Vidaara.orgClass 12 · Mathematics
CodeVID-M12-05-CT
Continuity and Differentiability — 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.
$f$ is continuous at $x=c$ requires $\lim_{x\to c}f(x)$ to equal:
- A.$0$
- B.$f(c)$
- C.$c$
- D.$\infty$
2.
$\dfrac{d}{dx}\sin(x^2)=$
- A.$\cos(x^2)$
- B.$2x\cos(x^2)$
- C.$2x\sin(x^2)$
- D.$\cos(2x)$
3.
For $x^2+y^2=25$, $\dfrac{dy}{dx}=$
- A.$-\tfrac{x}{y}$
- B.$\tfrac{x}{y}$
- C.$-\tfrac{y}{x}$
- D.$\tfrac{y}{x}$
4.
$f(x)=\dfrac{1}{x-2}$ is discontinuous at:
- A.$x=0$
- B.$x=2$
- C.every point
- D.nowhere
5.
If $f$ is differentiable at $c$, then $f$ is:
- A.discontinuous
- B.continuous at $c$
- C.constant
- D.zero at $c$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Is $f(x)=x^2$ continuous at $x=1$?
7.
Differentiate $(2x+1)^3$.
8.
Find $\dfrac{dy}{dx}$ if $xy=1$.
9.
Find $k$ so that $f(x)=\begin{cases}kx,&x\le1\\2,&x>1\end{cases}$ is continuous at $x=1$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find $k$ so that $f(x)=\begin{cases}kx+1,&x\le5\\3x-5,&x>5\end{cases}$ is continuous at $x=5$.
11.
Differentiate $(x^2+1)^5$.
12.
Find $\dfrac{dy}{dx}$ if $x^2+xy+y^2=0$.
13.
Discuss the continuity of $f(x)=|x-3|$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Find $a,b$ so that $f(x)=\begin{cases}x+a,&x<1\\3,&x=1\\bx+2,&x>1\end{cases}$ is continuous at $x=1$.
15.
If $y=\sin(\sqrt{x})$, find $\dfrac{dy}{dx}$.
Answer Key
Section A — Multiple Choice Questions
- (B) $f(c)$
- (B) $2x\cos(x^2)$
- (A) $-\tfrac{x}{y}$
- (B) $x=2$
- (B) continuous at $c$
Section B — Short Answer (2 marks)
- Yes.
- $6(2x+1)^2$.
- $-\dfrac{y}{x}$ (i.e. $-\tfrac{1}{x^2}$).
- $k=2$.
Section C — Short Answer (3 marks)
- $k=\tfrac95$.
- $10x(x^2+1)^4$.
- $-\dfrac{2x+y}{x+2y}$.
- Continuous for all $x$.
Section D — Long Answer (5 marks)
- $a=2,\ b=1$.
- $\dfrac{\cos\sqrt{x}}{2\sqrt{x}}$.
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