Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY1 Mark
Question
If $\text{f(x)}=\frac{1-\sin\text{x}}{(\pi-2\text{x})^2},$ when $\text{x}\neq\frac{\pi}{2}=\lambda$ then f(x) will be continuous function at $\text{x}=\frac{\pi}{2},$ where $\lambda=$
  1. $\frac{1}{8}$
  2. $\frac{1}{4}$
  3. $\frac{1}{2}$
  4. none of these
✓

Answer

  1. $\frac{1}{8}$
Solution:
If f(x) is continuous at $\text{x}=\frac{\pi}{2},$ then
$\lim\limits_{\text{x}\rightarrow\frac{\text{x}}{2}}\text{f}\text{(x)}=\text{f}\Big(\frac{\pi}{2}\Big)$
$\lim\frac{1-\sin\text{x}}{\text{x}\rightarrow\frac{\pi}{2}(\pi-2\text{x})^2}=\text{f}\Big(\frac{\pi}{2}\Big) \ ...(\text{i})$$$
suppose $\Big(\frac{\pi}{2}-\text{x}\Big)=\text{t},$ then
$\lim\limits_{\text{t}\rightarrow0}\begin{bmatrix}\frac{1-\sin\Big(\frac{\pi}{2}-\text{t}\Big)}{(2\text{t})^2} \end{bmatrix}=\text{f}\Big(\frac{\pi}{2}\Big)$ [From eq.(i)]
$\Rightarrow\lim\limits_{\text{t}\rightarrow0}\begin{bmatrix}\frac{1-\cos\text{t}}{(2\text{t})^2} \end{bmatrix}=\text{f}\Big(\frac{\pi}{2}\Big)$
$\Rightarrow\frac{1}{4}\lim\limits_{t\rightarrow0}\begin{bmatrix}\frac{2\sin^2\Big(\frac{\text{t}}{2}\Big)}{\text{t}^2} \end{bmatrix}=\text{f}\Big(\frac{\pi}{2}\Big)$
$\Rightarrow \frac{1}{4}\lim\limits_{\text{t} \rightarrow0} \begin{bmatrix}\frac{\frac{2}{4}\sin^{2}\big(\frac{\text{t}}{2}\big)}{\frac{\text{t}^{2}}{4}} \end{bmatrix} = \text{f} \Big(\frac{\pi}{2}\Big)$
$\Rightarrow \frac{1}{8}\lim\limits_{\text{t} \rightarrow0} \begin{bmatrix}\frac{\frac{2}{4}\sin^{2}\big(\frac{\text{t}}{2}\big)}{\frac{\text{t}^{2}}{4}} \end{bmatrix} = \text{f} \Big(\frac{\pi}{2}\Big)$
$\Rightarrow\frac{1}{8}\lim\limits_{\text{t}\rightarrow0}\begin{bmatrix}\frac{\sin\Big(\frac{\text{t}}{2}\Big)}{\frac{\text{t}}{2}} \end{bmatrix}^2=\text{f}\Big(\frac{\pi}{2}\Big)$
$\Rightarrow\text{f}\Big(\frac{\pi}{2}\Big)=\lambda=\frac{1}{8}$

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 CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

The vector $(\cos\text{a}\cos\beta)\hat{\text{i}}+(\cos\text{a}\sin\beta)\hat{\text{j}}+(\sin\text{a})\hat{\text{k}}$is a:
  1. Null vector
  2. Unit vector
  3. Constant vector
  4. None of these
→
If $\text{f(x)}=\begin{cases}\frac{\sin(\text{a}+1)}{\text{x}},&\text{x}<0\\\text{c},&\text{x}=0\\\frac{\sqrt{\text{x+bx}^2}-\sqrt{\text{x}}}{\text{bx}\sqrt{\text{x}}},&\text{x}>0&\end{cases}$ is continuouse at x = 0, then:
  1. $\text{a}=-\frac{3}{2},\text{b}=0,\text{c}=\frac{1}{2}$
  2. $\text{a}=-\frac{3}{2},\text{b}=1,\text{c}=-\frac{1}{2}$
  3. $\text{a}=-\frac{3}{2},\text{b}\in\text{R}-\{0\},\text{c}=\frac{1}{2}$
  4. $\text{None of these}.$
→
The function $\text{f}(\text{x})=\log_\text{e}\Big(\text{x}^3+\sqrt{\text{x}^6+1}\Big)$ is of the following type:
  1. Even and increasing.
  2. Odd and increasing.
  3. Even and decreasing.
  4. Odd and decreasing.
→
The value of $\int\limits^\pi_{-\pi}\sin^3\text{x}\cos^2\text{x}\text{ dx}$ is:
  1. $\frac{\pi^4}{2}$
  2. $\frac{\pi^4}{4}$
  3. $0$
  4. none of these
→
$P$ is a point on the line joining the points $A(0,5,-2)$ and $B(3,-1,2)$. If the $x$-coordinate of $P$ is 6 , then its $z$-coordinate is
→
For what valuse of $x$ is the rate of increase of $x^3 - 5x^2 + 5x + 8$ is twice the rate of increase of $x?$
→
In Graphical solution the feasible solution is any solution to a LPP which satisfies.
  1. Only objective function.
  2. Non - negativity restriction.
  3. Only constraint.
  4. All the three
→
The binary operation × defind on set R, given by $\text{a}\times\text{b}=\frac{\text{a}+\text{b}}{2}$ for all a, b $\epsilon$ R is:
  1. Commutative.
  2. Associative.
  3. Both (a) and (b).
  4. None of these.
→
The value of $\tan\bigg[\frac{1}{2}\cos^{-1}\Big(\frac{2}{3}\Big)\bigg]:$
  1. $ \frac { 1 }{ \sqrt { 5 } }$
  2. $ \frac { 1 }{ \sqrt { 7 } }$
  3. $ \frac { 1 }{ \sqrt { 8 } }$
  4. $ \frac { 1 }{ \sqrt { 9 } }$
→
System of linear equation 5x + ky = 5, 3x + 3y = 5 is consistent, if :
→