$\mathrm{If} \mathrm{u} = {\sin }^{-1}\left(\frac{{\mathrm{x}}^2 + {\mathrm{y}}^2}{\mathrm{x} + \mathrm{y}}\right) \mathrm{then} \mathrm{x}\frac{\partial \mathrm{u}}{\partial \mathrm{x}} + \mathrm{y}\frac{\partial \mathrm{u}}{\partial \mathrm{y}} \mathrm{is}$

• sin(u)

• tan(u)

• cos(u)

• cot(u)

$\mathrm{If} \mathrm{f}(\mathrm{x}, \mathrm{y}) = \frac{\cos \left(\mathrm{x} - 4\mathrm{y}\right)}{\cos \left(\mathrm{x} + 4\mathrm{y}\right)}, \mathrm{then} {\frac{\partial \mathrm{f}}{\partial \mathrm{x}}}_{\mathrm{y} = \frac{\mathrm{x}}{2}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• - 1

• 0

• 1

• 2

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0} \mathrm{and} \left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 4 \mathrm{and} \\ \left|\overrightarrow{\mathrm{c}}\right| = \sqrt{37}, \mathrm{then} \mathrm{the} \mathrm{angle} \mathrm{between} \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{is} : \end{gathered}$$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

$\mathrm{If} \hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} \mathrm{\lambda }\hat{\mathrm{i}} + 3\hat{\mathrm{j}} \mathrm{are} \mathrm{coplaner}, \mathrm{then} \mathrm{\lambda } = ?$

• - 1

• $\frac{1}{2}$

• $\mathit{-}\mathit{ }\frac{\mathit{3}}{\mathit{2}}$

• 2

65.

The position vector of a point lying on the line joining the points whose positions vectors are $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{is} :$

• $\hat{\mathrm{j}}$

• $\hat{\mathrm{i}}$

• $\hat{\mathrm{k}}$

• $\overrightarrow{0}$

If the volume of parallelopiped with conterminus edges $4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \hat{\mathrm{k}}, - \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 3\hat{\mathrm{i}} + 9\hat{\mathrm{j}} + \mathrm{p}\hat{\mathrm{k}}$ is 34 cubic units, then p is equal

• 4

• - 13

• 13

• 6

$\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}} = \overrightarrow{\mathrm{a}}\left(2\hat{\mathrm{i}} + \hat{\mathrm{j}}\right) = \overrightarrow{\mathrm{a}}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) = 1, \mathrm{then} \overrightarrow{\mathrm{a}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\hat{\mathrm{i}} - \hat{\mathrm{k}}$

• $\frac{1}{3}\left(3\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\frac{1}{3}\left(3\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

If A and B are two independent events such that

P(B) = $\frac{2}{7}, \mathrm{P}\left(\mathrm{A} \cup {\mathrm{B}}^{\mathrm{c}}\right)$= 0.8, then P(A) is equal to:

• 0.1

• 0.2

• 0.3

• 0.4

A number n is chosen at random from {1, 2, 3, 4, . . . , 1000}. The probability that n is a number that leaves remainder 1 when divided by 7, is :

• $\frac{71}{500}$

• $\frac{143}{1000}$

• $\frac{72}{500}$

• $\frac{71}{1000}$

In the random experiment of tossing two unbiased dice let E be the event of getting the sum 8 and F be the event of getting even numbers on both the dice. Then :

I. P(E) = $\frac{7}{36}$ II. p (F) = $\frac{1}{3}$

Which of the following is a correct statement ?

• Both I and II are true

• Neither I nor II is true

• I is true, II is false

• I is false, II is true