Continuity - JEE Advanced Previous Year Questions with Solutions

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Q. For every integer $n$, let $a_{n}$ and $b_{n}$ be real numbers. Let function $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by

$f(\mathrm{x})=\left\{\begin{array}{ll}{\mathrm{a}_{\mathrm{n}}+\sin \pi \mathrm{x},} & {\text { for } \quad \mathrm{x} \in[2 \mathrm{n}, 2 \mathrm{n}+1]} \\ {\mathrm{b}_{\mathrm{n}}+\cos \pi \mathrm{x},} & {\text { for } \quad \mathrm{x} \in(2 \mathrm{n}-1,2 \mathrm{n})}\end{array}, \text { for all integers } \mathrm{n}\right.$

If ƒ is continuous, then which of the following holds(s) for all n ?

(A) $a_{n-1}-b_{n-1}=0$

(B) $a_{n}-b_{n}=1$

(D) $a_{n-1}-b_{n}=-1$

[JEE 2012, 4M]

Sol. (B,D)

Q. For every pair of continuous function $f, g:[0,1] \rightarrow$ such that $\max \{f(x): x \in[0,1]\}=$ $\max \{g(x): x \in[0,1]\},$ the correct statement(s) is(are) :

(A) $(f(\mathrm{c}))^{2}+3 f(\mathrm{c})=(\mathrm{g}(\mathrm{c}))^{2}+3 \mathrm{g}(\mathrm{c})$ for some $\mathrm{c} \in[0,1]$

(B) $(f(\mathrm{c}))^{2}+f(\mathrm{c})=(\mathrm{g}(\mathrm{c}))^{2}+3 \mathrm{g}(\mathrm{c})$ for some $\mathrm{c} \in[0,1]$

(C) $(f(\mathrm{c}))^{2}+3 f(\mathrm{c})=(\mathrm{g}(\mathrm{c}))^{2}+\mathrm{g}(\mathrm{c})$ for some $\mathrm{c} \in[0,1]$

(D) $(f(\mathrm{c}))^{2}=(\mathrm{g}(\mathrm{c}))^{2}$ for some $\mathrm{c} \in[0,1]$

[JEE(Advanced)-2014, 3]

Sol. (A,D)

$f, \mathrm{g}[0,1] \rightarrow \mathrm{R}$

we take two cases.

Let $f \& \mathrm{g}$ attain their common maximum value at $\mathrm{P}$.

$\Rightarrow f(\mathrm{p})=\mathrm{g}(\mathrm{p})$ where $\mathrm{p} \in[0,1]$

let $f \& \mathrm{g}$ attain their common maximum value at different points.

$\Rightarrow f(\mathrm{a})=\mathrm{M} \& \mathrm{g}(\mathrm{b})=\mathrm{M}$

$\Rightarrow f(\mathrm{a})-\mathrm{g}(\mathrm{a})>0 \& f(\mathrm{b})-\mathrm{g}(\mathrm{b})<0$

$\Rightarrow f(\mathrm{c})-\mathrm{g}(\mathrm{c})=0$ for some $\mathrm{c} \in[0,1]$ as $\mathrm{f}^{\prime} \& \quad$ g’ are continuous functions

$\Rightarrow f(\mathrm{c})-\mathrm{g}(\mathrm{c})=0$ for some $\mathrm{c} \in[0,1]$ for all cases. $\ldots(1)$

Option $(\mathrm{A}) \Rightarrow f^{2}(\mathrm{c})-\mathrm{g}^{2}(\mathrm{c})+3(f(\mathrm{c})-\mathrm{g}(\mathrm{c}))=0$

which is true from ( 1)

Option (D) $\Rightarrow f^{2}(\mathrm{c})-\mathrm{g}^{2}(\mathrm{c})=0$ which is true from ( 1)

Now, if we take $f(\mathrm{x})=1 \& \mathrm{g}(\mathrm{x})=1 \forall \mathrm{x} \in[0,1]$

options $(\mathrm{B}) \&(\mathrm{C})$ does not hold. Hence $\quad$ option $(\mathrm{A}) \&(\mathrm{D})$ are correct.

Q. Let $[\mathrm{x}]$ be the greatest integer less than or equal to $\mathrm{x}$. Then, at which of the following point(s) the function $f(\mathrm{x})=\mathrm{x} \cos (\pi(\mathrm{x}+[\mathrm{x}]))$ is discontinuous?b

(A) x = –1 (B) x = 0 (C) x = 2 (D) x = 1

[JEE(Advanced)-2017, 4]

Sol. (A,C,D)

$f(\mathrm{x})=\mathrm{x} \cos (\pi \mathrm{x}+[\mathrm{x}] \pi)$

$\Rightarrow f(\mathrm{x})=(-1)^{[\mathrm{x}]} \mathrm{x} \cos \pi \mathrm{x}$

Discontinuous at all integers except zero.

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