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

• $\sqrt{2}$

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{\sqrt{2}}$

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

• 0

• 2r

• 3r

• 4r

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

• $\frac{35}{76}$

• $\frac{35}{38}$

• $\frac{7}{76}$

• $\frac{35}{72}$

$\mathrm{a} = \mathrm{i} + \mathrm{j} - 2\mathrm{k} \Rightarrow \sum _{}{\left\{\left(\mathrm{a} \times \mathrm{i}\right) \times \mathrm{j}\right\}}^2 = ?$

• $\sqrt{6}$

• 6

• 36

• $6\sqrt{6}$

Let a, b and c be three non-coplanar vectors and let p, q and r be the vectors defined by

$$\begin{gathered} \mathrm{p} = \frac{\mathrm{b} \times \mathrm{c}}{\left[\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right]}, \mathrm{q} = \frac{\mathrm{c} \times \mathrm{a}}{\left[\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right]}, \mathrm{r} = \frac{\mathrm{a} \times \mathrm{b}}{\left[\mathrm{abc}\right]} \mathrm{Then}, \\ \left(\mathrm{a} + \mathrm{b}\right) . \mathrm{p} + \left(\mathrm{b} + \mathrm{c}\right) . \mathrm{q} + \left(\mathrm{c} + \mathrm{a}\right) . \mathrm{r} = ? \end{gathered}$$

• 0

• 1

• 2

• 3

799.

Let a = i + 2j + k, b = i - j + k, c = i + j - k.

A vector in the plane of a and b has projection $\frac{1}{\sqrt{3}}$ on c. Then, one such vector is

• 4i + j - 4k

• $3\mathrm{i} + \mathrm{j} - 3\mathrm{k}$

• 4i - j + 4k

• 2i + j + 2k

The point if intersection of the lines

l₁ : r(t) = (i - 6j + 2k) + t(i + 2j + k)

l₂ : R(u) = (4j + k) + u(2i + j + 2k) is

• (10, 12, 11)

• (4, 4, 5)

• (6, 4, 7)

• (8, 8, 9)