Download App

RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS3 Marks
Question
Find the value of k, for which $\text{f}(\text{x}) = $ $ \begin{matrix} \frac{\sqrt{1 + \text{kx}} - \sqrt{1 - \text{kx}}}{\text{x}} , \text{if} - 1\leq\text{x} < 0\\ \frac{2\text{x} + 1}{\text{x} - 1} , \text{ if}0\leq\text{x}< 1 \end{matrix} $ is continuous at x = 0.
✓

Answer

$\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0^{-}}\text{f}(\text{x}) = \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{h}\rightarrow0}\text{f}(0 - \text{h}) [ \text{ Let x} = 0 - \text{h},\text{x}\rightarrow0^{-}\Rightarrow\text{h}\rightarrow0]$
$ = \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{h}\rightarrow0}\text{f}(-\text{h}) = \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{h}\rightarrow0}\frac{\sqrt{1 + \text{k}(-\text{h})} - \sqrt{1 - \text{k}(-\text{h})}}{-\text{h}}$
$ = \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{h}\rightarrow0}\frac{\sqrt{1 - \text{kh}} - \sqrt{1 + \text{kh}}}{-\text{h}}\times\frac{\sqrt{1 - \text{kh}} + \sqrt{1 + \text{kh}}}{\sqrt{1 - \text{kh}} + \sqrt{1 + \text{kh}}}$
$ = \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{h}\rightarrow0}\frac{(1 - \text{kh)} - (1 + \text{kh)}}{-\text{h}\left\{\sqrt{1 - \text{kh}} + \sqrt{1 + \text{kh}}\right\}} = \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{h}\rightarrow0}\frac{2\text{k}}{\left\{\sqrt{1 - \text{kh}} + \sqrt{1 + \text{kh}}\right\}}$
$ =\frac{2\text{k}}{2}$
$ \Rightarrow \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0^{-}}\text{f}(\text{x}) = \text{k}$ - - - - - - - (i)
Again $ \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0^{+}}\text{f}(\text{x}) = \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{h}\rightarrow0}\text{f}(0 + \text{h}) [ \text{Let x } = 0 \text{ h},\text{x}\rightarrow0^{+}\Rightarrow\text{h}\rightarrow0]$
$ \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{h}\rightarrow0}\text{f}( - \text{h}) = \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{h}\rightarrow0} \frac{2\text{h} + 1 }{\text{h} - 1 } = \frac{1}{-1}$
$ \Rightarrow\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0^{+}}\text{f}(\text{x}) = - 1 $ - - - - - - - - - (ii)
Also $\text{f}(0) = \frac{2\times0 + 1 }{0 - 1 } = - 1 $
$\because\text{ f is continuous at x } = 0$
$ \therefore\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0^{-}}\text{f}(\text{x}) = \DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0^{+}}\text{f}(\text{x}) = \text{f}(0)\Rightarrow\text{k} = - 1.$

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 "3 Marks Question" from RELATIONS AND FUNCTIONS→RELATIONS AND FUNCTIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A function f(x) is defined as, $​​​​​​​​​​\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-9}{\text{x}-3},&\text{if }\text{x}\neq3\\6,&\text{if }\text{ x}=3\end{cases}$ show that f(x) is continuous that x = 3
→
Write the difference between maximum and minimum values of $\sin^{-1}\text{x}$ for $\text{x}\in[-1,1].$
→
Find the maximum and minimum values, if any, of the function given by:
$f(x) = -(x - 1)^2 + 10$
→
show that $\text{y}=\text{be}^\text{x}+\text{ce}^{2\text{x}}$ is a solution of the differential equation, $\frac{\text{d}^2\text{y}}{\text{dx}^2}-3\frac{\text{dy}}{\text{dx}}+2\text{y}=0$
→
Evaluate the following intregals:
$\int\sqrt{\frac{1-\text{x}}{1+\text{x}}}\text{dx}$
→
ABCD is a parallelogram. If the coordinates of A, B, C are (-2, -1), (3, 0) and (1, -2) respectively, find the coordinates of D.
→
Using differentials, find the approximate value of each of the following up to 3 places of decimal:
$(0.999)^{\frac{1}{10}}$
→
Bag I contains $3$ white and $4$ black balls, while Bag $II$ contains $5$ white and $3$ black balls. One ball is transferred at random from Bag $I$ to Bag $II$ and then a ball is drawn at random from Bag $II.$ The ball so drawn is found to be white. Find the probability that the transferred ball is also white.
→
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point $'c'$ in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = x^2 - 3x + 2 $ on $[-1, 2]$
→
For the following matrices verify the distributivity of matrix, multiplication over matrix addtion i.e., A(B + C) = AB + AC.
$\text{A}=\begin{bmatrix}1&-1\\0&2\end{bmatrix},\text{B}=\begin{bmatrix}-1&0\\2&1\end{bmatrix}$ and $\text{C}=\begin{bmatrix}0&1\\1&-1\end{bmatrix}$
→