Application of Derivatives

166 CBSE Class 12 Mathematics past-year questions from 4 papers, 2023 to 2026, with answers and worked solutions.

Difficulty: 30 easy · 113 moderate · 23 hard.
Most-asked subtopics: Maxima and Minima (58), Increasing and Decreasing Functions (50), Rate of Change of Quantities (49).
Updated 19 August 2026.

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#1Rate of Change of Quantities
#2Rate of Change of Quantities
#3Rate of Change of Quantities
#4Rate of Change of Quantities
#5Rate of Change of Quantities
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#6Rate of Change of Quantities
#7Rate of Change of Quantities
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
#8Rate of Change of Quantities
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
#9Rate of Change of Quantities
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
#10Rate of Change of Quantities
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
#11Rate of Change of Quantities
#12Rate of Change of Quantities
#13Rate of Change of Quantities
#14Rate of Change of Quantities
#15Rate of Change of Quantities
#16Rate of Change of Quantities
#17Rate of Change of Quantities
#18Rate of Change of Quantities
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#22Rate of Change of Quantities
#23Rate of Change of Quantities
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
#24Rate of Change of Quantities
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
#25Rate of Change of Quantities
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
#26Rate of Change of Quantities
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
#27Rate of Change of Quantities
#28Rate of Change of Quantities
#29Rate of Change of Quantities
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
#30Rate of Change of Quantities
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
#31Rate of Change of Quantities
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
#32Rate of Change of Quantities
#33Rate of Change of Quantities
#34Rate of Change of Quantities
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
#35Rate of Change of Quantities
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
#36Rate of Change of Quantities
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
#37Rate of Change of Quantities
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
#38Rate of Change of Quantities
#39Rate of Change of Quantities
#40Rate of Change of Quantities
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#42Rate of Change of Quantities
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#45Rate of Change of Quantities
A ladder of fixed length ‘h’ is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions :
#46Rate of Change of Quantities
#47Rate of Change of Quantities
#48Rate of Change of Quantities
#49Rate of Change of Quantities
#50Increasing and Decreasing Functions

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Application of Derivatives

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