Calculus PYQ — Page 8

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

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:

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?

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

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{d^2y}{dx^2}$ equals.

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

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

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 function $f(x) = \sin 3x$, $x \in \left[0, \frac{\pi}{2}\right]$ (A) is increasing on $\left[0, \frac{\pi}{6}\right]$ (B) is decreasing on $\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$ (C) is increasing on $\left[0, \frac{\pi}{2}\right]$ (D) is decreasing on $\left[0, \frac{\pi}{2}\right]$ 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:

$\int\limits_{\sqrt{log_e 2}}^{\sqrt{log_e 4}} xe^{x^2} dx$ is equal to

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

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

If $x = t^3$, $y = t^2$ then $\frac{d^2y}{dx^2}$ is equal to:

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

The area (sq.units) bounded by the curve y = sinx, π ≤ x ≤ 2π and the x-axis is

$\int_{-\pi}^{\pi} \frac{e^{\sin x}}{e^{\sin x} + e^{-\sin x}}dx$ is equal to

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

If $y = e^{acos^{-1}x}, -1 < x < 1$, then $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx}$ is equal to

If $x = t^{1/2}$, $y = t^{3/2}$, then $\frac{dy}{dx}$ =

Let $f(x) = \log_e(\sin x), x \in (0, \pi)$, then which of the following statements is/are TRUE? (A) $f(x)$ is increasing on $(0, \pi/2)$ (B) $f(x)$ is decreasing on $(\pi/2, \pi)$ (C) $f(x)$ is increasing on $(0, \pi)$ (D) $f(x)$ is decreasing on $(0, \pi)$ Choose the correct answer from the options given below: