If A = $\left[\begin{array}{ccc}3& 2& 4\\ 1& 2& 1\\ 3& 2& 6\end{array}\right]$ and A_ij are the cofactors of a_ij then a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃ is equal to

• 8

• 6

• 4

• 0

A = $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$ and AB = BA = I, then B is equal to

• $\left[\begin{array}{cc}- \cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}- \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\\ \cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\sin \left(\mathrm{\theta }\right)& - \cos \left(\mathrm{\theta }\right)\\ - \cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\end{array}\right]$

The equation of motion of a particle moving along a straight line is s = 2t³ - 9t² + 12t, where the units of s and t are centimetre and second. The acceleration of the particle will be zero after

• $\frac{3}{2}\mathrm{s}$

• $\frac{2}{3}\mathrm{s}$

• $\frac{1}{2}\mathrm{s}$

• 1 s

The equation of the tangent to the curve y = 4xe^x at $\left(- 1, \frac{- 4}{\mathrm{e}}\right)$

• y = - 1

• $\mathrm{y} = - \frac{4}{\mathrm{e}}$

• x = - 1

• $\mathrm{x} = \frac{- 4}{\mathrm{e}}$

Given $\overrightarrow{\mathrm{p}} = 3\hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$, $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}}$, $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{p}} = \mathrm{x}\overrightarrow{\mathrm{a}} + \mathrm{y}\overrightarrow{\mathrm{b}} + \mathrm{z}\overrightarrow{\mathrm{c}}$ then x, y, z are respectively

• $\frac{3}{2}, \frac{1}{2}, \frac{5}{2}$

• $\frac{1}{2}, \frac{3}{2}, \frac{5}{2}$

• $\frac{5}{2}, \frac{3}{2}, \frac{1}{2}$

• $\frac{1}{2}, \frac{5}{2}, \frac{3}{2}$

A point on XOZ - plane divides the join of (5, - 3, - 2) and (1, 2, - 2) at

• $\left(\frac{13}{5}, 0, - 2\right)$

• $\left(\frac{13}{5}, 0, 2\right)$

• (5, 0, 2)

• (5, 0, - 2)

If the line $\overrightarrow{\mathrm{OR}}$ makes angles ${\mathrm{\theta }}_1, {\mathrm{\theta }}_2, {\mathrm{\theta }}_3$, with the planes XOY, YOZ, ZOX respectively, then ${\cos }^2\left({\mathrm{\theta }}_1\right) + {\cos }^2\left({\mathrm{\theta }}_2\right) + {\cos }^2\left({\mathrm{\theta }}_3\right)$, is equal to

• 1

• 2

• 3

• 4

Joint equation of pair of lines through (3, - 2) and parallel to x² - 4xy + 3y² = 0 is

• x² + 3y² - 4xy - 14x + 24y + 45 = 0

• x² + 3y² + 4xy - 14x + 24y + 45 = 0

• x² + 3y² + 4xy - 14x + 24y - 45 = 0

• x² + 3y² + 4xy - 14x - 24y - 45 = 0

Area bounded between the curve x² = y and the line y = 4x is

• $\frac{32}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{1}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{8}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{16}{3} \mathrm{sq} \mathrm{unit}$

   

Which of the following is true ?

• ${\int }_0^1{\mathrm{e}}^{\mathrm{x}}\mathrm{dx} = \mathrm{e}$

• ${\int }_0^1{2}^{\mathrm{x}}\mathrm{dx} = \log \left(2\right)$

• ${\int }_0^1\sqrt{\mathrm{x}}\mathrm{dx} = \frac{2}{3}$

• ${\int }_0^1\mathrm{xdx} = \frac{1}{3}$