Download App

Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
If $f(x)$ is continuous in $[0, \pi]$, where
$f(x)=\left\{\begin{array}{ll}x+a \sqrt{2} \sin x, & 0 \leq x < \frac{\pi}{4} \\2 x \cot x+b, & \frac{\pi}{4} \leq x \leq \frac{\pi}{2}, \text { then } \\a \cos 2 x-b \sin x, & \frac{\pi}{2} < x \leq \pi\end{array}\right.$
  • A
    $a=\frac{\pi}{6}, b=\frac{\pi}{12}$
  • B
    $a=-\frac{\pi}{6}, b=\frac{\pi}{12}$
  • ✓
    $a=\frac{\pi}{6}, b=-\frac{\pi}{12}$
  • D
    $a=-\frac{\pi}{6}, b=-\frac{\pi}{12}$
✓

Answer

Correct option: C.
$a=\frac{\pi}{6}, b=-\frac{\pi}{12}$
(C)
Sincc $f(x)$ is continuous in $[0, \pi]$.
$\therefore $ it is continuous at $x=\frac{\pi}{4}$ and $x=\frac{\pi}{2}$.
$\therefore \lim _{x \rightarrow\left(\frac{\pi}{4}\right)^{-}} f (x)=\lim _{x \rightarrow\left(\frac{\pi}{4}\right)^{+}} f (x)$
$\Rightarrow \lim _{x \rightarrow\left(\frac{\pi}{4}\right)^{-}}(x+ a \sqrt{2} \sin x)=\lim _{x \rightarrow\left(\frac{\pi}{4}\right)^{+}}(2 x \cot x+ b )$
$\Rightarrow \frac{\pi}{4}+ a \sqrt{2}\left(\frac{1}{\sqrt{2}}\right)=2\left(\frac{\pi}{4}\right)(1)+ b$
$\Rightarrow \frac{\pi}{4}+ a =\frac{\pi}{2}+ b$
$\Rightarrow a-b=\frac{\pi}{4}$ $\quad\ldots(i)$
Also, $\lim _{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} f (x)=\lim _{x \rightarrow\left(\frac{\pi}{2}\right)^{+}} f (x)$
$\Rightarrow \lim _{x \rightarrow\left(\frac{\pi}{2}\right)^{-}}(2 x \cot x+ b )=\lim _{x \rightarrow\left(\frac{\pi}{2}\right)^{+}}( a \cos 2 x- b \sin x)$
$\Rightarrow 2\left(\frac{\pi}{2}\right)(0)+ b = a (-1)- b (1)$
$\Rightarrow b =- a - b$
$\Rightarrow a+2 b=0$ $\quad\ldots(ii)$
From (i) and (ii), we get
$a=\frac{\pi}{6}$ and $b=\frac{-\pi}{12}$

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 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

Let $f (x)=\left\{\begin{array}{lc}5^{\frac{1}{x}} ; & x<0 \\ \lambda[x] ; & x \geq 0, \lambda \in R \end{array}\right.$, then at $x=0$
→
$\lim _{x \rightarrow 0} \frac{\sqrt{1+2 x}-1}{x}=$
→
Find the measure of the angle between hour- hand and the minute hand of a clock at twenty minutes past two.
→
The number of ways in which a host lady can invite for a party of $8$ out of $12$ people of whom two do not want to attend the party together is :
→
A line parallel to the straight line $2 x-y=0$ is tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{2}=1$ at the point $\left(x_1, y_1\right)$. Then $x_1^2+5 y_1^2$ is equal to
→
The letters of the word FATHER are written on separate cards, two cards are drawn at random Probability that both are vowels is
→
If $\cos2\text{x}+2\cos\text{x}=1$ then, $(2-\cos^2\text{x})\sin^2\text{x}$ is equal to :
→
If $\frac{c+i}{c-i}=a+i b$, where $a, b, c$ are real, then $a^2+b^2=$
→
The chance of getting a same number on 2 dice is
→
If $\tan A=\frac{1}{2}$, then $\tan 3 A=$
→