A wire of length $20 cm$ is bent in the form of a sector of a circle. The maximum area that can be enclosed by the wire is
View full solution →Question types
Application of Derivatives question types
273 questions across 8 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.
273
Questions
8
Question groups
5
Question types
01
M.C.Q (1 Marks)
70 Q→02Assertion (A) & Reason (B) MCQ
11 Q→031 Marks Question
74 Q→042 Marks Questions
38 Q→053 Marks Question
20 Q→065 Marks Questions
23 Q→07Case study (4 Marks)
31 Q→08Fill In The Blanks[1 Marks ]
6 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.
The value of $x$ for which $\left(x-x^2\right)$ is maximum, is
View full solution →The least value of the function $f(x)=2 \cos x+x$ in the closed interval $\left[0, \frac{\pi}{2}\right]$ is
View full solution →The value of $b$ for which the function $f(x)=x+\cos x+b$ is strictly decreasing over $R$ is
View full solution →The total surface area $(S)$ of the casted half cylinder will be
View full solution →Assertion $( A )$ : The maximum value of the function $f(x)$ $=x^5, x \in[-1,1]$, is attained at its critical point, $x=0$. Reason (R): The maximum of a function can only occur at points where derivative is zero.
View full solution →Consider the function $f(x)=x^{1 / 3}, x \in R$.
Assertion (A) : $f$ has a point of inflexion at $x=0$.
Reason $( R ): f^{\prime \prime}(0)=0$.
View full solution →Assertion (A) : $f$ has a point of inflexion at $x=0$.
Reason $( R ): f^{\prime \prime}(0)=0$.
Assertion (A) : $f(x)=\frac{1}{x-7}$ is decreasing $\forall x \in R-\{7\}$.
Reason (R) : $f^{\prime}(x)<0 \forall x \neq 7$.
View full solution →Reason (R) : $f^{\prime}(x)<0 \forall x \neq 7$.
Assertion $(A) :$ Let $f: R \rightarrow R$ be a function such that $f(x)=x^3+x^2+3 x+\sin x$. Then, $f$ is an increasing function.
Reason $(R) :$ If $f^{\prime}(x) < 0$, then $f(x)$ is a decreasing function.
View full solution →Reason $(R) :$ If $f^{\prime}(x) < 0$, then $f(x)$ is a decreasing function.
Assertion (A) : Both $\sin x$ and $\cos x$ are decreasing functions in $\left(\frac{\pi}{2}, \pi\right)$.
Reason (R): If a differentiable function decreases in $(a, b)$, then its derivative also decreases in $(a, b)$.
View full solution →Reason (R): If a differentiable function decreases in $(a, b)$, then its derivative also decreases in $(a, b)$.
The maximum value of $[x(x-1)+1]^{\frac{1}{3}}, 0 \leq x \leq 1$ is
View full solution →For all real values of $x,$ the minimum value of $\frac{1-x+x^{2}}{1+x+x^{2}}$ is
View full solution →The point on the curve $x^2 = 2y$ which is nearest to the point $(0, 5)$ is
View full solution →Find the interval of the function that is strictly increasing or decreasing: $(x + 1)^3 (x - 3)^{3 }$
View full solution →Find the interval in function $6 - 9x - x^2$ is increasing or decreasing.
View full solution →What is the maximum value of the function sin x + cos x?
At what points in the interval [0, 2$\pi$], does the function sin 2x attain its maximum value?
View full solution →Find both the maximum value and minimum value of $3{x^4} - 8{x^3} + 12{x^2} - 48x + 25$ on the interval [0, 3].
View full solution →Find the maximum profit that a company can make, if the profit function is given by $p(x) = 41 – 72x – 18x^2$
View full solution →Find the absolute maximum value and the absolute minimum value of the function: $f(x)=(x-1)^{2}+3, x \in[-3,1]$
View full solution →Find the intervals in which the function f given by $f(x) = x ^ { 3 } + \frac { 1 } { x ^ { 3 } } , x \neq 0$ is $x (i)$ increasing. $(ii)$ decreasing.
View full solution →Find the intervals in which the function f given by $f\left( x \right) = \frac{{4\sin x - 2x - x\cos x}}{{2 + \cos x}}$ is
View full solution →- increasing
- decreasing
Show that the function given by f(x) = $\frac{\log x}{x}$ has maximum at x = e.
View full solution →Find the local maxima and local minima of function,
$f(x)=x \sqrt{1-x}, \quad 0<x<1$
Find also the local maximum and the local minimum value.
View full solution →$f(x)=x \sqrt{1-x}, \quad 0<x<1$
Find also the local maximum and the local minimum value.
Find the local maxima and local minima of function
$g(x)=\frac{1}{x^{2}+2}$
Find also the local maximum and the local minimum value.
View full solution →$g(x)=\frac{1}{x^{2}+2}$
Find also the local maximum and the local minimum value.
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is$(a^{\frac{2}{3}} + b^{\frac{2}{3}})^{\frac{3}{2}}$
View full solution →A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their area is least when the side of square is double the radius of the circle.
A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2 \ m$ and volume is $8 \ m^3$. If building of tank costs $Rs. \ 70$ per sq.metres for the base and $Rs. \ 45$ per sq. metre for sides. What is the cost of least expensive tank ?
View full solution →Find the maximum area of an isosceles $\triangle $ inscribed in the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ with its vertex at one end of the major axis
View full solution →Read the following passage and answer the questions given below: elation between the height of the plant $\left(y^{\prime}\right.$ in cm $)$ with respect to its exposure to is governed by the following equation $y=4 x-\frac{1}{2} x^2$, where ' $x$ ' is the number of days exposed to the sunlight, for $x \leq 3$
View full solution →
(i) Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
(ii) Does the rate of growth of the plant increase or decrease in the first three days?
What will be the height of the plant after 2 days?
Ankit wants to construct a rectangular tank for his house that can hold $80 \mathrm{ft}^3$ of water. He wants to construct on one corner of terrace so that sufficient space is left after construction of tank. For that he has to keep width of tank constant $5 \mathrm{ft}$, but the length and heights are variables. The top of the tank is open. Building the tank cost ₹20 per sq. foot for the base and ₹10 per sq. foot for the side.
View full solution →
(i) Express cost of tank as a function of height(h).
(ii) Verify by second derivative test that cost is minimum at critical point.
(iii) Find the value of $\mathrm{h}$ at which $\mathrm{c}(\mathrm{h})$ is minimum.
OR
Find the minimum cost of tank?
On her birthday, Seema decided to donate some money to the children of an orphanage home. If there were $8$ children less, everyone would have got $₹ 10$ more. However, if there were $16$ children more, everyone would have got $₹10$ less. Let the number of children be $\mathrm{x}$ and the amount distributed by Seema for one child be $y\ ($in $₹)$ 
$(i)$ Represent given information in matrix algebra.
$(ii) $ Find the adjoint of Matrix containing information about of number of children and amount she paid?
$(iii)$ Find the number of children who were given some money by Seema?
OR
How much amount does Seema spend in distributing the money to all the students of the Orphanage?
View full solution →$(i)$ Represent given information in matrix algebra.
$(ii) $ Find the adjoint of Matrix containing information about of number of children and amount she paid?
$(iii)$ Find the number of children who were given some money by Seema?
OR
How much amount does Seema spend in distributing the money to all the students of the Orphanage?
The Government declare that farmers can get ₹ 300 per quintal for their onions on 1st July and after that, the price will be dropped by ₹ 3 per quintal per extra day. Govind's father has 80 quintals of onions in the field on 1st July and he estimates that the crop is increasing at the rate of 1 quintal per day.
View full solution →
(i) If $x$ is the number of days after $1^{\text {st }}$ July, then express price and quantity of onion and the revenue as a function of $x$.
(ii) Find the number of days after 1st July, when Govind's father attains maximum revenue.
(iii) On which day should Govind's father harvest the onions to maximize his revenue?
OR
Find the maximum revenue collected by Govind's father.
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 $.........$
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