NCERT Solutions for Class 12 Mathematics Chapter 11 Three Dimensional Geometry

NCERT Solutions for Class 12 Mathematics Chapter 11 - Three Dimensional Geometry

Welcome to the Chapter 11 - Three Dimensional Geometry, Class 12 Mathematics NCERT Solutions page. Here, we provide detailed question answers for Chapter 11 - Three Dimensional Geometry. The page is designed to help students gain a thorough understanding of the concepts related to natural resources, their classification, and sustainable development.

Our solutions explain each answer in a simple and comprehensive way, making it easier for students to grasp key topics Three Dimensional Geometry and excel in their exams. By going through these Three Dimensional Geometry question answers, you can strengthen your foundation and improve your performance in Class 12 Mathematics. Whether you’re revising or preparing for tests, this chapter-wise guide will serve as an invaluable resource.

Exercise 1

Q 1:

If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.

Let direction cosines of the line be l, m, and n.

\begin{align}l = cos90^0=0\end{align}

\begin{align}m = cos135^0=-\frac{1}{\sqrt{2}}\end{align}

\begin{align}n = cos45^0=\frac{1}{\sqrt{2}}\end{align}

\begin{align}Therefore, the\; direction\; cosines\; of \;the\; line\; are\;0, -\frac{1}{\sqrt{2}}\;and\;\frac{1}{\sqrt{2}}\end{align}

Q 2:

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Let the direction cosines of the line make an angle α with each of the coordinate axes.

∴ l = cos α, m = cos α, n = cos α

l²+m²+n² =1

⇒ cos²α + cos²α + cos²α = 1

⇒ 3cos²α =1

\begin{align}\Rightarrow cos^2α = \frac{1}{3}\end{align}

\begin{align}\Rightarrow cosα = \pm\frac{1}{\sqrt 3}\end{align}

Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are

\begin{align} \pm\frac{1}{\sqrt 3},\pm\frac{1}{\sqrt 3},and \pm\frac{1}{\sqrt 3}\end{align}

Q 3:

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

If a line has direction ratios of −18, 12, and −4, then its direction cosines are

\begin{align} \frac{-18}{\sqrt {(-18)^2 + (12)^2 + (-4)^2}},\frac{12}{\sqrt {(-18)^2 + (12)^2 + (-4)^2}},\frac{-4}{\sqrt {(-18)^2 + (12)^2 + (-4)^2}}\end{align}

\begin{align} i.e., \frac{-18}{22},\frac{12}{22},\frac{-4}{22}\end{align}

\begin{align} \frac{-9}{11},\frac{6}{11},\frac{-2}{11}\end{align}

Thus, the direction cosines are

\begin{align} \frac{-9}{11},\frac{6}{11} and \frac{-2}{11}\end{align}

Q 4:

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

The given points are A (2, 3, 4), B (− 1, − 2, 1), and C (5, 8, 7).

It is known that the direction ratios of line joining the points, (x₁, y₁, z₁) and (x₂, y₂, z₂), are given by, x₂ − x₁, y₂ − y₁, and z₂ − z₁.

The direction ratios of AB are (−1 − 2), (−2 − 3), and (1 − 4) i.e., −3, −5, and −3.

The direction ratios of BC are (5 − (− 1)), (8 − (− 2)), and (7 − 1) i.e., 6, 10, and 6.

It can be seen that the direction ratios of BC are −2 times that of AB i.e., they are proportional.

Therefore, AB is parallel to BC. Since point B is common to both AB and BC, points A, B, and C are collinear.

Q 5:

The vertices of ΔABC are A (3, 5, −4), B (−1, 1, 2), and C (−5, −5, −2).

The direction ratios of side AB are (−1 − 3), (1 − 5), and (2 − (−4)) i.e., −4, −4, and 6.

Therefore, the direction cosines of AB are

The direction ratios of BC are (−5 − (−1)), (−5 − 1), and (−2 − 2) i.e., −4, −6, and −4.

Therefore, the direction cosines of BC are

The direction ratios of CA are (−5 − 3), (−5 − 5), and (−2 − (−4)) i.e., −8, −10, and 2.

Therefore, the direction cosines of AC are

Frequently Asked Questions about Three Dimensional Geometry - Class 12 Mathematics

- 1. How many questions are covered in Three Dimensional Geometry solutions?
- All questions from Three Dimensional Geometry are covered with detailed step-by-step solutions including exercise questions, additional questions, and examples.
- 2. Are the solutions for Three Dimensional Geometry helpful for exam preparation?
- Yes, the solutions provide comprehensive explanations that help students understand concepts clearly and prepare effectively for both board and competitive exams.
- 3. Can I find solutions to all exercises in Three Dimensional Geometry?
- Yes, we provide solutions to all exercises, examples, and additional questions from Three Dimensional Geometry with detailed explanations.
- 4. How do these solutions help in understanding Three Dimensional Geometry concepts?
- Our solutions break down complex problems into simple steps, provide clear explanations, and include relevant examples to help students grasp the concepts easily.
- 5. Are there any tips for studying Three Dimensional Geometry effectively?
- Yes, practice regularly, understand the concepts before memorizing, solve additional problems, and refer to our step-by-step solutions for better understanding.

Exam Preparation Tips for Three Dimensional Geometry

The Three Dimensional Geometry is an important chapter of 12 Mathematics. This chapter’s important topics like Three Dimensional Geometry are often featured in board exams. Practicing the question answers from this chapter will help you rank high in your board exams.

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