Calculus PYQ — Page 25

The minimum value of $f(x) = |2x - 1|$ is

The given function $f(x) = [x]$ is discontinuous at :

Which of the following statements are correct ? (A) If $f : R \to R$ then $f(x) = |x|$ is continuous everywhere. (B) If $f : R \to R$ then $f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$. (C) Let $f : R - \{0\} \to R$ then $f(x) = \frac{1}{x}$ is continuous everywhere. (D) Let $f : R \to R$ then $f(x) = |x - 1| + |x - 2|$ is continuous everywhere but not differentiable at exactly 2 points. (E) If $f : R \to R$ then $f(x) = \cot x$ is continuous everywhere. Choose the correct answer from the options given below :

Area of the region bounded by the curve $|x| + |y| = 1$ and x-axis is :

Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :

The equation of tangent to the curve $x = a \cos^3 t, y = a \sin^3 t$ at t is :

The degree of the differential equation $\left(1 + \frac{dy}{dx}\right)^4 = \left(\frac{d^2y}{dx^2}\right)^2$ is:

Solution of $\frac{dy}{dx} = (1+x^2)(1+y^2)$ is:

Particular solution of the differential equation $\log\left(\frac{dy}{dx}\right) = x + y$, given that when $x = 0, y = 0$ is:

The area of the region bounded by the parabola $y^2 = 4ax$ and its latus rectum is:

If $\sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y)$, then $\frac{dy}{dx} =$

The maximum slope of the curve $y = -x^3 + 3x^2 + 9x - 27$ is:

The area enclosed between the curves $y = x^2$ and $x = y^2$ is

The value of $\int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 dx$ is:

The differential equation $\frac{dy}{dx} + \frac{x}{y} = 0$, represents the family of curves:

Solution of differential equation $x dy - y dx = 0$ respresents :

The value of integral $\int \sqrt{4x^2 + 9}\, dx$ is

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $x = 2at^2, y = at^4$ | (I) | Inverse trignometric function | | (B) | $f(x) = (2x + 3)^3$ | (II) | Implicit function | | (C) | $xy + y^2 = \tan(x + y)$ | (III) | Parametric function | | (D) | $y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right), -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}$ | (IV) | Composite function | Choose the **correct** answer from the options given below :

$\int_0^{\pi/2} \sqrt{1 - \sin 2x} \, dx$ is equal to :

Points of discontinuity of the greatest integer function $f(x) = [x]$, where $[x]$ denotes integer less than or equal to $x$, are

The rate of change of the area of a circular disc with respect to its circumference when radius is 3 is :

The function $f(x) = \frac{x-1}{x(x^2-1)}, x \neq 1, f(1) = 1$, is discontinuous at

The value of $\int_{0}^{3} |2x - 6| dx$ is :

The derivative of $\sec(\tan \sqrt{x})$ with respect to x is :