Download App

STD 12 - 5. continuity and differentiation — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 5. continuity and differentiation1 Mark
MCQ
Let $a, b \in \mathbb{R}$ and $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=a \cos \left(\left|x^3-x\right|\right)+b|x| \sin \left(\left|t^3+x\right|\right)$. Then $f$ is

($A$) differentiable at $x=0$ if $a=0$ and $b=1$

($B$) differentiable at $x=1$ if $a=1$ and $b=0$

($C$) $NOT$ differentiable at $x=0$ if $a=1$ and $b=0$

($D$) $NOT$ differentiable at $x=1$ if $a=1$ and $b=1$

  • A
    $A,D$
  • B
    $A,C$
  • ✓
    $A,B$
  • D
    $A,B,C$
✓

Answer

Correct option: C.
$A,B$
c
The correct options are

A differentiable at $x=0$ if $a=0$ and $b=1$

$B$ differentiable at $x=1$ if $a=1$ and $b=0$

$f(x)=a \cos \left(\left|x^3-x\right|\right)+b|x| \sin \left(\left|x^3+x\right|\right)$

As we know that $\cos \theta=\cos (-\theta)$,

$f(x)=\left\{\begin{array}{ll}a \cos \left(x^3-x\right)-b x \sin \left(-x^3-x\right) & x < 0 \\ a \cos \left(x^3-x\right)+b x \sin \left(x^3+x\right) & x \geq 0\end{array}\right.$

$\therefore f ( x )= a \cos \left( x ^3- x \right)+ bx \sin \left( x ^3+ x \right) \quad \forall x \in R$

Hence, $f(x)$ is differentiable at all $x \in R$ for any $a$ and $b$.

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 STD 12 - 5. continuity and differentiation→STD 12 - 5. continuity and differentiation — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

If $\vec{a},\vec{b},\vec{c}$are three non-zero, non-coplanar vectrors and $\overrightarrow {{b_1}} \, = \,\overrightarrow {b\,} \, - \,\frac{{\overrightarrow b \,.\,\overrightarrow a }}{{{{\left| {\overrightarrow a \,} \right|}^2}}}\overrightarrow a \,,\,\overrightarrow {{b_2}} \, = \overrightarrow b \, + \,\frac{{\overrightarrow b \,.\,\overrightarrow a }}{{{{\left| {\overrightarrow a \,} \right|}^2}}}\overrightarrow a \, $ and $ \overrightarrow {{c_1}} \, = \,\overrightarrow c \, - \,\frac{{\overrightarrow c \,.\,\overrightarrow a }}{{{{\left| {\overrightarrow a \,} \right|}^2}}}\overrightarrow a \, + \,\frac{{\overrightarrow c \,.\,\overrightarrow b }}{{{{\left| {\overrightarrow b \,} \right|}^2}}}\overrightarrow {{b_1}} \, $, $\overrightarrow {{c_2}} \, = \,\overrightarrow c \, - \,\frac{{\overrightarrow c \,.\,\overrightarrow a }}{{{{\left| {\overrightarrow a \,} \right|}^2}}}\overrightarrow a \, - \,\frac{{\overrightarrow c \,.\,\overrightarrow b }}{{{{\left| {\overrightarrow {{b_1}} \,} \right|}^2}}}\overrightarrow {{b_1}} \, ,$ $ \overrightarrow {{c_3}} \, = \,\overrightarrow c \, - \,\frac{{\overrightarrow c \,.\,\overrightarrow a }}{{{{\left| {\overrightarrow c \,} \right|}^2}}}\overrightarrow a \, + \,\frac{{\overrightarrow c \,.\,\overrightarrow {{b_2}} }}{{{{\left| {\overrightarrow c \,} \right|}^2}}}\overrightarrow {{b_1}} \, $ $, \overrightarrow {{c_4}} \, = \,\overrightarrow c \, - \,\frac{{\overrightarrow c \,.\,\overrightarrow a }}{{{{\left| {\overrightarrow c \,} \right|}^2}}}\overrightarrow a \, - \,\frac{{\overrightarrow b \,.\,\overrightarrow c }}{{{{\left| {\overrightarrow b \,} \right|}^2}}}\overrightarrow {{b_1}} \,.$ Then, which of the following is a set of mutually orthogonal vectors ?
→
Evaluate: $\int_0^{\pi / 4} \tan ^3 x d x$
→
Area bounded by the curve $\text{y}=\cos\text{x}$ between $\text{x}=0$ and $\text{x}=\frac{3\pi}{2}$ is:
→
If $\text{g(f(x))}=|\sin\text{x}|$ and $\text{f(g(x))}=(\sin\sqrt{\text{x}})^2,$ then
→
If the constraints in a linear programming problem are changed
→
If $\int_0^{{t^2}} {xf(x)dx = } \frac{2}{5}{t^5},\,\,t > 0,$ then $f\left( {\frac{4}{{25}}} \right) = $
→
$\text{f}(\text{x})=1+2\sin\text{x}+3\cos^{2}\text{x}, 0<\text{x}<\frac{2\pi}{3}$ is :
→
In transportation models designed in linear programming, points of demand is classified as:
→
The number of positive integral solutions of the equation $\left| {\begin{array}{*{20}{c}}{{x^3} + 1}&{{x^2}y}&{{x^2}z}\\{x{y^2}}&{{y^3} + 1}&{{y^2}z}\\{x{z^2}}&{y{z^2}}&{{z^3} + 1}\end{array}} \right|$ $= 11$ is
→
The area enclosed by the closed curve $C$ given by the differential equation $\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0$ is $4 \pi$.

Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is

→