Sample QuestionsContinuity and Differentiability questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $y=x \log x$ then value of $\frac{d^2 y}{d x^2}$ :
- A
$\frac{1}{1+x}$
- ✓
$\frac{1}{x}$
- C
$\log (1+x)$
- D
$1+\log x$
Answer: B.
View full solution →If $3 x+2 y=\sin x$ then $\frac{d y}{d x}$ :
- A
$\frac{\cos x+3}{2}$
- B
$\frac{\cos x-2}{3}$
- ✓
$\frac{\cos x-3}{2}$
- D
$\frac{\cos x+2}{3}$
Answer: C.
View full solution →If $y=\sin \left(m \sin ^{-1} x\right)$ in which of the option is correct :
- A
$\left(1-x^2\right) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}+m^2 y=0$
- ✓
$\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}+m^2 y=0$
- C
$\left(1+x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}-m^2 y=0$
- D
$\left(1+x^2\right) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}-m^2 x=0$
Answer: B.
View full solution →If $x=2 \cos \theta-\cos 2 \theta$ and $y=2 \sin \theta-\sin 2 \theta$ then $\frac{d y}{d x}$ :
- A
$\frac{\cos \theta+\cos 2 \theta}{\sin \theta-\sin 2 \theta}$
- ✓
$\frac{\cos \theta-\cos 2 \theta}{\sin 2 \theta-\sin \theta}$
- C
$\frac{\cos \theta-\cos 2 \theta}{\sin \theta-\sin 2 \theta}$
- D
$\frac{\cos 2 \theta-\cos \theta}{\sin 2 \theta+\sin \theta}$
Answer: B.
View full solution →If function $f(x)=\left\{\begin{array}{cl}\frac{e^{3 x}-e^{-5 x}}{x} & , \text { if } x \neq 0 \\ k & , \text { if } x=0\end{array}\right.$ is continuous then value of $k$ :
Answer: D.
View full solution →Show that function $f(x)=x^2, x=0$ is continuous.
View full solution →Show that function $F ( x )=\frac{1}{(x-a)}$, is discontinuous at $x=a$.
View full solution →If $f(x)=x \sin x$ then find $f^{\prime}\left(\frac{\pi}{2}\right)$.
View full solution →If function $F (x)=\frac{\sin (10 x)}{x}, x \neq 0$, is continuous at $x=0$. Then find the value of $F (0)$.
View full solution →If $x=t^2, y=t^3$ then find $\frac{d^2 y}{d x^2}$.
View full solution →If $y=\sin ^{-1} x$ then find $\frac{d^2 y}{d x^2}$.
View full solution →Find differentiation of $\log (1+\theta)$ w.r.t. $\sin ^{-1} \theta$.
View full solution →If $y=\sqrt{\sin x+y}$, then find $\frac{d y}{d x}$.
View full solution →If $y \sqrt{1-x^2}=\sin ^{-1} x$, then find $\frac{d y}{d x}$
View full solution →If $y=3 \cos x-2 \sin x$, then prove that $\frac{d^2 y}{d x^2}-y=0$
View full solution →If function $f(x)=\left\{\begin{array}{cc}x^5 \sin \frac{1}{x}, & x \neq 0 \\ k & x=0\end{array}\right.$, is continuous at $x =0$, find the value of $k$.
View full solution →Examine the continuity of function.$
f(x)=\left\{\begin{array}{ll}
1+x, & x \leq 3 \\
7-x, & x>3
\end{array} \text { at } x=3\right.
$
View full solution →Differentiation w.r.t. $x$ of $\tan ^{-1}\left[\frac{\sin x+\cos x}{\cos x-\sin x}\right]$.
View full solution →Prove that $\frac{d}{d x}\left[\frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1} \frac{x}{a}\right]=\sqrt{a^2-x^2}$
View full solution →Find $\frac{d y}{d x}$
(a) $y=\sin x^{\sin x^{\sin x ...... \infty}}$
(b) $y=\sqrt{\log _e x+\sqrt{\log _e x+\sqrt{\log _e x+\ldots \ldots \ldots \infty}}}$
(c) $y=e^{x+e^{x+e^{x+\ldots \infty}}}$
View full solution →If $x=a \cos ^3 \theta, y=a \sin ^3 \theta$ then find $\left(\frac{d^2 y}{d x^2}\right)_{\theta=\frac{\pi}{4}}$
View full solution →If $x=a \cos \theta+b \sin \theta$ and $y=a \sin \theta-b \cos \theta$ then prove that $y^2 y_2-x y_1+y=0$.
View full solution →If given function is continuous at $x=1$, then find $a$ and $b$.$
f(x)=\left\{\begin{array}{cl}
3 a x+b, & \text { if } x>1 \\
11, & \text { if } x=1 \\
5 a x-2 b, & \text { if } x<1
\end{array}\right.
$
View full solution →