CUET Mathematics Calculus > Application of Derivatives PYQs

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:

Function $f(x) = x^x, x > 0$ decreases on the interval

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 sides of an equilateral triangle are increasing at the rate of 5 cm/sec. The rate at which the area increases when the side is 20 cm, is

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:

The length of a rectangle is decreasing at the rate of 4 cm/minute and the width is increasing at the rate of 3 cm/minute, then the rate of change of the perimeter is

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

The demand for a certain product is represented by the function $p = 150 + 10x - x^2$ (in Rs.) where $x$ is the number of units demanded and $p$ is the price per unit, then the value of marginal revenue, when 10 units are sold is

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

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

Let $f(x) = x^2 + \frac{250}{x}$ be any function defined on $\mathbb{R} - \{0\}$, where $\mathbb{R}$ is the set of real numbers. Then which of the following are TRUE? (A) $f'(x) = 2x + \frac{250}{x^2}$ (B) $x = 5$ in the only critical point of $f(x)$ (C) minimum value of $f(x)$ is 75 (D) maximum value of $f(x)$ is 50. Choose the **correct** answer from the options given below:

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:

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:

If $x$ is real, the minimum value of $x^2 - 8x + 20$ is

For the function $f(x) = -2x^3 + 3x^2 + 36x - 10$, which of the following is/are true? (A) $f$ is increasing in $(-\infty, -2)$ (B) $f$ is increasing in $(-2, 3)$ (C) $f$ is decreasing in $(-\infty, -2)$ (D) $f$ is decreasing in $(3, \infty)$ Choose the correct answer from the options given below:

Consider a closed cylinder of radius $r$ with a fixed surface area. The volume of the cylinder is maximum when its height is

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

If $x = -1$ and $x = -2$ are the extreme points of $f(x) = \alpha\log|x| + \beta x^2 + x$ then

If maximum value of $f(x) = 2x^3 + 3x^2 - 6ax + 10$ occurs at $x = -3$, then the value of $\alpha$ is ____

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

For $x > 0$, the minimum value of $\frac{x}{\log_e x}$ is

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

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

For the function $f(x) = e^x + e^{-x}$ (A) $f'(x) = e^x - e^{-x}$ (B) The critical point is $x = 0$ (C) The minimum value is 1 (D) $x = 0$ is the point of local minimum. Choose the correct answer from the options given below:

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

If the cost function of a product is given by $C(x) = \frac{3}{4}x^2 - 5x + 21$, then the marginal cost when $x = 10$ is

The function $f: R \rightarrow R$ (where $R$ is set of real numbers) defined as $f(x) = x^2 + 2x$ is

The point of local maxima of the function $f(x) = (x - 2)^5(x + 2)^2$ is

A car is moving along the curve $y = x^3 + 12$. The point(s) on the curve at which the rate of change of its y-coordinate at a certain time is 3 times the rate of change of its x-coordinate is/are

The rate of change of area of a circle with respect to its circumference when radius in 6 cm, is

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) The maximum value of $f(x) = \sin(3x) + 6$ | (I) 2 | | (B) The maximum value of $f(x) = -\vert x + 2\vert + 4$ | (II) 5 | | (C) The minimum value of $f(x) = (3x + 1)^2 + 5$ | (III) 7 | | (D) The minimum value of $f(x) = 2 \cos x + 4$ | (IV) 4 | Choose the correct answer from the options given below:

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

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:

Match List-I with List-II | List-I | List-II | | --- | --- | | Function f(x) | Interval for increasing/decreasing of function f(x) | | --- | --- | | (A) $f(x) = x\vert x\vert $ | (I) Decreases on $(0, \infty)$ | | (B) $f(x) = x^2 + 2x - 5$ | (II) Increases on $(3, \infty)$ | | (C) $f(x) = x^2 - 6x + 9$ | (III) Decreases on $(-\infty, -1)$ | | (D) $f(x) = -x^2$ | (IV) Increases on $(-\infty, \infty)$ | Choose the correct answer from the options given below:

If $f(x) = 2x^3 - 15x^2 + 36x + 1$, $x \in [1, 5]$, then the absolute minimum value of $f(x)$ is:

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

The function $f(x) = x^2 - x + 1$ is

Let $f(x) = 4x^3 - 18x^2 + 27x - 5$, $x \in R$. Then which of the following statements are TRUE? (A) $f''(x) = 24x - 36$ (B) f has local maxima at $x = \frac{3}{2}$ but no minima (C) f has neither maxima nor minima (D) f has both maxima and minima Choose the correct answer from the options given below:

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

The total cost function is given by $c(x) = \frac{1}{3}x^3 - 5x^2 + 30x - 15$ and selling price per unit is Rs.6. The profit is maximum if the value of x is:

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

For the function $f(x) = e^{-2x}(2-x)^2$, the point of local maxima is:

For $x \in \mathbb{R}$, if $f(x) = -(x-1)^2 + 2$, then (A) $f$ is an increasing function on $(-\infty, 1]$ (B) $f$ has no critical points (C) $f$ has a maximum value at $x = 1$ (D) $f$ has a minimum value at $x = 1$ Choose the correct answer from the options given below:

A balloon which always remains spherical, has a variable diameter $\frac{3}{2}(5x+7)$. Then the rate of change of its volume with respect to x is

If $f(x) = \sin x - \cos x$, $x \in [0, 2\pi]$ then (A) $f(x)$ is increasing in $(0, \frac{3\pi}{4})$ (B) $f(x)$ is decreasing in $(0, \frac{3\pi}{4})$ (C) $f(x)$ is decreasing in $(\frac{3\pi}{4}, \frac{7\pi}{4})$ (D) $f(x)$ is decreasing in $(\frac{7\pi}{4}, 2\pi)$ Choose the correct answer from the options given below:

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

A cylindrical drum of radius 7 cm and height 2 m is being kept in a vertical position filled with milk. If the milk is leaking at 14 cm³/sec from its lower base, then the rate of decrease in the level of milk is: [Take $\pi = \frac{22}{7}$]

The function $f(x) = x^3 + 3x^2 + 4x + 4$, $x \in \mathbb{R}$ (set of real numbers) :

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

If it is given that at $x = 1$, the function $f(x) = x^4 - 62x^2 + 2ax + b$ attains its maximum value on the interval [0, 2], then the value of a is:

The volume of spherical balloon is increasing at the rate of $4 \text{ cm}^3/ \text{sec}$. The rate of increase of its surface area, when the radius is 3cm will be :-

The interval(s), where the function $f(x) = \begin{cases} \frac{1-e^x}{e^{2x}-1} & : x \neq 0 \\ \frac{-1}{2} & : x = 0 \end{cases}$ is increasing, is/ are:

A square board of side 36cm is made into a box without top by cutting a square from each corner and folding up the flaps to form a box then maximum volume of the box is

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

The maximum value of $\left(\frac{1}{x}\right)^x$ for $x > 0$ is

A boat 10 m high floating at a uniform speed of 13 meters per minute(m/min) away from a lamp post 15 m high. Then the rate at which the length of shadow of the boat increases is:

If the function $f(x) = 2x^2 - kx + 7$, is increasing on $[1,2]$, then $k$ lies in the interval

Which of the following statements are true? (A) The function $f(x) = \frac{x^4}{4} - \frac{4}{3}x^3 + \frac{x^2}{2} + 6x$ has 3 critical points. (B) The function $f(x) = |x| + 3$ has no minimum value. (C) A local maximum value is always the absolute maximum value. (D) $f(x) = x^2$ has minima at $x=0$. Choose the **correct** answer from the options given below:

The rate of change of the area of a circle with respect to its radius $r$, when $r = 3$cm, is:

The curve $x = y^2$ and $xy = k$ cut orthogonally, then $k^2$ is equal to:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Marginal average cost if cost function $C(x) = \frac{50}{\sqrt{x}}$ | (I) $50\sqrt{x}$ | | (B) Marginal average cost if cost function $C(x) = 50\sqrt{x}$ | (II) $-\frac{75}{x^2\sqrt{x}}$ | | (C) Revenue function if demand function $P=\frac{50}{\sqrt{x}}$ | (III) $\frac{-25}{x\sqrt{x}}$ | | (D) Marginal revenue if demand function $P=50\sqrt{x}$ | (IV) $75\sqrt{x}$ | Choose the **correct** answer from the options given below:

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

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

The rate of change of volume of a sphere with respect to its surface area, when radius is 4 cm, is equal to

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 demand function for a certain product is represented by the equation: $p = 20 + 5x - 3x^2$, where $x$ is the number of units demanded and $p$ is the price per unit (in Rs.), then the marginal revenue when 2 units are sold 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

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Function** | **Increasing on the interval** | | (A) $f(x) = -x^2 - 2x + 1$ | (I) $(-\infty, -1)$ | | (B) $f(x) = x^2 + 1$ | (II) $(1, \infty)$ | | (C) $f(x) = x^2 - 2x + 3$ | (III) $(-\infty, 0)$ | | (D) $f(x) = -x^2$ | (IV) $(0, \infty)$ | Choose the correct answer from the options given below:

The function $f(x) = 6 - 6x - 2x^2$

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

If the maximum value of the function $f(x) = \frac{2\log_e x}{x}$, $x > 0$ occurs at $x = e$, then $e^3 f''(e)$ is equal to

For the function $f(x) = 2x^3 - 3x^2 - 12x + 5$, the difference of maximum and minimum value of $f(x)$ is

Interval in which the function $f$ given by $f(x) = \tan x - 4x$, $x \in (0, \frac{\pi}{2})$ is strictly decreasing is

The marginal cost (MC) and marginal revenue (MR) functions of a product are $MC = 20 + \frac{x}{20}$ and $MR = 30$ respectively. If the fixed cost is 200, then the maximum value of the profit is:

For the function, $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 - 350$, which of the following statements are correct? (A) $x = -3$ and $x = -5$ are the only critical points of the given function. (B) $x = -3$ is a point of local minimum. (C) The local minimum value at $x = -3$ is 23.1. (D) $x = -5$ is a point of local maximum. Choose the correct answer from the options given below:

Match List-I with List-II | List-I (Curve) | List-II (Slope of tangent at $x = 4$) | |---|---| | (A) $y = \sqrt{x^3}$ | (I) -1 | | (B) $y = \sqrt{x}$ | (II) 1 | | (C) $y = x^3 - 47x$ | (III) 1/4 | | (D) $xy = 16$ | (IV) 3 | Choose the correct answer from the options given below:

The real valued function $f(x) = x^{15} + 5x^9 + 10$ is increasing for___________.

If $y = ax^2 + bx$ has minima at $x = 2$ and the minimum value is -12, then which of the following are correct? (A) $a = 3$ (B) $a = -3$ (C) $b = 12$ (D) $b = -12$ Choose the correct answer from the options given below:

Match List-I with List-II | List-I | List-II | |---|---| | (A) Maximum value of $f(x) = \sin^2 x - \cos^2 x$, $\forall x \in (\pi, 2\pi)$ is | (I) 0 | | (B) Minimum value of $f(x) = \sin x \cos x$ | (II) 1 | | (C) Point of Minima of $f(x) = x^x$ $(x > 0)$ | (III) $-\frac{1}{2}$ | | (D) Maximum value of $f(x) = -x^{2026}$ | (IV) $\frac{1}{e}$ | Choose the correct answer from the options given below:

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

The point on the curve $y^2 = 8x$ for which the abscissa and ordinate change at the same rate, is

The point on the curve $\frac{x^2}{4} + \frac{y^2}{9} = 1$ at which the tangent to the curve is parallel to the x-axis is

The demand for a certain product is represented by the function $p = 300 + 25x - x^2$ (in rupees), where x is the number of units demanded and p is the price per unit, then the marginal revenue when 15 units are sold, is

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:

Function $f(x) = x^3 - 3x + 3$ is (A) Increasing in the interval $(-1, 1)$ (B) Increasing in the interval $(1, \infty)$ (C) Decreasing in the interval $(-1, 1)$ (D) Increasing in the interval $(-\infty, -1) \cup (1, \infty)$ 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 semi vertical angle of a right circular cone of maximum volume of a given slant height is

The minimum value of the function $f(x) = x^3 + (10-x)^3$ occurs at:

If a revenue function is given by $R(x) = 2027x - 1013x^2 - 675x^3$, then the marginal revenue function (MR) is:

For the function $f(x) = x^{1/x}$, $x > 0$, which of the following are correct? (A) $x = 0$ is the only point where extremum may occur. (B) The given function is maximum at $x = e$. (C) The function has no extreme value for $x > 0$. (D) The maximum value of the function $f(x)$ is $e^{1/e}$. Choose the correct answer from the options given below:

Match List-I with List-II Consider the function f(x) = 2x³ - 21x² + 36x + 80, x∈[0, 6]. Then | List-I | List-II | |---|---| | (A) one of its critical points is at x = | (I) -28 | | (B) Its absolute maximum value is | (II) -42 | | (C) Its absolute minimum value is | (III) 97 | | (D) Its second derivative at x = 0 is | (IV) 6 | Choose the correct answer from the options given below:

If the interval in which f(x) = $\frac{x}{4}$ + $\frac{4}{x}$, x ≠ 0 is strictly increasing is (-∞, a) ∪ (b, ∞), then

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

The maximum value of f(x) = $\left(\frac{1}{x}\right)^x$ is

If the interval in which the function f(x) = $\frac{x}{x^2+1}$ is strictly increasing is (-a, a), then a is equal to

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:

The real valued function $f(x) = 12x^\frac{4}{3} - 6x^\frac{1}{3}, x \in [-8, 8]$ has absolute maximum value equal to

If the interval in which the function $f(x) = 4x^3 - 6x^2 - 72x + 30$ is strictly decreasing, is (a,b) then a+b is equal to

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases when the side is 10 cm, is

Match List-I with List-II The function $f(x) = (x - 1)(x + 1)^2$ has | List-I | List-II | |---|---| | (A) A local maxima at $x = $ ____ | (I) $\frac{1}{3}$ | | (B) A local minima at $x = $ ____ | (II) 0 | | (C) The local minimum value of $f(x) = $ ____ | (III) -1 | | (D) The local maximum value of $f(x) = $ ____ | (IV) $-\frac{32}{27}$ | Choose the correct answer from the options given below:

The interval on which the function $f(x) = x^4 - \frac{x^3}{3}$ is strictly decreasing, is:

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:

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

If $f(x) = a \log_e|x| + bx^2 + x$ has critical points at $x = -2$ and $x = 1$, then

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:

Which of the following functions are increasing on $x \in \left(0, \frac{\pi}{2}\right)$? (A) $f(x) = \sin x$ (B) $f(x) = \cos x$ (C) $f(x) = \tan x$ (D) $f(x) = \cos 3x$ Choose the correct answer from the options given below:

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

If the area of an equilateral triangle is increasing at the rate of $4\sqrt{3}$ cm²/sec, then the rate of increase of its perimeter when the side is 4cm, is

Which of the following are correct? (A) The function $f(x) = 3x+12$ is increasing on R. (B) The function $f(x) = e^{2x}$ is decreasing on R. (C) The function $f(x) = x^2-x-1$ is neither increasing nor decreasing on (-1, 1). (D) The function $f(x) = x^3-3x^2+4x$ is increasing on R. Choose the correct answer from the options given below:

If the function $f(x) = 2x^3 + 9x^2 + 12x-1$ is given,then $f(x)$ have

Match **List-I** with **List-II** The function $f(x) = 2x^3 - 15x^2 + 36x + 5$ for $x \in [2,5]$ has | List-I | List-II | |---|---| | (A) absolute maximum value | (I) 5 | | (B) absolute minimum value | (II) 60 | | (C) point of absolute maxima | (III) 3 | | (D) point of absolute minima | (IV) 32 | Choose the **correct** answer from the options given below:

The greatest possible value of '$a$' such that the function $f(x) = x^2 + a x + 1$ is always decreasing in the interval [1, 2] is:

The function $f(x) = \frac{x - 2}{x + 1}, x \neq -1$ is increasing when (Where $\mathbb{R}$ is a set of real numbers)

Consider the function $f(x) = \sin x$ in the interval $[\pi, 2\pi]$ then which of the following statements are correct? (A) $x = \frac{3\pi}{2}$ is its stationary point. (B) Its maximum value is 1 (C) Its minimum value is -1 (D) It attains its maximum value at $\pi$ and $2\pi$ Choose the **correct** answer from the options given below:

The radius of spherical balloon is decreasing at the rate of 0.1cm/sec, the rate at which its volume is decreasing, when its radius is 0.5cm is

Which of the following are NOT correct regarding the equation of tangent and normal to the curve $y = \frac{x-11}{(x-2)(x-3)}$ at the point, where it cuts the $x$-axis? (A) The point of contact is (11, 0). (B) The equation of tangent is $x - 72y - 11 = 0$ (C) The equation of normal is $72x + y - 11 = 0$ (D) The slope of the tangent at the given point of contact is $\frac{1}{88}$ Choose the **correct** answer from the options given below:

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

The total cost $c(x)$ associated with the production of $x$ units of an item is given by $c(x) = 0.001x^3 + 0.06x^2 + 20x + 500$. The marginal cost when 10 units are produced is:

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

If the maximum value of the function $f(x) = \frac{\log_ex}{x}$, $x > 0$ occurs at $x = a$, then $a^2f''(a)$ is equal to

The rate of change of area of a circle with respect to its circumference when radius is 4cm, is

Match List-I with List-II $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) The minimum value of } f(x) = (2x - 1)^2 + 3 & \text{(I) } 4 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) The maximum value of } f(x) = -|x + 1| + 4 & \text{(II) } 10 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) The minimum value of } f(x) = \sin(2x) + 6 & \text{(III) } 3 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) The maximum value of } f(x) = -(x - 1)^2 + 10 & \text{(IV) } 5 \\[1.2ex] \hline \end{array}$ Choose the correct answer from the options given below:

The function $f(x) = tanx - x$

The slope of the normal to the curve $y = 2x^2$ at $x = 1$ is:

The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 + 26x^2 + 15x + 400$. The marginal cost when 10 items are produced is:

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

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

The interval in which the function $g(x) = x^2 e^{-x}$ is increasing is:

The number of tangents to the curve $xy - 3y + 2 = 0$ having slope 2 is:

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

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

Consider the function $f(x) = x^3 - 3x$. Then Match List-I with List-II | List-I | List-II | |---|---| | (A) Point of local Maxima | (I) 1 | | (B) Point of local Minima | (II) -1 | | (C) Local maximum value | (III) 2 | | (D) Local minimum value | (IV) -2 | Choose the correct answer from the options given below:

For the function $f(x) = sinx + cosx, x \in [0, \pi]$, which one of the following is correct?

In which of the following intervals, the function $f(x) = -x^2 - 2x + 15$ is decreasing?

In which of the following interval the function $f(x) = x^x, x > 0$ is strictly increasing?

If a function $f(x)=x^{2}+b x+1$ is increasing in the interval $[1,2]$, then the least value of $b$ is :

For the function $f(x) = 2x^3 - 9x^2 + 12x - 5$, $x \in [0, 3]$, match List-I with List-II : | List-I | List-II | | --- | --- | | A. Absolute maximum value | (I) $ 3 $ | | B. Absolute minimum value | (II) $ 0 $ | | C. Point of maxima | (III) $ -5 $ | | D. Point of minima | (IV) $ 4 $ |

The rate of change (in $\mathrm{cm}^{2} / \mathrm{s}$ ) of the total surface area of a hemisphere with respect to radius $r$ at $r=\sqrt[3]{1.331} \mathrm{~cm}$ is :

$f(x)=\sin x+\frac{1}{2} \cos 2 x \text { in }\left[0, \frac{\pi}{2}\right]$ (A) $f^{\prime}(x)=\cos x-\sin 2 x$ (B) The critical points of the function are $x=\frac{\pi}{6}$ and $x=\frac{\pi}{2}$ (C) The minimum value of the function is $2$ (D) The maximum value of the function is $\frac{3}{4}$ Choose the correct answer from the options given below :

If $f(x)=2\left(\tan ^{-1}\left(e^{x}\right)-\frac{\pi}{4}\right)$, then $f(x)$ is :

If the lengths of the three sides of a trapezium other than the base are $10 \mathrm{~cm}$ each, then the maximum area of the trapezium is:

The critical points of $f(x) = x^3 + x^2 + x + 1$ are

The volume of a cube is increasing at the rate of 27 cm$^3$/s. How fast is the surface area increasing when the length of the cube is 12 cm.

The intervals for which $f(x) = x^4 - 2x^2$ is increasing are :

The slope of the normal to the curve $y = 2x^2 - 4$ at P (1, -2) is :

The minimum value of $f(x) = |2x - 1|$ is

Interval in which the function $f(x) = 2x^3 - 3x^2 - 12x + 10$ is decreasing is :

The angle of intersection between the curves $y = 4 - x^2$ and $y = x^2$ is :

The value of C, in Rolle's theorem for the function $f(x) = e^x \sin x$, when $x \in [0, \pi]$ is :

The equation of tangent to the curve $x = a \cos^3 t, y = a \sin^3 t$ at t is :

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to :

The appropriate change in the volume V of a cube of side $x$ metres caused by increasing the side by 2% is :

The maximum value of $2x^3 - 24x + 107$ in the interval $[1, 3]$ is :

The slope of the tangent to the curve $x = at^2$, $y = 2at$ at 't' is :

If the function $f(x) = x^4 - 62x^2 + ax + 9$ attains its local maximum value at $x = 1$, then a is equal to :

The interval in which the function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is strictly decreasing is :

The two curves $x^3 - 3xy^2 + 15 = 0$ and $3x^2 y - y^3 + 17 = 0$ :

The equation of tangent to the curve given by $x = a\sin^3 t$, $y = b\cos^3 t$ at a point where $t = \frac{\pi}{2}$ is :

The rate of change in area of a triangle having sides 10 cm and 12 cm when the variable angle between them is $\theta = 60°$, is :

The interval in which the function $f(x) = 10 - 6x - 2x^2$ is decreasing is :

The value of C which satisfies Rolle's Theorem for $f(x) = \sin^4 x + \cos^4 x$ in $\left[0, \frac{\pi}{2}\right]$. Then C is :

Angle between tangents to the curve $y = x^2 - 5x + 6$ at the points (2, 0) and (3, 0) is :

The rate of change of the area of a circular disc with respect to its circumference when radius is 3 is :

The interval in which the $f(x) = \sin x - \cos x$, $0 \leq x \leq 2\pi$ is strictly decreasing is :

If $2x + y = 6$ then the maximum value of $x^2 y$ is :

If the function $f(x) = x^2 - ax - 2$ is strictly decreasing on $(2, 3)$ then $a$ lies in the interval.

The slope of the tangent to the curve $y = 3x^2 + 2kx - 5$ at $x=1$ is $9$. The value of $k$ is :

The function $f(x) = 6(2x^4 - x^2)$ is strictly increasing in

The maximum slope of the tangents to the curve $y(x) = -x^3 + 3x^2 + 9x - 30$ is

The line $ax + by = 7$ is a tangent to the curve $y = \frac{x-7}{(x-2)(x-3)}$ at the point where it cuts the x-axis A. The y-intercept of the line is $-0.7$ B. $b = -7$ C. $a = 1$ D. the line passes through the point $(-13, -1)$ E. $b = -20$ Choose the correct answer from the options given below:

The slope of normal to the curve $y = 3x^2 + 3 \sin 3x$, at $x = 0$ is:

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

If x is real, then minimum value of $x^2 - 8x + 17$ is :

The tangent to the curve $y^2 + 2x - 5 = 0$ at the point (h, k) is parallel to the line $x + 2y = 4$, then the value of 'k' is:

The slope of normal to the curve $y = kx^2 - 3x + 2$ at $x = \frac{1}{2}$ is 5. The value of 'k' is

The point(s) on the curve $\frac{x^2}{9} + \frac{y^2}{64} = 1$, at which the tangents are parallel to x-axis are

An energy DRONE is flying along the curve $y = x^2 + 7$. A soldier is placed at $(3, 7)$. The nearest distance of the DRONE from soldier's position is

The maximum value of $x^{-x}$ is

The absolute maximum value of $y = x^3 - 3x + 2$, $0 \leq x \leq 2$, is

The interval in which the function, $f(x) = 7 - 4x - x^2$ is strictly increasing is

Consider the function $f(x) = x^{\frac{1}{x}}$. Its

The given function $f(x) = x^5 - 5x^4 + 5x^3 - 1$; has/have (a) local maxima at $x = 1$ (b) local maximum value is 0 (c) local minimum at $x = 3$ (d) local minimum value is $-28$ (e) The point of inflexion is $x = 1$ Choose the correct answer from the options given below

Match List-I with List-II | List-I | List-II | |---|---| | (a) If $x = t^2$ and $y = t^3$, then $\frac{d^2y}{dx^2}$ at $t = 1$ | (i) $-2$ | | (b) If $f(x) = \sqrt{x} + 1$, then $f''(1)$ | (ii) $-1$ | | (c) The minimum value of $f(x) = 9x^2 + 12x + 2$ is | (iii) $\frac{3}{4}$ | | (d) The point of inflexion of the function $f(x) = (x-2)^4 (x+1)^3$ is | (iv) $-\frac{1}{4}$ | Choose the correct answer from the options given below

The equation of the normal to the curve $y = x - \frac{1}{x}$ at $(1,0)$ is:

The minimum value of $x^2 - 8x + 17$ on the set $\mathbb{R}$ of all real numbers is: