For n = 4, using trapezoidal rule, the value of ${\int }_0^2\frac{\mathrm{dx}}{1 + \mathrm{x}}$ will be

• 1.116625

• 1.1176

• 1.1180

• None of these

The value of ${\int }_0^5\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}$ by chosing six sub-intervals and by using Simpson's rule will be

• 1.3562

• 1.3662

• 1.3456

• 1.2662

If G and G' are respectively centroid of $∆$ABC and $∆$A' B' C', then AA' + BB' + CC' is equal to

• 2GG'

• 3GG'

• $\frac{2}{3}\mathrm{GG}\text{'}$

• $\frac{1}{3}\mathrm{GG}\text{'}$

If a = $3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}}$, b = $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and c = $- 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 5\hat{\mathrm{k}}$, and if [·] is the least integer function, then [a + b + c] is equal to

• 1

• 2

• 3

• 0

If a = $- \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and b = $2\hat{\mathrm{i}} + \hat{\mathrm{k}}$, then the vector satisfyin the following conditions

(i) it is coplanar witha and b,

(ii) it is perpendicular to b and

(iii) a · c = 7, is

• $- \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $- \frac{3}{2}\hat{\mathrm{i}} + \frac{5}{2}\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$

• $- 3\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$

• $- 6\hat{\mathrm{i}} + \hat{\mathrm{k}}$

If the vectors $\mathrm{b} = \left(\tan \left(\mathrm{\alpha }\right), - 1, 2\sqrt{\sin \left(\frac{\mathrm{\alpha }}{2}\right)}\right)$ and $\mathrm{c} = \left(\tan \left(\mathrm{\alpha }\right), \tan \left(\mathrm{\alpha }\right), - \frac{3}{\sqrt{\sin \left({\displaystyle \raisebox{1ex}{$\mathrm{\alpha }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}}\right)$ are orthagonal and  a vector a = $\left(1, 3, \sin \left(2\mathrm{\alpha }\right)\right)$ makes an obtuse angle with the Z-axis, then the value of $\mathrm{\alpha }$ is

• $\left(4\mathrm{n} + 2\right)\mathrm{\pi } + {\tan }^{-1}\left(2\right)$

• $\left(4\mathrm{n} + 2\right)\mathrm{\pi } - {\tan }^{-1}\left(2\right)$

• $\left(4\mathrm{n} + 1\right)\mathrm{\pi } + {\tan }^{-1}\left(2\right)$

• $\left(4\mathrm{n} + 1\right)\mathrm{\pi } - {\tan }^{-1}\left(2\right)$

If $\int \frac{\cos \left(4\mathrm{x}\right) + 1}{\cot \left(\mathrm{x}\right) - \tan \left(\mathrm{x}\right)}\mathrm{dx} = \mathrm{Acos}\left(4\mathrm{x}\right) + \mathrm{B}$, then the value of A is

• $\frac{1}{2}$

• $\frac{1}{8}$

• $- \frac{1}{8}$

• $\frac{1}{4}$

48.

If f(x) = $\mathrm{f}\left(\mathrm{x}\right) = \sqrt{\mathrm{x}}, \mathrm{g}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}} - 1$ and $\int \mathrm{fog}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{Afog}\left(\mathrm{x}\right) + {\mathrm{Btan}}^{-1}\left(\mathrm{fog}\left[\mathrm{x}\right]\right) + \mathrm{C}$, then the value of A + B is

• 1

• 2

• 3

• None of these

The value of ${\int }_0^1{\tan }^{-1}\left(\frac{2\mathrm{x} - 1}{1 + \mathrm{x} - {\mathrm{x}}^2}\right)\mathrm{dx}$ is

• 0

• 1

• - 1

• None of these

If 0 < P(A) < 1, 0 < P(B) < 1 and P(A $\cup$ B) = P(A) + P(B) - P(A)P(B),then

• $\mathrm{P}{\left(\mathrm{A} \cup \mathrm{B}\right)}^{\mathrm{C}} = \mathrm{P}\left({\mathrm{A}}^{\mathrm{C}}\right)\mathrm{P}\left({\mathrm{B}}^{\mathrm{C}}\right)$

• P(A/B) = P(A)

• Both (a) and (b) are  true

• None of the above