Download App

Continuity — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsContinuity1 Mark
MCQ
If $\text{f(x)}=\begin{cases}\frac{1-\sin^2\text{x}}{3\cos^2\text{x}},&\text{if}\text{ x}<\frac{\pi}{2}\\\text{a},&\text{if}\text{ x}=\frac{\pi}{2}\\\frac{\text{b}(1-\sin\text{x})}{(\pi-2\text{x})^2},&\text{if}\text{ x }>\frac{\pi}{2}\end{cases}$ Then $f(x)$ is continuous at $\text{x}=\frac{\pi}{2},$ if:
  • A
    $\text{a}=\frac{1}{3},\text{ b}=2$
  • ✓
    $\text{a}=\frac{1}{3},\text{ b}=\frac{8}{3}$
  • C
    $\text{a}=\frac{2}{3},\text{ b}=\frac{8}{3}$
  • D
    none of these
✓

Answer

Correct option: B.
$\text{a}=\frac{1}{3},\text{ b}=\frac{8}{3}$
Given, $\text{f(x)}=\begin{cases}\frac{1-\sin^2\text{x}}{3\cos^2\text{x}},&\text{if}\text{ x}<\frac{\pi}{2}\\\text{a},&\text{if}\text{ x}=\frac{\pi}{2}\\\frac{\text{b}(1-\sin\text{x})}{(\pi-2\text{x})^2},&\text{if}\text{ x }>\frac{\pi}{2}\end{cases}$
We have
$\Big(\text{LHL at x}=\frac{\pi}{2}\Big)=\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}^-}\text{f(x)}=\lim\limits_{\text{h}\rightarrow0}\text{f}\Big(\frac{\pi}{2}-\text{h}\Big)$
$=\lim\limits_{\text{h}\rightarrow0}\Bigg(\frac{1-\sin^2\big(\frac{\pi}{2}-\text{h}\big)}{3\cos^2\big(\frac{\pi}{2}-\text{h}\big)}\Bigg)$
$=\lim\limits_{\text{h}\rightarrow0}\Big(\frac{1-\cos^2\text{h}}{3\sin^2\text{h}}\Big)$
$=\frac{1}{3}\lim\limits_{\text{h}\rightarrow0}\Big(\frac{\sin^2\text{h}}{\sin^2\text{h}}\Big)$
$=\frac{1}{3}$
$\Big(\text{RHL at x}=\frac{\pi}{2}\Big)=\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}^+}\text{f(x)}=\lim\limits_{\text{h}\rightarrow0}\text{f}\Big(\frac{\pi}{2}+\text{h}\Big)$
$=\lim\limits_{\text{h}\rightarrow0}\Bigg(\frac{\text{b}\big[1-\sin\big(\frac{\pi}{2}+\text{h})\big]}{\big[\text{x}-2\big(\frac{\pi}{2}+\text{h}\big)\big]^2}\Bigg)$
$=\lim\limits_{\text{h}\rightarrow0}\bigg(\frac{\text{b}(1-\cos\text{h})}{[-2\text{h}]^2}\bigg)$
$=\lim\limits_{\text{h}\rightarrow0}\bigg(\frac{2\text{b}\sin^{2}\frac{\text{h}}{2}}{4\text{h}^2}\bigg)$
$=\lim\limits_{\text{h}\rightarrow0}\bigg(\frac{2\text{b}\sin^2\frac{\text{h}}{2}}{16\frac{\text{h}^2}{4}}\bigg)$
$=\frac{\text{b}}{8}\lim\limits_{\text{h}\rightarrow0}\bigg(\frac{\sin\frac{\text{h}}{2}}{\frac{\text{h}}{2}}\bigg)^2$
$=\frac{\text{b}}{8}\times1$
$=\frac{\text{b}}{8}$
Also, $\text{f}\big(\frac{\pi}{2}\big)=\text{a}$
If $f(x)$ is continuous at $\text{x}=\frac{\pi}{2},$ then
$\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}^-}\text{f(x)}=\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}^+}\text{f(x)}=\text{f}\big(\frac{\pi}{2}\big)$
$\Rightarrow\frac{1}{3}=\frac{\text{b}}{8}=\text{a}$
$\Rightarrow\text{a}=\frac{1}{3}$ and $\text{b}=\frac{8}{3}$

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 "MCQ" from Continuity→Continuity — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $f (x)$ is defined on $[-2,2]$ by $f (x)=4 x^2-3 x+1$ and $g(x)=\frac{f(-x)-f(x)}{x^2+3}$, then $\int_{-2}^2 g(x) d x=$
→
Number of lines which make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$ with any two of $OX , OY$ and OZ are
→
If $\int_0^1 x \log \left(1+\frac{x}{2}\right) d x= a + b \log \frac{2}{3}$, then
→
If $\vec{\text{a}}$ and $\vec{\text{b}}$ be two unit vectors and $\theta$ the angle between them, than $\vec{\text{a}}+\vec{\text{b}}$ is a unit vector if $\theta=$
→
If $A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{array}\right]$, then the sum of all the diagonal entries of $A ^{-1}$ is
→
If $g(x)$ is the inverse function of $f(x)$ and $f^{\prime}(x)=$ $\frac{1}{1+x^4}$, then $g^{\prime}(x)$ is
→
A fair die is thrown twenty times. The probability that on the tenth throw the fourth six appears is:
→
The number of values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\theta \neq \frac{ n \pi}{5}$ for $n =0, \pm 1, \pm 2$ and $\tan \theta=\cot 5 \theta$ as well as $\sin 2 \theta=\cos 4 \theta$ is
→
If any ∆ABC, if a cos B = b cos A, then the triangle is ________.
→
If $A = \{1, 2, 3\}, B = \{1, 4, 6, 9\}$ and $R$ is a relation from $A$ to $B$ defined by $'x$ is greater than $y'$. The range of $R$ is:
→