$\int 5^{5^{5^{\mathrm{x}}}} 5^{5^{\mathrm{x}}} 5^{\mathrm{x}} \mathrm{dx}$ is equal to

• $5^{5^{5^{\mathrm{x}}}}{\left(\log \left(5\right)\right)}^3 + \mathrm{c}$

• $\frac{5^{5^{5^{\mathrm{x}}}}}{{\left(\log \left(5\right)\right)}^3} + \mathrm{c}$

• $\frac{5^{{}^{5^{\mathrm{x}}}}}{{\left(\log \left(5\right)\right)}^3} + \mathrm{c}$

• $5^{{}^{5^{\mathrm{x}}}}{\left(\log \left(5\right)\right)}^3 + \mathrm{c}$

$\overrightarrow{\mathrm{A}}, \overrightarrow{\mathrm{B}}, \overrightarrow{\mathrm{C}}$ are three non-zero vectors; no two of them are parallel. If $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}$ is collinear to $\overrightarrow{\mathrm{C}}$ and $\overrightarrow{\mathrm{B}} + \overrightarrow{\mathrm{C}}$ is collinear to $\overrightarrow{\mathrm{A}}$, then $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}} + \overrightarrow{\mathrm{C}}$ is equal to

• $\overrightarrow{\mathrm{A}}$

• $\overrightarrow{\mathrm{B}}$

• $\overrightarrow{\mathrm{C}}$

• $\overrightarrow{0}$

In tossing of a coin (m + n)(m > n) times, the probability of coming consecutive heads at least m times is

• $\frac{\mathrm{n} + 2}{2^{\mathrm{m} + 1}}$

• $\frac{\mathrm{m} - \mathrm{n}}{2^{\mathrm{m} + \mathrm{n}}}$

• $\frac{\mathrm{m} + \mathrm{n}}{2^{\mathrm{m} + \mathrm{n}}}$

• $\frac{\mathrm{mn}}{2^{\mathrm{m} + \mathrm{n}}}$

Let $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}$. If c is vector such that $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{c}} = \left|\overrightarrow{\mathrm{c}}\right| \left|\overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{a}}\right| = 2\sqrt{2}$ the angle between $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}} \mathrm{is} 30°$ is, then $\left|\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) \times \overrightarrow{\mathrm{c}}\right|$ is equal to

• $\frac{2}{3}$

• $\frac{3}{2}$

• 2

• 3

25.

The value of ${\int }_2^4\left\{\left|\mathrm{x} - 2\right| + \left|\mathrm{x} - 3\right|\right\}\mathrm{dx}$ is

• 1 sq unit

• 2 sq unit

• 3 sq unit

• 5 sq unit

The differential equation of all circles touching the axis of y at origin and centre on the x-axis is given by

• $\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} - {\mathrm{x}}^2 + {\mathrm{y}}^2 = 0$

• $2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} - {\mathrm{x}}^2 - {\mathrm{y}}^2 = 0$

• $\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} - 2\mathrm{xy} = 0$

• None of these

The solution of the differential equation

$\left({\mathrm{e}}^{- 2\sqrt{\mathrm{x}}} - \frac{\mathrm{y}}{\sqrt{\mathrm{x}}}\right)\frac{\mathrm{dx}}{\mathrm{dy}} = 1$ is given by

• ${\mathrm{e}}^{2\sqrt{\mathrm{x}}} = 2\sqrt{\mathrm{x}} + \mathrm{c}$

• ${\mathrm{ye}}^{- 2\sqrt{\mathrm{x}}} = \sqrt{\mathrm{x}} + \mathrm{c}$

• $\mathrm{y} = \sqrt{\mathrm{x}}$

• None of these

The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \sqrt{\frac{1 - {\mathrm{y}}^2}{1 - {\mathrm{x}}^2}}$ is

• ${\sin }^{-1}\left(\mathrm{y}\right) - {\sin }^{-1}\left(\mathrm{x}\right) = \mathrm{c}$

• ${\sin }^{-1}\left(\mathrm{y}\right) + {\sin }^{-1}\left(\mathrm{x}\right) = \mathrm{c}$

• ${\sin }^{-1}\left(\mathrm{xy}\right) = 2$

• None of these

If the vectors $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, - 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \mathrm{\lambda }\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ are linearly dependent, then the value of $\mathrm{\lambda }$ is equal to

• 0

• 1

• 2

• 3

If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ are two non-zero non-collinear vectors, then $2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \hat{\mathrm{i}}\right]\hat{\mathrm{i}} + 2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \hat{\mathrm{j}}\right]\hat{\mathrm{j}} + 2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \hat{\mathrm{k}}\right]\hat{\mathrm{k}} + 2\left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{a}}\right]$ is equal to

• $2\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$

• $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}$

• None of these