Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Find the value of k if f(x) is continuous at $\text{x}=\frac{\pi}{2},$ where
$\text{f}\text{(x)}=\begin{cases}\frac{\text{k}\cos\text{x}}{\pi-2\text{x}}, &\text{ x}\neq\frac{\pi}{2}\\3, &\text{ x}=\frac{\pi}{2}\end{cases}$
✓

Answer

Since f(x) is continuous at $\text{x}=\frac{\pi}{2},$ L.H.Limit = R.H.Limit.
$\Rightarrow\lim\limits_{\text{x} \rightarrow \frac{\pi^-}{2}}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow \frac{\pi^+}{2}}\text{f}\text{(x)}=\lim\limits_{\text{x} \rightarrow \frac{\pi}{2}}\text{f}\text{(x)}=\text{f}\Big(\frac{\pi}{2}\Big)$
$\Rightarrow\lim\limits_{\text{x} \rightarrow \frac{\pi^-}{2}}\frac{\text{k}\cos\text{x}}{\pi-2\text{x}}=3$
$\Rightarrow\text{k}\lim\limits_{\text{x} \rightarrow \frac{\pi}{2}}\frac{\sin\Big(\frac{\pi}{2}-\text{x}\Big)}{2\Big(\frac{\pi}{2}-\text{x}\Big)}=3$
$\Rightarrow\frac{\text{k}}{2}\lim\limits_{\text{x} \rightarrow \frac{\pi}{2}}\frac{\sin\Big(\frac{\pi}{2}-\text{x}\Big)}{2\Big(\frac{\pi}{2}-\text{x}\Big)}=3$
$\Rightarrow\frac{\text{k}}{2}=3$
$\Rightarrow\text{k}=6$

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 "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Check whether the $\ast$ operation defined on the set $ \text{A = R} \times \text{R as} $
$\text{(a, b)} \ast \text{(c, d)} = \text{(a + c, b + d)}$
is a binary operation or not, where R is the set of all real numbers. If it is a binary operation, is it commutative and associative too? Also find the identity element of $\ast$.
→
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that,
  1. Both balls are red,
  2. First ball is black and second is red,
  3. One of them is black and other is red.
→
If $\text{y}=\sin\Big[2\tan^{-1}\Big\{\sqrt{\frac{1-\text{x}}{1+\text{x}}}\Big\}\Big],$ find $\frac{\text{dy}}{\text{dx}}.$
→
If $y=x^x+x^a+a^x+a^a$ then find $\frac{d y}{d x}$.
→
Find the angle between the following pairs of lines:$\frac{5-\text{x}}{-2}=\frac{\text{y}+3}{1}=\frac{1-\text{z}}{3}$ and $\frac{\text{x}}{3}=\frac{1-\text{y}}{-2}=\frac{\text{z}+5}{-1}$
→
Find the area enclosed by the curve $3x^2 + 5y = 32$ and $y = |x - 2|$
→
Find a point on the parabola $y = (x - 4)^2$​​​​​​​, where the tangent is parallel to the chord joining (4, 0) and (5, 1).
→
Evaluate the following integrals:
$\int\frac{\text{x}^3-1}{\text{x}^3+\text{x}}\text{ dx}$
→
Find the particular solution of the differential equation$(1-\text{y}^2)(1+\log\text{x})\text{dx}+2\text{xy dy}=0,$ given that $\text{y}=0$ when $\text{x}=1.$
→
Find the values of $x$ for which $\text{y}=[\text{x}(\text{x}-2)]^2$ is an increasing function.
→