Let f'(x), be differentiable $\forall$a. If f(1) = - 2 and f'(x) $\ge$ 2 $\forall$ x $\in$ [1, 6], then

• f(6) < 8

• f(6) $\ge$ 8

• f(6) $\ge$ 5

• f(6) $\le$ 5

The minimum value of $\frac{\mathrm{x}}{\log \left(\mathrm{x}\right)}$ is

• e

• $\frac{1}{\mathrm{e}}$

• e²

• e³

23.

If the points (1, 2, 3) and (2, - 1, 0) lie on the opposite sides of the plane 2x + 3y - 2z = k, then

• k < 1

• k > 2

• k < 1 or k > 2

• 1 < k < 2

If $∆\mathrm{x} = \left|\begin{array}{ccc}1& \cos \left(\mathrm{x}\right)& 1 - \cos \left(\mathrm{x}\right)\\ 1 + \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& 1 + \sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\\ \sin \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)& 1\end{array}\right|$, then ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}∆\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{1}{4}$

• $\frac{1}{2}$

• 0

• $- \frac{1}{4}$

The triangle formed by the tangent to the curve f (x) = x² + bx - b at the point (1, 1) and the coordinate axes lies in the first quadrant. If its area is 2, then the value of b is

• - 1

• 3

• - 3

• 1

If a plane meets the coordinate axes at A, B and C such that the centroid of the triangle is (1, 2, 4), then the equation of the plane is

• x + 2y + 4z = 12

• 4x + 2y + z = 12

• x + 2y + 4z = 3

• 4x + 2y + z = 3

The volume of the tetrahedron included between the plane 3x + 4y - 5z - 60 = 0 and the coordinate planes is

• 60

• 600

• 720

• 400

${\int }_0^{2\mathrm{\pi }}\left(\sin \left(\mathrm{x}\right) + \left|\sin \left(\mathrm{x}\right)\right|\right)\mathrm{dx}$ is equal to

• 0

• 4

• 8

• 1

The value of ${\int }_0^{\sqrt{2}}\left[{\mathrm{x}}^2\right]\mathrm{dx}$, where [.] is the greatest integer function, is

• 2 - $\sqrt{2}$

• 2 + $\sqrt{2}$

• $\sqrt{2}$ - 1

• $\sqrt{2}$ - 2

If l (m,n) = ${\int }_0^1{\mathrm{t}}^{\mathrm{m}}{\left(1 + \mathrm{t}\right)}^{\mathrm{n}}\mathrm{dt}$, then the expression for l (m, n) in terms of l (m + 1, n + 1) is

• $\frac{2^{\mathrm{n}}}{\mathrm{m} + 1} - \frac{\mathrm{n}}{\mathrm{m} + 1} . \mathrm{l} \left(\mathrm{m} + 1, \mathrm{n} - 1\right)$

• $\frac{\mathrm{n}}{\mathrm{m} + 1} . \mathrm{l} \left(\mathrm{m} + 1, \mathrm{n} - 1\right)$

• $\frac{2\mathrm{n}}{\mathrm{m} + 1} + \frac{\mathrm{n}}{\mathrm{m} + 1} . \mathrm{l} \left(\mathrm{m} + 1, \mathrm{n} - 1\right)$

• $\frac{\mathrm{m}}{\mathrm{n} + 1} . \mathrm{l} \left(\mathrm{m} + 1, \mathrm{n} - 1\right)$