Maths · Vectors and 3-D geometry · chapter 11 of 13

Class 12 Maths Chapter 11: Three Dimensional Geometry MCQs with answers

10 multiple-choice questions on Three Dimensional Geometry, each with its answer and the reason, written from the NCERT chapter. For the CBSE board and JEE Main, which are set from the same NCERT chapter. Open a question's solution to check your working; then take the free test to find your next chapter.

Question 1
A line makes angles of 60° and 45° with the positive directions of the x-axis and the y-axis, and an acute angle γ with the positive direction of the z-axis. Then γ is
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Answer: (a) 60°

cos2 60° + cos2 45° + cos2 γ = 1 gives cos2 γ = 1 − 14 − 12 = 14. γ is acute, so cos γ = 12 and γ = 60°. Option (b) copies the angle with the y-axis; (c) takes cos γ = √32; (d) treats the line as perpendicular to the z-axis.

Question 2
Which of the following can be the direction cosines of a line?
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Answer: (b) 1√2, 12, 12

Direction cosines satisfy l2 + m2 + n2 = 1. For (b): 12 + 14 + 14 = 1. The sums for (a), (c) and (d) are 34, 13 and 23, so those are at most direction ratios.

Question 3
A line has direction ratios 2, −3, 6. Which of the following is NOT a set of direction ratios of the same line?
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Answer: (c) −2, −3, −6

Any set of direction ratios of the line is k(2, −3, 6) with k ≠ 0: (a) is k = −2, (b) is k = 12, (d) is k = 3. In (c) the first number needs k = −1 but the second needs k = 1, so (c) is not proportional to 2, −3, 6.

Question 4
If 2k, −k and 2k are the direction cosines of a line, where k > 0, then k is
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Answer: (a) 13

(2k)2 + (−k)2 + (2k)2 = 1 gives 9k2 = 1, so k = 13. Option (b) stops at k2; (c) uses 3k2 = 1; (d) inverts.

Question 5
The line joining the origin O to the point P(1, 1, √2) makes with the positive direction of the z-axis an angle of
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Answer: (b) 45°

OP = √(1 + 1 + 2) = 2, so cos γ = √22 = 1√2 and γ = 45°. Option (a) takes cos γ = 12 from the x or y coordinate.

Question 6
A line through the origin is perpendicular to the z-axis, makes an angle of 30° with the positive direction of the x-axis and an acute angle with the positive direction of the y-axis. Its direction cosines are
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Answer: (c) √32, 12, 0

Perpendicular to the z-axis: γ = 90°, n = 0. l = cos 30° = √32. Then m2 = 1 − 34 − 0 = 14 and m = 12 (acute angle). Option (b) swaps l and m; (d) puts n = cos 0°.

Question 7
The points A(1, −1, 2), B(3, 2, 0) and C(7, 8, p) are collinear. The value of p is
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Answer: (a) −4

Direction ratios of AB: 2, 3, −2; of AC: 6, 9, p − 2. Collinear points need these proportional: 62 = 93 = 3, so p − 2 = 3(−2) = −6 and p = −4. Option (c) forgets the 2 in p − 2; (b) is a sign slip on −4; (d) takes p − 2 = +6, dropping the minus of the direction ratio −2.

Question 8
The line →r = 3î − 2k̂ + λĵ
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Answer: (d) is parallel to the y-axis and passes through the point (3, 0, −2)

In →r = →a + λ→b, →a = 3î − 2k̂ is the position vector of a point on the line, (3, 0, −2), and →b = ĵ gives direction ratios 0, 1, 0, the direction of the y-axis. Option (c) swaps the roles of →a and →b; (a) is wrong because the direction is that of the y-axis itself.

Question 9
Which of the following points lies on the line x − 32 = y + 1−5 = z − 46?
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Answer: (d) (5, −6, 10)

A general point of the line is (3 + 2λ, −1 − 5λ, 4 + 6λ); λ = 1 gives (5, −6, 10). For (b) the three ratios are 0, −25, 0; (c) reads the direction ratios as a point; for (a) the ratios are 1, 35, 1.

Question 10
Two lines in space that are neither parallel nor intersecting are called
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Answer: (b) skew lines

Lines in space which are neither parallel nor intersecting are skew lines; they lie in different planes, so (a) is wrong. Perpendicular lines may intersect, and concurrent lines meet.

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