.
CHAPTER 10 STD 12 Date : 16/12/25
Vector Algebra Maths
//X Section A
• Write the answer of the following questions. [Each carries 2 Marks] [26]
® Ù Ù Ù ® Ù Ù Ù ® Ù Ù Ù
1. If a = i + j + k , b = 2 i - j + 3 k and c = i - 2 j + k , find a unit vector parallel to the vector
® ® ®
2a - b + 3c .
® ® ® ® ® Ù Ù Ù
2. Find a unit vector perpendicular to each of the vector a + b and a - b , where a = 3 i + 2 j + 2 k
® Ù Ù Ù
and b = i + 2 j - 2 k .
Ù Ù Ù Ù Ù Ù Ù Ù Ù
3. If vertices of triangle are A(2 i – j + k ), B( i – 3 j – 5 k ) and C(3 i – 4 j – 4 k ), determine the
type of triangle they form.
® ® ® ® ® ® ® ® ® ® ® ® ®
4. If a , b , c are unit vectors such that a + b c = 0 , find the value of a · b + b · c + c · a .
® ® ® ® ® ® ® ®
5. Find | a - b | , if two vectors a and b are such that | a | = 2, | b | = 3 and a × b = 4 .
® Ù Ù Ù
6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a = 2 i + 3 j - k and
® Ù Ù Ù
b = i - 2 j + k.
® ^ ^
7. Find a vector in the direction of vector a = i – 2 j that has magnitude 7 units.
8. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (– 1, 0, 0), (0, 1, 2) respectively then find ÐABC.
® ®
(ÐABC is the angle between vectors BA and BC )
9. Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
10. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
± æç ,
1 1 1ö
è 3 3 3÷ø
, .
® Ù Ù Ù ® Ù Ù Ù ® Ù Ù ® ®
11. If a = 2 i + 2 j + 3 k , b = - i + 2 j + k and c = 3 i + j are such that a + l b is perpendicular to
®
c , then find the value of l.
® ^ ^ ^
12. Find the area of a parallelogram whose adjacent sides are given by the vectors a = 3 i + j + 4 k and
® ^ ^ ^
b = i - j + k.
13. Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.
//X Section B
• Write the answer of the following questions. [Each carries 3 Marks] [42]
® Ù Ù Ù ® Ù Ù Ù ® Ù Ù Ù ®
14. Let a = i + 4 j + 2 k , b = 3 i - 2 j + 7 k and c = 2 i - j + 4 k . Find a vector d which is perpendicular
® ® ® ®
to both a and b and c · d = 15.
Ù Ù Ù
15. The scalar product of the vector i + j + k with a unit vector along the sum of vectors
Ù Ù Ù Ù Ù Ù
2 i + 4 j - 5 k and l i + 2 j + 3 k is equal to one. Find the value of l.
^ ^ ^
16. If with reference to the right handed system of mutually perpendicular unit vectors i , j and k ,
® ^ ^ ® ^ ^ ^ ® ® ® ® ® ®
a = 3 i - j , b = 2 i + j – 3 k , then express b in the form b = b 1 + b 2 , where b 1 is parallel to a
® ®
and b 2 is perpendicular to a .
Ù Ù Ù Ù Ù Ù
17. The two adjacent sides of a parallelogram are 2 i - 4 j + 5 k and i - 2 j - 3 k . Find the unit vector
parallel to its diagonal. Also, find its area.
, Ù Ù Ù Ù Ù Ù
The two adjacent sides of a parallelogram are 2 i - 4 j + 5 k and i - 2 j - 3 k . Find the unit vector
parallel to its diagonal. Also, find its area.
® ® ® ® ® ®
18. If a , b , c are mutually perpendicular vectors of equal magnitudes, show that the vector a + b + c
® ® ®
is equally inclined to a , b and c .
® ® ® ® ® ® ®
19. Three vectors a , b and c satisfy the condition a + b + c = 0 . Evaluate the quantity
® ® ® ® ® ® ® ® ®
m = a × b + b × c + c × a , if | a | = 1, | b | = 4 and | c | = 2.
® ® ® ® ® ®
20. Let a , b and c be three vectors such that | a | = 3, | b | = 4, | c | = 5 and each one of them being
® ® ®
perpendicular to the sum of the other two, find | a + b + c |.
21. Find the position vector of a point R which divides the line joining two points P and Q whose
® ® ® ®
position vectors are (2 a + b ) and ( a - 3 b ) externally in the ratio 1 : 2. Also, show that P is the
mid point of the line segment RQ.
22. Find the position vector of a point R which divides the line joining two points P and Q whose
Ù Ù Ù Ù Ù Ù
position vectors are i + 2 j - k and - i + j + k respectively, in the ratio 2 : 1.
(i) internally (ii) externally
Ù Ù Ù Ù Ù Ù Ù Ù Ù
23. Show that the vectors 2 i - j + k , i - 3 j - 5 k and 3 i - 4 j - 4 k form the vertices of a right
angled triangle.
® Ù Ù
p Ù p
24. If a unit vector a makes angles with i , with j and an acute angle q with k , then find q
3 4
®
and hence, the components of a .
25. Show that the points A(1, –2, –8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which
B divides AC.
® ® ® ® ® ® ®
26. Three vectors a , b and c satisfy the condition a + b + c = 0 . Evaluate the quantity
® ® ® ® ® ® ® ® ®
m = a × b + b × c + c × a , if | a | = 3, | b | = 4 and | c | = 2.
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
27. If i + j + k , 2 i + 5 j , 3 i + 2 j - 3k and i - 6 j - k are the position vectors of points A, B, C and
® ® ® ®
D respectively, then find the angle between AB and CD. Deduce that AB and CD are collinear..
, .
CHAPTER 10 STD 12 Date : 16/12/25
Vector Algebra Maths
Section [ A ] : 2 Marks Questions
No Ans Chap Sec Que Universal_QueId
1. - Chap 10 [Par... S5 7 QP25P11B1213_P2C10S5Q7
2. - Chap 10 [Par... S4 2 QP25P11B1213_P2C10S4Q2
3. - Chap 10 [Par... S11 7 QP25P11B1213_P2C10S11Q7
4. - Chap 10 [Par... S3 13 QP25P11B1213_P2C10S3Q13
5. - Chap 10 [Par... S6 17 QP25P11B1213_P2C10S6Q17
6. - Chap 10 [Par... S5 6 QP25P11B1213_P2C10S5Q6
7. - Chap 10 [Par... S6 7 QP25P11B1213_P2C10S6Q7
8. - Chap 10 [Par... S11 4 QP25P11B1213_P2C10S11Q4
9. - Chap 10 [Par... S4 9 QP25P11B1213_P2C10S4Q9
10. - Chap 10 [Par... S5 11 QP25P11B1213_P2C10S5Q11
11. - Chap 10 [Par... S3 10 QP25P11B1213_P2C10S3Q10
12. - Chap 10 [Par... S6 25 QP25P11B1213_P2C10S6Q25
13. - Chap 10 [Par... S3 16 QP25P11B1213_P2C10S3Q16
Section [ B ] : 3 Marks Questions
No Ans Chap Sec Que Universal_QueId
14. - Chap 10 [Par... S5 12 QP25P11B1213_P2C10S5Q12
15. - Chap 10 [Par... S5 13 QP25P11B1213_P2C10S5Q13
16. - Chap 10 [Par... S6 30 QP25P11B1213_P2C10S6Q30
17. - Chap 10 [Par... S5 10 QP25P11B1213_P2C10S5Q10
18. - Chap 10 [Par... S5 14 QP25P11B1213_P2C10S5Q14
19. - Chap 10 [Par... S11 10 QP25P11B1213_P2C10S11Q10
20. - Chap 10 [Par... S6 28 QP25P11B1213_P2C10S6Q28
21. - Chap 10 [Par... S5 9 QP25P11B1213_P2C10S5Q9
22. - Chap 10 [Par... S2 15 QP25P11B1213_P2C10S2Q15
23. - Chap 10 [Par... S3 17 QP25P11B1213_P2C10S3Q17
24. - Chap 10 [Par... S4 3 QP25P11B1213_P2C10S4Q3
25. - Chap 10 [Par... S5 8 QP25P11B1213_P2C10S5Q8
26. - Chap 10 [Par... S6 29 QP25P11B1213_P2C10S6Q29
27. - Chap 10 [Par... S6 27 QP25P11B1213_P2C10S6Q27
Welcome To Future - Quantum Paper
CHAPTER 10 STD 12 Date : 16/12/25
Vector Algebra Maths
//X Section A
• Write the answer of the following questions. [Each carries 2 Marks] [26]
® Ù Ù Ù ® Ù Ù Ù ® Ù Ù Ù
1. If a = i + j + k , b = 2 i - j + 3 k and c = i - 2 j + k , find a unit vector parallel to the vector
® ® ®
2a - b + 3c .
® ® ® ® ® Ù Ù Ù
2. Find a unit vector perpendicular to each of the vector a + b and a - b , where a = 3 i + 2 j + 2 k
® Ù Ù Ù
and b = i + 2 j - 2 k .
Ù Ù Ù Ù Ù Ù Ù Ù Ù
3. If vertices of triangle are A(2 i – j + k ), B( i – 3 j – 5 k ) and C(3 i – 4 j – 4 k ), determine the
type of triangle they form.
® ® ® ® ® ® ® ® ® ® ® ® ®
4. If a , b , c are unit vectors such that a + b c = 0 , find the value of a · b + b · c + c · a .
® ® ® ® ® ® ® ®
5. Find | a - b | , if two vectors a and b are such that | a | = 2, | b | = 3 and a × b = 4 .
® Ù Ù Ù
6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a = 2 i + 3 j - k and
® Ù Ù Ù
b = i - 2 j + k.
® ^ ^
7. Find a vector in the direction of vector a = i – 2 j that has magnitude 7 units.
8. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (– 1, 0, 0), (0, 1, 2) respectively then find ÐABC.
® ®
(ÐABC is the angle between vectors BA and BC )
9. Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
10. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
± æç ,
1 1 1ö
è 3 3 3÷ø
, .
® Ù Ù Ù ® Ù Ù Ù ® Ù Ù ® ®
11. If a = 2 i + 2 j + 3 k , b = - i + 2 j + k and c = 3 i + j are such that a + l b is perpendicular to
®
c , then find the value of l.
® ^ ^ ^
12. Find the area of a parallelogram whose adjacent sides are given by the vectors a = 3 i + j + 4 k and
® ^ ^ ^
b = i - j + k.
13. Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.
//X Section B
• Write the answer of the following questions. [Each carries 3 Marks] [42]
® Ù Ù Ù ® Ù Ù Ù ® Ù Ù Ù ®
14. Let a = i + 4 j + 2 k , b = 3 i - 2 j + 7 k and c = 2 i - j + 4 k . Find a vector d which is perpendicular
® ® ® ®
to both a and b and c · d = 15.
Ù Ù Ù
15. The scalar product of the vector i + j + k with a unit vector along the sum of vectors
Ù Ù Ù Ù Ù Ù
2 i + 4 j - 5 k and l i + 2 j + 3 k is equal to one. Find the value of l.
^ ^ ^
16. If with reference to the right handed system of mutually perpendicular unit vectors i , j and k ,
® ^ ^ ® ^ ^ ^ ® ® ® ® ® ®
a = 3 i - j , b = 2 i + j – 3 k , then express b in the form b = b 1 + b 2 , where b 1 is parallel to a
® ®
and b 2 is perpendicular to a .
Ù Ù Ù Ù Ù Ù
17. The two adjacent sides of a parallelogram are 2 i - 4 j + 5 k and i - 2 j - 3 k . Find the unit vector
parallel to its diagonal. Also, find its area.
, Ù Ù Ù Ù Ù Ù
The two adjacent sides of a parallelogram are 2 i - 4 j + 5 k and i - 2 j - 3 k . Find the unit vector
parallel to its diagonal. Also, find its area.
® ® ® ® ® ®
18. If a , b , c are mutually perpendicular vectors of equal magnitudes, show that the vector a + b + c
® ® ®
is equally inclined to a , b and c .
® ® ® ® ® ® ®
19. Three vectors a , b and c satisfy the condition a + b + c = 0 . Evaluate the quantity
® ® ® ® ® ® ® ® ®
m = a × b + b × c + c × a , if | a | = 1, | b | = 4 and | c | = 2.
® ® ® ® ® ®
20. Let a , b and c be three vectors such that | a | = 3, | b | = 4, | c | = 5 and each one of them being
® ® ®
perpendicular to the sum of the other two, find | a + b + c |.
21. Find the position vector of a point R which divides the line joining two points P and Q whose
® ® ® ®
position vectors are (2 a + b ) and ( a - 3 b ) externally in the ratio 1 : 2. Also, show that P is the
mid point of the line segment RQ.
22. Find the position vector of a point R which divides the line joining two points P and Q whose
Ù Ù Ù Ù Ù Ù
position vectors are i + 2 j - k and - i + j + k respectively, in the ratio 2 : 1.
(i) internally (ii) externally
Ù Ù Ù Ù Ù Ù Ù Ù Ù
23. Show that the vectors 2 i - j + k , i - 3 j - 5 k and 3 i - 4 j - 4 k form the vertices of a right
angled triangle.
® Ù Ù
p Ù p
24. If a unit vector a makes angles with i , with j and an acute angle q with k , then find q
3 4
®
and hence, the components of a .
25. Show that the points A(1, –2, –8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which
B divides AC.
® ® ® ® ® ® ®
26. Three vectors a , b and c satisfy the condition a + b + c = 0 . Evaluate the quantity
® ® ® ® ® ® ® ® ®
m = a × b + b × c + c × a , if | a | = 3, | b | = 4 and | c | = 2.
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
27. If i + j + k , 2 i + 5 j , 3 i + 2 j - 3k and i - 6 j - k are the position vectors of points A, B, C and
® ® ® ®
D respectively, then find the angle between AB and CD. Deduce that AB and CD are collinear..
, .
CHAPTER 10 STD 12 Date : 16/12/25
Vector Algebra Maths
Section [ A ] : 2 Marks Questions
No Ans Chap Sec Que Universal_QueId
1. - Chap 10 [Par... S5 7 QP25P11B1213_P2C10S5Q7
2. - Chap 10 [Par... S4 2 QP25P11B1213_P2C10S4Q2
3. - Chap 10 [Par... S11 7 QP25P11B1213_P2C10S11Q7
4. - Chap 10 [Par... S3 13 QP25P11B1213_P2C10S3Q13
5. - Chap 10 [Par... S6 17 QP25P11B1213_P2C10S6Q17
6. - Chap 10 [Par... S5 6 QP25P11B1213_P2C10S5Q6
7. - Chap 10 [Par... S6 7 QP25P11B1213_P2C10S6Q7
8. - Chap 10 [Par... S11 4 QP25P11B1213_P2C10S11Q4
9. - Chap 10 [Par... S4 9 QP25P11B1213_P2C10S4Q9
10. - Chap 10 [Par... S5 11 QP25P11B1213_P2C10S5Q11
11. - Chap 10 [Par... S3 10 QP25P11B1213_P2C10S3Q10
12. - Chap 10 [Par... S6 25 QP25P11B1213_P2C10S6Q25
13. - Chap 10 [Par... S3 16 QP25P11B1213_P2C10S3Q16
Section [ B ] : 3 Marks Questions
No Ans Chap Sec Que Universal_QueId
14. - Chap 10 [Par... S5 12 QP25P11B1213_P2C10S5Q12
15. - Chap 10 [Par... S5 13 QP25P11B1213_P2C10S5Q13
16. - Chap 10 [Par... S6 30 QP25P11B1213_P2C10S6Q30
17. - Chap 10 [Par... S5 10 QP25P11B1213_P2C10S5Q10
18. - Chap 10 [Par... S5 14 QP25P11B1213_P2C10S5Q14
19. - Chap 10 [Par... S11 10 QP25P11B1213_P2C10S11Q10
20. - Chap 10 [Par... S6 28 QP25P11B1213_P2C10S6Q28
21. - Chap 10 [Par... S5 9 QP25P11B1213_P2C10S5Q9
22. - Chap 10 [Par... S2 15 QP25P11B1213_P2C10S2Q15
23. - Chap 10 [Par... S3 17 QP25P11B1213_P2C10S3Q17
24. - Chap 10 [Par... S4 3 QP25P11B1213_P2C10S4Q3
25. - Chap 10 [Par... S5 8 QP25P11B1213_P2C10S5Q8
26. - Chap 10 [Par... S6 29 QP25P11B1213_P2C10S6Q29
27. - Chap 10 [Par... S6 27 QP25P11B1213_P2C10S6Q27
Welcome To Future - Quantum Paper
