Download App

Continuity — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity1 Mark
Question
The value of a for which the function $\text{f(x)}=\begin{cases}\frac{(4^\text{x}-1)^3}{\sin\Big(\frac{\text{x}}{\text{a}}\Big)\log\Big\{\Big(1+\frac{\text{x}^2}{3}\Big)\Big\}},&\text{x}\neq0\\12(\log4)^3,&\text{x}=0\end{cases}$ may be continuous at x = 0 is:
  1. 1
  2. 2
  3. 3
  4. None of these.
✓

Answer

  1. None of these.
Solution:
$\lim\limits_{\text{x}\rightarrow0}\frac{(4^\text{x}-1)^3}{\sin\Big(\frac{\text{x}}{\text{a}}\Big)\log\Big\{\Big(1+\frac{\text{x}^2}{3}\Big)\Big\}}=12(\log4)^3$
$\lim\limits_{\text{x}\rightarrow0}\frac{\frac{(4^\text{x}-1)^2}{\text{x}^3}}{\frac{\sin\Big(\frac{\text{x}}{\text{a}}\Big)}{\text{x}}}\times\frac{1}{\frac{\log\Big\{\Big(1+\frac{\text{x}}{3}\Big)\Big\}}{\text{x}^2}}=12(\log4)^3$
$\lim\limits_{\text{x}\rightarrow0}\frac{\frac{(4^\text{x}-1)^3}{\text{x}^3}}{\frac{\sin\Big(\frac{\text{x}}{\text{a}}\Big)}{\text{x}}}\times\frac{1}{\frac{\log\Big\{\Big(1+\frac{\text{x}^2}{3}\Big)\Big\}}{\text{x}^2}}=12(\log4)^3$
$\lim\limits_{\text{x}\rightarrow0}\frac{\Big(\frac{(4^\text{x}-1}{\text{x}^3}\Big)}{\frac{\sin\Big(\frac{\text{x}}{\text{a}}\Big)}{\frac{\text{x}}{\text{a}}}}\text{a}\text{x}\ {\times}\frac{\frac{1}{\log\Big\{\Big(1+\frac{\text{x}^2}{3}\Big)\Big\}}}{\frac{\text{x}^3}{3}}\text{x}^3=12(\log4)^3$
$3(\log4)^3=12(\log4)^3$
$3\text{a}=12$
$\text{a}=12$
Note: The question is incorrect, so it has been modified.

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→Continuity — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

The transpose of a row matrix is:
  1. zero matrix
  2. diagonal matrix
  3. column matrix
  4. row matrix
→
If a matrix $A$ is both symmetric and skew$-$symmetric, then:
→
The equation $\sin^{-1}\text{x}-\cos^{-1}\text{x}=\cos^{-1}(\frac{\sqrt3}{2})$ has:
  1. Nique solution.
  2. No solution.
  3. Infinitely many solution.
  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. $\text{None of these}.$
→
If $\text{f(x)}=\sqrt{1-\sqrt{1-\text{x}^2}},$ then f(x) is:
  1. Continuous on [-1, 1] and differentiable on (-1, 1)
  2. Continuous on [-1, 1] and differentiable on $(-1,0)\cup(0,1)$
  3. Continuous and differentiable on [-1, 1]
  4. None of these.
→
If $A, B$ are non-singular square matrices of the same order, then $\left(A B^{-1}\right)^{-1}=$
→
The distance of the point P(a, b, c) from the x-axis is:
  1. $\sqrt{\text{b}^2+\text{c}^2}$
  2. $\sqrt{\text{a}^2+\text{c}^2}$
  3. $\sqrt{\text{a}^2+\text{b}^2}$
  4. $\text{none of these}$
→
The value of $\int_{-\pi / 2}^{\pi / 2}\left(x^3+x \cos x+\tan 5 x+1\right) d x$ is
→
The equations in terms of x and y are:
  1. x – y = 50, 2x – y = 550
  2. x – y = 50, 2x + y = 550
  3. x + y = 50, 2x + y = 550
  4. x + y = 50, 2x – y = 550
→
Which of the following is not a convex set?
→