If a line in the space makes angle $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ with the coordinate axes, then

$\cos \left(2\mathrm{\alpha }\right) + \cos \left(2\mathrm{\beta }\right) + \cos \left(2\mathrm{\gamma }\right) + {\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right)$ equals

• - 1

• 0

• 1

• 2

Y-axis cuts the line joining the points (- 3, - 4) and (1, - 2) in the ratio

• 1 : 3

• 2 : 3

• 3 : 1

• 3 : 2

If the points (1, 1), (- 1, - 1), $\left(- \sqrt{3}, \sqrt{3}\right)$ are the vertices of a triangle, then this triangle is

• a right-angled triangle

• an isosceles triangle

• an equilateral triangle

• None of the above

The point of intersection of the lines $\frac{\mathrm{x} - 5}{3} = \frac{\mathrm{y} - 7}{- 1} = \frac{\mathrm{z} + 2}{1}$ and $\frac{\mathrm{x} + 3}{- 36} = \frac{\mathrm{y} - 3}{2} = \frac{\mathrm{z} - 6}{4}$ is

• (2, 10, 4)

• (21, 5/3, 10/3)

• (5, 7, - 2)

• (- 3, 3, 6)

The distance between the point (1, - 5, - 10) and the point of intersection of the plane x - y + z = 5 and the line $\frac{\mathrm{x} - 2}{3} = \frac{\mathrm{y} + 1}{4} = \frac{\mathrm{z} - 2}{12}$, will be

• 10

• 11

• 12

• 13

A point is in motion along the curve 12y = x³. The x-coordinate, changes faster than y-coordinate, if

• $\mathrm{x} \in \left[- 2, 2\right]$

• $\mathrm{x} \in \left[- \infty , - 2\right] \cup \left[2, \infty \right]$

• $\mathrm{x} \in \left[- 2, \infty \right]$

• $\mathrm{x} \in \left[- \infty , 2\right] \cup \left[2, \infty \right]$

47.

The ratio in which yz-plane divides the line joining (2, 4, 5) and (3, 5, 7) is

• - 2 : 3

• 2 : 3

• 3 : 2

• - 3 : 2

The y - coordinate of a point P on the line joining A(7, 2,1) and B(10, 5, 7) is 4. Then, x and z - coordinates of the point are

• x = 9, z = 5

• x = 3, z = 7

• x = 2, z = 3

• None of these

If point D divides the base BC of a $∆$ABC in the ratio n : m, then the value of mBD² + nCD² + (m + n)AD² is

• mAC² + nAB²

• (m + n)(AC² + AB²)

• nAC² + mAB²

• None of these

The graph of the equation y² + z² = 0 in three dimensional space is

• YZ-plane

• Z-axis

• Y-axis

• X-axis