31.

The value of $\sqrt{12}$. upto three places of decimals using the method of Newton-Raphson, will be

• 3.463

• 3.462

• 3.467

• None of these

If the total cost C(x) in rupees associated with the production of x units of an item is given by C (x) = 3x³ - 2x² + x + 100. Then, the marginal change in cost, when x = 5, is

• 200

• 225

• 206

• 226

The area of the parallelogram having the diagonals 3i + j - 2k and i - 3j + 4k is

• 5 sq units

• $10\sqrt{3} \mathrm{sq} \mathrm{units}$

• $5\sqrt{3} \mathrm{sq} \mathrm{units}$

• 10 sq units

The value of $\left(\mathrm{i} + \mathrm{j}\right) . \left[\left(\mathrm{j} + \mathrm{k}\right) \times \left(\mathrm{k} + \mathrm{i}\right)\right]$ is

• 0

• 1

• - 1

• 2

The solution of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y} - \mathrm{xtan}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)$ is

• $\mathrm{xsin}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \mathrm{C} = 0$

• $\mathrm{xsin}\left(\mathrm{y}\right) + \mathrm{C} = 0$

• $\mathrm{xsin}\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{C}$

• None of these

If $\hat{\mathrm{a}} \mathrm{and} \hat{\mathrm{b}}$ are unit vectors and 0 is the angle between them, then $\sin \left(\raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$ is equal to

• $\frac{\left|\hat{\mathrm{a}} + \hat{\mathrm{b}}\right|}{2}$

• $\frac{\hat{\mathrm{a}} - \hat{\mathrm{b}}}{2}$

• $\frac{\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|}{2}$

• $\left|\hat{\mathrm{a}} - \hat{\mathrm{b}}\right|$

The particular solution of $\cos \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \mathrm{a}\left(\mathrm{where}, \mathrm{a} \in \mathrm{R}\right)$, (y = 2 when x = 0), is

• $\cos \left(\frac{\mathrm{y} - 2}{\mathrm{x}}\right) = \mathrm{a}$

• $\sin \left(\frac{\mathrm{y} - 2}{\mathrm{x}}\right) = \mathrm{a}$

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

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

The ratio in which yz-plane divide the line joining the points A(3, 1, - 5) and B(1, 4, - 6) is

• - 3 : 1

• 3 : 1

• - 1 : 3

• 1 : 3

If ${}^{\mathrm{n}}\mathrm{P}_5 = 20{}^{\mathrm{n}}\mathrm{P}_3$, then the value of n is

• 7

• 5

• 8

• 9

The order and degree of the differential equation $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = {\left\{\mathrm{y} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ are given by

• 4 and 2

• 1 and 2

• 1 and 4

• 2 and 4