Vidaara.orgClass 12 · Mathematics
CodeVID-M12-06-CT
Application of Derivatives — Full Chapter Test
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- This is a full-length test covering the whole chapter — every topic is included.
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
If $A=\pi r^2$ and $\tfrac{dr}{dt}=2$, then $\tfrac{dA}{dt}$ at $r=3$ is:
- A.$6\pi$
- B.$12\pi$
- C.$9\pi$
- D.$3\pi$
2.
$f$ is strictly increasing on $I$ if, for all $x\in I$:
- A.$f'(x)<0$
- B.$f'(x)=0$
- C.$f'(x)>0$
- D.$f''(x)>0$
3.
At a critical point with $f''(c)>0$, $f$ has a local:
- A.maximum
- B.minimum
- C.inflection
- D.none
4.
Marginal cost is:
- A.$C(x)/x$
- B.$C'(x)$
- C.$\int C\,dx$
- D.$xC(x)$
5.
$f(x)=x^2-4x+1$ is increasing on:
- A.$(-\infty,2)$
- B.$(2,\infty)$
- C.$\mathbb{R}$
- D.$(0,2)$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
The radius of a circle increases at $3$ cm/s. Find $\dfrac{dA}{dt}$ when $r=5$ cm.
7.
Find the interval where $f(x)=x^2-2x$ is increasing.
8.
Find the critical point of $f(x)=x^2-4x$.
9.
The side of a square grows at $2$ cm/s. Find $\dfrac{dA}{dt}$ when the side is $10$ cm.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
The volume of a sphere increases. Find $\dfrac{dV}{dt}$ when $r=2$ and $\dfrac{dr}{dt}=0.5$.
11.
Find the intervals of increase and decrease of $f(x)=x^2-4x+3$.
12.
Find the local maximum and minimum of $f(x)=x^3-3x$.
13.
The edge of a cube increases at $3$ cm/s. Find the rate of change of volume when the edge is $5$ cm.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
A spherical balloon's radius increases at $2$ cm/s. Find the rate of increase of its volume and surface area when $r=10$ cm.
15.
Find the intervals in which $f(x)=x^4-2x^2$ is increasing and decreasing.
Answer Key
Section A — Multiple Choice Questions
- (B) $12\pi$
- (C) $f'(x)>0$
- (B) minimum
- (B) $C'(x)$
- (B) $(2,\infty)$
Section B — Short Answer (2 marks)
- $30\pi$ cm$^2$/s.
- $(1,\infty)$.
- $x=2$.
- $40$ cm$^2$/s.
Section C — Short Answer (3 marks)
- $8\pi$ cubic units/s.
- Decreasing on $(-\infty,2)$, increasing on $(2,\infty)$.
- Max $2$ at $x=-1$; min $-2$ at $x=1$.
- $225$ cm$^3$/s.
Section D — Long Answer (5 marks)
- $\dfrac{dV}{dt}=800\pi$ cm$^3$/s, $\dfrac{dS}{dt}=160\pi$ cm$^2$/s.
- Increasing on $(-1,0)\cup(1,\infty)$; decreasing on $(-\infty,-1)\cup(0,1)$.
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