CUET UG Mathematics Applied Continuity Previous Year Questions (PYQs)
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Identify the correct option(s):
A. A modulus function is continuous at every point in its domain.
B. A modulus function may or may not be continuous at every point in its domain.
C. Every rational function is continuous in its domain.
D. If a function $f$ is differentiable at a point then it is also continuous at that point.
E. If a function $f$ is continuous at a point then it is also differentiable at that point.
Choose the correct answer from the options given below:
1
2
3
4
Solution
A. A modulus function $|x|$ is continuous at every point in its domain.
Hence A is true.
B. Since modulus function is continuous everywhere in its domain, this statement is false.
C. A rational function $\frac{p(x)}{q(x)}$ is continuous at every point where $q(x)\neq0$.
Thus it is continuous in its domain. Hence C is true.
D. If a function is differentiable at a point, then it must be continuous at that point.
Hence D is true.
E. Continuity does not necessarily imply differentiability (example: $f(x)=|x|$ at $x=0$).
Hence E is false.
Correct statements are A, C and D.
If
$f(x)=
\begin{cases}
\frac{\sqrt{1-\cos2x}}{x\sqrt2}, & x\ne0 \\
k, & x=0
\end{cases}$
then the value of $k$ will make function $f$ continuous at $x=0$ is:
1
2
3
4
Solution
For continuity at $x=0$,
$\lim_{x\to0}f(x)=f(0)=k$
So evaluate limit:
$\lim_{x\to0}\frac{\sqrt{1-\cos2x}}{x\sqrt2}$
Using identity:
$1-\cos2x=2\sin^2x$
So,
$\sqrt{1-\cos2x}=\sqrt{2\sin^2x}=\sqrt2|\sin x|$
Hence,
$\frac{\sqrt{1-\cos2x}}{x\sqrt2}=\frac{\sqrt2|\sin x|}{x\sqrt2}=\frac{|\sin x|}{x}$
As $x\to0$,
$\frac{|\sin x|}{x}\to1$
Therefore,
$k=1$
CUET UG Mathematics Applied
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