Class 11 Maths Chapter 12 Introduction to Three-dimensional Geometry MCQs

1. The distance between the points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) in 3D space is given by:

1. a) \( \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \)
2. b) \( (x_2 – x_1) + (y_2 – y_1) + (z_2 – z_1) \)
3. c) \( \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2} \)
4. d) \( \sqrt{x_1^2 + y_1^2 + z_1^2} \)

Answer: c

2. The coordinates of a point in space are written as:

1. a) (x, y)
2. b) (x, y, z)
3. c) (r, θ)
4. d) (x, z)

Answer: b

3. The distance of the point P(x, y, z) from the origin is:

1. a) x + y + z
2. b) \( \sqrt{x^2 + y^2} \)
3. c) \( \sqrt{x^2 + y^2 + z^2} \)
4. d) z

Answer: c

4. The coordinates of the midpoint of the line segment joining A(x₁, y₁, z₁) and B(x₂, y₂, z₂) are:

1. a) \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \)
2. b) (x₁ + x₂, y₁ + y₂, z₁ + z₂)
3. c) (x₂ – x₁, y₂ – y₁, z₂ – z₁)
4. d) \( \left( \frac{x_1 – x_2}{2}, \frac{y_1 – y_2}{2}, \frac{z_1 – z_2}{2} \right) \)

Answer: a

5. If l, m, n are the direction cosines of a line, then:

1. a) \( l^2 + m^2 + n^2 = 0 \)
2. b) \( l^2 + m^2 + n^2 = 1 \)
3. c) \( l^2 – m^2 – n^2 = 1 \)
4. d) \( l^2 + m^2 = 1 \)

Answer: b

6. Which of the following represents direction ratios of a line?

1. a) Proportional to direction cosines
2. b) Satisfies \( l^2 + m^2 + n^2 = 1 \)
3. c) Always integers
4. d) None of these

Answer: a

7. The direction cosines of a line parallel to the x-axis are:

1. a) (1, 0, 0)
2. b) (0, 1, 0)
3. c) (0, 0, 1)
4. d) (1, 1, 1)

Answer: a

8. If the direction ratios of a line are 3, -6, 2, then the direction cosines are proportional to:

1. a) (3, -6, 2)
2. b) \( \left( \frac{3}{7}, \frac{-6}{7}, \frac{2}{7} \right) \)
3. c) (9, 36, 4)
4. d) (-3, 6, -2)

Answer: b

9. The equation of a plane passing through (x₁, y₁, z₁) and normal to vector \( \vec{n} = \langle a, b, c \rangle \) is:

1. a) \( ax + by + cz + d = 0 \)
2. b) \( a(x – x_1) + b(y – y_1) + c(z – z_1) = 0 \)
3. c) \( ax + by + cz = d \)
4. d) \( x + y + z = 0 \)

Answer: b

10. A plane parallel to the xy-plane has the equation:

1. a) \( z = c \)
2. b) \( z = 0 \)
3. c) \( x + y = 0 \)
4. d) \( ax + by = c \)

Answer: a

1. a) \( x – 2y + 3z = 0 \)
2. b) \( x + 2y – 3z = 0 \)
3. c) \( x – y + z = 0 \)
4. d) None of these

Answer: a

1. a) \( \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \)
2. b) \( \frac{ax_1 + by_1 + cz_1 + d}{\sqrt{a^2 + b^2 + c^2}} \)
3. c) \( \sqrt{x_1^2 + y_1^2 + z_1^2} \)
4. d) None of these

Answer: a

1. a) \( \vec{r} \cdot \vec{n} = d \)
2. b) \( ax + by + cz + d = 0 \)
3. c) \( \vec{r} \times \vec{n} = 0 \)
4. d) None of these

Answer: a