Application of Derivatives

160 questions · Mathematics · JEE Main

Application of Derivatives

160 questions · Mathematics · JEE Main

If the function $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1$ , where $a > 0$ , attains its local maximum and local minimum values at p and q , respectively, such that $p^{2} = q$ , then $f (3)$ is equal to :2025 · 2 Apr · Shift 1 · Q32 · MCQ
Let $A (4, - 2), B (1, 1)$ and $C (9, - 3)$ be the vertices of a triangle ABC . Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and $A B$ of the triangle $A B C$ …2025 · 2 Apr · Shift 2 · Q50 · Numerical
The shortest distance between the curves $y^{2} = 8 x$ and $x^{2} + y^{2} + 12 y + 35 = 0$ is:2025 · 3 Apr · Shift 2 · Q31 · MCQ
Let $f : R \to R$ be a function defined by $f (x) = ∣∣ x + 2∣ - 2∣ x ∥$ . If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$ , then $m + n$ is2025 · 3 Apr · Shift 2 · Q43 · MCQ
Let $a > 0$ . If the function $f (x) = 6 x^{3} - 45 a x^{2} + 108 a^{2} x + 1$ attains its local maximum and minimum values at the points $x_{1}$ and $x_{2}$ respectively such that $x_{1} x_{2} = 54$ , then $a + x_{1} + x_{2}$ is equal…2025 · 4 Apr · Shift 2 · Q30 · MCQ
Let $x = - 1$ and $x = 2$ be the critical points of the function $f (x) = x^{3} + a x^{2} + b lo g_{e} ∣ x ∣ + 1, x e q 0$ . Let $m$ and M respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $[- 2, - \frac{1}{2}]$ …2025 · 7 Apr · Shift 1 · Q43 · MCQ
Let f : ℝ $\to$ ℝ be a polynomial function of degree four having extreme values at x = 4 and x = 5. If $x \to 0 lim \frac{f ( x )}{x ^{2}} = 5$ , then f(2) is equal to :2025 · 7 Apr · Shift 2 · Q26 · MCQ
Let the function $f (x) = \frac{x}{3} + \frac{3}{x} + 3, x e q 0$ be strictly increasing in $(- \infty, α_{1}) \cup (α_{2}, \infty)$ and strictly decreasing in $(α_{3}, α_{4}) \cup (α_{4}, α_{5})$ . Then $i = 1 \sum 5 α_{i}^{2}$ …2025 · 8 Apr · Shift 2 · Q44 · MCQ
Let $f (x) = \int_{0}^{x^{2}} \frac{t ^{2} - 8 t + 15}{e ^{t}} dt, x \in R$ . Then the numbers of local maximum and local minimum points of $f$ , respectively, are :2025 · 22 Jan · Shift 2 · Q40 · MCQ
If the set of all values of $a$ , for which the equation $5 x^{3} - 15 x - a = 0$ has three distinct real roots, is the interval $(α, β)$ , then $β - 2 α$ is equal to .2025 · 23 Jan · Shift 1 · Q47 · Numerical
A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of $81 cm^{3} / min$ and the thickness of the…2025 · 23 Jan · Shift 2 · Q43 · MCQ
Consider the region $R = {(x, y) : x \leq y \leq 9 - \frac{11}{3} x^{2}, x \geq 0}$ . The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is:2025 · 24 Jan · Shift 1 · Q39 · MCQ
Let $(2, 3)$ be the largest open interval in which the function $f (x) = 2 lo g_{e} (x - 2) - x^{2} + a x + 1$ is strictly increasing and (b, c) be the largest open interval, in which the function $g (x) = (x - 1)^{3} (x + 2 - a)^{2}$ is…2025 · 24 Jan · Shift 2 · Q28 · MCQ
The sum of all local minimum values of the function $f (x) = {1 - 2 x, x 2$ is2025 · 28 Jan · Shift 1 · Q40 · MCQ
If $5 f (x) + 4 f (\frac{1}{x}) = x^{2} - 2, \forall x e q 0$ and $y = 9 x^{2} f (x)$ , then $y$ is strictly increasing in :2024 · 1 Feb · Shift 1 · Q49 · MCQ
Let the sum of the maximum and the minimum values of the function $f (x) = \frac{2 x ^{2} - 3 x + 8}{2 x ^{2} + 3 x + 8}$ be $\frac{m}{n}$ , where $gcd (m, n) = 1$ . Then $m + n$ is equal to :2024 · 4 Apr · Shift 1 · Q47 · MCQ
Let $f (x) = 3 x - 2 + 4 - x$ be a real valued function. If $α$ and $β$ are respectively the minimum and the maximum values of $f$ , then $α^{2} + 2 β^{2}$ is equal to2024 · 4 Apr · Shift 2 · Q36 · MCQ
Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then…2024 · 5 Apr · Shift 1 · Q31 · MCQ
Let $f (x) = x^{5} + 2 x^{3} + 3 x + 1, x \in R$ , and $g (x)$ be a function such that $g (f (x)) = x$ for all $x \in R$ . Then $\frac{g ( 7 )}{g ^{'} ( 7 )}$ is equal to :2024 · 5 Apr · Shift 1 · Q38 · MCQ
For the function $f (x) = sin x + 3 x - \frac{2}{π} (x^{2} + x), where x \in [0, \frac{π}{2}],$ consider the following two statements : (I) $f$ is increasing in $(0, \frac{π}{2})$ . (II) $f^{'}$ …2024 · 5 Apr · Shift 1 · Q48 · MCQ
Let the maximum and minimum values of $(8 x - x^{2} - 12 - 4)^{2} + (x - 7)^{2}, x \in R$ be $M$ and $m$ , respectively. Then $M^{2} - m^{2}$ is equal to .2024 · 5 Apr · Shift 2 · Q53 · Numerical
The interval in which the function $f (x) = x^{x}, x > 0$ , is strictly increasing is2024 · 6 Apr · Shift 1 · Q49 · MCQ
Let $f (x) = 4 cos^{3} x + 33 cos^{2} x - 10$ . The number of points of local maxima of $f$ in interval $(0, 2 π)$ is2024 · 8 Apr · Shift 1 · Q33 · MCQ
The number of critical points of the function $f (x) = (x - 2)^{2/3} (2 x + 1)$ is2024 · 8 Apr · Shift 1 · Q38 · MCQ
For the function $f (x) = (cos x) - x + 1, x \in R$ , between the following two statements (S1) $f (x) = 0$ for only one value of $x$ in $[0, π]$ . (S2) $f (x)$ is decreasing in $[0, \frac{π}{2}]$ and increasing in $[\frac{π}{2}, π]$ …2024 · 8 Apr · Shift 1 · Q41 · MCQ
If the function $f (x) = 2 x^{3} - 9 ax^{2} + 12 a^{2} x + 1, a > 0$ has a local maximum at $x = α$ and a local minimum at $x = α^{2}$ , then $α$ and $α^{2}$ are the roots of the equation :2024 · 8 Apr · Shift 2 · Q47 · MCQ
Let $A$ be the region enclosed by the parabola $y^{2} = 2 x$ and the line $x = 24$ . Then the maximum area of the rectangle inscribed in the region $A$ is .2024 · 8 Apr · Shift 2 · Q56 · Numerical
Let the set of all positive values of $λ$ , for which the point of local minimum of the function $(1 + x (λ^{2} - x^{2}))$ satisfies $\frac{x ^{2} + x + 2}{x ^{2} + 5 x + 6} < 0$ , be $(α, β)$ . Then $α^{2} + β^{2}$ is equal to …2024 · 9 Apr · Shift 1 · Q58 · Numerical
Let the set of all values of $p$ , for which $f (x) = (p^{2} - 6 p + 8) (sin^{2} 2 x - cos^{2} 2 x) + 2 (2 - p) x + 7$ does not have any critical point, be the interval $(a, b)$ . Then $16 ab$ is equal to .2024 · 9 Apr · Shift 2 · Q57 · Numerical
Let for a differentiable function $f : (0, \infty) \to R, f (x) - f (y) ⩾ lo g_{e} (\frac{x}{y}) + x - y, \forall x, y \in (0, \infty)$ . Then $n = 1 \sum 20 f^{'} (\frac{1}{n ^{2}})$ …2024 · 27 Jan · Shift 1 · Q55 · Numerical
Let $g (x) = 3 f (\frac{x}{3}) + f (3 - x)$ and $f^{''} (x) > 0$ for all $x \in (0, 3)$ . If $g$ is decreasing in $(0, α)$ and increasing in $(α, 3)$ , then $8 α$ is :2024 · 27 Jan · Shift 2 · Q44 · MCQ
Consider the function $f : [\frac{1}{2}, 1] \to R$ defined by $f (x) = 42 x^{3} - 32 x - 1$ . Consider the statements (I) The curve $y = f (x)$ intersects the $x$ -axis exactly at one point. (II) The curve $y = f (x)$ …2024 · 29 Jan · Shift 1 · Q46 · MCQ
Let $f (x) = 2^{x} - x^{2}, x \in R$ . If $m$ and $n$ are respectively the number of points at which the curves $y = f (x)$ and $y = f^{'} (x)$ intersect the $x$ -axis, then the value of $m + n$ is .2024 · 29 Jan · Shift 1 · Q59 · Numerical
The function $f (x) = \frac{x}{x ^{2} - 6 x - 16}, x \in R - {- 2, 8}$ 2024 · 29 Jan · Shift 2 · Q36 · MCQ
The function $f (x) = 2 x + 3 (x)^{\frac{2}{3}}, x \in R$ , has2024 · 29 Jan · Shift 2 · Q45 · MCQ
The maximum area of a triangle whose one vertex is at $(0, 0)$ and the other two vertices lie on the curve $y = - 2 x^{2} + 54$ at points $(x, y)$ and $(- x, y)$ , where $y > 0$ , is :2024 · 30 Jan · Shift 1 · Q35 · MCQ
Let $f (x) = (x + 3)^{2} (x - 2)^{3}, x \in [- 4, 4]$ . If $M$ and $m$ are the maximum and minimum values of $f$ , respectively in $[- 4, 4]$ , then the value of $M - m$ is2024 · 30 Jan · Shift 2 · Q44 · MCQ
$If f (x) = x^{3} 3 x^{2} + 2 x^{3} - x 2 x^{2} + 1 2 x 4 1 + 3 x x^{3} + 6 x^{2} - 2 for all x \in R, then 2 f (0) + f^{'} (0) is equal to$ …2024 · 31 Jan · Shift 1 · Q48 · MCQ
Let $f :\to R \to (0, \infty)$ be strictly increasing function such that $lim_{x \to \infty} \frac{f ( 7 x )}{f ( x )} = 1$ . Then, the value of $lim_{x \to \infty} [\frac{f ( 5 x )}{f ( x )} - 1]$ …2024 · 31 Jan · Shift 2 · Q41 · MCQ
If the function $f : (- \infty, - 1] \to (a, b]$ defined by $f (x) = e^{x^{3} - 3 x + 1}$ is one - one and onto, then the distance of the point $P (2 b + 4, a + 2)$ from the line $x + e^{- 3} y = 4$ is :2024 · 31 Jan · Shift 2 · Q43 · MCQ
The sum of the absolute maximum and minimum values of the function $f (x) = x^{2} - 5 x + 6 - 3 x + 2$ in the interval $[- 1, 3]$ is equal to :2023 · 1 Feb · Shift 2 · Q26 · MCQ
The number of points, where the curve $y = x^{5} - 20 x^{3} + 50 x + 2$ crosses the $x$ -axis, is .2023 · 6 Apr · Shift 2 · Q44 · Numerical
If $a_{α}$ is the greatest term in the sequence $α_{n} = \frac{n ^{3}}{n ^{4} + 147}, n = 1, 2, 3, \dots$ , then $α$ is equal to .2023 · 8 Apr · Shift 1 · Q44 · Numerical
A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm $^{2}$ ) is equal to :2023 · 10 Apr · Shift 1 · Q32 · MCQ
Let $g (x) = f (x) + f (1 - x)$ and $f^{''} (x) > 0, x \in (0, 1)$ . If $g$ is decreasing in the interval $(0, a)$ and increasing in the interval $(α, 1)$ , then $tan^{- 1} (2 α) + tan^{- 1} (\frac{1}{α}) + tan^{- 1} (\frac{α + 1}{α})$ …2023 · 10 Apr · Shift 2 · Q24 · MCQ
If the local maximum value of the function $f (x) = (\frac{3 e}{2 s i n x})^{s i n^{2} x}, x \in (0, \frac{π}{2})$ , is $\frac{k}{e}$ , then $(\frac{k}{e})^{8} + \frac{k ^{8}}{e ^{5}} + k^{8}$ is equal to2023 · 12 Apr · Shift 1 · Q32 · MCQ
$max_{0 \leq x \leq π} {x - 2 sin x cos x + \frac{1}{3} sin 3 x} =$ 2023 · 13 Apr · Shift 1 · Q36 · MCQ
Consider the triangles with vertices $A (2, 1), B (0, 0)$ and $C (t, 4), t \in [0, 4]$ . If the maximum and the minimum perimeters of such triangles are obtained at $t = α$ and $t = β$ respectively, then $6 α + 21 β$ is equal to …2023 · 15 Apr · Shift 1 · Q42 · Numerical
Let $x = 2$ be a local minima of the function $f (x) = 2 x^{4} - 18 x^{2} + 8 x + 12, x \in (- 4, 4)$ . If M is local maximum value of the function $f$ in ( $- 4, 4)$ , then M =2023 · 25 Jan · Shift 1 · Q26 · MCQ
Let $f : (0, 1) \to R$ be a function defined $f (x) = \frac{1}{1 - e ^{- x}}$ , and $g (x) = (f (- x) - f (x))$ . Consider two statements (I) g is an increasing function in (0, 1) (II) g is one-one in (0, 1) Then,2023 · 25 Jan · Shift 1 · Q38 · MCQ
Let the function $f (x) = 2 x^{3} + (2 p - 7) x^{2} + 3 (2 p - 9) x - 6$ have a maxima for some value of $x 0$ . Then, the set of all values of p is2023 · 25 Jan · Shift 2 · Q30 · MCQ
If the functions $f (x) = \frac{x ^{3}}{3} + 2 b x + \frac{a x ^{2}}{2}$ and $g (x) = \frac{x ^{3}}{3} + a x + b x^{2}, a e q 2 b$ have a common extreme point, then $a + 2 b + 7$ is equal to :2023 · 30 Jan · Shift 2 · Q30 · MCQ
A wire of length $20 m$ is to be cut into two pieces. A piece of length $l_{1}$ is bent to make a square of area $A_{1}$ and the other piece of length $l_{2}$ is made into a circle of area $A_{2}$ . If $2 A_{1} + 3 A_{2}$ is minimum…2023 · 31 Jan · Shift 1 · Q32 · MCQ
The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :2022 · 24 Jun · Shift 1 · Q24 · MCQ
For the function $f (x) = 4 lo g_{e} (x - 1) - 2 x^{2} + 4 x + 5, x > 1$ , which one of the following is NOT correct?2022 · 24 Jun · Shift 1 · Q29 · MCQ
The sum of absolute maximum and absolute minimum values of the function $f (x) = ∣2 x^{2} + 3 x - 2∣ + sin x cos x$ in the interval [0, 1] is :2022 · 24 Jun · Shift 1 · Q30 · MCQ
Let $λ^{*}$ be the largest value of $λ$ for which the function $f_{λ} (x) = 4 λ x^{3} - 36 λ x^{2} + 36 x + 48$ is increasing for all x $\in$ R. Then $f_{λ^{*}} (1) + f_{λ^{*}} (- 1)$ is…2022 · 24 Jun · Shift 2 · Q36 · MCQ
If the absolute maximum value of the function $f (x) = (x^{2} - 2 x + 7) e^{(4 x^{3} - 12 x^{2} - 180 x + 31)}$ in the interval $[- 3, 0]$ is $f (α)$ , then :2022 · 25 Jul · Shift 1 · Q26 · MCQ
The curve $y (x) = a x^{3} + b x^{2} + c x + 5$ touches the $x$ -axis at the point $P (- 2, 0)$ and cuts the $y$ -axis at the point $Q$ , where $y^{'}$ is equal to 3 . Then the local maximum value of $y (x)$ is:2022 · 25 Jul · Shift 1 · Q27 · MCQ
The sum of the maximum and minimum values of the function $f (x) = ∣5 x - 7∣ + [x^{2} + 2 x]$ in the interval $[\frac{5}{4}, 2]$ , where $[t]$ is the greatest integer $\leq t$ , is .2022 · 25 Jul · Shift 2 · Q40 · Numerical
Water is being filled at the rate of 1 cm3 / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm2 / sec) at which the wet conical…2022 · 25 Jun · Shift 2 · Q30 · MCQ
Let $f (x) = ∣ (x - 1) (x^{2} - 2 x - 3) ∣ + x - 3, x \in R$ . If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to .2022 · 25 Jun · Shift 2 · Q44 · Numerical
Let the function $f (x) = 2 x^{2} - lo g_{e} x, x > 0$ , be decreasing in $(0, a)$ and increasing in $(a, 4)$ . A tangent to the parabola $y^{2} = 4 a x$ at a point $P$ on it passes through the point $(8 a, 8 a - 1)$ …2022 · 26 Jul · Shift 1 · Q44 · Numerical
If the maximum value of $a$ , for which the function $f_{a} (x) = tan^{- 1} 2 x - 3 a x + 7$ is non-decreasing in $(- \frac{π}{6}, \frac{π}{6})$ , is $\overset{a}{ˉ}$ , then $f_{\overset{a}{ˉ}} (\frac{π}{8})$ is equal to :2022 · 26 Jul · Shift 2 · Q24 · MCQ
The sum of the absolute minimum and the absolute maximum values of the function f(x) = |3x $-$ x2 + 2| $-$ x in the interval [ $-$ 1, 2] is :2022 · 26 Jun · Shift 1 · Q27 · MCQ
Let $f (x) = 2 cos^{- 1} x + 4 cot^{- 1} x - 3 x^{2} - 2 x + 10$ , $x \in [- 1, 1]$ . If [a, b] is the range of the function f, then 4a $-$ b is equal to :2022 · 26 Jun · Shift 1 · Q32 · MCQ
Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r, for which the sum of their volumes is maximum, is :2022 · 26 Jun · Shift 2 · Q29 · MCQ
A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $tan^{- 1} \frac{3}{4}$ . Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter…2022 · 27 Jul · Shift 2 · Q36 · Numerical
If the minimum value of $f (x) = \frac{5 x ^{2}}{2} + \frac{α}{x ^{5}}, x > 0$ , is 14 , then the value of $α$ is equal to :2022 · 28 Jul · Shift 1 · Q35 · MCQ
The function $f (x) = x e^{x (1 - x)}, x \in R$ , is :2022 · 28 Jul · Shift 2 · Q26 · MCQ
The number of real solutions of $x^{7} + 5 x^{3} + 3 x + 1 = 0$ is equal to .2022 · 28 Jun · Shift 1 · Q33 · MCQ
Let $f (x) = 3^{(x^{2} - 2)^{3} + 4}, x \in R$ . Then which of the following statements are true? $P : x = 0$ is a point of local minima of $f Q : x = 2$ is a point of inflection of $f R : f^{'}$ is…2022 · 29 Jul · Shift 1 · Q40 · MCQ
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square…2022 · 29 Jun · Shift 1 · Q30 · MCQ
If xy4 attains maximum value at the point (x, y) on the line passing through the points (50 + $α$ , 0) and (0, 50 + $α$ ), $α$ > 0, then (x, y) also lies on the line :2022 · 30 Jun · Shift 1 · Q28 · MCQ
Let $f (x) = 4 x^{3} - 11 x^{2} + 8 x - 5, x \in R$ . Then f :2022 · 30 Jun · Shift 1 · Q29 · MCQ
A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of…2022 · 30 Jun · Shift 1 · Q38 · Numerical
The function $f (x) = x^{3} - 6 x^{2} + a x + b$ is such that $f (2) = f (4) = 0$ . Consider two statements : Statement 1 : there exists x1, x2 $\in$ (2, 4), x1 < x2, such that f'(x1) = $-$ 1 and f'(x2) = 0. Statement 2 : there exists x3, x4 $\in$ …2021 · 1 Sep · Shift 2 · Q29 · MCQ
Let f be a real valued function, defined on R $-$ { $-$ 1, 1} and given by f(x) = 3 loge $\frac{x - 1}{x + 1} - \frac{2}{x - 1}$ . Then in which of the following intervals, function f(x) is increasing?2021 · 16 Mar · Shift 2 · Q25 · MCQ
The maximum value of $f (x) = sin^{2} x 1 + sin^{2} x sin^{2} x 1 + cos^{2} x cos^{2} x cos^{2} x cos 2 x cos 2 x sin 2 x, x \in R$ …2021 · 16 Mar · Shift 2 · Q40 · MCQ
Consider the function $f : R \to R$ defined by $f (x) = ⎩ ⎨ ⎧ (2 - sin (\frac{1}{x})) ∣ x ∣ 0 x \neq = 0 x = 0$ Then $f$ is :2021 · 17 Mar · Shift 2 · Q30 · MCQ
Let f : [ $-$ 1, 1] $\to$ R be defined as f(x) = ax2 + bx + c for all x $\in$ [ $-$ 1, 1], where a, b, c $\in$ R such that f( $-$ 1) = 2, f'( $-$ 1) = 1 for x $\in$ ( $-$ 1, 1) the maximum value of f ''(x) is $\frac{1}{2}$ . If f(x) $\le…2021 · 17 Mar · Shift 2 · Q41 · Numerical
Let $A = [a_{ij}]$ be a 3 $\times$ 3 matrix, where $a_{ij} = ⎩ ⎨ ⎧ 1 - x 2 x + 1,,, i f i = j i f ∣ i - j ∣ = 1 o t h er w i se .$ …2021 · 20 Jul · Shift 1 · Q30 · MCQ
Let 'a' be a real number such that the function f(x) = ax2 + 6x $-$ 15, x $\in$ R is increasing in $(- \infty, \frac{3}{4})$ and decreasing in $(\frac{3}{4}, \infty)$ . Then the function g(x) = ax2 $-$ 6x…2021 · 20 Jul · Shift 1 · Q33 · MCQ
The sum of all the local minimum values of the twice differentiable function f : R $\to$ R defined by $f (x) = x^{3} - 3 x^{2} - \frac{3 f ^{''} ( 2 )}{2} x + f^{''} (1)$ is :2021 · 20 Jul · Shift 2 · Q32 · MCQ
Let f : R $\to$ R be defined as $f (x) = {- \frac{4}{3} x^{3} + 2 x^{2} + 3 x, 3 x e^{x}, x > 0 x \leq 0$ . Then f is increasing function in the interval2021 · 22 Jul · Shift 2 · Q25 · MCQ
The function f(x) = $\frac{4 x ^{3} - 3 x ^{2}}{6} - 2 sin x + (2 x - 1) cos x$ :2021 · 24 Feb · Shift 1 · Q27 · MCQ
The minimum value of $α$ for which the equation $\frac{4}{s i n x} + \frac{1}{1 - s i n x} = α$ has at least one solution in $(0, \frac{π}{2})$ is .......2021 · 24 Feb · Shift 1 · Q37 · Numerical
Let $f : R \to R$ be defined as $f (x) = ⎩ ⎨ ⎧ - 55 x, 2 x^{3} - 3 x^{2} - 120 x, 2 x^{3} - 3 x^{2} - 36 x - 336, i f x < - 5 i f - 5 \leq x \leq 4 i f x > 4,$ Let…2021 · 24 Feb · Shift 2 · Q29 · MCQ
Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = $-$ 1 and x = 1. If $x \to 0 lim \frac{f ( x )}{x ^{3}} = 1$ , then $5. f (2)$ is equal to …2021 · 25 Feb · Shift 1 · Q45 · Numerical
Let $f (x) = 3 sin^{4} x + 10 sin^{3} x + 6 sin^{2} x - 3$ , $x \in [- \frac{π}{6}, \frac{π}{2}]$ . Then, f is :2021 · 25 Jul · Shift 1 · Q24 · MCQ
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then $(\frac{4}{π} + 1) k$ …2021 · 26 Aug · Shift 1 · Q37 · Numerical
The local maximum value of the function $f (x) = (\frac{2}{x})^{x^{2}}$ , x > 0, is2021 · 26 Aug · Shift 2 · Q25 · MCQ
The maximum slope of the curve $y = \frac{1}{2} x^{4} - 5 x^{3} + 18 x^{2} - 19 x$ occurs at the point :2021 · 26 Feb · Shift 1 · Q34 · MCQ
Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to .2021 · 26 Feb · Shift 2 · Q47 · Numerical
A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the…2021 · 27 Aug · Shift 1 · Q33 · MCQ
The number of distinct real roots of the equation 3x4 + 4x3 $-$ 12x2 + 4 = 0 is .2021 · 27 Aug · Shift 1 · Q35 · Numerical
A box open from top is made from a rectangular sheet of dimension a $\times$ b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :2021 · 27 Aug · Shift 2 · Q30 · MCQ
The number of real roots of the equation $e^{4 x} + 2 e^{3 x} - e^{x} - 6 = 0$ is :2021 · 31 Aug · Shift 1 · Q22 · MCQ
If 'R' is the least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the function f(x) = x2 + ax + 1 is decreasing on [1, 2], then the value of |R $-$ S| is …2021 · 31 Aug · Shift 1 · Q39 · Numerical
Let f(x) be a cubic polynomial with f(1) = $-$ 10, f( $-$ 1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = $-$ 1. Then f(3) is equal to .2021 · 31 Aug · Shift 2 · Q42 · Numerical
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :2020 · 2 Sep · Shift 1 · Q31 · MCQ
Let f : (–1, $\infty$ ) $\to$ R be defined by f(0) = 1 and f(x) = $\frac{1}{x} lo g_{e} (1 + x)$ , x $e$ 0. Then the function f :2020 · 2 Sep · Shift 2 · Q42 · MCQ
The function, f(x) = (3x – 7)x2/3, x $\in$ R, is increasing for all x lying in :2020 · 3 Sep · Shift 1 · Q27 · MCQ
If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape; then the rate of change of its volume (in cm3/sec), when the length of a side of the cube is 10 cm, is :2020 · 3 Sep · Shift 2 · Q34 · MCQ
Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $\geq$ 1 and f''(x) $\geq$ 4, for all x $\in$ (1, 6), then :2020 · 4 Sep · Shift 1 · Q33 · MCQ
The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x2–1 below the x-axis, is :2020 · 4 Sep · Shift 2 · Q27 · MCQ
If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, -4), then PQ2 is equal to:2020 · 5 Sep · Shift 1 · Q33 · MCQ
If x = 1 is a critical point of the function f(x) = (3x2 + ax – 2 – a)ex , then :2020 · 5 Sep · Shift 2 · Q34 · MCQ
The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time interval [t1 , t2 ] is attained at the point :2020 · 6 Sep · Shift 1 · Q29 · MCQ
The set of all real values of $λ$ for which the function $f (x) = (1 - cos^{2} x) (λ + sin x), x \in (- \frac{π}{2}, \frac{π}{2})$ has exactly one maxima and exactly one…2020 · 6 Sep · Shift 2 · Q25 · MCQ
Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If $x \to 0 lim (2 + \frac{f ( x )}{x ^{3}}) = 4$ , then which one of the following is not true?2020 · 7 Jan · Shift 2 · Q30 · MCQ
Let ƒ(x) = xcos–1(–sin|x|), $x \in [- \frac{π}{2}, \frac{π}{2}]$ , then which of the following is true?2020 · 8 Jan · Shift 1 · Q25 · MCQ
Let ƒ(x) be a polynomial of degree 3 such that ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point at x = –1 and ƒ'(x) has a critical point at x = 1. Then ƒ(x) has a local minima at x = .2020 · 8 Jan · Shift 2 · Q20 · Numerical
A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness the melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate (in cm/min.) at which of the thickness of ice decreases, is :2020 · 9 Jan · Shift 1 · Q27 · MCQ
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function, ƒ(x) = 9x4 + 12x3 – 36x2 + 25, x $\in$ R, then :2019 · 8 Apr · Shift 1 · Q31 · MCQ
Let ƒ : [0, 2] $\to$ R be a twice differentiable function such that ƒ''(x) > 0, for all x $\in$ (0, 2). If $ϕ$ (x) = ƒ(x) + ƒ(2 – x), then $ϕ$ is :2019 · 8 Apr · Shift 1 · Q45 · MCQ
The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is2019 · 8 Apr · Shift 2 · Q41 · MCQ
If ƒ(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set S = {x $\in$ R : ƒ(x) = ƒ(0)} Contains exactly :2019 · 9 Apr · Shift 1 · Q26 · MCQ
A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $tan^{- 1} (\frac{1}{2})$ . Water is poured into it at a constant rate of 5 cubic meter per minute. The the rate (in m/min.),…2019 · 9 Apr · Shift 2 · Q40 · MCQ
The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :2019 · 9 Jan · Shift 1 · Q27 · MCQ
A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3 /min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is :2019 · 10 Apr · Shift 2 · Q24 · MCQ
The shortest distance between the point $(\frac{3}{2}, 0)$ and the curve y = $x$ , (x > 0), is -2019 · 10 Jan · Shift 1 · Q40 · MCQ
A helicopter is flying along the curve given by y – x3/2 = 7, (x $\geq$ 0). A soldier positioned at the point $(\frac{1}{2}, 7)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is -2019 · 10 Jan · Shift 2 · Q42 · MCQ
The maximum value of the function f(x) = 3x3 – 18x2 + 27x – 40 on the set S = ${x \in R : x^{2} + 30 \leq 11 x}$ is :2019 · 11 Jan · Shift 1 · Q28 · MCQ
Let f(x) = $\frac{x}{a ^{2} + x ^{2}} - \frac{d - x}{b ^{2} + ( d - x ) ^{2}},$ x $\in$ R, where a, b and d are non-zero real constants. Then :2019 · 11 Jan · Shift 2 · Q25 · MCQ
A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when…2019 · 12 Apr · Shift 1 · Q36 · MCQ
If m is the minimum value of k for which the function f(x) = x $k x - x^{2}$ is increasing in the interval [0,3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :2019 · 12 Apr · Shift 1 · Q44 · MCQ
If the function f given by f(x) = x3 – 3(a – 2)x2 + 3ax + 7, for some a $\in$ R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, $\frac{f ( x ) - 14}{( x - 1 ) ^{2}} = 0 (x e 1)$ …2019 · 12 Jan · Shift 2 · Q41 · MCQ
If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm2) of this cone is :2018 · 15 Apr · Shift 1 · Q43 · MCQ
Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f(x) = 2x3 $-$ 9x2 + 12x + 5 in the interval [0, 3]. Then M $-$ m is equal to :2018 · 16 Apr · Shift 1 · Q30 · MCQ
Let $f (x) = x^{2} + \frac{1}{x ^{2}}$ and $g (x) = x - \frac{1}{x}$ , $x \in R - {- 1, 0, 1}$ . If $h (x) = \frac{f ( x )}{g ( x )}$ , then the local…2018 · Shift 0 · Q43 · MCQ
The function f defined by f(x) = x3 $-$ 3x2 + 5x + 7 , is :2017 · 9 Apr · Shift 1 · Q40 · MCQ
Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :2017 · Shift 0 · Q41 · MCQ
The minimum distance of a point on the curve y = x2−4 from the origin is :2016 · 9 Apr · Shift 1 · Q44 · MCQ
Let f(x) = sin4x + cos4 x. Then f is an increasing function in the interval :2016 · 10 Apr · Shift 1 · Q40 · MCQ
A wire of length $2$ units is cut into two parts which are bent respectively to form a square of side $= x$ units and a circle of radius $= r$ units. If the sum of the areas of the square and the circle so formed is minimum, then:2016 · Shift 0 · Q30 · MCQ
Let $f (x)$ be a polynomial of degree four having extreme values at $x = 1$ and $x = 2$ . If $x \to 0 lim [1 + \frac{f ( x )}{x ^{2}}] = 3$ , then f $(2)$ is equal to :2015 · Shift 0 · Q34 · MCQ
If $x = - 1$ and $x = 2$ are extreme points of $f (x) = α lo g ∣ x ∣ + β x^{2} + x$ then2014 · Shift 0 · Q33 · MCQ
The real number $k$ for which the equation, $2 x^{3} + 3 x + k = 0$ has two distinct real roots in $[0, 1]$ 2013 · Shift 0 · Q43 · MCQ
Let $a, b \in R$ be such that the function $f$ given by $f (x) = I n ∣ x ∣ + b x^{2} + a x, x e 0$ has extreme values at $x = - 1$ and $x = 2$ Statement-1 : $f$ has local maximum at $x = - 1$ and at $x = 2$ . Statement-2 : $a = \frac{1}{2}$ …2012 · Shift 0 · Q27 · MCQ
A line is drawn through the point $(1, 2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle $O P Q,$ where $O$ is the origin. If the area of the triangle $O P Q$ is least, then the slope of the line $P Q$ is :2012 · Shift 0 · Q34 · MCQ
A spherical balloon is filled with $4500 π$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 π$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the…2012 · Shift 0 · Q36 · MCQ
For $x \in (0, \frac{5 π}{2}),$ define $f (x) = 0 \int x t sin t d t .$ Then $f$ has2011 · Shift 0 · Q34 · MCQ
Let $f : R \to R$ be defined by $f (x) = {k - 2 x, i f 2 x + 3, i f x \leq - 1 x > - 1$ If $f$ has a local minimum at $x = - 1$ , then a possible value of $k$ …2010 · Shift 0 · Q30 · MCQ
Let $f : R \to R$ be a continuous function defined by $f (x) = \frac{1}{e ^{x} + 2 e ^{- x}}$ Statement - 1 : $f (c) = \frac{1}{3},$ for some $c \in R$ . Statement - 2 : $0 < f (x) \leq \frac{1}{2 2},$ …2010 · Shift 0 · Q39 · MCQ
Given $P (x) = x^{4} + a x^{3} + b x^{2} + c x + d$ such that $x = 0$ is the only real root of $P^{'} (x) = 0.$ If $P (- 1) < P (1),$ then in the interval $[- 1, 1] :$ 2009 · Shift 0 · Q36 · MCQ
Suppose the cubic $x^{3} - p x + q$ has three distinct real roots where $p > 0$ and $q > 0$ . Then which one of the following holds?2008 · Shift 0 · Q40 · MCQ
How many real solutions does the equation $x^{7} + 14 x^{5} + 16 x^{3} + 30 x - 560 = 0$ have?2008 · Shift 0 · Q41 · MCQ
If $p$ and $q$ are positive real numbers such that $p^{2} + q^{2} = 1$ , then the maximum value of $(p + q)$ is2007 · Shift 0 · Q52 · MCQ
The function $f (x) = tan^{- 1} (sin x + cos x)$ is an incresing function in2007 · Shift 0 · Q53 · MCQ
The function $f (x) = \frac{x}{2} + \frac{2}{x}$ has a local minimum at2006 · Shift 0 · Q57 · MCQ
A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$ . The maximum area enclosed by the park is2006 · Shift 0 · Q58 · MCQ
A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 cm / s^{2}$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 cm / s$ …2005 · Shift 0 · Q115 · MCQ
A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?2005 · Shift 0 · Q62 · MCQ
Area of the greatest rectangle that can be inscribed in the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ 2005 · Shift 0 · Q72 · MCQ
A spherical iron ball $10$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50$ cm $^{3}$ /min. When the thickness of ice is $5$ cm, then the rate at which the thickness of ice decreases is2005 · Shift 0 · Q73 · MCQ
A point on the parabola $y^{2} = 18 x$ at which the ordinate increases at twice the rate of the abscissa is2004 · Shift 0 · Q78 · MCQ
The real number $x$ when added to its inverse gives the minimum sum at $x$ equal :2003 · Shift 0 · Q103 · MCQ
If the function $f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1,$ where $a > 0,$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2} = q$ , then $a$ equals2003 · Shift 0 · Q78 · MCQ
The maximum distance from origin of a point on the curve $x = a sin t - b sin (\frac{a t}{b}) y = a cos t - b cos (\frac{a t}{b}),$ both $a, b > 0$ is2002 · Shift 0 · Q77 · MCQ