If the vectors ii - 2xj - 3yk and i + 3xj + 2yk are orthogonal to each other, then the locus of the point (x, y) is

• a circle

• an ellipse

• a parabola

• a straight line

The magnitude of the projection of the vector a = 4i - 3j + 2k on the line which makes equal angles with the coordinate axes is

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{\sqrt{2}}$

$$\begin{gathered} \mathrm{For} \mathrm{any} \mathrm{vector} \mathrm{r} \\ \mathrm{i} \times \left(\mathrm{r} \times \mathrm{i}\right) + \mathrm{j} \times \left(\mathrm{r} \times \mathrm{j}\right) + \mathrm{k} \times \left(\mathrm{r} \times \mathrm{k}\right) = \end{gathered}$$

• 0

• 2r

• 3r

• 4r

74.

If the vectors AB = - 3i + 4k and AC = 5i - 2j + 4k are the sides of a $∆$ABC, then the length of the median through A

• $\sqrt{14}$

• $\sqrt{18}$

• $\sqrt{25}$

• $\sqrt{29}$

$\mathrm{If} \left|\mathrm{a}\right| = 1, \left|\mathrm{b}\right| = 2 \mathrm{and} \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{a} \mathrm{and} \mathrm{b} \mathrm{is} 120°, \mathrm{then} {\left\{\left(\mathrm{a} +\hspace{0.17em}3\mathrm{b}\right) \times \left(3\mathrm{a} - \mathrm{b}\right)\right\}}^2 \mathrm{is} \mathrm{equal} \mathrm{to}$

• 425

• 375

• 325

• 300

$$\begin{gathered} \mathrm{Let} \mathrm{v} = 2\mathrm{i} + \mathrm{j} - \mathrm{k} \mathrm{and} \mathrm{w} = \mathrm{i} + 3\mathrm{k}. \mathrm{If} \mathrm{u} \mathrm{is} \mathrm{any} \mathrm{unit} \mathrm{vector}, \\ \mathrm{then} \mathrm{the} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{scalar} \mathrm{triple} \mathrm{product} \left[\begin{array}{ccc}\mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right] \mathrm{is} \end{gathered}$$

• 1

• $\sqrt{10} + \sqrt{6}$

• $\sqrt{59}$

• $\sqrt{60}$

A class has fifteen boys and five girls.Suppose three students are selected at random from the class. The probability that there are two boys and one girl is

• $\frac{35}{76}$

• $\frac{35}{38}$

• $\frac{7}{76}$

• $\frac{35}{72}$

$\underset{\mathrm{x}\to 0}{\lim }\frac{\sqrt{1 + \sqrt{1 + {\mathrm{x}}^2 }} - 2}{\mathrm{x} - 8} = ?$

• $\frac{3}{2}$

• $\frac{1}{4}$

• $\frac{1}{24}$

• $\frac{1}{12}$

If f : R $\to$ R is defined by f(x) = $\left[\frac{\mathrm{x}}{5}\right]$ for x ∈ R, where [y] denotes the greatest integer not exceeding y, then $\left\{\mathrm{f}\left(\mathrm{x}\right) : \left|\mathrm{x}\right| < 71\right\}$ is equal to

• $\left\{- 14, - 13, . . . 0, . . . 13, 14\right\}$

• $\left\{- 14, - 13, . . . 0, . . . 14, 15\right\}$

• $\left\{- 15, - 14, . . . 0, . . . 14, 15\right\}$

• $\left\{- 15, - 14, . . . 0, . . . 13, 14\right\}$

$\mathrm{If} {\cos }^{-1}\left(\frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right) = \mathrm{k} \left(\mathrm{a} \mathrm{constant}\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $\frac{\mathrm{x}}{\mathrm{y}}$

• $\frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}$

• $\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}$