Calculus PYQ — Page 26

The area of the region $\{(x, y) : y \geq x^2 \text{ and } y \leq |x|\}$ is

The slope of the tangent to the curve $x = at^2$, $y = 2at$ at 't' is :

If m and n are respectively the order and degree of the differential equation : $\left(\frac{d^2 y}{dx^2}\right)^5 + 6 \frac{\left(\frac{d^2 y}{dx^2}\right)^3}{\frac{d^3 y}{dx^3}} + \frac{d^3 y}{dx^3} = x^2 + 5$, then :

The interval in which the function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is strictly decreasing is :

$\int \frac{\sqrt{\tan x}}{\sin x \cos x} dx$ equals :

$\int e^x \sec x (1 + \tan x) dx$ equals :

The rate of change in area of a triangle having sides 10 cm and 12 cm when the variable angle between them is $\theta = 60°$, is :

Which of the following regions will represent the shaded area in the given figure ?

The angle of intersection between the curves $y = 4 - x^2$ and $y = x^2$ is :

Area of the region bounded by the curve $y = \cos x$ and x-axis between $x = 0$ and $x = \pi$ is :

The equation of the tangent, to the curve $y = x^2 - 2x - 3$ which is perpendicular to the line $x + 2y + 3 = 0$, is

$\int e^x (\tan x + \log_e \sec x) \, dx =$

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $y = \log(\sin x)$ | (I) | $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$ | | (B) | $y = e^{(1 + \log x)}$ | (II) | $\frac{d^2y}{dx^2} = 2$ | | (C) | $y = \log\lvert x \rvert$ | (III) | $\frac{d^2y}{dx^2} = 0$ | | (D) | $y = x^2 + 4x - 1$ | (IV) | $\frac{d^2y}{dx^2} = -\csc^2 x$ | Choose the **correct** answer from the options given below :

The critical points of $f(x) = x^3 + x^2 + x + 1$ are

If $y = x^x$, $\frac{dy}{dx}$ will be:

The solution of the differentiable equation $2x\frac{dy}{dx} + y = 14x^3, x > 0$, is

$\int_1^2 \frac{x \, dx}{(x+1)(x+2)} =$

Interval in which the function $f(x) = 2x^3 - 3x^2 - 12x + 10$ is decreasing is :

$\int \left(x + \frac{1}{x}\right)^2 dx$ equals :

Angle between tangents to the curve $y = x^2 - 5x + 6$ at the points (2, 0) and (3, 0) is :

The interval in which the $f(x) = \sin x - \cos x$, $0 \leq x \leq 2\pi$ is strictly decreasing is :

Which of the following differential equation represents the family of circles touching the x-axis at the origin ?

If $y = \log\left[\frac{x^2}{e^2}\right]$ then value of $\frac{d^2y}{dx^2}$ is :

The equation of tangent to the curve given by $x = a\sin^3 t$, $y = b\cos^3 t$ at a point where $t = \frac{\pi}{2}$ is :