Calculus PYQ — Page 16

$\int \frac{e^{7\log_e x} - e^{6\log_e x}}{e^{4\log_e x} - e^{3\log_e x}} dx$ is equal to: (Here, c is an arbitrary constant)

The integrating factor of the differential equation, $x^2 \frac{dy}{dx} + xy = log_e x$ is equal to

$\int (e^{x\log a} + e^{a\log x}) dx$ is equal to (where $a > 1$)

The order of $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \left[a \frac{d^2y}{dx^2}\right]^{\frac{1}{2}}$ is

If the function $f(x) = \begin{cases} ax + 2, & x \leq 1 \\ x^2 + 3x + b, & x > 1 \end{cases}$ is differentiable at $x = 1$, then the value of $(2a + b)$ is

If $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$, then $\frac{dy}{dx}$ is equal to

The semi vertical angle of a right circular cone of maximum volume of a given slant height is

For the function $f(x) = x^x, x > 0$, which of the following are TRUE? (A) $f'(x) = x^x(1 + \log x)$ (B) $x = e$ is the critical point (C) $f$ is increasing in $(\frac{1}{e}, \infty)$ (D) $f$ is increasing in $(0, \infty)$ Choose the *correct* answer from the options given below:

The function $f(x) = \begin{cases} \frac{(\sin 2x)}{x} + \cos x & , if \ x \neq 0 \\ K & , if \ x = 0 \end{cases}$ is continuous at $x = 0$, then the value of K is:

If $f(x) = \begin{cases}\frac{1- \tan x}{4x-\pi}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4}\end{cases}$ is continuous at $x = \frac{\pi}{4}$, then the value of k is

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) Point of minima of $f(x) = \vert x+1\vert $ | (I) 1 | | (B) Minimum value of $f(x) = \vert x\vert $ | (II) -1 | | (C) Maximum value of $f(x) = 1 - x^2$ | (III) 2 | | (D) Minimum value of $f(x) = 2 + \sin^2 x$ | (IV) 0 | Choose the correct answer from the options given below:

A spherical ice ball is melting at the rate of 100 $\pi$ cm³/min. The rate at which its radius is decreasing when its radius is 15 cm, is

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\int \dfrac{dx}{x^2 - 16}$ | (I) $\dfrac{1}{8} \log \left\vert \dfrac{4 + x}{4 - x} \right\vert + c$, Where C is an arbitrary constant, | | (B) $\int \dfrac{dx}{x^2 + 16}$ | (II) $\log \left\vert x + \sqrt{x^2 - 16} \right\vert + c$, Where C is an arbitrary constant, | | (C) $\int \dfrac{dx}{16 - x^2}$ | (III) $\dfrac{1}{8} \log \left\vert \dfrac{x - 4}{x + 4} \right\vert + c$, Where C is an arbitrary constant, | | (D) $\int \dfrac{dx}{\sqrt{x^2 - 16}}$ | (IV) $\dfrac{1}{4} \tan^{-1} \left( \dfrac{x}{4} \right) + c$, Where C is an arbitrary constant, | Choose the correct answer from the options given below:

The number of arbitrary constants in the general solution of a differential equation with degree 1 and order 3, is

If $\int \frac{(1 + x \log x)}{xe^{-x}} dx = e^x f(x) + C$, where C is constant of integration, then $f(x)$ is

In which of the following interval, the function $f(x) = \frac{x}{\log x}$ is decreasing?

Match List-I with List-II | List-I | List-II | |---|---| | (A) Degree of the differential equation $\frac{d^2y}{dx^2} = e^{dy/dx}$ is | (I) 2 | | (B) Order of the differential equation $(\frac{dy}{dx})^2 + \frac{d^3y}{dx^3} = 0$ is | (II) not defined | | (C) Degree of the differential equation $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 5x^2 = 0$ | (III) 3 | | (D) If p is the order and q is the degree of the differential equation $\frac{dy}{dx} + 3y = e^x$, then p + q is | (IV) 1 | Choose the correct answer from the options given below:

Area of the region bounded by the curve $y^2 = 4x$, $y$-axis and the line $y = 3$ is equal to

$\int \frac{dx}{e^x + e^{-x}}$ is equal to

Which of the following functions has a local minima at $x = 0$? (A) $f(x) = x^3$ (B) $f(x) = |x|$ (C) $f(x) = x^2$ (D) $f(x) = x^{-2}$ Choose the correct answer from the options given below:

The two positive numbers whose sum is 16 and the sum of whose squares is minimum then the positive numbers are:

The function $f(x) = 4x^3 - 7x^2$ has point(s) of local minima at

$\int \tan^{-1}\sqrt{x} $ $dx$ equals to: (Here C is an arbitrary constant)

The demand function (in Rs.) for a product is given by $P = 20 - 0.25x$, where P is the price per unit and x is the number of units sold, then the price of one unit, when the revenue is maximized, is: