Summary - Mathematics

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CHAPTER 11 STD 12 Date : 16/12/25
Three Dimensional Geometry Maths
//X Section A
• Write the answer of the following questions. [Each carries 2 Marks] [48]
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1. Find the distance between the lines l1 and l2 given by r = i + 2 j - 4 k + l(2 i + 3 j + 6 k ) and
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r = 3 i + 3 j - 5 k + m(2 i + 3 j + 6 k ) .
2. Find the vector equation of the line passing through the point (1, 2, –4) and perpendicular to the
x - 8 y + 19 z - 10 x - 15 y - 29 z - 5
two lines : 3 = - 16 = and = = .
7 3 8 -5
1 – x 7 y – 14 z – 3 7 – 7x y – 5 6 – z
3. Find the values of p so that the lines 3 = 2p
=
2 and 3p = 1 = 5 are at
right angles.
4. Find the equation of the line in vector and in cartesian form that passes through the point with
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position vector 2 i – j + 4 k and is in the direction i + 2 j - k .
5. Find the direction cosines of a line which makes equal angles with the coordinate axes.
6. Find the vector and the Cartesian equations of the line through the point (5, 2, – 4) and which is
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parallel to the vector 3 i + 2 j - 8 k .
7. Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.
8. Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points
(–1, –2, 1), (1, 2, 5).
x – 5 y + 4 z – 6
9. The cartesian equation of a line is = = . Write its vector form.
3 7 2
10. Show that the line through the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the line through the
points (0, 3, 2) and (3, 5, 6).
x + 3 y -1 z + 3 x +1 y - 4 z -5
11. Find the angle between the pair of lines = = and = = .
3 5 4 1 1 2
12 –3 –4 4 12 3 3 –4 12
12. Show that the three lines with direction cosines , , ; , , ; , , are mutually
13 13 13 13 13 13 13 13 13
perpendicular.
13. If a line has direction ratios 2, –1, –2, determine its direction cosines.
14. Find the cartesian equation of the line which passes through the point (–2, 4, –5) and parallel to the
x+3 y-4 z+8
line given by = = .
3 5 6
15. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.
x –5 y + 2 z x y z
16. Show that the lines = = and = = are perpendicular to each other..
7 –5 1 1 2 3
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17. ) Find the angle between the following pairs of lines : r = 2 i - 5 j + k + l (3 i + 2 j + 6 k ) and
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r = 7 i - 6 k + m ( i + 2 j + 2 k)
18. Find the shortest distance between the lines whose vector equations are
® Ù Ù Ù Ù Ù Ù ® Ù Ù Ù Ù Ù Ù
r = (i + 2 j + 3 k ) + l (i – 3 j + 2 k ) and r = 4 i + 5 j + 6 k + m (2 i + 3 j + k ) .
19. If a line has the direction ratios –18, 12, – 4, then what are its direction cosines ?
20. Show that the points A(2, 3, – 4), B(1, – 2, 3) and C(3, 8, – 11) are collinear.
x - 2 y -1 z +3 x+2 y-4 z-5
21.1) Find the angle between the following pair of lines : = = and = =
2 5 -3 -1 8 4
x –1 y – 2 z – 3 x –1 y –1 z –6
22. If the lines = = and = = are perpendicular, find the value of k.
–3 2k 2 3k 1 –5
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23.1) Find the angle between the following pairs of lines : r = 3 i + j - 2 k + l ( i - j - 2 k ) and
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r = 2 i - j - 56 k + m ( 3 i - 5 j - 4 k )

, ® Ù Ù Ù Ù Ù Ù
1) Find the angle between the following pairs of lines : r = 3 i + j - 2 k + l ( i - j - 2 k ) and
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r = 2 i - j - 56 k + m ( 3 i - 5 j - 4 k )
24. If a line makes angles 90°, 135°, 45° with the x, y and z- axes respectively, find its direction cosines.

//X Section B
• Write the answer of the following questions. [Each carries 3 Marks] [21]
25. Find the shortest distance between the lines whose vector equations are
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r = (1 – t ) i + (t – 2) j + (3 – 2t ) k and r = ( s + 1) i + ( 2s – 1) j – ( 2s + 1) k .
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26. . Find the shortest distance between the lines r = ( l – 1)i + ( l + 1) j – (1 + l )k and
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r = (1 – m ) i + (2m – 1) j + (m + 2)k .
x +1 y +1 z +1 x – 3 y – 5 z – 7
27. Find the shortest distance between the lines = = and = = .
7 -6 1 1 –2 1
28. Find the shortest distance between the lines l1 and l2 whose vector equations are
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r = i + j + l(2 i - j + k ) and
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r = 2 i + j - k + m(3 i - 5 j + 2 k ).
® æÙ Ù Ùö æÙ Ù Ùö
29. Find the shortest distance between the lines r = çè i + 2 j + k ÷ø + l çè i – j + k ÷ø and

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r = 2 i - j - k + m ç 2 i + j + 2k÷ .
è ø
30. Find the shortest distance between lines
® Ù Ù Ù Ù Ù Ù ® Ù Ù Ù Ù Ù
r = 6 i + 2 j + 2 k + l (i - 2 j + 2 k ) and r = – 4 i – k + m (3 i – 2 j – 2 k ) .
31. Find the direction cosines of the sides of the triangle whose vertices are (3, 5, – 4), (– 1, 1, 2) and
(– 5, – 5, – 2).

, .
CHAPTER 11 STD 12 Date : 16/12/25
Three Dimensional Geometry Maths

Section [ A ] : 2 Marks Questions

No Ans Chap Sec Que Universal_QueId
1. - Chap 11 [Par... S4 10 QP25P11B1213_P2C11S4Q10
2. - Chap 11 [Par... S3 5 QP25P11B1213_P2C11S3Q5
3. - Chap 11 [Par... S2 10 QP25P11B1213_P2C11S2Q10
4. - Chap 11 [Par... S2 5 QP25P11B1213_P2C11S2Q5
5. - Chap 11 [Par... S1 2 QP25P11B1213_P2C11S1Q2
6. - Chap 11 [Par... S4 6 QP25P11B1213_P2C11S4Q6
7. - Chap 11 [Par... S1 4 QP25P11B1213_P2C11S1Q4
8. - Chap 11 [Par... S2 3 QP25P11B1213_P2C11S2Q3
9. - Chap 11 [Par... S2 7 QP25P11B1213_P2C11S2Q7
10. - Chap 11 [Par... S2 2 QP25P11B1213_P2C11S2Q2
11. - Chap 11 [Par... S4 8 QP25P11B1213_P2C11S4Q8
12. - Chap 11 [Par... S2 1 QP25P11B1213_P2C11S2Q1
13. - Chap 11 [Par... S4 2 QP25P11B1213_P2C11S4Q2
14. - Chap 11 [Par... S2 6 QP25P11B1213_P2C11S2Q6
15. - Chap 11 [Par... S3 1 QP25P11B1213_P2C11S3Q1
16. - Chap 11 [Par... S2 11 QP25P11B1213_P2C11S2Q11
17. - Chap 11 [Par... S2 8.1 QP25P11B1213_P2C11S2Q8.1
18. - Chap 11 [Par... S2 14 QP25P11B1213_P2C11S2Q14
19. - Chap 11 [Par... S1 3 QP25P11B1213_P2C11S1Q3
20. - Chap 11 [Par... S4 5 QP25P11B1213_P2C11S4Q5
21. - Chap 11 [Par... S2 9.1 QP25P11B1213_P2C11S2Q9.1
22. - Chap 11 [Par... S3 3 QP25P11B1213_P2C11S3Q3
23. - Chap 11 [Par... S2 8.1 QP25P11B1213_P2C11S2Q8.1
24. - Chap 11 [Par... S1 1 QP25P11B1213_P2C11S1Q1


Section [ B ] : 3 Marks Questions

No Ans Chap Sec Que Universal_QueId
25. - Chap 11 [Par... S2 15 QP25P11B1213_P2C11S2Q15
26. - Chap 11 [Par... S7 4.4 QP25P11B1213_P2C11S7Q4.4
27. - Chap 11 [Par... S2 13 QP25P11B1213_P2C11S2Q13
28. - Chap 11 [Par... S4 9 QP25P11B1213_P2C11S4Q9
29. - Chap 11 [Par... S2 12 QP25P11B1213_P2C11S2Q12

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