Calculus PYQ — Page 10

The general solution of the differential equation $e^x dy + (y e^x + 2x)dx = 0$ is

$\int \frac{dx}{9x^2 - 16}$ is equal to

Area (in sq. units) of the region bounded by curves $y^2 = x$ and $x = 4$ is

Match List-I with List-II The function $f(x) = (x - 1)(x + 1)^2$ has | List-I | List-II | |---|---| | (A) A local maxima at $x = $ ____ | (I) $\frac{1}{3}$ | | (B) A local minima at $x = $ ____ | (II) 0 | | (C) The local minimum value of $f(x) = $ ____ | (III) -1 | | (D) The local maximum value of $f(x) = $ ____ | (IV) $-\frac{32}{27}$ | Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Integrating factor of $xdy - (y + x^2)dx = 0$ | (I) $x^2$ | | (B) Integrating factor of $xdy + (2y + x^2)dx = 0$ | (II) $x^3$ | | (C) Integrating factor of $(3y - x^2)dx + xdy = 0$ | (III) $x$ | | (D) Integrating factor of $(y + 3x^2)dx + xdy = 0$ | (IV) $\frac{1}{x}$ | Choose the correct answer from the options given below:

If $f(x) = 2x^3 - 15x^2 + 36x + 1$, $x \in [1, 5]$, then the absolute minimum value of $f(x)$ is:

If $f(x) = \begin{cases} mx + 1,\ x \geq \pi/2 \\sin x + n, x \leq \pi/2, & \end{cases}$ is continuous at $x = \pi/2$, where $m \in \mathbb{Z}$ (set of integers), then $\sin 2n =$

Function $f(x) = x^3 - 3x + 3$ is (A) Increasing in the interval $(-1, 1)$ (B) Increasing in the interval $(1, \infty)$ (C) Decreasing in the interval $(-1, 1)$ (D) Increasing in the interval $(-\infty, -1) \cup (1, \infty)$ Choose the correct answer from the options given below:

In which of the following interval the function $f(x) = x^x, x > 0$ is strictly increasing?

Area of the bounded region between the curve $y = |x - 2|$ and the line $y = 2$ is:

The area enclosed between the graph of y = x³ and the lines x = 0, y = 1, y = 8 is

If it is given that at $x = 1$, the function $f(x) = x^4 - 62x^2 + 2ax + b$ attains its maximum value on the interval [0, 2], then the value of a is:

If $y = 3^x + e^x + x^x + x^3$, then the value of $\frac{dy}{dx}$ at $x = 3$ is

Which of the following statements is/are true? (A) $(\tan^{-1} y - x)dy = (1 + y^2)dx$ is a differential equation where variables are separable. (B) $(1 + x^2)dy + 2xydy = \cot x \ dx (x \neq 0)$ is a first order linear differential equation. (C) $(4x + 6y + 5)dy - (3y + 2x + 4)dx = 0$ is not a homogeneous differential equation. (D) $(xy)dx - (x + y^2)dy = 0$ is a homogeneous differential equation. Choose the correct answer from the options given below:

$\int \left(\frac{1}{\log_e x} - \frac{1}{(\log_e x)^2}\right)dx$ is equal to

Match List-I with List-II | List-I | List-II | |---|---| | Differential equation | Integrating factor | | (A) $x\frac{dy}{dx} - y = 2x^2$ | (I) $e^{-y}$ | | (B) $\frac{dy}{dx} + \frac{y}{x} = 2x$ | (II) $\frac{1}{x}$ | | (C) $x\frac{dy}{dx} + 2y = x^2logx$ | (III) $x$ | | (D) $\frac{dx}{dy} - x = y$ | (IV) $x^2$ | Choose the correct answer from the options given below:

The area bounded by $y = 3x + 1$, $x = 0$, $y = 0$ and $x = a$ is 8 Sq.units. Then value of $a$ (where $a > 0$) is

Match List-I with List-II | List-I | List-II | |---|---| | (A) Marginal average cost if cost function $C(x) = \frac{50}{\sqrt{x}}$ | (I) $50\sqrt{x}$ | | (B) Marginal average cost if cost function $C(x) = 50\sqrt{x}$ | (II) $-\frac{75}{x^2\sqrt{x}}$ | | (C) Revenue function if demand function $P=\frac{50}{\sqrt{x}}$ | (III) $\frac{-25}{x\sqrt{x}}$ | | (D) Marginal revenue if demand function $P=50\sqrt{x}$ | (IV) $75\sqrt{x}$ | Choose the **correct** answer from the options given below:

The particular solution of the differential equation $\left[x \sin^2\left(\frac{y}{x}\right) - y\right]dx + xdy = 0$, $y = \frac{\pi}{4}$ when $x = 1$ is

The edge of a cube is increasing at a rate of 7 cm/s. The rate of change of area of the cube when its side is 3 cm is:

If $e^x + e^y = e^{x+y}$, then $\frac{dy}{dx}$ equals

Area (in sq. units) of the region bounded by the curves $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is

If $f(x)$ and $g(x)$ are continuous functions in [0, a] such that $f(x) = f(a - x)$ and $g(x) + g(a - x) = a$ then $\int_{0}^{a} f(x)g(x)dx =$

A cylindrical drum of radius 7 cm and height 2 m is being kept in a vertical position filled with milk. If the milk is leaking at 14 cm³/sec from its lower base, then the rate of decrease in the level of milk is: [Take $\pi = \frac{22}{7}$]