Karnataka 2nd PUC Maths Chapter 11 Important Questions 2027: Three-Dimensional Geometry

Three-Dimensional Geometry is an important chapter in Karnataka 2nd PUC Mathematics. It covers direction cosines, direction ratios, equations of lines, angles between lines, shortest distance, projections, and related concepts.

The Karnataka 2nd PUC Maths Chapter 11 Important Questions 2027 are useful for practising these concepts through objective questions. The Karnataka Government Mathematics CET material contains 52 multiple-choice questions from Three-Dimensional Geometry, along with questions from KCET and JEE Main.

The questions are classified as Easy, Average, and Difficult. 

Karnataka 2nd PUC Three-Dimensional Geometry Important Questions

The chapter material begins with direction cosines and direction ratios. It also covers lines in three-dimensional space, angles between lines, shortest distance, distance of a point from a line, projections, and skew lines.

The 2nd PUC Three-Dimensional Geometry Important Questions cover topics such as:

• Direction Cosines

• Direction Ratios

• Equations of Lines

• Angles Between Lines

• Perpendicular Lines

• Shortest Distance Between Lines

• Distance of a Point from a Line

• Projection of a Line Segment

• Area of a Quadrilateral

• Skew Lines

The PDF contains 52 MCQs covering these concepts.

3D Geometry 1 Mark Questions

Question 1: If the direction cosines of a line is k, k, k, then

(a) k > 0
(b) 0 < k < 1
(c) k = 1
(d) k = 1/√3 or −1/√3

Question 2: The direction cosines of a normal to the yz-plane are

(a) (0, 0, 1)
(b) (1, 0, 0)
(c) (0, 1, 0)
(d) (1, 1, 0)

Question 3: If (1/3, 1/4, k) are the direction cosines of a line, then the value of k is

(a) 119/12
(b) √119/12
(c) 119/114
(d) 11/12

3D Geometry 2 Mark Questions

Question 1: OX, OY are the positive x-axis and positive y-axis respectively where O = (0, 0, 0). The direction cosines of the line which bisects xOy are

(a) 1, 1, 0
(b) 1/√2, 1/√2, 0
(c) 1/√2, 0, 1/√2
(d) 0, 0, 1

Question 2: The direction ratios of a line are 1, −1, 0. Then the angle of inclination of the line with the y-axis is

(a) 3π/4
(b) −π/4
(c) π/3
(d) −π/6

Question 3: If a, b, c are direction ratios and l, m and n are the direction cosines of a line, such that

l/a = m/b = n/c = k,

then k is equal to

(a) ±1/√(a² + b² + c²)
(b) ±√(a² + b² + c²)
(c) ±(a² + b² + c²)
(d) ±1/(a² + b² + c²)

3D Geometry 3/5 Mark Questions

Question 1: The measure of the angle between the lines

x = k + 1, y = 2k − 1, z = 2k + 3, k ∈ R

and

(x − 1)/2 = (y − 2)/1 = (z − 3)/(−1)

is

(a) cos⁻¹(2/3√6)
(b) cos⁻¹(2/3)
(c) tan⁻¹√2
(d) cos⁻¹(1/2)

Question 2: The three points A(2, 4, 3), B(4, a, 9) and C(10, −1, 7) form a right-angled triangle with ∠B = 90°. Then the value of a is

(a) 1 or 4
(b) −1 or 4
(c) 1 or −4
(d) −1 or −4

Question 3: The shortest distance between the lines

r = −(i + j + k) + λ(2i + 3j + 4k)

and

r = −i + μ(3i + 4j + 5k)

is

(a) 1
(b) 1/√2
(c) 1/√3
(d) 1/√6

Direction Cosines and Ratios Questions

The PDF explains that if a line makes angles α, β and γ with the positive directions of the x, y and z axes, then cos α, cos β and cos γ are its direction cosines, denoted by l, m and n. It also gives the relation l² + m² + n² = 1.

Important Direction Cosines Questions from the PDF include:

Question 1: If the direction cosines of a line is k, k, k, then

(a) k > 0
(b) 0 < k < 1
(c) k = 1
(d) k = 1/√3 or −1/√3

Question 2: If (1/3, 1/4, k) are the direction cosines of a line, then the value of k is

(a) 119/12
(b) √119/12
(c) 119/114
(d) 11/12

Equation of a Line Important Questions

The PDF includes questions on lines represented through direction ratios and vector forms, along with questions involving perpendicular and intersecting lines.

Question: The value of P so that the lines

(1 − x)/3 = (7y − 14)/(2p) = (z − 3)/2

and

(7 − 7x)/(3p) = (y − 5)/1 = (6 − z)/5

intersect at right angle is

(a) 10/11
(b) 70/11
(c) 10/11
(d) 70/9

Another question asks:

Question: If the line joining (2, 3, −1) and (3, 5, −3) is perpendicular to the line joining (1, 2, 3) and (3, 5, λ), then λ =

(a) −3
(b) 2
(c) 5
(d) 7

3D Geometry Formula-Based Questions

The PDF provides formulas and relationships for direction cosines and direction ratios. It states that for direction cosines l, m, and n,

l² + m² + n² = 1

For direction ratios a, b and c, the source gives

l/a = m/b = n/c = k

and

k = ±1/√(a² + b² + c²).

The material also gives the relation for the projection of a line segment. If the projections of the line segment on the x, y and z axes are a, b and c, then

PQ = √(a² + b² + c²).

Questions based directly on these relationships are included throughout the MCQ set.2

Three-Dimensional Geometry Important Questions PDF

The Three-Dimensional Geometry Important Questions PDF contains 52 MCQs from the chapter along with concept explanations, formulas, difficulty levels and an answer key. The material is issued by the Karnataka Government Department of School Education (Pre-University) Mathematics II PUC CET material.

How to Prepare Karnataka 2nd PUC Maths Chapter 11 Important Questions 2027?

1. Revise direction cosines and direction ratios first: Start with their definitions and the relationships given in the chapter.

2. Practise equation-based questions: Work on questions involving lines, direction ratios and perpendicularity.

3. Revise angle-related concepts: Practise questions involving angles between lines and mutually perpendicular lines.

4. Practise shortest-distance questions: Solve the questions involving skew lines, parallel lines and distance from a point to a line.

5. Attempt formula-based MCQs: Use the formulas provided in the chapter while solving direct questions.

6. Solve previous-year questions: Practise the KCET and JEE Main questions included in the PDF after revising the related concepts.

Regular practice of Karnataka 2nd PUC Maths Chapter 11 Important Questions 2027 can help students become familiar with different types of Three-Dimensional Geometry problems.

Three-Dimensional Geometry covers direction cosines, direction ratios, equations of lines, angles, shortest distance, projections, and skew lines. The uploaded material provides 52 MCQs covering these concepts and includes questions from KCET and JEE Main.

Students can revise the concepts and formulas first and then practise the MCQs and previous-year questions included in the material. This approach provides focused practice for Karnataka 2nd PUC Maths Chapter 11 Important Questions 2027.

Related Resources

• Karnataka 2nd PUC Mid Term Time Table 2026 

• Karnataka 2nd PUC Mid Term Question Paper

• Karnataka 2nd PUC Previous Year Question Paper 

• Karnataka 2nd PUC Exam Pattern 2027

• Karnataka 2nd PUC Syllabus 2026-27