Download App

Continuity — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsContinuity1 Mark
MCQ
If $\text{f(x)}=\begin{cases}\frac{\sin(\cos\text{x})-\cos\text{x}}{(\pi-2\text{x})^2},&\text{x}\neq\frac{\pi}{2}\\\text{k},&\text{x}=\frac{\pi}{2}\end{cases}$ is continuous at $\text{x}=\frac{\pi}{2},$ then $k$ is equal to:
  • ✓
    $0$
  • B
    $\frac{1}{2}$
  • C
    $1$
  • D
    $-1$
✓

Answer

Correct option: A.
$0$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}}\frac{\sin(\cos\text{x})-\cos\text{x}}{(\pi-2\text{x})^2}$
$\text{x}\rightarrow\frac{\pi}{2}+\text{h}$
$\text{x}\rightarrow\frac{\pi}{2}\Rightarrow\text{h}\rightarrow0$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{h}\rightarrow0}\frac{\sin\Big(\cos\big(\frac{\pi}{2}+\text{h}\big)\Big)-\cos\big(\frac{\pi}{2}+\text{h}\big)}{\Big(\pi-2\big(\frac{\pi}{2}+\text{h}\big)\Big)^2}$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{h}\rightarrow0}\frac{\sin(-\sin\text{h})+\sin\text{h}}{4\text{h}^2}$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{h}\rightarrow0}\frac{\sin\text{h}-\sin(\sin\text{h})}{4\text{h}^2}$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{h}\rightarrow0}\frac{\sin\text{h}-\sin(\sin\text{h})}{4\text{h}^2}$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{h}\rightarrow0}\frac{2\cos\big(\frac{\text{h}+\sin\text{h}}{2}\big)\sin\big(\frac{\text{h}-\sin\text{h}}{2}\big)}{4\text{h}^2}$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{h}\rightarrow0}\frac{2\cos\big(\frac{\text{h}+\sin\text{h}}{2}\big)}{4\text{h}^2}\times\frac{\sin\big(\frac{\text{h}-\sin\text{h}}{2}\big)}{\frac{\text{h}-\sin\text{h}}{2}}\times\frac{\text{h}-\sin\text{h}}{2}$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{h}\rightarrow0}\frac{2\cos\big(\frac{\text{h}+\sin\text{h}}{2}\big)}{4\text{h}^2}\times\frac{\text{h}-\sin\text{h}}{2}$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{h}\rightarrow0}\frac{2\cos\big(\frac{\text{h}+\sin\text{h}}{2}\big)}{4\text{h}^2}\times(\text{h}-\sin\text{h})$
$\text{f}\Big(\frac{\pi}{2}\Big)=\lim\limits_{\text{h}\rightarrow0}\frac{1}{4}\frac{\cos\big(\frac{\text{h}+\sin\text{h}}{2}\big)}{\text{h}^2}\times(\text{h}-\sin\text{h})=0$
$\text{k}=0$

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

The value of $\int_{-\pi / 4}^{\pi / 4} \log \left(\frac{2+\sin \theta}{2-\sin \theta}\right) \cdot d \theta$ is
→
If $y=\sec \left(\tan ^{-1} x\right)$, then $\frac{d y}{d x}$ at $x=1$, is equal to
→
If a coin is tossed twice and $X$ is the number of tails, then var $(X)=$
→
The separate equations of the lines represented by $3 x^2-10 x y-8 y^2=0$ are
→
The vector equation of the plane passing through the intersection of the planes $\bar{r} \cdot(\hat{i}-\hat{j}+2 \hat{k})=3$ and $\bar{r} \cdot(3 \hat{i}-\hat{j}-\hat{k})=4$ is
→
If the slope of one of the lines given by $a x^2+2 h x y+$ $b y^2=0$ is two times the other, then
→
If $f: R \rightarrow R$ is given by $f(x)=x^3+3$, then $f^{-1}(x)$ is equal to:
→
If the length of a vector be 7 and direction ratios be $2,-3,6$, then its direction cosines are
→
If $\int_1^k(2 x-3) d x=12$, then $k$ =
→
A cylindrical vessel of radius $0.5m$ is filled with oil at the rate of $0.25\pi/\text{minute}$. The rate at which the surface of the oil is rising, is:
→