VECTOR LEVEL−I
1.
OA and OB are two vectors such that | OA OB | = | OA 2 OB | . Then (A) BOA = 90 (B) BOA > 90 (C) BOA < 90 (D) 60 BOA 90
2.
If b and c are two non-collinear vectors such such that a. b c 4 and
a b c x 2 2x 6 b sin y c , then the point ( x, y) lies on (A) x =1 (B) y =1 (C) y = (D) x + y = 0
3.
The scalar a . b c a b c equals (A) 0
(B) 2 a b c
(C) a b c
(D) None of these
4.
If aˆ, bˆ, cˆ are three unit vectors, such that aˆ bˆ cˆ is also a unit vector, and 1, 2, 3 are angle between the vectors, aˆ, bˆ; bˆ, cˆ and cˆ, aˆ respectively then cos1 + cos2 + cos3 equals (A) 3 (B) -3 (C) 1 (D) -1
5.
If angle between a and b is
/3
(A) 6.
7.
a
3, b
-/3
(C)
2/3
(D)
-2/3
a
b
5 such that each is perpendicular perpendicular to sum of the other other two, then
4, c
c =
(B)
5
(C) 10 2
2
(D) 5 3
If x and y are two vectors and is the angle between them, then
If u
(B) iˆ
a iˆ
ˆj ( a
(A) u is unit vector (C) u = 2a 10.
3
, then angle between 2a and 3b is
The vectors 2ˆi m jˆ 3mkˆ and 1 m ˆi 2m jˆ kˆ include an acute angle for (A) all real m (B) m < –2 or m > –1/2 (C) m = –1/2 (D) m [–2, –1/2]
(A) 0
9.
(B)
(A) 5 2
8.
2 ˆj )
(C) sin kˆ
2
1 x 2
y is equal to
(D) cos
2
( a kˆ ) , then
(B) u = a + i + j + k (D) none of these
Let aˆ and bˆ be two unit vectors such that aˆ bˆ is also a unit vector. Then the angle between aˆ and bˆ is (A) 30 (C) 90
(B) 60 (D) 120
ˆ If a ˆi jˆ kˆ , b 4ˆi 3 jˆ 4kˆ and c ˆi jˆ kˆ are linearly dependent vectors and c =
11.
3. (A) =1, = -1 (C) = -1, 1
(B) = 1, 1 (D) = 1, = 1
Let a 2ˆi jˆ 2kˆ and b ˆi jˆ . If c is a vector such that a c = c , c a 2 2 and the
12.
angle between a b and c is 30, then a b c =
2 3 (C) 2
3 2 (D) 3
(A)
(B)
13.
Let a i k , b x i j 1 x k and b y i x j 1 x y k . Then a b c depends on (A) only x (B) only y (C) NEITHER x NOR y (D) both x and y
14.
If | a b || a | , then b. 2a b equals (A) 0 (C) 2a.b
(B) 1 (D) none of these
If | a | = 3, | b | = 5, | c | = 7 and a b c = 0, then angle between a and b is
15.
(A) (C)
4
(B)
3
(D) none of these
2
Given that angle between the vectors a ˆi 3 jˆ kˆ and b 2 ˆi jˆ kˆ is acute, whereas
16.
the vector b makes with the co-ordinate axes on obtuse angle then belongs to (A) (-, 0) (B) (0, ) (C) R (D) none of these
17. If a, b and c are unit coplanar vectors then the scalar triple product
2a b, 2b c, 2c a = (A) 0
(B) 1
(C) 3
(D)
3
a b a b , then the angle between a and b is
18. If
(A) acute (C) /2
(B) obtuse (D) none of these
b c and r 2b y c b intersect at a point with position vector |b| | c |
19. If the lines r x
b c , then | b | | c |
z
(A) z is the AM between | b | and | c |
(B) z is the GM between | b | & | c |
(C) z is the HM between | b | and | c | 20.
(D) z = | b | + | c |
25.
` 26.
(B) Two (D) Infinite
If p and d are two unit vectors and is the angle between them, then 2 1 sin = sin (A) p d (B) p d 2 2 2 2 1 1 (C) p d (D) pˆ d ˆ 1 cos 2 1 cos 2 2
The value of k for which the points A(1, 0, 3) , B(-1, 3,4) ,C(1, 2, 1) and D(k, 2, 5) are coplanar is (2)2 (D) -1
a
a2
1 a3
If b
b2
1 b3
c
c2
1 c3
0 and the vectors A = (1, a, a2), B = (1, b, b2), C = (1,c,c2) are
non - coplanar, then the value of abc will be (A) –1 (B) 1 (C) 0 (D) None of these ˆ , ˆi k ˆ , cˆi c jˆ bk ˆ lie in Let a, b, c be distinct non-negative numbers. If the vectors aˆi a jˆ ck a plane, then c is (A) the arithmetic mean of a and b (B) the geometric mean of a and b (C) the harmonic mean of a and b (D) equal to zero The unit vector perpendicular to the plane determined by P(1, -1, 2), Q(2, 0, -1), R(0, 2, 1) is i 2 j k i j 2k (A) (B) 6 6 2i j k (C) (D) None of these 6
27.
(A) 1 (C) 0
24.
The number of unit vectors perpendicular to vectors a 1,1,0 and b = 0,1,1 is
23.
(B) a b (D) c a
(A) One (C) Three 22.
(A) a b c (C) b c 21.
Let ABCDEF be a regular hexagon and AB a , BC b , CD c then AE is
If A, B, C are non-coplanar vectors then (A) 3 (B) 1
A . B C B . A C
is equal to
C A. B C . A B (B) 0 (D) None of there
28. of
If the vector aiˆ jˆ k ˆ, iˆ b jˆ k ˆ and iˆ jˆ ck ˆ (a b c1) are coplanar, then the value 1 1 a (A) 1 (C) 2
29.
1
1 b
1 1 c
is equal to (B) 0 (D) None of these
If a , b , c are vectors such that a . b =0 and a b
2
2
(A) a (C) b
2
b
a
2
2
c . Then
c
(B) a
2
2
b
2
2
c
(D) None of these
c
30.
The points with position vector 60i + 3j, 40i – 8j and ai –52j are collinear if (A) a = -40 (B) a = 40 (C) a = 20 (D) none of these .
31.
Let aˆ and bˆ be two unit vectors such that aˆ bˆ is also a unit vector. Then the angle between aˆ and bˆ is (A) 30
(B) 60
(C) 90
(D) 120
32.
If vectors ax ˆi 3 jˆ 5kˆ and x ˆi 2 jˆ 2axkˆ make an acute angle with each other, for all x R, then a belongs to the interval 1 6 3 (A) ,0 (B) ( 0, 1) (C) 0, (D) ,0 4 25 25
33.
A vector of unit magnitude that is equally inclined to the vectors ˆi jˆ , jˆ kˆ and ˆi kˆ is; 1 ˆ ˆ ˆ 1 ˆ ˆ ˆ i j k i j k (A) (B) 3 3 1 ˆ ˆ ˆ i j k (C) (D) none of these 3
34.
Let a, b, c be three distinct positive real numbers. If p, q, r lie in plane, where p a ˆi a jˆ bkˆ , q ˆi kˆ and r c ˆi c jˆ b kˆ then b is (A) A.M of a, c (C) the H.M of a, c
(B) the G.M of a, c (D) equal to c
85.
The scalar A . B C A B C is equal to ______________________
36.
If a, b, c are unit coplanar vectors, then the scalar triple product 2a b, 2b c, 2c a is equal to _____________________
37.
The area of a parallelogram whose diagonals represent the vectors 3ˆi jˆ 2kˆ and ˆi 3 jˆ 4kˆ is (A) 10 3 (C) 8
(B) 5 3 (D) 4
38.
The value of a b b c c a is equal to (A) 2 a b c
(B) 3 a b c
(C) a b c
(D) 0
LEVEL −II
1.
If a is any vector in the plane of unit vectors bˆ and cˆ , with bˆ cˆ = 0, then the magnitude of the vector a bˆ cˆ is
2.
(A) | a | (B) 2 (C) 0 (D) none of these . If a and b are two unit vectors and is the angle between them, then the unit vector
along the angular bisector of a and b will be given by
(A)
2 cos
(C)
2 cos
2
2
(D) none of these.
2 If a is a unit vector and projection of x along a is 2 units and a x b x , then x is given by 1 1 a b a b (A) (B) 2a b a b 2 2 (C) a a b (D) none of these.
4.
ab
(B)
ab 2 sin
3.
ab
If 4 a 5b 9c 0 , then ( a b ) [ ( b c ) ( c a ) ]is equal to (A) A vector perpendicular to plane of a, b and c
(B)
A scalar quantity
(C) 0
(D)
None of these
5.
The shortest distance of the point (3, 2, 1) from the plane, which passes through a(1, 1, 1) and which is perpendicular to vector 2ˆi 3kˆ , is 4 1 (A) (B) 2 (C) 3 (D) 3 13
6.
Let a 2ˆi jˆ kˆ , b ˆi 2 jˆ kˆ and a unit vector c be coplanar. If c is perpendicular to
a then c = 1 (A) jˆ kˆ 2 1 ˆ (C) i ˆ 2 j 5
7.
1 ˆ ˆ ˆ i j k 3 1 ˆ ˆ ˆ i j k 2
(B) (D)
Let a and b be the two non–collinear unit vector. If u a a b b and v a b , then v is
(A) u
(C) u u a b
(B) u u a (D) none of these
2
8.
2
(A) 4 (C) 8 9.
2
If a, b and c are unit vectors, then a b b c c a does NOT exceed (B) 9 (D) 6
If a r b ta and a.r 3, where a 2ˆi jˆ kˆ and b ˆi 2 jˆ kˆ then r equals 7 2 7 1 (A) ˆi jˆ (B) ˆi jˆ 6 5 6 3 7ˆ 2ˆ 1ˆ (C) i j k (D) none of these 6 3 3
10. If a b b c c a = 0 and at least one of the numbers , and is non-zero,
then the vectors a, b and c are (A) perpendicular (C) co-planar
(B) parallel (D) none of these
11. The vectors a and b are non-zero and non-collinear. The value of x for which vector c = (x –2) a + b and d = (2x +1) a – b are collinear. (A) 1 (B) 1/2 (C) 1/3 (D) 2 12
a b
c,
(A) a = 1, b
(C) b = 2, b 13.
b c
a , then
c
(B) c = 1, a = 1
2a
(D) b = 1, b
a
If a , b , c are three non - coplanar vectors and p , q , r are vectors defined by the relations p
b c abc
, q
c
a
abc
, r
a b a bc
then the value of expression
(a + b).p + (b + c).q + (c + a).r is equal to
(A) 0 (C) 2 14.
15.
(B) 1 (D) 3
The value of |a iˆ |2 + |a (A) a2 (C) 3a2
is (B) 2a2 (D) None of these
If a ˆi jˆ, b 2 jˆ kˆ and r a b a, r b a b , then a unit vector in the direction of r is; 1 ˆ 1 ˆ i 3 jˆ kˆ i 3 jˆ kˆ (A) (B) 11 11 1 ˆ ˆ ˆ i j k (C) (D) none of these 3
16.
ˆj|2 + |a kˆ | 2
a.ˆi a ˆi a. jˆ a jˆ a.kˆ a kˆ is equal to; (A) 3 a
(B) r
(C) 2 r
(D) none of these
17.
If the vertices of a tetrahedron have the position vectors 0, ˆi jˆ, 2 jˆ kˆ and ˆi kˆ then the volume of the tetrahedron is (A) 1/6 (B) 1 (C) 2 (D) none of these
18.
A = (1, -1, 1), C = (-1, -1, 0) are given vectors; then the vector B which satisfies A B C and A.B 1 is ___________________________________
19.
If a, b, c are given non-coplanar unit vectors such that a (b c )
bc , then the angle 2
between a and c is ________________________________ 20.
Vertices of a triangle are (1, 2, 4) (3, 1, -2) and (4, 3, 1) then its area is_______________
21.
A unit vector coplanar with i j 2k and i 2 j k and perpendicular to i j k _______________________
is
LEVEL−III
1. If a, b, c are coplanar vectors and a is not parallel to b then c b a b a a c a b b is equal to (A) a b a b c (B) a b a b c
(C) a b a b c
(D) none of these
2.
The projection of ˆi jˆ kˆ on the line whose equation is r = (3 + ) ˆi + (2 -1) jˆ + 3 kˆ , being the scalar parameter is; 1 (A) (B) 6 14 6 (C) (D) none of these 14
3.
If p, q are two non-collinear and non-zero vectors such that (b –c) p q +(c –a) p + (a –b) q = 0 where a, b, c are the lengths of the sides of a triangle, then the triangle is (A) right angled (B) obtuse angled (C) equilateral (D) isosceles
L−I 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37.
B A C A C B C B A C B D B B A D C O B
2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 22. 24. 26. 28. 30. 32. 34. 36. 38.
A D B D B A A A C C A C A A C C O A
A B A A D C D A A
2. 4. 6. 8. 10. 12. 14. 16. 18. 20.
B C A B C D B D K 5 5/2
L−II 1. 3. 5. 7. 9. 11. 13. 15. 17. 19.
/ 3
21.
−
J K 2
ON
J K 2
L−III 1. 3.
2. C
C
