Download App

STD 12 - 5. continuity and differentiation — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 5. continuity and differentiation1 Mark
MCQ
Let $f: R \rightarrow R$ be a function defined by

$f(x)=\left\{\begin{array}{cc}x^2 \sin \left(\frac{\pi}{x^2}\right) & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$

Then which of the following statements is $TRUE$?

  • A
    $f(x)=0$ has infinitely many solutions in the interval $\left[\frac{1}{10^{10}}, \infty\right)$.
  • B
    $f(x)=0$ has no solutions in the interval $\left[\frac{1}{\pi}, \infty\right)$.
  • C
     The set of solutions of $f(x)=0$ in the interval $\left(0, \frac{1}{10^{10}}\right)$ is finite.
  • ✓
    $f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^2}, \frac{1}{\pi}\right)$.
✓

Answer

Correct option: D.
$f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^2}, \frac{1}{\pi}\right)$.
d
Option-$A$: $f ( x )= x ^2 \sin \frac{\pi}{ x ^2}$

$f ( x )=0 \Rightarrow \sin \frac{\pi}{ x ^2}=0 \Rightarrow \frac{\pi}{ x ^2}= n \pi, n \in N$

$x ^2=\frac{1}{ n } \Rightarrow x =\frac{1}{\sqrt{ n }}$

$\frac{1}{\sqrt{ n }} \geq \frac{1}{10^{10}} \Rightarrow 10^{10} \geq \sqrt{ n } \Rightarrow n \leq 10^{20}$, finite number of solutions

Option-$B$ : $x =\frac{1}{\sqrt{ n }} \Rightarrow \frac{1}{\sqrt{ n }}>\frac{1}{\pi} \Rightarrow \pi>\sqrt{ n } \Rightarrow n <\pi^2$, Number of solutions is 9

Option-$C$: $x =\frac{1}{\sqrt{ n }}, \frac{1}{\sqrt{ n }}<\frac{1}{10^{10}} \Rightarrow \sqrt{ n }>10^{10} \Rightarrow n >10^{20}$. Infinite number of solutions

Option-$D$ : $\frac{1}{\pi^2}<\frac{1}{\sqrt{ n }}<\frac{1}{\pi} \Rightarrow \sqrt{ n } \in\left(\pi, \pi^2\right) \Rightarrow n \in\left(\pi^2, \pi^4\right)$, Definitely more than $25$ solutions

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

More "M.C.Q (1 Marks)" from STD 12 - 5. continuity and differentiation→STD 12 - 5. continuity and differentiation — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

$\int_0^{\pi /2} {\sqrt {\cos \theta } {{\sin }^3}\theta } \,d\theta = $
→
If the line $y = x$ touches the curve $y = x^2+ bx + c$ at a point $(1, 1)$ then:
→
The minimum value of the function $y = 2{x^3} - 21{x^2} + 36x - 20$ is
→
The value of the $\int\limits_0^{\frac{\pi }{2}} {\left( {\frac{{1 + \sin 3y}}{{1 + 2\sin y}}} \right)} \,dy$ is  equal to
→
Area of the region bounded by the curve $y = \tan x,$ tangent drawn to the curve at $x = \frac{\pi }{4}$ and the $x -$ axis is
→
Compute the area of the figure bounded by straight lines $x = 0, x = 2$ and the curves $y=2^x$ and $y=2 x-x^2$
→
Let $f : R \rightarrow R$ be a differentiable function with $f(0)=1$ and satisfying the equation $f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)$ for all $x, y \in R$ Then, the value of $\log _c(f(4))$ is. . . . . .
→
$2\tan^{-1}(\cos\text{x})=\tan^{-1}(2\ \text{cosec x})$
→
Let $ A$ be a $2$$ \times $$2$ matrix with non-zero entries and let ${A^2} = I$ where $I$  is $2\times 2$ identity matrix. Define $tr(A) =$ sum of diagonal elements of $A$ and $|A|=$ determinant of matrix $A$ 

Statement $-1 :$ ${\rm{tr}}\left( A \right) = 0$

Statement $-2 :$  $\det \left( A \right) = 1$

→
If $y = 1 + x + {{{x^2}} \over {2\,!}} + {{{x^3}} \over {3\,!}} + ..... + {{{x^n}} \over {n\,!}}$, then ${{dy} \over {dx}} = $
→