Available Questions 43 found Page 1 of 3
Standalone Questions
#1477
Mathematics
Applications of Derivatives
SA
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.
Key:
Sol:
Sol:
#1446
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2025
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Determine the values of x for which $f(x)=\frac{x-4}{x+1}$, $x\ne-1$ is an increasing or a decreasing function.
Key:
Sol:
Sol:
#1428
Mathematics
Applications of Derivatives
SA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
3 Marks
Find the value of 'a' for which $f(x)=\sqrt{3}\sin x-\cos x-2ax+6$ is decreasing in R.
Key:
Sol:
Sol:
#1422
Mathematics
Applications of Derivatives
VSA
REMEMBER
2025
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Surface area of a balloon (spherical), when air is blown into it, increases at a rate of $5\text{ mm}^{2}/\text{s}$. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.
Key:
Sol:
Sol:
#1416
Mathematics
Applications of Derivatives
LA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
5 Marks
The relation between the height of the plant (y cm) with respect to exposure to sunlight is governed by the equation $y=4x-\frac{1}{2}x^{2}$, where x is the number of days exposed to sunlight. (i) Find the rate of growth of the plant with respect to sunlight. (ii) In how many days will the plant attain its maximum height? What is the maximum height?
Key:
Sol:
Sol:
#1405
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
2 Marks
For the curve $y=5x-2x^{3}$ if x increases at the rate of $2\text{ units/s}$, then how fast is the slope of the curve changing when $x=2$?
Key:
Sol:
Sol:
#1401
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2025
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
If $f(x)=x+\frac{1}{x}$, $x\ge1$, show that f is an increasing function.
Key:
Sol:
Sol:
#1400
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2025
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Find the least value of 'a' so that $f(x)=2x^{2}-ax+3$ is an increasing function on $[2, 4]$.
Key:
Sol:
Sol:
#1380
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2025
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Find the values of 'a' for which $f(x)=\sin x-ax+b$ is increasing on R.
Key:
Sol:
Sol:
#1373
Mathematics
Applications of Derivatives
LA
UNDERSTAND
2025
AISSCE(Board Exam)
KNOWLEDGE
5 Marks
Find the absolute maximum and absolute minimum of function $f(x)=2x^{3}-15x^{2}+36x+1$ on $[1, 5]$.
Key:
Sol:
Sol:
#1362
Mathematics
Applications of Derivatives
SA
REMEMBER
2025
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side of the triangle is 15 cm?
Key:
Sol:
Sol:
#1359
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2025
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Find the intervals in which function $f(x)=5x^{\frac{3}{2}}-3x^{\frac{5}{2}}$ is (i) increasing (ii) decreasing.
Key:
Sol:
Sol:
#1336
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2024
AISSCE(Board Exam)
Competency
2 Marks
The area of the circle is increasing at a uniform rate of $2~cm^{2}/sec$. How fast is the circumference of the circle increasing when the radius $r=5$ cm?
Key:
Sol:
Sol:
#1315
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2024
AISSCE(Board Exam)
Competency
2 Marks
The volume of a cube is increasing at the rate of $6~cm^{3}/s.$ How fast is the surface area of cube increasing, when the length of an edge is 8 cm?
Key:
Sol:
Sol:
#1314
Mathematics
Applications of Derivatives
VSA
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Find the interval in which the function $f(x)=x^{4}-4x^{3}+10$ is strictly decreasing.
Key:
Sol:
Sol:
#1290
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Show that the function $f(x)=4x^{3}-18x^{2}+27x-7$ has neither maxima nor minima.
Key:
Sol:
Sol:
#1272
Mathematics
Applications of Derivatives
VSA
ANALYZE
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Show that $f(x)=e^{x}-e^{-x}+x-tan^{-1}x$ is strictly increasing in its domain.
Key:
Sol:
Sol:
#1270
Mathematics
Applications of Derivatives
VSA
REMEMBER
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
If M and m denote the local maximum and local minimum values of the function $f(x)=x+\frac{1}{x}(x\ne0)$ respectively, find the value of $(M-m)$
Key:
Sol:
Sol:
#1262
Mathematics
Applications of Derivatives
LA
UNDERSTAND
2024
AISSCE(Board Exam)
Competency
5 Marks
The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that volume of cylinder so formed is maximum.
Key:
Sol:
Sol:
#1261
Mathematics
Applications of Derivatives
LA
REMEMBER
2024
AISSCE(Board Exam)
Competency
5 Marks
It is given that function $f(x)=x^{4}-62x^{2}+ax+9$ attains local maximum value at $x=1$ Find the value of 'a', hence obtain all other points where the given function f(x) attains local maximum or local minimum values.
Key:
Sol:
Sol:
Case-Based Questions
CASE ID: #109
Cl: CBSE Class 12
Mathematics
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections.
Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be $y$ metres.,
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of
x and y..
x and y..
Key:
Sol:
Sol:
k
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
Write the area of the solar panel as a function of $x$
Key:
Sol:
Sol:
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.
Key:
Sol:
Sol:
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
2 Marks
Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.
Key:
Sol:
Sol: