Are these practice questions for Three Dimensional Geometry based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Three Dimensional Geometry free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Three Dimensional Geometry mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Three Dimensional Geometry for full coverage.

How many practice questions are there for Three Dimensional Geometry?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Three Dimensional Geometry Class 12 Mathematics Practice Questions with Answers

easy

Describe the condition for two lines with direction ratios

a_1, b_1, c_1

and

a_2, b_2, c_2

to be (i) perpendicular and (ii) parallel.

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easy

Define skew lines.

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easy

State the vector and Cartesian equations for a line passing through two given points. Identify each component in the equations.

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easy

Identify the direction ratios of the line given by the Cartesian equation

\frac{x-5}{2} = \frac{y+3}{-4} = \frac{z}{6}

.

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easy

Determine the vector equation of a line that passes through the point

(5, -2, 4)

and is parallel to the vector

2\hat{i} - \hat{j} + 3\hat{k}

.

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easy

Formulate a condition, using the dot product, to critique whether two lines, given in vector form

\vec{r} = \vec{a}_1 + \lambda \vec{b}_1

and

\vec{r} = \vec{a}_2 + \mu \vec{b}_2

, are perpendicular. Justify your formulation.

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easy

Justify why the direction cosines of a line uniquely determine its direction in space, whereas direction ratios do not.

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easy

Explain the difference between direction cosines and direction ratios of a line.

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easy

A line has direction ratios

6, 2, -3

. Calculate its direction cosines.

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easy

State the relationship that exists between the direction cosines (

l, m, n

) of any line.

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easy

Calculate the vector and Cartesian equations for the line passing through the points

A(-1, 0, 2)

and

B(3, 4, 6)

.

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easy

Find the Cartesian form of the equation of the line given by the vector equation

\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 2\hat{j} + 2\hat{k})

.

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easy

Define direction cosines of a directed line in space.

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medium

Create the vector equation of a line that passes through the origin and is equally inclined to the positive coordinate axes.

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medium

Calculate the shortest distance between the skew lines

l_1: \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}

and

l_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}

.

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medium

If the direction ratios of a line are

2, -3, 6

, explain the process to find its direction cosines.

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medium

Evaluate the statement: "The shortest distance between any two lines in space is always found along the line segment that is perpendicular to both lines." Is this statement universally true? Justify your answer.

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medium

Show that the line passing through the points

(0, 3, 2)

and

(3, 5, 6)

is perpendicular to the line passing through the points

(1, -1, 2)

and

(3, 4, -2)

.

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medium

List the direction cosines of a line that is equally inclined to the positive coordinate axes.

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medium

Examine if the points

A(1, 2, 3)

,

B(4, 0, 4)

, and

C(-5, 6, 1)

are collinear.

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medium

Write the vector equation of the z-axis.

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medium

Summarize the derivation of the formula for the direction cosines of a line passing through two points

\mathrm{P}(x_1, y_1, z_1)

and

\mathrm{Q}(x_2, y_2, z_2)

.

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medium

Explain how to derive the Cartesian form of the equation of a line from its vector form

\vec{r} = \vec{a} + \lambda \vec{b}

. Let

\vec{a} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}

and

\vec{b} = a\hat{i} + b\hat{j} + c\hat{k}

.

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medium

Evaluate whether the line joining points A(1, 2, 3) and B(-1, -2, -3) is perpendicular to the line joining points C(4, 1, 5) and D(3, 5, 2). Justify your conclusion.

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medium

Describe the concept of the angle between two lines in space. State the formula to find this angle in terms of (i) their direction cosines and (ii) their direction ratios.

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medium

Propose a method to determine if three points A, B, and C are collinear using the concept of direction ratios.

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medium

Justify that the points A(1, -1, 3), B(2, -4, 5), and C(5, -13, 11) are collinear. Further, formulate the ratio in which point B divides the line segment AC.

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medium

If a line makes angles of

90^\circ

and

60^\circ

with the positive x and y-axes respectively, find the angle it makes with the positive z-axis.

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medium

State the formula for the shortest distance between two skew lines

\vec{r} = \vec{a}_1 + \lambda \vec{b}_1

and

\vec{r} = \vec{a}_2 + \mu \vec{b}_2

.

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medium

Find the value of

p

so that the lines

\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{p}

and

\frac{x-4}{2} = \frac{y-5}{1} = \frac{z-6}{-3}

are at right angles.

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medium

Derive the formula for the shortest distance between two parallel lines

\vec{r} = \vec{a}_1 + \lambda \vec{b}

and

\vec{r} = \vec{a}_2 + \mu \vec{b}

. Justify each step of your derivation using vector concepts.

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medium

A line L passes through the point P(1, 2, 3) and is perpendicular to two other lines L1 and L2 with direction ratios

<1, -2, 3>

and

<2, 1, -1>

respectively. Formulate the Cartesian equation of the line L.

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medium

Calculate the angle between the pair of lines given by the equations:

L_1: \frac{x-2}{2} = \frac{y-1}{5} = \frac{z+3}{-3}

and

L_2: \frac{x+2}{-1} = \frac{y-4}{8} = \frac{z-5}{4}

.

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medium

Calculate the shortest distance between the lines

l_1

and

l_2

whose vector equations are

\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})

and

\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k})

.

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medium

A line makes angles

45^\circ

,

135^\circ

, and

60^\circ

with the positive x, y, and z-axes respectively. Recall the formula for direction cosines and find them.

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hard

Design a method to find the coordinates of the foot of the perpendicular drawn from a point P(

x_1, y_1, z_1

) to the line

\frac{x-a}{l} = \frac{y-b}{m} = \frac{z-c}{n}

. Justify your proposed steps.

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hard

The vertices of a triangle are

A(1, 2, 3)

,

B(-1, 0, 4)

, and

C(0, 1, 1)

. Find the direction cosines of the median through vertex A.

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hard

Calculate the distance between the parallel lines

L_1: \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})

and

L_2: \vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(2\hat{i} + 3\hat{j} + 4\hat{k})

.

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hard

Design a comprehensive algorithm to determine the relationship between two lines given in Cartesian form. Your algorithm must be able to classify the lines as intersecting, parallel, skew, or coincident and, where applicable, calculate the point of intersection or the shortest distance. Justify each decision point in your algorithm.

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hard

Find the vector equation of the line passing through the point

(2, -1, 3)

and perpendicular to the lines

\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k})

and

\vec{r} = (2\hat{i} - \hat{j} - 3\hat{k}) + \mu(\hat{i} + 2\hat{j} + 2\hat{k})

.

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hard

Derive the formula for the shortest distance between two skew lines in Cartesian form:

L_1: \frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}

and

L_2: \frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}

. Justify the use of the scalar triple product in your derivation.

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hard

A variable line in two adjacent positions has direction cosines

(l, m, n)

and

(l+\delta l, m+\delta m, n+\delta n)

. If

\delta\theta

is the small angle between the two positions, prove that

(\delta\theta)^2 = (\delta l)^2 + (\delta m)^2 + (\delta n)^2

.

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hard

Formulate the equation of a line passing through the point P(2, -1, 3) and perpendicular to the lines

\vec{r} = (\hat{i}+\hat{j}-\hat{k}) + \lambda(2\hat{i}-2\hat{j}+\hat{k})

and

\vec{r} = (2\hat{i}-\hat{j}-3\hat{k}) + \mu(\hat{i}+2\hat{j}+2\hat{k})

. Then, create a procedure to find the shortest distance between this newly formulated line and the x-axis.

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hard

A line makes angles

\alpha, \beta, \gamma, \delta

with the four diagonals of a cube. Prove that

\cos^2\alpha + \cos^2\beta + \cos^2\gamma + \cos^2\delta = \frac{4}{3}

.

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Three Dimensional GeometryClass 12 Mathematics Practice Questions