Try each question on your own first. Use the Show Answer button only after you have selected your option, then read the explanation and revise the related concept from your chapter notes.
1. The distance between two points (x1, y1, z1) and (x2, y2, z2) in three-dimensional space is given by:
A. √[(x2-x1)² + (y2-y1)² + (z2-z1)²]
B. (x2-x1) + (y2-y1) + (z2-z1)
C. x2y2z2 - x1y1z1
D. The formula is undefined in 3D
Answer: A. √[(x2-x1)² + (y2-y1)² + (z2-z1)²]
Explanation: This is the direct three-dimensional extension of the standard distance formula.
2. Direction cosines of a line are the cosines of the angles the line makes with the:
A. Coordinate axes
B. Coordinate planes
C. Origin only
D. Another given line only
Answer: A. Coordinate axes
Explanation: Direction cosines l, m, n represent the cosines of the angles a line makes with the x, y, and z axes respectively.
3. For the direction cosines l, m, n of a line, the relation that always holds true is:
A. l² + m² + n² = 1
B. l + m + n = 1
C. l = m = n always
D. lmn = 1
Answer: A. l² + m² + n² = 1
Explanation: Since direction cosines come from a unit vector along the line, the sum of their squares must equal 1.
4. The equation of a line in three dimensions, passing through a given point with known direction ratios, is expressed in:
A. Symmetric (Cartesian) or vector form
B. Slope-intercept form only
C. Matrix form only
D. Quadratic form
Answer: A. Symmetric (Cartesian) or vector form
Explanation: In 3D, lines are typically represented either in vector form or in the symmetric Cartesian form using direction ratios.
5. Two lines in space are perpendicular if the sum of the products of their corresponding direction ratios is:
A. Zero
B. One
C. Equal to their magnitudes
D. Undefined
Answer: A. Zero
Explanation: This condition is analogous to the dot product of the two direction vectors being zero, indicating a 90° angle between them.
6. The general equation of a plane in three-dimensional space is of the form:
A. Ax + By + Cz + D = 0
B. Ax + By = C only
C. x + y + z = 0 always
D. Ax² + By² + Cz² = 0
Answer: A. Ax + By + Cz + D = 0
Explanation: This is the standard Cartesian form of a plane's equation, where A, B, C are the direction ratios of its normal.
7. The angle between two planes is found using the angle between their:
A. Normal vectors
B. Direction ratios of associated lines only
C. Position vectors of a chosen point
D. Intercepts on the axes
Answer: A. Normal vectors
Explanation: Since each plane is characterized by its normal vector, the angle between two planes equals the angle between their normals.
8. The perpendicular distance of a point from a plane is calculated using a formula that primarily involves the plane's:
A. Normal vector
B. Distance between two points formula only
C. Direction cosines alone, without the plane's equation
D. Angle formula only
Answer: A. Normal vector
Explanation: The standard distance formula uses the coefficients of the plane's equation (which form its normal vector) along with the given point's coordinates.
9. Two lines in space that neither intersect nor are parallel to each other are called:
A. Skew lines
B. Coincident lines
C. Perpendicular lines
D. Parallel lines
Answer: A. Skew lines
Explanation: Skew lines exist only in three (or more) dimensions and do not share a common plane.
10. The shortest distance between two skew lines is measured along their:
A. Common perpendicular
B. Any arbitrary connecting line
C. The x-axis
D. Their (non-existent) point of intersection
Answer: A. Common perpendicular
Explanation: The shortest distance between skew lines is always along the unique line segment that is perpendicular to both.
11. The equation of a plane passing through three non-collinear points can be found using a condition based on a:
A. Determinant (scalar triple product) involving the points
B. Simple addition of the coordinates
C. Direction cosines alone
D. Distance formula only
Answer: A. Determinant (scalar triple product) involving the points
Explanation: Setting up a determinant using the coordinates of the three points and a general point (x, y, z) gives the equation of the plane.
12. If a line is parallel to a plane, then the line's direction vector and the plane's normal vector are:
A. Perpendicular to each other, so their dot product is zero
B. Parallel to each other
C. Equal in magnitude
D. Related in an undefined way
Answer: A. Perpendicular to each other, so their dot product is zero
Explanation: A line lying parallel to a plane cannot have any component along the plane's normal direction, making the dot product zero.
This page contains a chapter-wise objective practice set. Try the questions without checking the answer first, then use the explanation to understand the point you missed. The current page contains 12 questions, so you can practise in smaller sets when you are revising.
Quick revision tip
After the practice set, write down the question numbers you missed and revise only those concepts once more. This makes MCQ practice more useful than repeatedly attempting the same questions without reviewing mistakes.
