JEE (Main) Mathematics - Three Dimensional Geometry MCQs
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Q61. The position vector of a point P in space is \( \mathbf{r} = 3\hat{i} - 2\hat{j} + \hat{k} \). What are the coordinates of point P?
- a) (3, -2, 1)
- b) (3, 2, -1)
- c) (-3, 2, 1)
- d) (-3, -2, -1)
Answer: a) (3, -2, 1)
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Q62. The distance between the points A(1, 2, 3) and B(4, 5, 6) is:
- a) \( \sqrt{14} \)
- b) \( \sqrt{27} \)
- c) \( \sqrt{34} \)
- d) \( \sqrt{50} \)
Answer: c) \( \sqrt{34} \)
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Q63. A point P divides the line segment joining A(1, 2, 3) and B(4, 5, 6) in the ratio 2:3. The coordinates of point P are:
- a) (2.6, 3.6, 4.6)
- b) (2.8, 3.8, 4.8)
- c) (3.4, 4.4, 5.4)
- d) (3.6, 4.6, 5.6)
Answer: c) (3.4, 4.4, 5.4)
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Q64. The direction cosines of a line with direction ratios (2, 3, 6) are:
- a) \( \left( \frac{2}{7}, \frac{3}{7}, \frac{6}{7} \right) \)
- b) \( \left( \frac{2}{6}, \frac{3}{6}, \frac{6}{6} \right) \)
- c) \( \left( \frac{2}{9}, \frac{3}{9}, \frac{6}{9} \right) \)
- d) \( \left( \frac{2}{5}, \frac{3}{5}, \frac{6}{5} \right) \)
Answer: a) \( \left( \frac{2}{7}, \frac{3}{7}, \frac{6}{7} \right) \)
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Q65. If the direction ratios of two lines are \( (1, 2, 3) \) and \( (4, -1, 2) \), then the angle \( \theta \) between them is:
- a) \( \cos^{-1} \left( \frac{1}{\sqrt{14}} \right) \)
- b) \( \cos^{-1} \left( \frac{2}{\sqrt{14}} \right) \)
- c) \( \cos^{-1} \left( \frac{3}{\sqrt{14}} \right) \)
- d) \( \cos^{-1} \left( \frac{4}{\sqrt{14}} \right) \)
Answer: b) \( \cos^{-1} \left( \frac{2}{\sqrt{14}} \right) \)
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Q66. The parametric equations of a line are given by \( x = 1 + t, y = 2 - 3t, z = 4 + t \). The direction ratios of this line are:
- a) (1, -3, 1)
- b) (1, 3, 1)
- c) (-1, 3, -1)
- d) (1, -1, 1)
Answer: a) (1, -3, 1)
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Q67. The lines \( L_1: x = 1 + t, y = 2 + 2t, z = 3 + t \) and \( L_2: x = 4 + s, y = 5 + s, z = 6 + 2s \) are:
- a) Parallel
- b) Intersecting
- c) Coincident
- d) Skew
Answer: d) Skew
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Q68. The shortest distance between the skew lines \( L_1: \mathbf{r} = (1, 2, 3) + t(1, 2, 3) \) and \( L_2: \mathbf{r} = (4, 5, 6) + s(3, 2, 1) \) is:
- a) \( 3 \)
- b) \( 2 \)
- c) \( 1 \)
- d) \( \sqrt{3} \)
Answer: b) \( 2 \)
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Q69. The direction ratios of the line joining the points P(2, 3, 4) and Q(5, 6, 7) are:
- a) (3, 3, 3)
- b) (4, 4, 4)
- c) (1, 1, 1)
- d) (2, 2, 2)
Answer: a) (3, 3, 3)
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Q70. The equation of the line passing through the points A(1, 2, 3) and B(4, 5, 6) is:
- a) \( \frac{x - 1}{3} = \frac{y - 2}{3} = \frac{z - 3}{3} \)
- b) \( \frac{x - 1}{2} = \frac{y - 2}{2} = \frac{z - 3}{2} \)
- c) \( \frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3} \)
- d) \( \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} \)
Answer: a) \( \frac{x - 1}{3} = \frac{y - 2}{3} = \frac{z - 3}{3} \)
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Q71. The direction cosines of the line with direction ratios (4, 3, 5) are:
- a) \( \left( \frac{4}{7}, \frac{3}{7}, \frac{5}{7} \right) \)
- b) \( \left( \frac{4}{6}, \frac{3}{6}, \frac{5}{6} \right) \)
- c) \( \left( \frac{4}{9}, \frac{3}{9}, \frac{5}{9} \right) \)
- d) \( \left( \frac{4}{5}, \frac{3}{5}, \frac{5}{5} \right) \)
Answer: a) \( \left( \frac{4}{7}, \frac{3}{7}, \frac{5}{7} \right) \)
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Q72. The angle \( \theta \) between the lines \( L_1: x = 2 + t, y = 1 + 2t, z = 3 + t \) and \( L_2: x = 3 + s, y = 4 + s, z = 5 + 2s \) is:
- a) \( 45^\circ \)
- b) \( 60^\circ \)
- c) \( 90^\circ \)
- d) \( 120^\circ \)
Answer: b) \( 60^\circ \)
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Q73. The equations of the skew lines \( L_1: \mathbf{r} = (1, 2, 3) + t(1, 2, 1) \) and \( L_2: \mathbf{r} = (4, 5, 6) + s(2, -1, 3) \) are:
- a) Parallel
- b) Coincident
- c) Intersecting
- d) Skew
Answer: d) Skew
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Q74. The midpoint of the line joining A(1, 2, 3) and B(4, 5, 6) is:
- a) (2.5, 3.5, 4.5)
- b) (2, 3, 4)
- c) (3, 4, 5)
- d) (1.5, 2.5, 3.5)
Answer: a) (2.5, 3.5, 4.5)
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Q75. If the direction ratios of two lines are (2, 1, 3) and (4, -1, 5), the angle between them is:
- a) \( \cos^{-1} \left( \frac{8}{\sqrt{35} \cdot \sqrt{42}} \right) \)
- b) \( \cos^{-1} \left( \frac{4}{\sqrt{35} \cdot \sqrt{42}} \right) \)
- c) \( \cos^{-1} \left( \frac{6}{\sqrt{35} \cdot \sqrt{42}} \right) \)
- d) \( \cos^{-1} \left( \frac{2}{\sqrt{35} \cdot \sqrt{42}} \right) \)
Answer: a) \( \cos^{-1} \left( \frac{8}{\sqrt{35} \cdot \sqrt{42}} \right) \)
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