Calculus PYQ — Page 7

The function $f(x) = 2\log_e(x-2) - x^2 + 4x + 1, (x > 2)$ is increasing on the interval:

The function $f(x) = kx^3 + 6kx^2 + 18x + 17$ is increasing on $\mathbb{R}$(set of real numbers) if:

The function $f(x) = \frac{x}{2} + \frac{2}{x}, x \neq 0$ is increasing on (A) $(-\infty, -2)$ (B) $(-2, 2)$ (C) $(2, \infty)$ (D) $(-1, 1)$ Choose the correct answer from the options given below:

The function $f(x) = x^4 - 2x^2$ is increasing on

The function $f(x) = \frac{x}{3} + \frac{3}{x}$ is increasing in the interval:

The function $f(x) = 4 - 3x + 3x^2 - x^3$ is (Here $\mathbb{R}$ is set of real numbers)

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:

The function $f(x) = \log_e(\sin x), x \in (0, \pi)$ is (A) strictly increasing on $\left(0, \frac{\pi}{2}\right)$ (B) strictly decreasing on $\left(0, \frac{\pi}{2}\right)$ (C) strictly increasing on $\left(\frac{\pi}{2}, \pi\right)$ (D) strictly decreasing on $\left(\frac{\pi}{2}, \pi\right)$ (E) strictly increasing on $(0, \pi)$ Choose the correct answer from the options given below:

The function $f(x) = x^2 - 4x + 6$ is (A) Strictly decreasing on $(-\infty, 2) \cup (2, \infty)$ (B) Strictly increasing on $(2, \infty)$ (C) Strictly increasing on $(-\infty, \infty)$ (D) Strictly decreasing on $(-\infty, 2)$ Choose the correct answer from the options given below:

The function $f(x) = x^2e^{-2x}$ increases on

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

The function $f(x) = x + \frac{a^2}{x}$, $a > 0$, $x \neq 0$ has a local maxima at

The function $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 + 163$ has a local maxima at

The function $f(x) = x + \frac{1}{x}$ has

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \begin{cases} x^2, & x \ge 1 \\ x, & x < 1 \end{cases}$ is

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 function $f(x) = tanx - x$

The equation of the tangent to the curve $y = \frac{(x - 3)}{(x - 1)(x - 2)}$ at the point, where it cuts x-axis is:

The equation of the tangent line to the curve $y = x^2 - 2x + 5$ which is parallel to the line $4x - y + 1 = 0$ is

The equation of tangent line to $y = 2x^2 + 7$, which is parallel to the line $4x - y + 3 = 0$ is

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:

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

The differential equation representing the family of curves $y = Ax + \frac{B}{x}$, $x \neq 0$ where A and B are arbitrary constants, is given by

The differential equation representing the curve $y = e^{2x}(a + bx)$, where a, b are arbitrary constants is