${∆}^2{\mathrm{y}}_0$ is equal to

• 2y₂ - 2y₁ - y₀

• y₂ - 2y₁ - y₀

• 2y₂ - 2y₁ + y₀

• y₂ - 2y₁ + y₀

${∆}^2\left(3{\mathrm{e}}^{\mathrm{x}}\right)$ is equal to

• 3e^x

• 3(h - 1)e^x

• 3(e^h - 1)²e^x

• None of the above

If vectors $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}, \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}}$ are coplanar, then $\mathrm{\lambda }$ is equal to

• - 2

• 3

• 2

• - 3

Given, $\overrightarrow{\mathrm{a}} \perp \overrightarrow{\mathrm{b}}$, $\left|\overrightarrow{\mathrm{a}}\right| = 1$ and if $\left(\overrightarrow{\mathrm{a}} + 3\overrightarrow{\mathrm{b}}\right)\left(2\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right) = - 10$, $\left|\overrightarrow{\mathrm{b}}\right|$ is equal to

• 1

• 3

• 2

• 4

15.

$\int \frac{\mathrm{dx}}{\mathrm{x} +\hspace{0.17em}\sqrt{\mathrm{x}}}$ is equal to

• $\frac{1}{2}\log \left(1 +\hspace{0.17em}\sqrt{\mathrm{x}}\right) + \mathrm{c}$

• $2\log \left(1 +\hspace{0.17em}\sqrt{\mathrm{x}}\right) + \mathrm{c}$

• $\frac{1}{4}\log \left(1 +\hspace{0.17em}\sqrt{\mathrm{x}}\right) + \mathrm{c}$

• $3\log \left(1 +\hspace{0.17em}\sqrt{\mathrm{x}}\right) + \mathrm{c}$

   

The value of $\int \frac{{\mathrm{x}}^2}{1 +\hspace{0.17em}{\mathrm{x}}^6}\mathrm{dx}$ is

• x³ + c

• $\frac{1}{3}{\tan }^{-1}\left({\mathrm{x}}^3\right) + \mathrm{c}$

• log(1 + x³) + c

• None of these

What are the DR's of vector parallel to (2, - 1, 1) and (3 4, - 1) ?

• (1, 5, - 2)

• (- 2, - 5, 2)

• (- 1, 5, 2)

• (- 1, - 5, - 2)

$\int \left(1 - \cos \left(\mathrm{x}\right)\right){\csc }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\tan \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

• $- \cot \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

• $2\tan \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

• $- 2\cot \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

$\int \frac{\cos \left(\mathrm{x}\right)}{\sqrt{1 + \sin \left(\mathrm{x}\right)}}\mathrm{dx}$ is equal to

• $\sin \left(\frac{\mathrm{x}}{2}\right) - \cos \left(\frac{\mathrm{x}}{2}\right)$

• $\sin \left(\frac{\mathrm{x}}{2}\right) + \cos \left(\frac{\mathrm{x}}{2}\right)$

• $2\left[\sin \left(\frac{\mathrm{x}}{2}\right) - \cos \left(\frac{\mathrm{x}}{2}\right)\right] +\hspace{0.17em}\mathrm{c}$

• $2\left[\sin \left(\frac{\mathrm{x}}{2}\right) + \cos \left(\frac{\mathrm{x}}{2}\right)\right] +\hspace{0.17em}\mathrm{c}$

$\left[\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}\right] = \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$, then

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right] = 1$

• $\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}$ are coplanar

• $\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right] = - 1$

• $\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}$ are mutually perpendicular