List of top Mathematics Questions on Lines

The line \( x = 1 + 5\mu, y = -5 + \mu, z = -6 - 3\mu \) passes through which of the following points?

Find a unit vector in the direction of \( \overrightarrow{DA} \).

If the lines \[ \frac{x - 1}{-3} = \frac{y - 2}{2k} = \frac{z - 3}{2} \] and \[ \frac{x - 1}{3k} = \frac{y - 1}{1} = \frac{z - 6}{-7} \] are perpendicular to each other, find the value of \( k \) and hence write the vector equation of a line perpendicular to these two lines passing through the point \( (3, -4, 7) \).

Find the distance between the line \[ \frac{x}{2} = \frac{2}{y - 6} = \frac{1}{-z - 1} \] and another line parallel to it passing through the point \( (4, 0, -5) \).

The lines \[ \frac{1 - x}{2} = \frac{y - 1}{3} = \frac{z}{1} \] and \[ \frac{2x - 3}{2p} = \frac{y - 1}{1} = \frac{z - 4}{7} \] are perpendicular to each other for \( p \) equal to:

Two vertices of the parallelogram ABCD are given as \( A(-1, 2, 1) \) and \( B(1, -2, 5) \). If the equation of the line passing through \( C \) and \( D \) is: \[ \frac{x - 4}{1} = \frac{y + 7}{-2} = \frac{z - 8}{2}, \] then find the distance between sides \( AB \) and \( CD \). Hence, find the area of parallelogram ABCD.

Find the equation of the line passing through the point of intersection of the lines \[ \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \] and \[ \frac{x - 1}{0} = \frac{y}{-3} = \frac{z - 7}{2}, \] and perpendicular to these given lines.

If a line makes an angle of \( \frac{\pi}{4} \) with the positive directions of both \( x \)-axis and \( z \)-axis, then the angle which it makes with the positive direction of \( y \)-axis is:

Self-study helps students to build confidence in learning. It boosts the self-esteem of the learners. Recent surveys suggested that close to 50% learners were self-taught using internet resources and upskilled themselves.

A student may spend 1 hour to 6 hours in a day in upskilling self. The probability distribution of the number of hours spent by a student is given below: \[ P(X = x) = \begin{cases} kx^2, & \text{for } x = 1, 2, 3, \\ 2kx, & \text{for } x = 4, 5, 6, \\ 0, & \text{otherwise.} \end{cases} \] where \( x \) denotes the number of hours. Based on the above information, answer the following questions:
1. Express the probability distribution given above in the form of a probability distribution table.
2. Find the value of \( k \).
3. (a) Find the mean number of hours spent by the student. (b) Find \( P(1 < X < 6) \).

Find the equation of the line passing through the point of intersection of the lines: \[ \frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 2}{3}, \quad \frac{x - 1}{0} = \frac{y - 3}{-3} = \frac{z - 7}{2}, \] and perpendicular to these given lines.