Calculus PYQ

The derivative of x³ + 2x² - 5x + 1 is:

The value of $\int \left\{ \frac{1}{\log_e x} - \frac{1}{(\log_e x)^2} \right\} dx$ is

The largest open interval, in which the function $f(x) = \frac{x}{x^2 + 1}$ increases, is

The value of derivative of the function $\cot^{-1}\{(\cos 2x)^{1/2}\}$ at $x = \frac{\pi}{6}$ is

If $y = 5e^{2x} + 4e^{3x}$, then $\frac{d^2y}{dx^2}$ equals:

The area of the smaller region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and the straight line $3x + 4y = 12$ is:

The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is

If under pure competition demand and supply functions are given by $p = \sqrt{10 - x}$ and $p = \frac{1}{2}(x-2)$ respectively, where $p$ is price per unit and $x$ is quantity, then the consumer surplus is:

The interval, on which the function $f(x) = x^2e^{-x}$ is increasing, is equal to

The value of $\int_0^1 \log_e\left(\frac{1}{x} - 1\right)dx$ is:

Let $f(x) = x^3 - 6x^2 + 9x - 8$ be a function, then which of the following statements are TRUE? (A) $f'(x) = 3(x - 1)(x - 3)$ (B) The critical points of the function are $x = 1$ and $x = 3$ (C) $x = 1$ is the point of local minimum (D) The local maximum value is $-4$ Choose the correct answer from the options given below:

The area (in sq.units) of region bounded by $y^2 = 9x$, $x = 2$, $x = 4$ and the $x$-axis in the first quadrant is

For $x > e$, $\int \frac{dx}{x - \sqrt{x}}$ is equal to

The area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant, is

The area of the region bounded by $y = -1$, $y = 2$, $x = y^3$ and $x = 0$ is $\frac{m}{n}$ sq. units, where $\gcd(m, n) = 1$, then $m - n$ is equal to:

If the minimum value of $a$ is $-\frac{k}{2}$ such that the function $f(x) = x^2 + ax + 5$ is increasing in [1, 2]. Then value of $k$ is

The largest open interval in which the function $f(x) = 4x^3 - 5x^2 - 8x + 12$ increases, is:

The value of $\int_0^1 x e^x dx$ is:

In the following differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ order and degree is:

$\int e^{2x}(\sin x + \frac{1}{2}\cos x) dx$ is equal to

$\int \sqrt{1 + \frac{x^2}{9}} dx$ is equal to (Where C is an arbitrary constant)

The point on the curve y = (x - 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4) is:

$\int \frac{1}{(x + 1)(x + 2)} dx$ is equal to

Let f be a function defined by $f(x) = 2x^3 - 3x^2 - 36x + 2$, then which of the following are correct? (A) The critical points of f(x) are -2 and 3. (B) The function f(x) increases in the interval $(3, \infty)$ (C) The function f(x) decreases in the interval (-2,3) (D) The function f(x) increases in the interval (-2,3) Choose the **correct** answer from the options given below: