right)^2$ | (III) order = 2 and degree = 2 | | (D) $\sqrt\frac{d^2y}{dx^2} + 1 = \frac{dy}{dx}$ | (IV) order = 2 and degree = 4 | Choose the correct answer from the options given below:

The area of the region bounded by the parabola $y^2 = 8x$ and its latus rectum in the first quadrant, is