Embibe Experts Solutions for Chapter: Mathematical Tools in Physics, Exercise 2: Exercise - 2

Find the first derivative and second derivative of given functions with respect to the independent variable $x$.

$y=\ln x^2+\sin x$.

Find the first derivative and second derivative of the given function with respect to the independent variable $x$.

$y=\sqrt[7]{x}+\tan x$.

Integrate by using the substitution suggested in the bracket, 
$\int 12{\left(y^4+4y^2+1\right)}^2\left(y^3+2y\right)dy$.  [Use, $u=y^4+4y^2+1$]

If $\overrightarrow{A}=3\hat{\mathrm{i}}+4\hat{\mathrm{j}}$ and $\overrightarrow{B}=7\hat{\mathrm{i}}+24\hat{\mathrm{j}}$, then the vector having the same magnitude as $\overrightarrow{B}$ and parallel to $\overrightarrow{A}$ has sum of coefficient of $\hat{\mathrm{i}}, \hat{\mathrm{j}}$ as _____.
 

If $\overrightarrow{A}\Vert \overrightarrow{B}$, then magnitude of $\overrightarrow{A}\times \overrightarrow{B}$ is equal to _____.

The magnitude of area of the parallelogram formed by the adjacent sides of vectors $\overrightarrow{A}=3\hat{\mathrm{i}}+2\hat{\mathrm{j}}$ and $\overrightarrow{B}=2\hat{\mathrm{i}}-4\hat{\mathrm{k}}$ is $\sqrt{a}$ units. The value of $a$ is _____

 

If $\overrightarrow{A}$ is _____to $\overrightarrow{B}$, then $\overrightarrow{A}· \overrightarrow{B}=0$.

The vector $\overrightarrow{A}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$, where $\hat{\mathrm{i}}$ and $\hat{\mathrm{j}}$ are unit vectors along $x$-axis and $y$-axis, respectively, makes an angle of _____ degree with the $x$-axis.