Let f : R → R be defined by f(x) = 3x² – 5 and g : R → R be defined by g(x) =${\frac{x}{x^{2}+1}}.$ find gof(x) .

Find $\stackrel{\rightarrow}{a}$.$\stackrel{\rightarrow}{b}$,If $\stackrel{\rightarrow}{a}$ = $-\stackrel{\wedge}{i}+\stackrel{\wedge}{j} -2\stackrel{\wedge}{k}$ and $\stackrel{\rightarrow}{b}$ = $2\stackrel{\wedge}{i}+3\stackrel{\wedge}{j} -\stackrel{\wedge}{k}$.

If $\stackrel{\rightarrow}{a}\$and $\stackrel{\rightarrow}{b}\$ are two unit vectors such that $\stackrel{\rightarrow}{a}\$ +$\stackrel{\rightarrow}{b}\$ is also a unit vector, then find the angle between $\stackrel{\rightarrow}{a}\$ and $\stackrel{\rightarrow}{b}\$

If $|\stackrel{\rightarrow}{a}|$=2, $|\stackrel{\rightarrow}{b}|$=3 and $\stackrel{\rightarrow}{a}$.$\stackrel{\rightarrow}{b}$=3 then find the projection of $|\stackrel{\rightarrow}{b}|$ on $|\stackrel{\rightarrow}{a}|$.

Write the integrating factor of the following differential equation. $(1+y^{2})+(2x y-\mathrm{cot}\;y){\frac{d y}{d x}}\;=0$

Find $|\stackrel{\rightarrow}{x}|\$, if for a unit vector $\stackrel{\rightarrow}{a}\$ and ($\stackrel{\rightarrow}{x}\$ + $\stackrel{\rightarrow}{a}\$ )($\stackrel{\rightarrow}{x}\$ -$\stackrel{\rightarrow}{a}\$ ) = 15.

Find the integrating factor of the differential equation

For what value of λ are the vectors $\stackrel{\wedge}{i}+2λ \stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ and $2\stackrel{\wedge}{i}+\stackrel{\wedge}{j} -3\stackrel{\wedge}{k}$ perpendicular ?

If $\stackrel{\wedge}{\mathrm{a}}$, $\stackrel{\wedge}{\mathrm{b}}$ and $\stackrel{\wedge}{\mathrm{c}}$ are mutually perpendicular unit vectors, then find value of $|2\stackrel{\wedge}{\mathrm{a}}$ + $\stackrel{\wedge}{\mathrm{b}}$ + $\stackrel{\wedge}{\mathrm{c}}|$.

Write the integrating factor of the differential equations ${\sqrt{x}}{\frac{d y}{d x}}+y=e^{-2{\sqrt{x}}}\,.$

Write the value of λ so that the vectors $\stackrel{\rightarrow}{a}\$ = $2 \stackrel{\wedge}{i}+λ\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$ and $\stackrel{\rightarrow}{b}\$ = $\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +3\stackrel{\wedge}{k}$ are perpendicular to each other ?

Find the projection of vector $\stackrel{\wedge}{i}+3\stackrel{\wedge}{j} +7\stackrel{\wedge}{k}$ on the vector $2\stackrel{\wedge}{i}-3\stackrel{\wedge}{j} +6\stackrel{\wedge}{k}$.

Find ‘λ’ when the projection of $\stackrel{\rightarrow}{a}\$ = $λ\stackrel{\wedge}{i}+\stackrel{\wedge}{j} +4\stackrel{\wedge}{k}$ on $\stackrel{\rightarrow}{b}\$ = $2\stackrel{\wedge}{i}+6\stackrel{\wedge}{j} +3\stackrel{\wedge}{k}$ is 4 units.

If $\stackrel{\rightarrow}{a}\$ and $\stackrel{\rightarrow}{b}\$ are perpendicular vectors, $|\stackrel{\rightarrow}{a}\$ + $\stackrel{\rightarrow}{b}|\$ = 13 and $|\stackrel{\rightarrow}{a}|\$ =5,find the value of $|\stackrel{\rightarrow}{b}|\$.

Write the projection of $\stackrel{\rightarrow}{b}\$ +$\stackrel{\rightarrow}{c}\$ on $\stackrel{\rightarrow}{a}\$,where $\stackrel{\rightarrow}{a}\$ = $2\stackrel{\wedge}{i}-2\stackrel{\wedge}{j} +\stackrel{\wedge}{k}$, $\stackrel{\rightarrow}{b}\$ = $\stackrel{\wedge}{i}+2\stackrel{\wedge}{j} -2\stackrel{\wedge}{k}$ and $\stackrel{\rightarrow}{c}\$ = $2\stackrel{\wedge}{i}-\stackrel{\wedge}{j} +4\stackrel{\wedge}{k}$