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 function $f$ is defined such that
$
f(x)=\left\{\begin{array}{cc}
\frac{k \cos x}{\pi-2 x}, & \text { if } x \neq \frac{\pi}{2} \\
3, & \text { if } x=\frac{\pi}{2}
\end{array}\right.
$is continuous at $x=\frac{\pi}{2}$, then value of $k$ :
Answer: C.
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 →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 $y=\sqrt{\sin x+y}$, then find $\frac{d y}{d x}$.
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 $y \sqrt{1-x^2}=\sin ^{-1} x$, 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 →Show that function$
f(x)=\left\{\begin{array}{ll}
3-x, & \text { if } x<1 \\
2, & \text { if } x=1 \\
1+x, & \text { if } x>1
\end{array}\right.
$
is continuous at $x=1$.
View full solution →If $y= P e^{a x}+ Q e^{b x}$ then show that$
\frac{d^2 y}{d x^2}-(a+b) \frac{d y}{d x}+a b y=0
$
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 →If $y=x^x+x^a+a^x+a^a$ then find $\frac{d y}{d x}$.
View full solution →If $(a x+b) e^{y / x}=x$ then show that : $ x^3 \frac{d^2 y}{d x^2}=\left(x \frac{d y}{d x}-y\right)^2$
View full solution →If $x^y+y^x=b^a+a^b$ then find $\frac{d y}{d x}$.
View full solution →If $y^3+3 a x^2+x^3=0$ then prove that:$
\frac{d^2 y}{d x^2}+\frac{2 a^2 x^2}{y^5}=0
$
View full solution →If $y=x \cos (\log x)$ then prove that :$
x^2 y_2-x y_1+2 y=0
$
View full solution →If $y=\cos \sqrt{x}$, then the value of $\frac{d y}{d x}$ will be ________ .
View full solution →Dfferentiation of $\cos (\sqrt{x})$ w.r.t. $x$ is ________ .
View full solution →If $3 x+2 y=\cos y$, then $\frac{d y}{d x}$ __________ .
View full solution →If $y=\log _a x$ then $\frac{d y}{d x}=$ __________ .
View full solution →If $y=x+e^x$ then $\frac{d^2 y}{d x^2}=$ _________ .
View full solution →Examine the continuity and differentiability of function $f(x)=|x|+|x-1|$ in interval $(-1,2)$.
View full solution →Find the value of $\frac{d y}{d x}$ at $\theta=\frac{\pi}{4}$ when :
$x=a e^\theta(\sin \theta-\cos \theta)$
$y=a e^\theta(\sin \theta+\cos \theta)$
View full solution →If $y=\frac{\sin ^{-1} x}{x}$ then prove that :$
x\left(1-x^2\right) y_2+\left(2-3 x^2\right) y_1-x y=0
$
View full solution →If $y=\sin \left[2 \tan ^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right)\right]$ then find $\frac{d y}{d x}$.
View full solution →If function $f(x)=\frac{\sqrt{x}-\sqrt{a}}{x-a}$, is continuous at $x=a$, then find value of $f(a)$.
View full solution →