Vidaara.orgClass 12 · Mathematics
CodeVID-M12-05-CON-01
Continuity — Assignment
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- 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.
$f(x)=\dfrac{1}{x-2}$ is discontinuous at:
- A.$x=0$
- B.$x=2$
- C.every point
- D.nowhere
3.
Continuity at $c$ needs:
- A.LHL $=$ RHL only
- B.LHL $=$ RHL $=f(c)$
- C.$f(c)$ defined only
- D.limit $=0$
4.
Which is continuous for all real $x$?
- A.$\tan x$
- B.$\tfrac1x$
- C.$\sin x$
- D.$[x]$
5.
The greatest-integer function $[x]$ is discontinuous at:
- A.all reals
- B.every integer
- C.no point
- D.$x=0$ only
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Is $f(x)=x^2$ continuous at $x=1$?
7.
Find $k$ so that $f(x)=\begin{cases}kx,&x\le1\\2,&x>1\end{cases}$ is continuous at $x=1$.
8.
State the points of discontinuity of $\tan x$.
9.
Is $f(x)=|x|$ continuous at $x=0$?
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.
Discuss the continuity of $f(x)=|x-3|$.
12.
Is $f(x)=\begin{cases}x^2,&x\le0\\x,&x>0\end{cases}$ continuous at $0$?
13.
Find $a$ if $f(x)=\begin{cases}ax^2,&x\le2\\8,&x>2\end{cases}$ is continuous at $x=2$.
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.
Show that $f(x)=|x|+|x-1|$ is continuous on $\mathbb{R}$.
Answer Key
Section A — Multiple Choice Questions
- (B) $f(c)$
- (B) $x=2$
- (B) LHL $=$ RHL $=f(c)$
- (C) $\sin x$
- (B) every integer
Section B — Short Answer (2 marks)
- Yes.
- $k=2$.
- $x=(2n+1)\tfrac{\pi}{2},\ n\in\mathbb{Z}$.
- Yes.
Section C — Short Answer (3 marks)
- $k=\tfrac95$.
- Continuous for all $x$.
- Yes.
- $a=2$.
Section D — Long Answer (5 marks)
- $a=2,\ b=1$.
- Continuous everywhere.
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