Three Dimensional Geometry · Mathematics · NDA

If (1, −1, 2) and (2, 1, −1) are the end points of a diameter of a sphere $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz − 1 = 0$, then what is $u + v + w$ equal to?

If $\langle l, m, n \rangle$ are the direction cosines of a normal to the plane $2x − 3y + 6z + 4 = 0$, then what is the value of $49(7l^2 + m^2 − n^2)$?

A line through $(1, −1, 2)$ with direction ratios $\langle 3, 2, 2 \rangle$ meets the plane $x + 2y + 3z = 18$. What is the point of intersection of line and plane?

If $p$ is the perpendicular distance from origin to the plane passing through $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$, then what is $3p^2$ equal to?

Consider the points A(2, 4, 6), B(−2, −4, −2), C(4, 6, 4), and D(8, 14, 12). Which of the following statements is/are correct?

1. The points are the vertices of a rectangle ABCD.

2. The mid-point of A C is the same as that of BD.

Select the correct answer using the code given below :

Consider the equation of a sphere x2 + y2 + z2 − 4x − 6y − 8z − 16 = 0.

Which of the following statements is/are correct ?

1. z-axis is tangent to the sphere.

2. The centre of the sphere lies on the plane x + y + z − 9 = 0.

Select the correct answer using the code given below :

Consider the following statements :

1. The direction ratios of y-axis can be <0, 4, 0>

2. The direction ratios of a line perpendicular to z-axis can be <5, 6, 0>

Which of the statements given above is/are correct?