If u = ( x^2 + y^2 + z^2 )^ 3/2 , then ( u x )^2 + ( u y )^2 + ( … - Vidyadip

MCQ

If $u = {({x^2} + {y^2} + {z^2})^{3/2}}$, then ${\left( {{{\partial u} \over {\partial x}}} \right)^2} + {\left( {{{\partial u} \over {\partial y}}} \right)^2} + {\left( {{{\partial u} \over {\partial z}}} \right)^2} = $

A
$9u$
✓
$9{u^{4/3}}$
C
$9{u^2}$
D
${u^{4/3}}$

✓

Answer

Correct option: B.

$9{u^{4/3}}$

b
(b) $\frac{{\partial u}}{{\partial x}} = \frac{3}{2}{({x^2} + {y^2} + {z^2})^{1/2}}.2x$

$\therefore $ ${\left( {\frac{{\partial u}}{{\partial x}}} \right)^2} = \frac{9}{4}({x^2} + {y^2} + {z^2})4{x^2} = 9{x^2}({x^2} + {y^2} + {z^2})$

$\therefore $ ${\left( {\frac{{\partial u}}{{\partial x}}} \right)^2} + {\left( {\frac{{\partial u}}{{\partial y}}} \right)^2} + {\left( {\frac{{\partial u}}{{\partial z}}} \right)^2}$

$= 9\,({x^2} + {y^2} + {z^2})\,({x^2} + {y^2} + {z^2})$

$= 9\,{({x^2} + {y^2} + {z^2})^2}$ = $9.{u^{4/3}}$.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let a circle $C:(x-h)^{2}+(y-k)^{2}=r^{2}, k>0$, touch the $x$-axis at $(1,0)$. If the line $x + y =0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is $2$ , then the value of $h + k + r$ is equal to

The real part of ${(1 - i)^{ - i}}$is

If the sum of the coefficients in the expansion of $(x - 2y + 3 z)^n,$ $n \in N$ is $128$ then the greatest coefficie nt in the exp ansion of $(1 + x)^n$ is

If $\mathrm{A}, \,\mathrm{B}$ are symmetric matrices of same order, then $\mathrm{A B}-\mathrm{B A}$ is a

Shortest dist ance between the lines

${L_1}:\bar r = \hat i + \hat j + \lambda \left( {\hat i + \hat j - \hat k} \right)$

${L_2}:\bar r = \hat j + \hat k + \mu \left( {\hat j + 2\hat k - \hat i} \right)$ equal to

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\beta \hat{\mathrm{k}}, \alpha, \beta \in \mathrm{R}$. Let a vector $\overrightarrow{\mathrm{b}}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and $|\vec{b}|^2=6$, If $\vec{a} \cdot \vec{b}=3 \sqrt{2}$, then the value of $\left(\alpha^2+\beta^2\right)|\vec{a} \times \vec{b}|^2$ is equal to

If $\alpha$, $\beta$,$\gamma$ are positive number such that $\alpha + \beta = \pi$ and $\beta + \gamma = \alpha$, then $tan\ \alpha$ is equal to - (where $\gamma \ne n\pi ,n \in I$ )

If the roots of the equation ${x^2} - 2ax + {a^2} + a - 3 = 0$are real and less than $3$, then

$\tan \left( {2{{\cos }^{ - 1}}\frac{3}{5}} \right) = $

Corner points of the feasible region for an $\operatorname{LPP}$ are $(0,2),(3,0),(6,0),(6,8)$ and $(0,5)$ Let $F=4 x+6 y$ be the objective function. The Minimum value of $F$ occurs at $....$