If a line segment OP makes angles of $\frac{\mathrm{\pi }}{4} \mathrm{and} \frac{{\displaystyle \mathrm{\pi }}}{{\displaystyle 3}}$ with X-axis and Y-axis, respectively. Then, the direction cosines are

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

• $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{\sqrt{2}}$

• $1, \sqrt{3}, 1$

• $1, \frac{1}{\sqrt{3}}, 1$

If a plane passing through the point (2, 2, 1) and is perpendicular to the planes 3x + 2y + 4z + 1 = 0 and 2x + y + 3z + 2 = 0. Then, the equation of the plane is

• 2x - y - z - 1 = 0

• 2x + 3y + z - 1 = 0

• 2x + y + z + 3 = 0

• x - y + z - 1 = 0

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

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

The length of longer diagonal of the parallelogram constructed on 5a + 2b and a - 3b, if it is given that $\left|\mathrm{a}\right| = 2\sqrt{2}, \left|\mathrm{b}\right| = 3$and the angle between a and b is $\frac{\mathrm{\pi }}{4}$, is

• 15

• $\sqrt{113}$

• $\sqrt{593}$

• $\sqrt{369}$

If the gradient of the tangent at any point (x, y) of acurve which passes through the point $\left(1, \frac{\mathrm{\pi }}{4}\right)$ is $\left\{\frac{\mathrm{y}}{\mathrm{x}} - {\sin }^2\left(\frac{{\displaystyle \mathrm{y}}}{{\displaystyle \mathrm{x}}}\right)\right\}$, then the equation of the curve is

• $\mathrm{y} = {\cot }^{-1}\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$

• $\mathrm{y} = {\cot }^{-1}\left({\log }_{\mathrm{e}}\left(\frac{\mathrm{x}}{\mathrm{e}}\right)\right)$

• $\mathrm{y} = {\mathrm{xcot}}^{-1}\left({\log }_{\mathrm{e}}\left(\mathrm{ex}\right)\right)$

• $\mathrm{y} = {\cot }^{-1}\left({\log }_{\mathrm{e}}\left(\frac{\mathrm{e}}{\mathrm{x}}\right)\right)$

If a plane meets the coordinate axes at A, B and C in such a way that the centroid of $∆$ABC is at the point (1, 2, 3), then equation of the plane is

• $\frac{\mathrm{x}}{1} + \frac{\mathrm{y}}{2} + \frac{\mathrm{z}}{3} = 1$

• $\frac{\mathrm{x}}{3} + \frac{\mathrm{y}}{6} + \frac{\mathrm{z}}{9} = 1$

• $\frac{\mathrm{x}}{1} + \frac{\mathrm{y}}{2} + \frac{\mathrm{z}}{3} = \frac{1}{3}$

• None of these

If 3p and 4p are resultant of force 5p,then angle between 3p and 5p is

• ${\sin }^{-1}\left(\frac{3}{5}\right)$

• ${\sin }^{-1}\left(\frac{4}{5}\right)$

• $90°$

• None of these