Karnataka 2nd PUC Mathematics Three Dimensional Geometry MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – II PUC
NAME OF THE CHAPTER: THREE DIMENSIONAL GEOMETRY

Direction Cosines:
If a line makes an angle 𝛼, 𝛽, 𝛾 with positive direction of 𝑥, 𝑦, 𝑧 axes respectively then
cos𝛼, 𝑐𝑜𝑠𝛽, 𝑐𝑜𝑠𝛾 are called direction cosines of the line denoted by 𝑙, 𝑚, 𝑛.
i.e 𝑙 = 𝑪𝒐𝒔𝜶 , 𝒎 = 𝒄𝒐𝒔𝜷, 𝒏 = 𝒄𝒐𝒔𝜸 and 𝒍𝟐 + 𝒎𝟐 + 𝒏𝟐 = 𝟏
Direction ratio’s:
The three numbers a, b, c which are proportional to Direction cosines 𝑙, 𝑚, 𝑛 are called
direction ratio’s
𝑙 𝑚 𝑛
∴ 𝑎 = 𝑏 = 𝑐 = 𝑘 => 𝑙 = 𝑎𝑘, 𝑚 = 𝑏𝑘, 𝑛 = 𝑐𝑘 => 𝑙 2 + 𝑚2 + 𝑛2 = 𝑘 2 (𝑎2 + 𝑏 2 + 𝑐 2 )

=> 1 = 𝑘 2 (𝑎2 + 𝑏 2 + 𝑐 2 )
1 𝑎 𝑏 𝑐
 𝑘 = ± √𝑎2 ∴ 𝑙 = ± √𝑎2 , 𝑚 = ± √𝑎2 , 𝑛 = ± √𝑎2
+𝑏 2 +𝑐 2 +𝑏2 +𝑐 2 +𝑏2 +𝑐 2 +𝑏2 +𝑐 2

 Direction ratio’s of a line segment joining two points P( 𝑥1 , 𝑦1 , 𝑧1 ) Q (𝑥2 , 𝑦2 , 𝑧2 )
⃗⃗⃗⃗⃗
𝑃𝑄 = 𝑃𝑉𝑜𝑓 𝑄 − 𝑃𝑉 𝑜𝑓 𝑃 i.e ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ − ⃗⃗⃗⃗⃗
𝑃𝑄 = 𝑂𝑄 𝑂𝑃
∴ ⃗⃗⃗⃗⃗
𝑃𝑄 = (𝑥2 − 𝑥1 )𝑖 + (𝑦2 − 𝑦1 )𝑗 + (𝑧2 − 𝑧1 )𝑘
∴ Direction ratios are 𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 , 𝑧2 − 𝑧1
𝑥2 −𝑥1 𝑦2 −𝑦1 𝑧2 −𝑧1
∴ Direction cosines are ⃗⃗⃗⃗⃗ |
|𝑃𝑄
, ⃗⃗⃗⃗⃗⃗⃗ , ⃗⃗⃗⃗⃗⃗⃗
|𝑃𝑄| |𝑃𝑄|

 Direction cosines of 𝑥, 𝑦, 𝑧 axes are (1, 0, 0), (0, 1, 0) & (0, 0, 1).
EQUATION OF A LINE IN SPACE
1) Equation of a line through a given point & parallel to a given vector 𝐛
Let A(𝑥1 𝑦1 𝑧1 ) be a point on a line & 𝑏⃗ = 𝑎𝑖 + 𝑏𝑗 + 𝑐𝑘 be the vector parallel to a line
⃗⃗⃗⃗⃗⃗
𝑂𝐴 = 𝑎, ⃗⃗⃗⃗⃗ = 𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘
𝑂𝑃
(i) Vector form: 𝑟 = 𝑎 + 𝜆 𝑏⃗
𝑥−𝑥1 𝑦−𝑦1 𝑧−𝑧1
(ii) Cartesian form: = =
𝑎 𝑏 𝑐

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ANGLE BETWEEN TWO LINES:
Let 𝑟 = ⃗⃗⃗⃗
𝑎1 + 𝜆𝑏1 & 𝑟 = ⃗⃗⃗⃗ ⃗⃗⃗⃗2 be the two lines then the angle between two line is
𝑎2 + 𝜆𝑏
⃗⃗⃗⃗ .𝑏
𝑏 ⃗⃗⃗⃗
cos𝜃 = ⃗⃗⃗⃗⃗⃗⃗1 ⃗⃗⃗⃗2
|𝑏1 | |𝑏2 |

𝑎1 𝑎2 +𝑏1 𝑏2 +𝑐1 𝑐2
Let ⃗⃗⃗
𝑏1 = 𝑎1 𝑖 + 𝑏1 𝑗 + 𝑐1 𝑘 ⃗⃗⃗⃗
𝑏2 = 𝑎2 𝑖 + 𝑏2 𝑗 + 𝑗2 𝑘 then 𝐶𝑜𝑠𝜃 = | |
√𝑎12 +𝑏12 +𝑐12 √𝑎22 +𝑏22 +𝑐22

𝑵𝒐𝒕𝒆: Two lines with direction ratio’s a1, b1, c1 & a2, b2, c2 are
𝑎 𝑏 𝑐
(𝑖) Perpendicular if 𝑎1 𝑎2 + 𝑏1 𝑏2 + 𝑐1 𝑐2 = 0 (ii) Parallel if 𝑎1 = 𝑏1 = 𝑐1
2 2 2

Skew lines:
The lines in a space which are neither interesting nor parallel are called skew lines ,
Skew lines are non-coplanar
Shortest distance between two skew lines:
Let 𝑟 = ⃗⃗⃗⃗ ⃗⃗⃗1 & 𝑟 = ⃗⃗⃗⃗
𝑎1 + 𝜆𝑏 ⃗⃗⃗⃗2 be two skew lines.
𝑎2 + 𝜆𝑏
⃗⃗⃗⃗1 x ⃗b2 ).(a
(𝑏 ⃗⃗⃗⃗1 )
⃗ 2 −a
Then the distance between two lines is given by d=| ⃗ 2|
⃗ 1x b
|
|𝑏

Distance between two parallel lines:
𝑎1 + 𝜆𝑏⃗ & 𝑟 = ⃗⃗⃗⃗
Let 𝑟 = ⃗⃗⃗⃗ 𝑎2 + 𝜆𝑏⃗ be two parallel lines then the distance between two parallel
⃗ ×(a
𝑏 ⃗ 2 −a⃗⃗⃗⃗1 )
lines is given by 𝑑 = | ⃗
|𝑏 |
|

Golden points
𝑥 𝑦 𝑧
1) i) Equation of 𝑋 − 𝑎𝑥𝑖𝑠 is y = 0, z = 0 or 1 = 0 = 0
𝑥 𝑦 𝑧
ii) Equation of Y = axis is 𝑥 = 0, 𝑧 = 0 or 0 = 1 = 0
𝑥 𝑦 𝑧
iii) Equation of Z – axis is 𝑥 = 0, y = 0 or 0 = 0 = 1

2) i) Equation of XY plane is z = 0
ii) Equation of YZ plane is 𝑥 = 0
iii) Equation of 𝑍𝑋 plane is y = 0
1
3) i) Angle between two diagonals of a cube is Cos-1 3

ii) The angle between a diagonal of a cube & the diagonal of a faces of the cube is
2
Cos-1(√3)

iii) If a straight line makes angles 𝛼, 𝛽, √ , 𝛿 with diagonals of a cube then
4
Cos2𝛼+cos2𝛽 + 𝑐𝑜𝑠2√+cos2𝛿 = 3

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4) The ratio in which the line joining the points (𝑥1 , 𝑦1 , 𝑧1 )&(𝑥2 , 𝑦2 , 𝑧2 )𝑖𝑠 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑏𝑦
−𝑧 𝑧 𝑥 −𝑧 𝑥 𝑧 𝑦 −𝑧 𝑦
(i) 𝑋𝑌 𝑝𝑙𝑎𝑛𝑒 is 𝑧 1 & the point of division ( 1 𝑧2−𝑧2 1 , 1 𝑧2−𝑧2 1 , 0)
2 1 2 1 2

−𝑥1 𝑥1 𝑦2 −𝑥2 𝑦1 𝑥1 𝑧2 −𝑥2 𝑧1
(ii) YZ plane is & the point of division (0, , 𝑥 −𝑥 )
𝑥2 𝑥1 −𝑥2 1 2

𝑦 𝑦 𝑥 −𝑦 𝑥 𝑦 𝑧 −𝑦 𝑧
(iii) ZX 𝑝𝑙𝑎𝑛𝑒 is − 𝑦1 &𝑡ℎ𝑒 point of division ( 1𝑦2−𝑦2 1 , 0, 1𝑦2 −𝑦2 1 )
2 1 2 1 2

𝑬𝒙: Find the ratio in which the line joining (3, -2, 4) & (4, −1, −2) is divided by
+4 11 4
a) 𝑋𝑌 𝑝𝑙𝑎𝑛𝑒 is – (−2) = 2: 1, Point ( 3 , − 3 , 0)
3
b) YZ plane is – 4 = 3: 4 externally, point (0, -5, 22)
−2
c) 𝑍𝑋 plane is – (−1) = 2 ∶ 1 externally point (5, 0, -8)

Ex: Find the distance of the point (3, −4, 5) from 𝑥 − 𝑎𝑥𝑖𝑠, 𝑦 − 𝑎𝑥𝑖𝑠 & z – axis.
=> 𝑥 − 𝑎𝑥𝑖𝑠 = √16 + 25 = √41 , 𝑦 − 𝑎𝑥𝑖𝑠 = √9 + 25 = √34 , 𝑧 − 𝑎𝑥𝑖𝑠 = √9 + 16 = 5
5) If the projection of the line joining P(𝑥1 , 𝑦1 , 𝑧1 ) & Q (𝑥2 , 𝑦2 , 𝑧2 ) on the Co-ordinate axes are
a, b, c then PQ = √𝑎2 + 𝑏 2 + 𝑐 2

Direction Cosines and Direction Ratios of a Line
1. If the direction cosines of a line is 𝐤, 𝐤, 𝐤, then (Average)
1 1
(a) k > 0 (b) 0 < k < 1 (c) k = 1 (d) k = or −
√3 √3

2. Let a, b, c are direction ratios and l, m and n be the direction cosines
𝒍 𝒎 𝒏
of a line, such that 𝒂 = 𝒃 = 𝒄 = 𝒌 , k being a constant then k is equal to

(Average)
1
(a) ± √𝑎2 (b) ±√𝑎2 + 𝑏 2 + 𝑐 2
+𝑏 2 +𝑐 2
1
(c) ±(𝑎2 + 𝑏 2 + 𝑐 2 ) (d)± 𝑎2 +𝑏2 +𝑐 2

3.The direction Cosines of a normal to the yz-plane are, (Easy)
(a) (0, 0, 1) (b) (1, 0, 0) (c) (0, 1, 0) (d) (1, 1, 0)
4.A vector 𝒓
⃗ is inclined at equal angles to 𝑶𝑿, 𝑶𝒀 & OZ. If the magnitudes of 𝒓
⃗ is 6 units.
Then 𝒓
⃗ = (Average)
(a) 2√3 (𝑖 + 𝑗 + 𝑘) (b) −2√3 (−𝑖 + 𝑗 + 𝑘)
(c) 2√3(−𝑖 + 𝑗 + 𝑘) (d) 2√3 (𝑖 − 𝑗 + 𝑘)

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𝟏 𝟏
5.If (𝟑 , 𝟒 , 𝒌) are the direction cosines of a line then the value of k is (Easy)
119 √119 119 11
(a) 12 (b) (c) 114 (d) 12
12

6.OX, OY are positive 𝒙 − 𝒂𝒙𝒊𝒔, positive 𝒚 − 𝒂𝒙𝒊𝒔 respectively where 0= (𝟎, 𝟎, 𝟎) the
direction cosines of the line which bisects 𝒙𝒐𝒚 are (Average)
1 1 1 1
(a) 1,1,0 (b) , ,0 (c) , 0, (d) 0,0,1
√2 √2 √2 √2

7.The direction ratio’s of a line are 1,-1, 0 then the angle of inclination of the line with
𝒚 − 𝒂𝒙𝒊𝒔 is (Easy)
3𝜋 −𝜋 𝜋 −𝜋
(a) 4 (b) 4 (c) 3 (d) 6

8. If a line makes an angles α, β, γ with the positive direction of the
co-ordinate axes. Then 𝒔𝒊𝒏𝟐 𝛂 + 𝒔𝒊𝒏𝟐 𝛃 + 𝒔𝒊𝒏𝟐 𝛄 = (Average)
(a) 2 (b) 1 (c) 0 (d) -1
9. If a line makes an angles α, β, γ with the positive direction of the
coordinate axes. Then cos2α + cos2β + cos2γ = (Average)
(a) 2 (b) 1 (c) 0 (d) -1
10. The direction cosines of a line which makes equal angles with coordinate axes are
(Easy)
1 1 1 1 1 −2
(a) , , (b) √3, √3, √3 (c) , , (d) 1 , 2 , 3
√3 √3 √3 √6 √6 √6
𝝅 𝟑𝝅 𝝅
11. If 𝟐 , 𝟒 and 𝟒 are direction angles of a directed line ⃗⃗⃗⃗⃗⃗
𝑶𝑷, then direction angles of the

⃗⃗⃗⃗⃗⃗ are
directed line 𝑷𝑶 (Easy)
𝜋 3𝜋 𝜋 𝜋 𝜋 3𝜋
(a) − 2 , − 4 , − 4 (b) 2 , 4 , 4
𝜋 −3𝜋 −𝜋 𝜋 3𝜋 𝜋
(c) 2 , 4 , 4 (d) 2 , 4 , 4 .

12. If the directed line ⃗⃗⃗⃗⃗⃗
𝑶𝑷 makes angles 𝟗𝟎∘ , 𝟔𝟎∘ and 𝟒𝟓∘ with the positive direction of
𝐱, 𝐲 and 𝒛-axes respectively. Then its direction cosines of the directed line ⃗⃗⃗⃗⃗⃗
𝑷𝑶
(Average)
1 1 1 1 1 1 1 1
(a) 0, 2 , (b) 1, − 2 , − (c) 1, 2 , (d) 0, − 2 , − .
√2 √2 √2 √2

13. The points (𝟔, −𝟏, 𝟐), (𝟓, 𝟐, 𝟒)&(𝟖, −𝟕, 𝒌) are collinear then K = (Average)
(a) 2 (b) 3 (c) -2 (d) 1
14. The direction ratio’s of a line are 1,-1, 0 then the angle of inclination of the line
with 𝒚 − 𝒂𝒙𝒊𝒔 is (Easy)
3𝜋 −𝜋 𝜋 −𝜋
(a) 4 (b) 4 (c) 3 (d) 6

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Equation of a Line in Space
15. The equation of a line which passes through the point (𝟏, 𝟐, 𝟑) & is parallel to the
vector 𝟑𝒊 + 𝟐𝒋 − 𝟐𝒌 is (Easy)
(a) 𝑟 = (3𝑖 + 2𝑗 − 2𝑘) + 𝜆(𝑖 + 2𝑗 + 3𝑘) (b) 𝑟 = (𝑖 + 2𝑗 + 3𝑘) + 𝜆(3𝑖 + 2𝑗 − 2𝑘)
(c)𝑟 = (𝑖 + 2𝑗 + 3𝑘) + 𝜆(2𝑖 − 5𝑘) (d) 𝑟 = (𝑖 + 2𝑗 + 3𝑘) + 𝜆(4𝑖 + 4𝑗 + 𝑘)
16. In the figure the equation of straight line AB is (Difficult)
2(𝑥−1) √2(𝑦−2) 𝑧−3
(a) = = 0
√3 1
𝑥−1 𝑦−2 𝑧−3
(b) = =
√3 √2 0
𝑥+1 𝑦+2 𝑧+3
(c) √3
= =
1 1
2

(𝑥−1) √2(𝑦−2) 𝑧−3
(d) = = 0
√3 1

17. The equation of a line passing through the point (-3, 2, -4) & equally inclined to the
axes are (Difficult)
(a) 𝑥 − 3 = 𝑦 + 2 = 𝑧 − 4 (b) 𝑥 + 3 = 𝑦 − 2 = 𝑧 + 4
𝑥+3 𝑦−2 𝑧+4 𝑥−3 𝑦+2 𝑧−4
(c) = = (d) = =
1 2 3 1 2 3

18. The equation of straight line passing through the point (a, b, c) & parallel to Z –axis
(Easy)
𝑥−𝑎 𝑦−𝑏 𝑧−𝑐 𝑥−𝑎 𝑦−𝑏 𝑧−𝑐 𝑥−𝑎 𝑦−𝑏 𝑧−𝑐 𝑥−𝑎 𝑦−𝑏 𝑧−𝑐
(a) = = (b) = = (c) = = (d) = =
1 1 0 0 1 1 1 0 0 0 0 1

19. The equation of the line passing through origin with direction angles
𝝅 𝝅 𝟑𝝅
, 𝟒 and is (Easy)
𝟐 𝟒
x y z x y z x y z x y z
(a) 1 = = −√2 (b) 0 = −1 = −1 (c) = 1 = −1 (d) 0 = 1 = 1
√2 0

20. Equation of a line passing through (𝟏, 𝟐, −𝟑)& parallel to the line
𝒙−𝟐 𝒚+𝟏 𝒛−𝟏
= = is (Average)
𝟏 𝟑 𝟒
𝑥−1 𝑦−2 𝑧+3 𝑥−2 𝑦+1 𝑧−1 𝑥−1 𝑦−3 𝑧−4 𝑥+1 𝑦+3 𝑧−3
(a) = = (b) = = −3 (c) = = −3 4) 1 = =
1 3 4 1 2 1 2 3 4
𝐱+𝟑 𝐲−𝟓 𝐳+𝟔
21. If Cartesian equation of a line is = = , then its vector form is given by
𝟐 𝟒 𝟐

(Easy)

(a) r = 3î − 5ĵ + 6k̂ + λ( 2î + 4ĵ + 2k̂)

(b) r = −3î + 5ĵ − 6k̂ + λ( 2î + 4ĵ + 2k̂)

(c) r = 3î − 5ĵ + 6k̂ + λ( −2î − 4ĵ − 2k̂)

(d) r = 3î + 5ĵ + 6k̂ − λ(2î − 4ĵ − 2k̂)

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22. The equation of the line through the point ( 𝟎, 𝟏, 𝟐 ) and perpendicular to the line
𝒙−𝟏 𝒚+𝟏 𝒛−𝟏
= = −𝟐 (KCET-2025) (Average)
𝟐 𝟑
𝑥 𝑦−1 𝑧−2 𝑥 𝑦−1 𝑧−2
(a) 3 = = (b) 3 = −4 =
4 3 3
𝑥 𝑦−1 𝑧−2 𝑥 𝑦−1 𝑧−2
(c) 3 = = −3 (d) −3 = =
4 4 3
𝒙+𝟏 𝒚+𝟐 𝒛+𝟏
23. The unit vector perpendicular to both the lines = = &
𝟑 𝟏 𝟐
𝒙−𝟐 𝒚+𝟐 𝒛−𝟑
= = is (Difficult)
𝟏 𝟐 𝟑
–𝑖+7𝑗+7𝑘 –𝑖−7𝑗+5𝑘 –𝑖+7𝑗+5𝑘 7𝑖−7𝑗−𝑘
(a) (b) (c) (d)
√99 5√3 5√3 √99

Angle Between Two Lines
24. The condition for parallelism of two lines with direction ratios 𝐚𝟏 , 𝐛𝟏 , 𝐜𝟏 and 𝐚𝟐 , 𝐛𝟐 , 𝐜𝟐
is (Easy)
a b c
(a) a1 a2 + b1 b2 + c1 c2 = 0 (b) a1 = b1 = c1
2 2 2

(c) a1 = a2 , b1 = b2 , c1 = c2 (d) a1 a2 + b1 b2 + c1 c2 = −1
25. The condition for perpendicularity of two lines with direction ratios 𝒂𝟏 , 𝒃𝟏 , 𝒄𝟏 and
𝒂𝟐 , 𝒃𝟐 , 𝒄𝟐 is (Easy)
a b c
(a) a1 a2 + b1 b2 + c1 c2 = 0 (b) a1 = b1 = c1
2 2 2

(c) a1 = a2 , b1 = b2 , c1 = c2 (d) a1 a2 + b1 b2 + c1 c2 = −1
26. The direction ratios of two lines are K, 3, 5 & 2, -1, 2. If the angle between the lines
is 450 then K = (Average)
(a) 2 (b) 3 (c) 1 (d) 4
𝒙+𝟏 𝒚−𝟐 𝒛+𝟑 𝒙−𝟏 𝒚+𝟐 𝒛−𝟑
27. The angle between the straight lines = = & = = −𝟑 is (Average)
𝟐 𝟓 𝟒 𝟏 𝟐

(a) 450 (b) 300 (c) 600 (d) 900
28. The angle between the lines
⃗ = (𝟒𝒊 − 𝒋) + 𝝀(𝟐𝒊 + 𝒋 − 𝟑𝒌 & 𝒓
𝒓 ⃗ = (𝒊 − 𝒋 + 𝟐𝒌) + 𝝁(𝒊 − 𝟑𝒋 + 𝟐𝒌) is (Average)
3𝜋 𝜋 2𝜋 𝜋
(a) 2 (b) 3 (c) 3 (d) 6

29. The angle between the lines 𝟐𝒙 = 𝟑𝒚 = −𝒛 & 𝟔𝒙 = −𝒚 = −𝟒𝒛 is
(Average) (KCET-2018)
(a) 300 (b) 450 (c) 900 (d) 00
30. The angle between the lines 𝒙 = 𝟏, 𝒚 = 𝟐 & 𝒚 = −𝟏, 𝒛 = 𝟎 is (Average)
(a) 900 (b) 300 (c) 600 (d) 00

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31. If a line makes an angle of 𝝅/𝟑 with each 𝑿 and 𝒀 axis then the acute angle made
by 𝒁-axis is (KCET-2023) (Easy)
𝜋 𝜋 𝜋 𝜋
(a) (b) (c) (d)
2 6 4 3

32. The angle between the lines passing through the points (4, 7, 8), (2, 3, 4) & (−𝟏, −𝟐, 𝟏),
(𝟏, 𝟐, 𝟓) is (Average)
𝜋 𝜋 𝜋
(a) 0 (b) 2 (c) 4 (d) 6

33. The angle between the lines whose direction ratios are 𝒂, 𝒃, 𝒄 𝒂𝒏𝒅
𝒃 − 𝒄, 𝒄 − 𝒂 , 𝒂 − 𝒃 is (Average)
a) 450 b) 300 c) 600 d) 900
𝒙−𝟏 𝒚−𝟏 𝒛 𝒙−𝟐 𝒚−𝟏 𝟏−𝒛
34. Two lines = = 𝟓 and = = are (Difficult)
𝟑 𝟒 𝟑 𝟒 𝟓

(a) parallel (b) intersecting at an angle of 900
(c) skew lines (d) intersecting in an acute angle.
35. The measure of the angle between the lines x = k +1, y = 2k - 1, z = 2k+ 3, k
𝐱−𝟏 𝐲−𝟐 𝐳−𝟑
∈R and = = −𝟏 is (Difficult)
𝟐 𝟏

2 2 1
(a)cos−1 (3√6) (b)cos −1 (3) (c)tan−1 √2 (d)cos −1 (2)

36. The angle between two diagonals of a cube is (KCET-2014) (Average)
1 1
(a) cos−1 ( ) (b) 30∘ (c) cos−1 ( ) (d) 45∘
3 √3

√𝟑 𝟏 √𝟑 √𝟑 𝟏 −√𝟑
37. The angle between the lines whose direction cosines are ( 𝟒 , 𝟒 , 𝟐 ) and ( 𝟒 , 𝟒 , 𝟐 ) is

(KCET-2021) (Easy)
𝜋 𝜋 𝜋
(a) 𝜋 (b) 2 (c) 3 (d) 6

38. The angle between the pair of lines (KCET-2023) (Easy)
𝒙+𝟑 𝒚−𝟏 𝒛+𝟑 𝒙+𝟏 𝒚−𝟒 𝒛−𝟓
= = and = = is
𝟑 𝟓 𝟒 𝟏 𝟏 𝟐
27 19 8√3 5√3
(a) 𝜃 = cos −1 [ 5 ] (b) 𝜃 = cos −1 [21] (c) 𝜃 = cos −1 [ 15 ] (d) 𝜃 = cos −1 [ 16 ]
𝒙−𝟏 𝒚−𝟐 𝒛−𝟑 𝒙−𝟏 𝒚−𝟓 𝒛−𝟔
39. If lines −𝟑 = 𝟐𝒌 = & 𝟑𝒌 = = −𝟓 are mutually perpendicular, then
𝟐 𝟏

K= (KCET-2025) (Average)
−10 −7
(a) (b) 10 (c) −10 (d) -7
7

40. The three points 𝑨( 𝟐 , 𝟒 , 𝟑 ) , 𝑩( 𝟒 , 𝒂 , 𝟗)𝒂𝒏𝒅 𝑪(𝟏𝟎 , −𝟏 , 𝟕)form a right angled triangle
with ∠𝑩 = 𝟗𝟎𝟎 then the value of “a” is (KCET-2026) (Average)
(a) 1 or 4 (b) -1 or 4 (c) 1 or -4 (d) -1 or -4

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41. If the line joining (2, 3, -1) & (3, 5, -3) is perpendicular to the line joining (1, 2, 3)
& (3, 5, 𝝀) then 𝝀 = (Average)
(a) -3 (b) 2 (c) 5 (d) 7
𝟏−𝒙 𝟕𝒚−𝟏𝟒 𝒛−𝟑 𝟕−𝟕𝒙 𝒚−𝟓 𝟔−𝒛
42. The value of ‘P’ so that the lines = = & = = intersect at right
𝟑 𝟐𝒑 𝟐 𝟑𝒑 𝟏 𝟓

angle is (Difficult)
10 70 10 70
(a) 11 (b) 11 (c) 11 (d) 9
𝐱 𝐲 𝐳 𝟓−𝐱 𝟕𝐲−𝟏𝟒 𝐳−𝟑 𝟐
43. The angle between the lines , 𝟐 = 𝟐 = 𝟏 and = = is 𝐜𝐨𝐬 −𝟏 (𝟑) , then p is
−𝟐 𝐩 𝟒

equal to (JEE MAINS) (Difficult)
4 7 7 2
(a) − 7 (b) 2 (c) − 4 (d) 7

Shortest Distance Between Two Lines
44. The shortest distance between the lines
⃗ = −(𝒊 + 𝒋 + 𝒌) + 𝝀(𝟐𝒊 + 𝟑𝒋 + 𝟒𝒌) & 𝒓
𝒓 ⃗ = −𝒊 + 𝝁(𝟑𝒊 + 𝟒𝒋 + 𝟓𝒌) is (Difficult)
1 1 1
(a) 1 (b) (c) (d)
√2 √3 √6

45. The distance between the lines given by
⃗ = 𝒊 + 𝒋 + 𝝀(𝒊 − 𝟐𝒋 + 𝟑𝒌) &
𝒓 ⃗ = (𝟐𝒊 − 𝟑𝒌) + 𝝁(𝒊 − 𝟐𝒋 + 𝟑𝒌) is
𝒓 (Difficult)
59 59 115 √59
(a) √14 (b) √ 7 (c) √ 7 (d) 7

𝒙−𝟑 𝒚−𝟑 𝒛+𝟓
46. The distance of the point (1, 2, -4) from the line = = is
𝟐 𝟑 𝟔

(KCET-2021) (Difficult)
√293 293 √293 293
(a) (b) (c)) (d) 49
49 7 7
𝐱 𝐲−𝟐 𝐳−𝟑
47. The length of perpendicular drawn from the point (3,-1,11) to the line 𝟐 = =
𝟑 𝟒

is (KCET- 2024) (Difficult)
(a) √33 (b) √66 (c) √53 (d) √29
𝐱 𝐲 𝐳 𝐱+𝟐 𝐲−𝟒 𝐳−𝟓
48. The shortest distance between the lines 𝟐 = 𝟐 = 𝟏 and −𝟐 = = lies in the
𝟖 𝟒

interval (JEE MAINS) (Difficult)
(a) (3 , 4] (b) ( 2 , 3 ] (c) [ 1 , 2 ) (d) [ 0 , 1)
49. The distance of the point P( 4 , 6 , -2) from the line passing through the point
( -3 , 2, 3) and parallel to a line with direction ratios 3 , 3 -1 is equal to
(JEE MAINS) (Difficult)
(a) 2√3 (b) √14 (c) 3 (d) √6

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50. The projection of a line segment on 𝒙, 𝒚, 𝒛 axes are respectively √𝟐, 5, 3 the length
of the line segment is (Average)
(a) 6 (b) 11 (c) 8 (d) 5
51. The area of the quadrilateral 𝑨𝑩𝑪𝑫, when 𝑨(𝟎, 𝟒, 𝟏), 𝑩(𝟐, 𝟑, −𝟏), 𝑪(𝟒, 𝟓, 𝟎) and 𝑫(𝟐, 𝟔, 𝟐)
is equal to (KCET-2022) (Average)
(a) 9 sq. units (b) 18 sq. units
(c) 27 sq. units (d) 81 sq. units
52. Statement 1: Skew lines are non - intersecting non - parallel lines
(𝑎 ⃗⃗⃗⃗1 ×𝑏
⃗⃗⃗⃗⃗1 )∙(𝑏
⃗⃗⃗⃗⃗2 −𝑎 ⃗ 2)
Statement 2: The distance between two Skew lines is | ⃗⃗⃗⃗1 ×𝑏
|𝑏 ⃗ 2|
|. (Average)

(a) Statement 1 is true and Statement 2 is false
(b) Statement 1 is false and Statement 2 is true
(c) Statement 1 is true and Statement 2 is true
(d) Statement 1 is false and Statement 2 is false

Three-Dimensional Geometry
1 2 3 4 5 6 7 8 9 10
d a b a b b a a d a
11 12 13 14 15 16 17 18 19 20
d a b a b b a a d a
21 22 23 24 25 26 27 28 29 30
b d b b a d d c c a
31 32 33 34 35 36 37 38 39 40
b d b b a d d c c a
41 42 43 44 45 46 47 48 49 50
d b b d b c c b b a
51 52
a c

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