Every invertible function is:
- ✓Monotonic function.
- BConstant function.
- CIdentity function.
- DNot necessarily monotonic function.
Answer: A.
View full solution →Question types
APPLICATION OF DERIVATIVES question types
1,028 questions across 8 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.
1,028
Questions
8
Question groups
5
Question types
01
M.C.Q (1 Marks)
182 Q→02Assertion (A) & Reason (B) MCQ
32 Q→031 Marks Question
33 Q→042 Marks Questions
89 Q→053 Marks Question
172 Q→065 Marks Questions
484 Q→07Case study (4 Marks)
26 Q→08Fill In The Blanks[1 Marks ]
10 Q→Sample Questions
APPLICATION OF DERIVATIVES questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
Choose the correct answer from the given four options$:\ f(x) = x^x$ has a stationary point at:
- A$\text{x}=\text{e}$
- ✓$\text{x}=\frac{1}{\text{e}}$
- C$\text{x}=1$
- D$\text{x}=\sqrt{\text{e}}$
Answer: B.
View full solution →If $s = t^3- 4t^2+ 5$ describes the motion of a particle, then its velocity when the acceleration vanishes, is:
- A$\frac{16}{2}\ \text{unit}/\text{sec}.$
- B$\frac{\text{-32}}{3}\ \text{unit}/\text{sec}.$
- C$\frac{4}{3}\ \text{unit}/\text{sec}.$
- ✓$-\frac{16}{3}\ \text{unit}/\text{sec}.$
Answer: D.
View full solution →If the function $\text{f}(\text{x})=\frac{-\text{x}}{2}+\sin\text{x}$ defined on $\Big[\frac{-\pi}{3},\frac{\pi}{3}\Big]$ is:
- ✓Increasing.
- BDecreasing.
- CConstant.
- DNone of these.
Answer: A.
View full solution →If $\text{f}(\text{x})=\frac{1}{4\text{x}^{2}+2\text{x}+1}$, then its maximum value is :
- ✓$\frac{4}{3}$
- B$\frac{2}{3}$
- C$1$
- D$\frac{3}{4}$
Answer: A.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A)$ : Let $\text{f(x)}=\text{e}^\frac{1}{\text{x}}$ is defined for all real values of $x.$
Reason $(R) : \text{f(x)}=\text{e}^\frac{1}{\text{x}}$ is always decreasing as $\text{f'(x)} < 0$ is $\text{x }\in\text{ R}$
Assertion $(A)$ : Let $\text{f(x)}=\text{e}^\frac{1}{\text{x}}$ is defined for all real values of $x.$
Reason $(R) : \text{f(x)}=\text{e}^\frac{1}{\text{x}}$ is always decreasing as $\text{f'(x)} < 0$ is $\text{x }\in\text{ R}$
- ✓$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C$A$ is true: $R$ is false.
- D$A$ is false: $R$ is true.
Answer: A.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) :$ if $\text{f(x)}=\text{a}(\text{x}+\sin\text{x})$ is increasing function if $a \in(0,\infty)$
Reason $(R) :$ The given function $\text{f(x) }$is increasing only if $a \in(0,\infty)$
Assertion $(A) :$ if $\text{f(x)}=\text{a}(\text{x}+\sin\text{x})$ is increasing function if $a \in(0,\infty)$
Reason $(R) :$ The given function $\text{f(x) }$is increasing only if $a \in(0,\infty)$
- A$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C$A$ is true: $R$ is false.
- ✓$A$ is false: $R$ is true.
Answer: D.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) :$ The curve $y = x^2$ represents a parabola with vertex at origin.
Reason $(R) :$ For a curve Tangent and Normal lines are always perpendicular at thepoint of contact.
Assertion $(A) :$ The curve $y = x^2$ represents a parabola with vertex at origin.
Reason $(R) :$ For a curve Tangent and Normal lines are always perpendicular at thepoint of contact.
- ABoth $A$ and $R $ are true and $R$ is the correct explanation of $A$
- ✓Both $A$ and $R$ are true but $R$ is $\text{NOT}$ the correct explanation of $A$
- C$A$ is true but $R$ is false.
- D$A$ is false but $R$ is true.
Answer: B.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A) :$ The function $\text{y}=\log(1+\text{x)}\ \frac{2\text{x}}{2+\text{x}}$ is decreasing throughout its domain
Reason $(R) :$ The domain of the function $\text{y}=\log(1+\text{x)} \ \frac{2\text{x}}{2+\text{x}}$ is $(-1,\infty).$
Assertion $(A) :$ The function $\text{y}=\log(1+\text{x)}\ \frac{2\text{x}}{2+\text{x}}$ is decreasing throughout its domain
Reason $(R) :$ The domain of the function $\text{y}=\log(1+\text{x)} \ \frac{2\text{x}}{2+\text{x}}$ is $(-1,\infty).$
- ABoth $A$ and $R$ are true and $R$ is the correct explanation of $A$
- BBoth $A$ and $R$ are true but $R$ is $\text{NOT}$ the correct explanation of $A$
- C$A$ is true but $R$ is false.
- ✓$A$ is false but $R$ is true.
Answer: D.
View full solution →Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) :$ The equation of tangent to the curve $ \text{y} = \sin\text{x}$ at the point $(0, 0)$ is $y = x.$
Reason $(R) :$ if $\text{y}=\sin$ then $\frac{\text{dy}}{\text{dx}}$ at $x = 0$ is $1.$
Assertion $(A) :$ The equation of tangent to the curve $ \text{y} = \sin\text{x}$ at the point $(0, 0)$ is $y = x.$
Reason $(R) :$ if $\text{y}=\sin$ then $\frac{\text{dy}}{\text{dx}}$ at $x = 0$ is $1.$
- ✓$A$ is true, $R$ is true: $R$ is a correct explanation for $A.$
- B$A$ is true $R$ is true; $R$ is not a correct explanation for $A.$
- C$A$ is true: $R$ is false.
- D$A$ is false: $R$ is true.
Answer: A.
View full solution →Find the distance between the planes $\overrightarrow{\text{r}}.(2\hat{i} - 3\hat{j} +6\hat{k}) - 4 = 0 \text{ and}\overrightarrow{\text{r}}.(6\hat{i} - 9\hat{j} +18\hat{k}) + 30 = 0.$
View full solution →$\text{If A} = \begin{bmatrix} 2 & 3 \\ 5 & -2 \\ \end{bmatrix}, \text{then write A}^{-1}. $
View full solution →If $\overrightarrow{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} + 3\hat{\text{k}}$ and $ \overrightarrow{\text{b}} = 3\hat{\text{i}}+ 5\hat{\text{j}} - 2\hat{\text{k}},$ then find $|\overrightarrow{\text{a}}\times|\overrightarrow{\text{b}|}.$
View full solution →$\text{If A} = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \\ \end{bmatrix}. \text{find } \alpha \text{ satisfying } 0 < \alpha < \frac{\pi}{2} \text{when A+ A}^{\text{T}} = \sqrt{2}\text{ I}_{2} : $ where $\text{A}^{\text{T}}$ is transpose of $\text{A}$
View full solution →If A is a matrix $3\times3$ If A is a and $\text{|3A| = K|A|,}$ then write the value of k.
View full solution →Show that the function $f(x) = x^3 – 3x^2 + 6x – 100$ is increasing on R.
View full solution →The volume of a sphere is increasing at the rate of 8cm3/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12cm.
The length x, of a rectangle is decreasing at the rate of 5 cm/minute and the width y, is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of the area of the rectangle.
The volume of a cube is increasing at the rate of $9 cm^3/s$. How fast is its surface area increasing when the length of an edge is $10 cm$?
View full solution →Show that the function f given by $f(x) = \tan^{–1} (sin\ x + cos\ x)$ is decreasing for all $\text{x} \in \bigg(\frac{\pi}{4}, \frac{\pi}{2}\bigg).$
View full solution →Find the equations of the tangent and normal to the given curves at the indicated points:
$y = x^3 $ at $(1, 1)$
View full solution →$y = x^3 $ at $(1, 1)$
If $y = \log_e x$, then find $\triangle\text{y}$ when $x = 3$ and $\triangle\text{x} = 0.03.$
View full solution →Find the point on the curve $y = x^2$ where the slope of the tangent is equal to the x-coordinate of the point.
View full solution →Find the equations of the tangent and normal to the parabola $y^2 = 4ax$ at the point $(at^2, 2a.t)$
View full solution →Write the coordinates of the point at which the tangent to the curve $y = 2x^2 - x + 1$ is parallel to the line $y = 3x + 9.$
View full solution →Solve the differential equation: $(x + 1)\frac{dy}{dx} -y = e^{3x} (x + 1)^{3}$
View full solution →Give that vectors $\overrightarrow{\text{a}}, \overrightarrow{\text{b}}, \overrightarrow{\text{c}}.$ form a triangle such that $\overrightarrow{\text{a}} = \overrightarrow{\text{b}} + \overrightarrow{\text{c}}.$ Find p, q, r, s such that area of triangle is $5\sqrt{6}$ where $\overrightarrow{\text{a}} = \text{p }\hat{i} + \text{q }\hat{j} + \text{r }\hat{k} = \overrightarrow{\text{b}} = \text{s }\hat{i} + \text{3 }\hat{j} + \text{4 }\hat{k} \text{ and} \overrightarrow{\text{c}} = \text{3 }\hat{i} + \hat{j} - \text{2}\hat{k}.$
View full solution →Differentiate $(\sin 2x)^{x} + \sin^{-1}\sqrt{3x}$ with respect to $x.$
View full solution →$\text{Find k, if}\ f(x) = \begin{cases} k\sin\frac{\pi}{2}(x + 1), & x\leq0 \\ \frac{\tan x - \sin x}{x^{3}}, & x\geq0 \end{cases}$ is continuous at $x = 0.$
View full solution →Using elementary row operations, find the inverse of the following matrix:
$\text{A} = \begin{pmatrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{pmatrix}$
View full solution →$\text{A} = \begin{pmatrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{pmatrix}$
A magazine company in a town has $5000$ subscribers on its list and collects fix charges of $₹ 3000$ per year from each subscriber. 'The company proposes to increase the annual charges, and it is believed that for every increase of $₹ 1$ one subscriber will discontinue service.
Based on the above information, answer the following questions.
View full solution →Based on the above information, answer the following questions.
- If $x$ denote the amount of increase in annual charges, then revenue $R,$ as a function of $x$ can be represented as.
- $R(x) = 3000 × 5000 × x$
- $R(x) = (3000 - 2x)(5000 + 2x)$
- $R(x) = (5000 + x)(3000 - x)$
- $R(x) = (3000 + x)(5000 - x)$
- If magazine company increases $₹\ 500$ as annual charges, then R is equal to.
- $₹\ 15750000$
- $₹\ 16750000$
- $₹\ 17500000$
- $₹\ 15000000$
- If revenue collected by the magazine company is $₹\ 15640000$, then value of amount increased as annual charges for each subscriber, is.
- $400$
- $1600$
- Both $(a)$ and $(b)$
- None of these
- What amount of increase in annual charges will bring maximum revenue?
- $₹\ 1000$
- $₹\ 2000$
- $₹\ 3000$
- $₹\ 4000$
- Maximum revenue is equal to.
- $₹\ 15000000$
- $₹\ 16000000$
- $₹\ 20500000$
- $₹\ 25000000$
Shreya got a rectangular parallelepiped shaped box and spherical ball inside it as return gift. Sides of the box are x, 2x, and $\frac{\text{x}}{3},$ while radius of the ball is r.
Based on the above information, answer the following questions.
View full solution →Based on the above information, answer the following questions.
- If S represents the sum of volume of parallelepiped and sphere, then Scan be written as.
- $\frac{4\text{x}^3}{3}+\frac{2}{2}\pi\text{r}^2$
- $\frac{2\text{x}^2}{3}+\frac{4}{3}\pi\text{r}^2$
- $\frac{2\text{x}^3}{3}+\frac{4}{3}\pi\text{r}^3$
- $\frac{2}{3}\text{x}+\frac{4}{3}\pi\text{r}$
- If sum of the surface areas of box and ball are given to be constant $k^2$ then x is equal to.
- $\sqrt{\frac{\text{k}^2-4\pi\text{r}^2}{6}}$
- $\sqrt{\frac{\text{k}^2-4\pi\text{r}}{6}}$
- $\sqrt{\frac{\text{k}^2-4\pi}{6}}$
- $\text{None of these}$
- The radius of the ball, when Sis minimum, is.
- $\sqrt{\frac{\text{k}^2}{54+\pi}}$
- $\sqrt{\frac{\text{k}^2}{54+4}}$
- $\sqrt{\frac{\text{k}^2}{64+3\pi}}$
- $\sqrt{\frac{\text{k}^2}{4\pi+3}}$
- Relation between length of the box and radius of the ball can be represented as.
- $\text{x} = \frac{2}{\text{r}}$
- $\text{x}=\frac{\text{r}}{2}$
- $\text{x}=\frac{2}{\text{r}}$
- $\text{x}=3\text{r}$
- Minimum value of S is.
- $\frac{\text{k}^2}{2(3\pi+54)^\frac{2}{3}}$
- $\frac{\text{k}}{2(3\pi+54)^\frac{3}{2}}$
- $\frac{\text{k}^3}{2(4\pi+54)^\frac{1}{2}}$
- $\text{None of these}$
A tin can manufacturer designs a cylindrical tin can for a company making sanitizer and disinfector. The tin can is made to hold 3 litres of sanitizer or disinfector.
Based on the above in formation, answer the following questions.
View full solution →Based on the above in formation, answer the following questions.
- If r cm be the radius and h cm be the height of the cylindrical tin can, then the surface area expressed as a function of r as.
- $2\pi\text{r}^2$
- $2\pi\text{r}^2+6000$
- $2\pi\text{r}^2+\frac{5000}{\text{r}}$
- $2\pi\text{r}^2+\frac{6000}{\text{r}}$
- The radius that will minimize the cost of the material to manufacture the tin can is.
- $\sqrt[3]{\frac{600}{\pi}}\text{cm}$
- $\sqrt{\frac{500}{\pi}}\text{cm}$
- $\sqrt[3]{\frac{1500}{\pi}}\text{cm}$
- $\sqrt{\frac{1500}{\pi}}\text{cm}$
- The height that will minimize the cost of the material to manufacture the tin can is.
- $\sqrt[3]{\frac{600}{\pi}}\text{cm}$
- $2\sqrt[3]{\frac{1500}{\pi}}\text{cm}$
- $\sqrt{\frac{1500}{\pi}}$
- $2\sqrt{\frac{1500}{\pi}}$
- If the cost of material used to manufacture the tin can is $₹\frac{100}{\text{m}^2}$ and $\sqrt[3]{\frac{1500}{\pi}}\approx7.8,$ then minimum cost is approximately.
- ₹ 11.538
- ₹ 12
- ₹ 13
- ₹ 14
- To minimize the cost of the material used to manufacture the tin can, we need to minimize the.
- Volume.
- Curved surface area.
- Total surface area.
- Surface area of the base.
An open water tank of aluminium sheet of negligible thickness, with a square base and vertical sides, is to be constructed in a farm for irrigation. It should hold $32000$ l of water, that comes out from a tube well.
Based on above information, answer the following questions.
View full solution →Based on above information, answer the following questions.
- If the length, width, and height of the open tank be $x, x$ and $y$ $m$ respectively, then total surface area of tank is.
- $(x^2 + 2xy)m^2$
- $(2x^2 + 4xy)m^2$
- $(2x^2 + 2xy)m^2$
- $(2x^2 + 8xy)m^2$
- The relation between $x$ and $y$ is.
- $x^2y = 32$
- $xy^2 = 32$
- $x^2y^2 = 32$
- $xy = 32$
- The outer surface area of tank will be minimum when depth of tank is equal to.
- Half of its width.
- Its width.
- $\big(\frac{1}{4}\big)^\text{th}$ of its Width
- $\big(\frac{1}{3}\big)^\text{rd}$ of its Width
- The cost of material will be least when width of tank is equal to.
- Half of its depth
- Twice of its depth
- $\big(\frac{1}{4}\big)^\text{th}$ of its depth
- Thrice of its depth
- If cost of aluminium sheet is $ ₹ \frac{360}{\text{m}^2}$ then the minimum cost for the construction of tank will be.
- $₹\ 15, 000$
- $₹\ 16, 280$
- $₹\ 17, 280$
- $₹\ 18, 280$
Western music concert is organised every year in the stadium that can hold 36000 spectators. With ticket price of ₹ 10, the average attendance has been 24000. Some financial expert estimated that price of a ticket should be determined by the function.
$\text{p}(\text{x})=15-\frac{\text{x}}{3000}$ where x is the number of tickets sold.
Based on the above information, answer the following questions.
View full solution →$\text{p}(\text{x})=15-\frac{\text{x}}{3000}$ where x is the number of tickets sold.
Based on the above information, answer the following questions.
- The revenue, R as a function of x can be represented as.
- $15\text{x}-\frac{\text{x}^2}{3000}$
- $15-\frac{\text{x}^2}{3000}$
- $15\text{x}-\frac{1}{3000}$
- $15\text{x}-\frac{\text{x}}{3000}$
- The range of x is.
- [24000, 36000]
- [0, 24000]
- [0, 36000]
- None of these
- The value of x for which revenue is maximum, is.
- 20000
- 21000
- 22500
- 25000
- When the revenue is maximum, the price of the ticket is.
- ₹ 5
- ₹ 5.5
- ₹ 7
- ₹ 7.5
- How many spectators should be present to maximize the revenue?
- 21500
- 21000
- 22000
- 22500
Fill in the blanks:
The equation of normal to the curve y = tan x at (0, 0) is ________.
The equation of normal to the curve y = tan x at (0, 0) is ________.
Fill in the blanks:
The values of a for which the function f(x) = sinx - ax + b increases on R are ______.
The values of a for which the function f(x) = sinx - ax + b increases on R are ______.
Fill in the blanks:
The function $\text{f(x)}=\frac{2\text{x}^2-1}{\text{x}^4},\text{ x}>0,$ decreases in the interval _______.
View full solution →The function $\text{f(x)}=\frac{2\text{x}^2-1}{\text{x}^4},\text{ x}>0,$ decreases in the interval _______.
Fill in the blanks:
The least value of the function $\text{f(x)}=\text{ax}+\frac{\text{b}}{\text{a}}(\text{a}>0,\text{b}>0,\text{x}>0)$ is ______.
View full solution →The least value of the function $\text{f(x)}=\text{ax}+\frac{\text{b}}{\text{a}}(\text{a}>0,\text{b}>0,\text{x}>0)$ is ______.
Fill in the blanks:
The curves $y = 4x^2+ 2x - 8$ and $y = x^3- x + 13$ touch each other at the point _____.
View full solution →The curves $y = 4x^2+ 2x - 8$ and $y = x^3- x + 13$ touch each other at the point _____.
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