Download App

STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
MCQ
Let $f:R \to R$ is a function defined by $f\left( x \right) = \left[ x \right]\cos \left( {\frac{{2x - 1}}{2}} \right)\pi $, where $\left[ x \right]$ denotes the greatest integer function , then $ f$ is. . .
  • A
    discontinuous only at  $ x=0$ 
  • B
    discontinuous only at non-zero intergral values of $x$
  • C
    continuous only at $x=0 $
  • ✓
    continuous for every real $ x$
✓

Answer

Correct option: D.
continuous for every real $ x$
d
$f(x)=[x] \cos \left(\frac{2 x-1}{2}\right) \pi$

Doubtful points are $x=0, n \in I$

${\rm{LHL}} = \mathop {\lim }\limits_{x \to {\pi ^ - }} [x]\cos \left( {\frac{{2x - 1}}{2}} \right)\pi $

$\Rightarrow(n-1) \cos \left(\frac{2 n-1}{2}\right) \pi$

$[x]$ is the greatest integer function.

${\rm{RHL}} = \mathop {\lim }\limits_{x \to {n^ + }} [x]\cos \left( {\frac{{2x - 1}}{2}} \right)\pi $

$ = n\cos \left( {\frac{{2n - 1}}{2}} \right)\pi $

Now values of the function at $x=0$ is $f(n)=0$

Since $\mathrm{LHL}=\mathrm{RHL}=\mathrm{f}(\mathrm{n})$

$f(x)=[x] \cos \left(\frac{2 x-1}{2}\right)$ is continuous for every real $x$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

A bag contains $30$ white balls and $10$ red balls. $16$ balls are drawn one by one randomly from the bag with replacement. If $X$ be the number of white balls drawn, then $\left( {\frac{{{\rm{mean\, of\, X}}}}{{{\rm{standard\, deviation\, of\, X}}}}} \right)$ is equal to
→
$\mathop {\lim }\limits_{x \to \pi /2} \frac{{{a^{\cot x}} - {a^{\cos x}}}}{{\cot x - \cos x}} = $
→
Let $z_0$ be a root of the quadratic equation, $x^2 + x + 1= 0.$ If $z = 3 + \,6iz_0^{81}\, - 3iz_0^{93}, $ then arg $z$ is equal to
→
Let $f$ and $g$ be increasing and decreasing functions, respectively from $[0 , \infty )$ to $[0 , \infty )$. Let $h (x) = f [g (x)]$ . If $h (0) = 0$, then $ h (x) - h (1)$ is
→
The area between the curve $y = 4 + 3x - {x^2}$ and $x -$ axis is
 
→
Let $a, b, c > 1, a^3, b^3$ and $c^3$ be in $A.P.$, and $\log _a b$, $\log _c a$ and $\log _b c$ be in G.P. If the sum of first $20$ terms of an $A.P.$, whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-444$, then abc is equal to
→
Let $a \in Z$ and $[t]$ be the greatest integer $\leq t$. Then the number of points, where the function $f(x)=[a$ $+13 \sin x], x \in(0, \pi)$ is not differentiable, is $........$.
→
Let $A$ be a matrix such that $A.\,\left[ {\begin{array}{*{20}{c}}
1&2\\
0&3
\end{array}} \right]$ is a scalar matrix and $\left| {3A} \right| = 108$ . Then $A^2$ equals
→
Let  $f(x) =\left| {\begin{array}{*{20}{c}}
{\cos \,x}&{\sin \,x}&{\cos \,x}\\
{\cos \,2x}&{\sin \,2x}&{2\,\cos \,2x}\\
{\cos \,3x}&{\sin \,3x}&{3\,\cos \,3x}
\end{array}} \right| $then $f ' \left( {\frac{\pi }{2}} \right)$=
→
The function $\mathrm{f}: \mathrm{N}-\{1\} \rightarrow \mathrm{N}$; defined by $\mathrm{f}(\mathrm{n})=$ the highest prime factor of $n$, is :
→