Calculus PYQ — Page 8

$\int \left(\frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}\right) dx =$ (Given that $c$ is an arbitrary constant)

If $y = \frac{1}{1+x^{b-a}+x^{c-a}} + \frac{1}{1+x^{c-b}+x^{a-b}} + \frac{1}{1+x^{a-c}+x^{b-c}}$ then $\frac{d^2y}{dx^2}$ is

The area (in sq. units) of the region bounded by the curve $y = 2x^3$, $x$ - axis and ordinates $x = -1$ and $x = 1$ is:

If $xy + \frac{x^2}{y} = x^3y + y$, then $\frac{dy}{dx}$ is equal to

The maximum value of the function $f(x) = x^2(60 - x)$ in [20, 80] is:

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

The interval in which the function $f(x) = 2x^3 + 3x^2 - 12x + 1$ is strictly increasing, is

The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by

The function, $f(x) = x - \frac{1}{x}$ is

Match List-I with List-II | List-I | List-II | |---|---| | (A) $\int_{-a}^a f(x) dx = 0$ | (I) 0 | | (B) $\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$ | (II) 1 | | (C) $\int_{-\pi}^{\pi} \cos x dx$ | (III) $f$ is an odd function | | (D) $\int_{-1}^1 x^{101} dx + 1$ | (IV) $f(2a-x) = f(x)$ | Choose the correct answer from the options given below:

The largest interval, in which the function $f(x) = x^3 + 2x^2 - 1$ is increasing, is:

For $x \in \mathbb{R} - \{-1,0,1\}$, $\int \frac{1}{x - x^5}dx$ is equal to

If $x^2 - y^2 = t - \frac{1}{t}$, and $x^4 + y^4 = t^2 + \frac{1}{t^2}$, then which of the following is correct?

For the differential equation $x\frac{dy}{dx} + 3y = x^2\log_e x$, which of the following statements are TRUE? (A) Product of order and degree is 1 (B) Integrating factor is $x^3$ (C) Integrating factor is $3x$ (D) General solution is $y = \frac{x^3}{36}(6\log_e|x| - 1) + Cx^{-3}$, C is an arbitrary constant. Choose the correct answer from the options given below:

The sum of two positive numbers is 60. If the sum of their squares in minimum, then the absolute value of the difference of their cubes is

For x ∈ ℝ - {0}, the function f(x) = $\frac{3}{x}$ + 7 is decreasing when

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

Match List-I with List-II | List-I | List-II | |---|---| | Definite integral | Value | | (A) $\int_0^1 \frac{2x}{1 + x^2} dx$ | (I) 2 | | (B) $\int_{-1}^1 sin^3 x \cos^4 x dx$ | (II) $log_e\left(\frac{3}{2}\right)$ | | (C) $\int_0^{\pi} \sin x dx$ | (III) $log_e 2$ | | (D) $\int_2^3 \frac{2}{x^2 - 1} dx$ | (IV) 0 | Choose the correct answer from the options given below:

The absolute maximum value of the function $f(x) = 4x - \frac{1}{2}x^2$ in the interval $\left[-2, \frac{9}{2}\right]$ is

If $f(x) = x^3 e^{-x}$, then the value of $f''(1)$ is equal to

If $f(x) = x^2 - 4x + 13, x \in \mathbb{R}$, then which of the following are correct? (A) $x = 2$ is a stationary point of $f(x)$. (B) $f(x)$ is increasing function on $(2, \infty)$ (C) $f(x)$ have maxima at $x = 2$ (D) $f(2) = 9$ Choose the correct answer from the options given below:

The rate of change of volume of a sphere with respect to its surface area, when the radius is 6cm is:

$\int e^{(x \log 5)}e^x dx$, is: Where $C$ is the constant of integration.

The interval on which the function $f(x) = x^3 + 2x^2 - 1$ is decreasing, is