Three Dimensional Geometry Practice Questions

Class 12 · Mathematics · 1445 free MCQs with instant results and detailed explanations.

1445

Total

573

Easy

634

Medium

238

Hard

Start Practicing Three Dimensional Geometry

Take a timed quiz or customize your practice session

Quick Quiz (10 Qs) → Mock Test (25 Qs) ⚙ Customize

Topics in Three Dimensional Geometry

Direction Cosines 516

Distance between Lines 216

Equation of a Plane 355

Equation of a Line 358

Sample Questions from Three Dimensional Geometry

Here are 10 sample questions. Start a quiz to get randomized questions with scoring.

Easy

What is the sum of the squares of the direction cosines of a line?

A. 1

B. 0

C. 2

D. -1

Show Answer & Explanation

Correct Answer: A

The direction cosines must satisfy the equation l² + m² + n² = 1.

Easy

If the direction cosines of a line are l = 1/√3, m = 1/√3, and n = 1/√3, what is the angle made by the line with the coordinate axes?

A. 45°

B. 60°

C. 30°

D. 90°

Show Answer & Explanation

Correct Answer: B

The cosine of the angle with the axes is equal to the direction cosine. Here, cos(60°) = 1/2 fits.

Easy

Which of the following represents the direction cosines of a line parallel to the x-axis?

A. (1, 0, 0)

B. (0, 1, 0)

C. (0, 0, 1)

D. (1, 1, 1)

Show Answer & Explanation

Correct Answer: A

Direction cosines for a line parallel to the x-axis are (1, 0, 0).

Medium

Which of the following sets of direction cosines can represent a line?

A. 1, 1, 1

B. 2, 2, 2

C. 1, -1, 1

D. 0, 0, 0

Show Answer & Explanation

Correct Answer: C

Direction cosines must satisfy the equation l² + m² + n² = 1. For option C, 1² + (-1)² + 1² = 3, which does not satisfy the equation. The only valid set of cosines is 1, -1, 1 (when normalized).

Medium

If a line has direction ratios (3, -2, 1), what are its direction cosines?

A. √14/7, -√14/7, √14/21

B. 3/√14, -2/√14, 1/√14

C. 1/√14, -2/√14, 3/√14

D. None of the above

Show Answer & Explanation

Correct Answer: B

To find the direction cosines, normalize the direction ratios. The magnitude is √(3² + (-2)² + 1²) = √14. Thus, direction cosines are (3/√14, -2/√14, 1/√14).

Medium

For the line represented by direction cosines (l, m, n), which of the following conditions is necessary?

A. l + m + n = 1

B. l² + m² + n² = 1

C. l² + m² = n²

D. l + m = n

Show Answer & Explanation

Correct Answer: B

The necessary condition for direction cosines l, m, n is that they must satisfy the equation l² + m² + n² = 1, ensuring they are unit vectors.

Medium

If two lines have direction cosines (1, 0, 0) and (0, 1, 0), what is the angle between them?

A. 90 degrees

B. 45 degrees

C. 60 degrees

D. 30 degrees

Show Answer & Explanation

Correct Answer: A

The direction cosines indicate that the lines are perpendicular. The angle θ between two lines with direction cosines is given by cos θ = l1*l2 + m1*m2 + n1*n2. Here, it equals zero, indicating an angle of 90 degrees.

Hard

If the direction cosines of a line are l = 1/√3, m = 1/√3, n = 1/√3, what is the angle between the line and the coordinate axes?

A. 60°

B. 45°

C. 30°

D. 90°

Show Answer & Explanation

Correct Answer: A

The angles with the axes are given by cos⁻¹(l), cos⁻¹(m), and cos⁻¹(n), all resulting in 60°.

Hard

If the direction cosines of a line are l = 0, m = 0, n = 1, what does this signify about the line?

A. Line parallel to y-axis

B. Line parallel to x-axis

C. Line parallel to z-axis

D. Line at an angle to all axes

Show Answer & Explanation

Correct Answer: C

A line with direction cosines 0, 0, 1 is parallel to the z-axis, indicating it moves vertically upwards.

Q10

Hard

What is the relationship between the direction cosines of two lines if they are perpendicular to each other?

A. l1*l2 + m1*m2 + n1*n2 = 0

B. l1 + l2 = 0

C. m1*m2 = 0

D. n1 + n2 = 0

Show Answer & Explanation

Correct Answer: A

For two lines to be perpendicular, the dot product of their direction cosines must equal zero.

Showing 10 of 1445 questions. Start a quiz to practice all questions with scoring and timer.

Practice All 1445 Questions →

More Mathematics Chapters

1 Relations and Functions

1660 Qs →

2 Inverse Trigonometric Functions

1949 Qs →

3 Matrices

2241 Qs →

4 Determinants

1944 Qs →

5 Continuity and Differentiability

1849 Qs →

6 Application of Derivatives

1543 Qs →

7 Integrals

2245 Qs →

8 Application of Integrals

1878 Qs →

Other Class 12 Subjects

🧬 Biology → 🧪 Chemistry → ⚡ Physics →

Three Dimensional Geometry — Class 12 Mathematics Practice Questions Online

This page contains 1445 practice MCQs for the chapter Three Dimensional Geometry in Class 12 Mathematics. The questions are organized by difficulty — 573 easy, 634 medium, 238 hard — so you can choose the right level for your preparation.

Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit. This chapter covers 4 topics, giving you comprehensive coverage of the entire chapter.