Applications of Derivatives - ISC Class 12 Mathematics Questions with Answers
87 past-paper questions on Applications of Derivatives from ISC Class 12 Mathematics papers (2027-2017), newest first. Open one to see its answer.
- Statement I: If a satellite's altitude function $h(t)$ has a positive derivative ($h'(t) > 0$), the satellite is moving away from… 2027 · 1 mark · MCQ
- A large industrial water tank is shaped like an inverted right circular cone with a semi-vertical angle of $\tan^{-1}(0.5)$… 2027 · 5 marks · Case based
- Given the function $f(x) = \log\left[\frac{\sqrt{1+x^2}+x}{\sqrt{1+x^2}-x}\right] + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$: Find… 2027 · 3 marks · Short answer
- For what value(s) of $k$ do the tangents of two curves $x = y^2$ and $xy = k$ cut at right angles? 2027 · 1 mark · MCQ
- Evaluate the nature of the point $(0,0)$ for the curve $y = x^4$, given that $f''(0) = 0$. 2027 · 1 mark · MCQ
- A particle moves along the curve $6y = x^3 + 2$. At what point(s) on the curve, is the y-coordinate changing 8 times as fast as… 2027 · 1 mark · MCQ
- An engineering team is designing a section of a new roller coaster track. The vertical profile of the track for a specific… 2027 · 5 marks · Case based
- Find the equation of the tangent to the curve $x^2 + xy - 3 = 0$ at $(1, 2)$. 2026 · 2 marks · Short answer
- Find the point on the curve $y = (x-2)^2$ at which the tangent is parallel to the chord joining the end points $(2,0)$ and… 2026 · 2 marks · Short answer
- A person has manufactured a water tank in the shape of a closed right circular cylinder. The volume of the cylinder is… 2026 · 6 marks · Short answer
- Observe the graph given below and answer the question that follows. Statement 1: $f(x)$ increases in $(-\infty, -1)$ and… 2026 · 1 mark · MCQ
- Prove that $y = -(8x^2 + 4x + 5)$ is an increasing function in $\left(-\infty, -\frac{1}{4}\right]$. 2026 · 2 marks · Derivation
- A Dolphin jumps and taken a path given by the equation $h(t) = \frac{1}{2}(-7t^2 + 3t + 2)$, ($t \ge 0$), $h(t)$ is the height of… 2026 · 6 marks · Short answer
- The surface of a spherical balloon is increasing at the rate of $4\text{ cm}^2/\text{sec}$. Find the rate of change of volume… 2026 · 2 marks · Short answer
- A manufacturing company produces airtight cylindrical containers for storing sensitive chemicals. For safety, each container is… 2026 · 6 marks · Case based
- A van is carrying a large amount of money in cash to deposit it in two ATM machines on a hill station. The location of these… 2026 · 6 marks · Case based
- Find the equation of the normal at $(1, 2)$ to the curve $x^2 = 4y$. 2026 · 2 marks · Short answer
- A water treatment plant uses a large cylindrical storage tank to collect treated water before distribution. The tank has a… 2026 · 2 marks · Numerical
- A kite is being pulled down by a string that goes through a ring on the ground 8 meters away from the person pulling it. If the… 2025 · 4 marks · Short answer
- Consider the functions $f(x) = -(x-h)^2 + 2k$ and $g(x) = e^{x-2} + k$, where $h, k \in \mathbb{R}$. Find $f'(x)$. The graphs of… 2025 · 2 marks · Short answer
- If $f(x) = \log(1 + x) + \frac{1}{1+x}$, show that $f(x)$ attains its minimum value at $x = 0$. 2025 · 2 marks · Derivation
- A cylindrical popcorn tub of radius $10\text{ cm}$ is being filled with popcorns at the rate of $314\text{ cm}^3$ per minute. The… 2025 · 1 mark · MCQ
- $y = x^5 + x^3, x \in \mathbb{R}$ is a function. Rina, Abha and Saurabh have given their opinions about the function in the… 2025 · 1 mark · MCQ
- Find the equation to the tangent at $(0,0)$ on the curve $y = 4x^2 - 2x^3$. [Analysis] 2025 · 2 marks · Short answer
- A point $x = c$ is called the critical point of a function if: 2025 · 1 mark · MCQ
- A cone of maximum volume is inscribed in a given sphere. Then prove that ratio of the height of the cone to the diameter of the… 2025 · 6 marks · Derivation
- The graph of $f(x) = -x^3 + 27x - 2$ is given below: Find the slope of the above graph. Find the co-ordinates of turning points… 2025 · 4 marks · Short answer
- Case Study: The length of the perimeter of a slice of a pizza in the form of a sector of circle is 20 cm. $r$ be the radius of… 2025 · 4 marks · Long answer
- Find the acute angle between the curves $y = |x^2 - 1|$ and $y = |x^2 - 3|$ at their point of intersection when $x > 0$. 2025 · 4 marks · Short answer
- Let $f(x) = \frac{\log 5x}{kx}$, where $x > 0, k \in \mathbb{R}^+$. Show that $f'(x) = \frac{1-\log 5x}{kx^2}$. The graph of $f$… 2025 · 4 marks · Long answer
- $f(x)$ is a quadratic function. 2025 · 1 mark · Assertion-reason
- Given $2f(x) = \log|x|^a - bx^2 + x$, if $f(x)$ has extreme values at $x = -1$ and $x = 2$, find the value of $a$ and $b$. 2025 · 2 marks · Short answer
- The displacement, $x\text{ m}$ of a particle from a fixed point at time '$t$' seconds is given by… 2025 · 1 mark · Short answer
- Find the point on the curve $y = 2x^2 - 6x - 4$ at which the tangent is parallel to the $x$-axis. 2025 · 2 marks · Short answer
- Given $f(x) = 2\log(x-2) - x^2 + 4x + 1$ and $f'(x) = \frac{-k(x-p)(x-q)}{(x-k)}$. Find $k + p + q$. The Statement “$f(x)$ is… 2025 · 4 marks · Long answer
- In a particular forest, the population density ($P$) of wild animals (in thousands of animals per sq. km.) at a distance '$r$' km… 2025 · 4 marks · Short answer
- Sonia watches a painting which has its bottom edge 2 meters (m) above eye level and its top edge is 3 m above eye level as shown… 2025 · 6 marks · Long answer
- Find the equation of the normal to the curve $y = x^2 - 3x + 1$ at the point $(3, 1)$. 2025 · 2 marks · Short answer
- Maximum value of the function is 0. Shown below the graph of $f(x) = -2|x-3|$. Assertion (A): Maximum value of the function is 0… 2025 · 1 mark · Assertion-reason
- $y = \ln(x+1) - \ln x$ is a curve. The tangent to the curve at the point $P(1, \ln 2)$ meets x-axis at A and y-axis at B. The… 2025 · 4 marks · Long answer
- Let $f(x)$ be a polynomial function of degree 7 such that $\frac{d}{dx}(f(x)) = (x-2)^3(x+1)^2(7x-2)$ has a local minimum at… 2025 · 1 mark · Assertion-reason
- Find the interval in which the function $f(x) = x^2 e^{-x}$ is strictly increasing or decreasing. [Understanding] 2025 · 2 marks · Short answer
- Rahul sits on a speed boat in an island to do water sports, which is moving along a curve $y = \frac{1}{\cos x \sin x}$ in water… 2025 · 2 marks · Short answer
- A given quantity of metal is to be cast into a solid half circular cylinder with a rectangular base and semi-circular ends. If… 2025 · 6 marks · Derivation
- Consider the graph of the function $f(x)$ shown below: Statement 1: The function $f(x)$ is increasing in $(\frac{1}{2}, 2)$… 2025 · 1 mark · MCQ
- Let $f(x) = \cos x + \sqrt{3}\sin x, 0 \le x \le 2\pi$. The following diagram shows the graph of $f$. The $y$-intercept is at… 2025 · 6 marks · Long answer
- Seema enjoys a roller coaster ride in Ferrari world by first going downwards and then upwards to the maximum height. The relation… 2025 · 2 marks · Short answer
- In which one of the following intervals is the function $f(x) = x^3 - 12x$ increasing? 2024 · 1 mark · MCQ
- Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is '$r$' cm and height is '$h$' cm. It has a volume of… 2024 · 6 marks · Short answer
- Given that $\frac{1}{y} + \frac{1}{x} = \frac{1}{12}$ and $y$ decreases at a rate of $1\text{ cm s}^{-1}$, find the rate of… 2024 · 1 mark · Short answer
- Find a point on the curve $y = (x - 2)^2$ at which the tangent is parallel to the line joining the chord through the points… 2024 · 2 marks · Numerical
- Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of… 2024 · 6 marks · Numerical
- For the curve $y^2 = 2x^3 - 7$, the slope of the normal at $(2, 3)$ is: 2023 · 1 mark · MCQ
- Find the coordinates of a point on the curve $y = x^2 + 7x + 2$ which is closest to the straight line $y = 3x - 3$. 2023 · 6 marks · Long answer
- The interval in which the function $f(x) = 5 + 36x - 3x^2$ increases will be: 2023 · 1 mark · MCQ
- A running track of $440\text{ m}$ is to be laid out enclosing a football field. The football field is in the shape of a rectangle… 2023 · 6 marks · Numerical
- The function $f$ is defined for all $x \in \mathbb{R}$. The line with equation $y = 6x - 1$ is the tangent to the graph of $f$ at… 2023 · 2 marks · Short answer
- Prove that the semi-vertical angle of the right circular cone of given volume and least curved area is $\cot^{-1}\sqrt{2}$. 2023 · 6 marks · Derivation
- An edge of a variable cube is increasing at the rate of $10\text{ cm/sec}$. How fast will the volume of the cube increase if the… 2023 · 1 mark · MCQ
- Find the maximum volume of the cylinder which can be inscribed in a sphere of radius $3\sqrt{3}\text{ cm}$. (find the answer in… 2023 · 6 marks · Long answer
- Find the point on the curve $y^2 = 4x + 8$ for which the abscissa and ordinate changes at the same rate. 2023 · 2 marks · Short answer
- Any tangent to the curve $y = 3x^7 + 5x + 3$ : 2022 · 2 marks · MCQ
- A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window as shown in the figure… 2022 · 8 marks · Case based
- Identify from the given options the slope of the normal to the curve $x^2 + 3y + y^2 = 5$ at $(1, 1)$: 2021 · 1 mark · MCQ
- A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height. 2021 · 6 marks · Short answer
- The function $f(x) = x^3 - 3x$ is strictly decreasing on: 2021 · 1 mark · MCQ
- The function $f(x) = \sin x + \cos x$, $x \in (0, \pi/2)$; Maximum value of $f(x)$ is: 2021 · 1 mark · MCQ
- Show that the radius of a closed right circular cylinder of given surface area and maximum volume is equal to half of its height. 2020 · 6 marks · Derivation
- The edge of a variable cube is increasing at the rate of $10\text{ cm/sec}$. How fast is the volume of the cube increasing when… 2020 · 2 marks · Numerical
- The equation of tangent at $(2, 3)$ on the curve $y^2 = px^3 + q$ is $y = 4x - 7$. Find the values of ‘$p$’ and ‘$q$’. 2020 · 4 marks · Short answer
- Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is isosceles. 2020 · 6 marks · Derivation
- A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the… 2019 · 4 marks · Numerical
- Prove that the function $f(x) = x^3 - 6x^2 + 12x + 5$ is increasing on $\mathbb{R}$. 2019 · 2 marks · Derivation
- Find the point on the straight line $2x + 3y = 6$, which is closest to the origin. 2019 · 6 marks · Numerical
- The volume of a closed rectangular metal box with a square base is $4096\text{ cm}^3$. The cost of polishing the outer surface of… 2019 · 6 marks · Numerical
- Find the equations of the normals to the curve $y = x^3 + 2x + 6$ which are parallel to the line $x + 14y + 4 = 0$. 2018 · 4 marks · Short answer
- Find the approximate change in the volume $V$ of a cube of side $x$ metres caused by decreasing the side by $1\%$. 2018 · 2 marks · Short answer
- Find the points on the curve $y = 4x^3 - 3x + 5$ at which the equation of the tangent is parallel to the x-axis. 2018 · 4 marks · Short answer
- A cone is inscribed in a sphere of radius $12\text{ cm}$. If the volume of the cone is maximum, find its height. 2018 · 6 marks · Short answer
- A circular disc of radius 3 cm. is heated. Due to expansion its radius increases at the rate of $0.05\text{ cm/s}$. Find the rate… 2018 · 4 marks · Short answer
- An open topped box is to be made by removing equal squares from each corner of a $3\text{ m}$ by $8\text{ m}$ rectangle sheet of… 2018 · 6 marks · Long answer
- Find the intervals in which the function $f(x)$ is strictly increasing where, $f(x) = 10 - 6x - 2x^2$. 2018 · 2 marks · Short answer
- Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is… 2018 · 6 marks · Derivation
- Water is dripping out from a conical funnel of semi-vertical angle $\frac{\pi}{4}$ at the uniform rate of… 2018 · 4 marks · Short answer
- If $y - 2x - k = 0$ touches the conic $3x^2 - 5y^2 = 15$, find the value of $k$. 2017 · 3 marks · Numerical
- Find the coordinates of the centre, foci and equation of directrix of the hyperbola $x^2 - 3y^2 - 4x = 8$. 2017 · 5 marks · Short answer
- Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube. 2017 · 5 marks · Derivation
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