VECTOR -3D and
Complex Numbers
With some miscellaneous problems
Question bank on Vector, 3D and Complex Number & Misc. Subjective Problem
Select the correct alternative : (Only one is correct)
Q.1


(A) 10
Q.2
 
 
If a = 11 , b = 23 , a  b = 30 , then a  b is :
(B) 20
(C) 30
(D) 40

The position vector of a point P moving in space is given by OP  R  (3 cos t )î  (4 cos t ) ĵ  (5 sin t )k̂ .
The time 't' when the point P crosses the plane 4x – 3y + 2z = 5 is



(B) sec
(C) sec
(A) sec
2
6
3
Q.3
(D)

sec
4
Indicate the correct order sequence in respect of the following :
x4 y6
y6
x 1 y  2 z  3
=
=
and
=
=
are orthogonal.
3
1
1
1
2
2
II.
The planes 3x – 2y – 4z = 3 and the plane x – y – z = 3 are orthogonal.
III.
The function f (x) = ln(e–2 + ex) is monotonic increasing ∀ x ∈ R.
IV.
If g is the inverse of the function, f (x) = ln(e–2 + ex) then g(x) = ln(ex – e–2).
(A) FTFF
(B) TFTT
(C) FFTT
(D) FTTT
I.
The lines
z1
z1  z 2
Q.4* If z is purely imaginary then z  z
2
1
2
(A) 1
Q.5
Q.6
(B) 2
(C) 3
(D) 0
In a regular tetrahedron, the centres of the four faces are the vertices of a smaller tetrahedron. The ratio
m
of the volume of the smaller tetrahedron to that of the larger is , where m and n are relatively prime
n
positive integers. The value of (m + n) is
(A) 3
(B) 4
(C) 27
(D) 28
  
  
1  
 
If a , b , c are non-coplanar unit vectors such that a x (b x c) 
( b  c) then the angle between a & b is
(A) 3 π/4
Q.7
is equal to :
(B) π/4
(C) π/2
2
(D) π
The sine of angle formed by the lateral face ADC and plane of the base ABC of the tetrahedron ABCD
where A ≡ (3, –2, 1); B ≡ (3, 1, 5); C ≡ (4, 0, 3) and D ≡ (1, 0, 0) is
(A)
2
29
(B)
5
29
(C)
3 3
29
(D)
2
29
Q.8* Let z be a complex number, then the region represented by the inequality ⏐z + 2⏐ < ⏐z + 4⏐ is
given by :
(A) Re (z) > − 3
(B) Im (z) < − 3
(C) Re (z) < − 3 & Im (z) > − 3
(D) Re (z) < − 4 & Im (z) > − 4
Q.9

The volume of the parallelpiped whose edges are represented by the vectors a  2 i  3 j  4 k ,


b  3 i  j  2 k , c  i  2 j  k is :
(A) 7
(B) 5
(C) 4
(D) none
[2]
  
  



Q.10 Let u , v , w be the vectors such that u  v  w  0 , if u  3 , v  4 & w  5 then the value of
   
 
u . v  v . w  w .u is :
(A) 47
(B) −25
(C) 0
(D) 25



  


 
Q.11 Let a = î  ˆj , b = ˆj  k̂ , c = k̂  î . If d is a unit vector such that a . d  0  [b , c , d] then d
(A) 
1
( î  ˆj 2 k̂ )
6
(B) 
1
( î  ˆj k̂ )
3
Q.12* If z be a complex number for which z 
(A)
(B)
2 1

(C) 
1
( î ˆj  k̂ )
3
(D) ± k̂
1
 2 , then the greatest value of ⏐z⏐ is :
z
3 1
(C) 2 2  1
(D) none

Q.13 If the non − zero vectors a & b are perpendicular to each other, then the solution of the equation ,
  
r  a  b is :


1
a .a

 
(C) r  x a  b



(A) r  x a    a  b





1
b.b
 

(D) r  x b  a

(B) r  x b    a  b

where x is any scalar.
x 2 y3 z 4
x 1 y  4 z  5




and
are coplanar if
k
2
1
1
1
k
(A) k = 1 or – 1
(B) k = 0 or – 3
(C) k = 3 or – 3
(D) k = 0 or – 1
Q.14 The lines
Q.15 Which one of the following statement is INCORRECT ?
 
 
 




(A) If n . a = 0 , n . b = 0 & n . c = 0 for some non zero vector n , then a b c = 0
(B) there exist a vector having direction angles α = 30º & β = 45º
(C) locus of point for which x = 3 & y = 4 is a line parallel to the z - axis whose distance from the
z-axis is 5



(D) In a regular tetrahedron OABC where ' O ' is the origin , the vector OA  OB  OC is perpendicular
to the plane ABC.
Q.16* Given that the equation, z2 + (p + iq) z + r + is = 0 has a real root where p, q, r, s ∈ R. Then which one
is correct
(A) pqr = r2 + p2s
(B) prs = q2 + r2p
(C) qrs = p2 + s2q
(D) pqs = s2 + q2r
Q.17 The distance of the point (3, 4, 5) from x-axis is
(D) 41
(B) 5
(C) 34
 

Q.18 Given non zero vectors A, B and C , then which one of the following is false?
 

  
(A) A vector orthogonal to A  B and C is ± (A  B)  C
 
 
 
(B) A vector orthogonal to A  B and A  B is ± A  B
 

  
(C) Volume of the parallelopiped determined by A, B and C is | A  B ·C |
 

 A ·B
(D) Vector projection of A onto B is 
| B|
(A) 3
[3]
  
Q.19 Given three vectors a , b , c such that they are non − zero, non − coplanar vectors, then which of the
following are non coplanar.














(A) a  b , b  c , c  a




(B) a  b , b  c , c  a
     
(D) a  b, b  c, c  a
(C) a  b , b  c , c  a
Q.20* The sum i + 2i2 + 3i3 + ........ + 2002i2002, where i =  1 is equal to
(A) – 999 + 1002i
(B) – 1002 + 999i
(C) – 1001 + 1000i (D) – 1002 + 1001i

Q.21 Locus of the point P , for which OP represents a vector with direction cosine
cos α =
1
( ' O ' is the origin) is :
2
(A) A circle parallel to y z plane with centre on the x − axis
(B) a cone concentric with positive x − axis having vertex at the origin and the slant height equal to the
magnitude of the vector
(C) a ray emanating from the origin and making an angle of 60º with x − axis
| |

(D) a disc parallel to y z plane with centre on x − axis & radius equal to O P sin 60º


Q.22 A line with direction ratios (2, 1, 2) intersects the lines r  ˆj   ( î  ˆj  k̂ ) and r  î  ( 2î  ĵ  k̂ ) at
A and B, then l (AB) is equal to
(A) 3
(B) 3
(C) 2 2
(D)
2
Q.23 The vertices of a triangle are A (1, 1, 2) , B (4, 3, 1) & C (2, 3, 5). The vector representing the internal
bisector of the angle A is :
(A) i  j  2 k
(B) 2 i  2 j  k
(C) 2 i  2 j  k
(D) 2 i  2 j  k
Q.24* Lowest degree of a polynomial with rational coefficients if one of its root is, 2  i is
(A) 2
(B) 4
(C) 6
(D) 8
Q.25 A plane vector has components 3 & 4 w.r.t. the rectangular cartesian system. This system is rotated
through an angle
(A) 4, 3

in anticlockwise sense. Then w.r.t. the new system the vector has components :
4
(B)
7
2
,
1
2
(C)
1
2
,
7
(D) none
2




Q.26 Let a = xi + 12j − k ; b = 2i + 2xj + k & c = i + k . If the ordered set b c a is left handed , then:

(A) x ∈ (2, ∞)
(B) x ∈ (− ∞, − 3)
(C) x ∈ (− 3, 2)

(D) x ∈ {− 3, 2}
Q.27 If cos  î  ˆj  k̂ , î  cos  ˆj  k̂ & î  ˆj  cos  k̂ (α ≠ β ≠ γ ≠ 2 n π) are coplanar then the value of

2
2
2
cos ec 2  cosec 2  cos ec 2  equal to


(A) 1
(B) 2
(C) 3
(D) none of these
Q.28* The straight line (1 + 2i)z + (2i – 1) z = 10i on the complex plane, has intercept on the imaginary axis
equal to
(A) 5
(B)
5
2
(C) –
5
2
(D) – 5
[4]
Q.29 The perpendicular distance of an angular point of a cube of edge 'a' from the diagonal which does not
pass that angular point, is
(B)
(A) 3 a
2a
(C)
3
a
2
(D)
2
a
3
   
Q.30 Which one of the following does not hold for the vector V = a x b x a ?


(A) perpendicular to a


(C) coplanar with a & b


(B) perpendicular to b
 
(D) perpendicular to a x b .
Q.31* Let z1, z2 & z3 be the complex numbers representing the vertices of a triangle ABC respectively. If P is
a point representing the complex number z0 satisfying ;
a (z1 − z0) + b (z2 − z0) + c (z3 − z0) = 0 , then w.r.t. the triangle ABC, the point P is its :
(A) centroid
(B) orthocentre
(C) circumcentre
(D) incentre

Q.32 Given the position vectors of the vertices of a triangle ABC, A  (a ) ; B  ( b ) ; C  ( c ) . A vector r is
parallel to the altitude drawn from the vertex A, making an obtuse angle with the positive Y-axis. If





| r | = 2 34 ; a  2î  ˆj  3k̂ ; b  î  2 ˆj  4k̂ ; c  3î  ˆj  2k̂ , then r is
(A)  6î  8ˆj  6k̂
(B) 6î  8ˆj  6k̂
(C)  6î  8ˆj  6k̂
(D) 6î  8ˆj  6k̂
Q.33* The complex number, z = 5 + i has an argument which is nearly equal to :
(A) π/32
(B) π/16
(C) π/12
(D) π/8
  
a.a a.b a.c





Q.34 If a  î  ˆj  k̂ , b  î  ˆj  k̂ , c  î  2ˆj  k̂ , then the value of b.a b.b b.c equal to
 

c.a c.b c.c
(A) 2
(B) 4
(C) 16
(D) 64
Q.35* If the equation x2 + ax + b = 0 where a, b ∈ R has a non real root whose cube is 343 then
(7a + b) has the value
(A) 98
(B) – 49
(C) – 98
(D) 49
Direction for Q.36 to Q.40.


Let
A  î  2 ˆj  3k̂ and B  3î  4 ˆj  5k̂
 
 
Q.36 The value of the scalar | A  B |2 (A ·B) 2 is equal to
(A) 8
(B) 7 10
(C) 10 7
(D) 64
Q.37 Equation of a line passing through the point with position vector 2î  3 ˆj and orthogonal to the plane


containing the vectors A and B , is


(B) r  (  2) î  ( 2  3)ˆj  k̂
(A) r  (  2) î  ( 2  3)ˆj  k̂

(C) r   î  ( 2  3)ˆj  k̂
(D) none
[5]
Q.38 Equation of a plane containing the point with position vector (î  ˆj  k̂ ) and parallel to the vectors


A and B , is
(A) x + 2y + z = 0
(B) x – 2y – z – 2 = 0
(C) x – 2y + z – 4 = 0
(D) 2x + y + z – 1 = 0
 

Q.39 Volume of the tetrahedron whose 3 coterminous edges are the vectors A , B and C  2î  ˆj  4k̂ is
(A) 1
(B) 4/3
(C) 8/3
(D) 8


Q.40 Vector component of A perpendicular to the vector B is given by
  
  
  
B  ( A  B)
A  (A  B)
B  ( A  B)



(B)
(C)
(A)
B2
B2
A2
  
A  ( A  B)

(D)
A2
Select the correct alternatives : (More than one are correct)
Q.41 If a, b, c are different real numbers and a i  b j  c k ; b i  c j  a k & ci  a j  b k are position
vectors of three non-collinear points A, B & C then :
(A) centroid of triangle ABC is

a bc   
i jk
3

(B) i  j  k is equally inclined to the three vectors
(C) perpendicular from the origin to the plane of triangle ABC meet at centroid
(D) triangle ABC is an equilateral triangle.
  


Q.42 The vectors a , b , c are of the same length & pairwise form equal angles. If a  i  j & b  j  k , the

pv's of c can be :
(B)   ,
4 1
4
, 
 3 3
3
(A) (1, 0, 1)
1
3
1
 1 4
, 
 3 3
3
4 1
3 3
(D)   ,
(C)  ,  , 
Q.43* Which of the following locii of z on the complex plane represents a pair of straight lines?
(A) Re z2 = 0
(B) Im z2 = 0
(C) ⏐z⏐ + z = 0
(D) ⏐z − 1⏐ = ⏐z − i⏐
  

  





Q.44 If a , b , c & d are linearly independent set of vectors & K1 a  K 2 b  K 3 c  K 4 d = 0 then :
(A) K1 + K2 + K3 + K4 = 0
(B) K1 + K3 = K2 + K4 = 0
(C) K1 + K4 = K2 + K3 = 0
(D) none of these
Q.45 If a , b , c & d are the pv's of the points A, B, C & D respectively in three dimensional space & satisfy




the relation 3 a  2 b  c  2 d = 0, then :
(A) A, B, C & D are coplanar
(B) the line joining the points B & D divides the line joining the point A & C in the ratio 2 : 1.
(C) the line joining the points A & C divides the line joining the points B & D in the ratio 1 : 1
  

(D) the four vectors a , b , c & d are linearly dependent.
Q.46* If z3 − i z2 − 2 i z − 2 = 0 then z can be equal to :
(A) 1 − i
(B) i
(C) 1 + i
(D) − 1 − i
[6]






  
 
Q.47 If a & b are two non collinear unit vectors & a , b , x a − y b form a triangle, then :
   

 a b

(A) x = − 1 ; y = 1 & ⏐ a + b ⏐ = 2 cos 

 2 


   




(B) x = − 1 ; y = 1 & cos  a b + ⏐ a + b ⏐ cos a ,  a  b  = − 1



   
   
 a b
 a b
 
(C) ⏐ a + b ⏐ = − 2 cot 
cos 

 & x = −1 , y = 1
 2 
 2 






(D) none
~

Q.48 The lines with vector equations are ; r1 =  3 i  6 j    4 i  3 j  2 k and


~

r2 =  2 i  7 j    4 i  j  k are such that :
(A) they are coplanar
(C) they are skew
(B) they do not intersect
(D) the angle between them is tan−1 (3/7)
Q.49* Given a, b, x, y ∈ R then which of the following statement(s) hold good?
(A) (a + ib) (x + iy)–1 = a – ib ⇒ x2 + y2 = 1
(B) (1–ix) (1 + ix)–1 = a – ib ⇒ a2 + b2 = 1
(C) (a + ib) (a – ib)–1 = x – iy ⇒ |x + iy| = 1
(D) (y – ix) (a + ib)–1 = y + ix ⇒ |a – ib| = 1
Q.50 The acute angle that the vector 2 i  2 j  k makes with the plane contained by the two vectors
2 i  3 j  k and i  j  2 k is given by :
 1 

 3
 1 

 3
(A) cos–1 
(B) sin–1 
(C) tan–1
 2
(D) cot–1
 2
Q.51 The volume of a right triangular prism ABCA1B1C1 is equal to 3. If the position vectors of the vertices of
the base ABC are A(1, 0, 1) ; B(2,0, 0) and C(0, 1, 0) the position vectors of the vertex A1 can be:
(A) (2, 2, 2)
(B) (0, 2, 0)
(C) (0, − 2, 2)
(D) (0, − 2, 0)
 
2
Q.52* If xr = CiS  r  for 1 ≤ r ≤ n r, n ∈ N then :
 n

r 1

 n

r 1

(A) Limit
Re   x r  = − 1
n
Im   x r  = 1
(C) Limit
n
 n

r 1

 n

r 1

(B) Limit
Re   x r  = 0
n
(D) Limit
Im   x r  = 0
n
[7]



Q.53 If a line has a vector equation , r  2 i  6 j   i  3 j then which of the following statements holds
good ?
(A) the line is parallel to 2 i  6 j
(B) the line passes through the point 3 i  3 j
(C) the line passes through the point i  9 j
(D) the line is parallel to xy plane
  
Q.54 If a , b , c are non-zero, non-collinear vectors such that a vector

  

p = a b cos 2  a b

(A) parallel to a


(C) coplanar with b & c

 c and a vector q = a c cos   a  c b then p + q is


(B) perpendicular to a


(D) coplanar with a and c
Q.55* The greatest value of the modulus of of the complex number ' z ' satisfying the equality z 
(A)
1  5
2
(B)
3 5
2
(C)
3 5
2
(D)
1
= 1 is
z
5 1
2
SUBJECTIVE:
Q.1
 1


3ˆ
 

j and x  a  (q 2  3) b , y   p a  qb . If x  y , then express p
Let a  3 î  ˆj and b  î 
2
2
as a function of q, say p = f (q), (p ≠ 0 & q ≠ 0) and find the intervals of monotonicity of f (q).
Q.2
xx  x
ex 1
ln x
= 1 and Lim
= 1. Evaluate Lim
.
Using only the limit theorems Lim
x 1 ln x  x  1
x 0
x
x 1 x  1
Q.3
The three vectors a  4 i  2 j  k ; b  2 i  j  k and c  2 i  k are all drawn from the point with



p.v. îˆj . Find the equation of the plane containing their end point in scalar dot product form.

2
Q.4
z

Q.5
(cos2 n 1 x  cos2 n 1 x) dx where n  N

2



Let points P, Q & R have position vectors, r1 = 3i − 2j − k; r2 = i + 3j + 4k & r3 = 2i+j−2k
respectively, relative to an origin O. Find the distance of P from the plane OQR.
3
Q.6
Evaluate:  ( x  1)( x  2)( x  3) dx
1
Q.7
  
  
Given that vectors A , B , C form a triangle such that A  B  C , find a,b,c,d such that the area of the



triangle is 5 6 where A = ai + bj + ck; B = di + 3j + 4k & C =3i + j − 2k.
[8]
n 1
Q.8
Lim n 
k 1
n

n 
k 0 k
k  k  1


 x  dx
 x  
n  n


n
Q.9
Find the distance of the point P(i + j + k) from the plane L which passes through the three points
A(2i + j + k), B(i + 2j + k), C(i + j + 2k) . Also find the pv of the foot of the perpendicular from P on the
plane L .
Q.10 Evaluate: (a) 
Q.11
 
sin 4 x  cos 4 x
 0,  ;
dx
,
x
∈
 2
sin 3 x cos x
(b) 
sin 4 x  cos 4 x
dx
sin x cos 3 x

Find the equation of the straight line which passes through the point with position vector a , meets the line
 
  
r  b  tc and is parallel to the plane r . n  1 .
Q.12 Integrate: 
dx
.
cos3 x  sin 3 x
Q.13 Find the equation of the line passing through the point (1, 4, 3) which is perpendicular to both of the lines
x 1
y3 z2
x2
y4
z 1
=
=
and
=
=
2
1
4
3
2
2
Also find all points on this line the square of whose distance from (1, 4, 3) is 357.
Q.14
 n 2  n 1

Lim 

n  
n


2 n 2  n 1
Q.15 If z-axis be vertical, find the equation of the line of greatest slope through the point (2, –1, 0) on the
plane 2x + 3y – 4z = 1.
2
2
sin x
cos x
dx , where a > 0 and b > 0.
Q.16 Let I = 
dx and J = 
a cos x  b sin x
a
cos
x
b
sin
x

0
0
Compute the values of I and J.
[9]
[10]
Select the correct alternative : (Only one is correct)
Q.37
Q.36 C
Q.30
Q.29 D
Q.23
Q.22 A
Q.16
Q.15 B
Q.9
A
Q.1
Q.2
B
Q.8
B
A
D
D
B
A
Q.3
Q.10
Q.17
Q.24
Q.31
Q.38
C
B
D
B
D
C
Q.4
Q.11
Q.18
Q.25
Q.32
Q.39
A
A
D
B
A
B
Q.5
Q.12
Q.19
Q.26
Q.33
Q.40
D
A
A
C
B
Q.6
A
Q.13 A
Q.20 D
Q.27 B
Q.34 C
Q.7
Q.14
Q.21
Q.28
Q.35
B
B
B
A
A
A
Select the correct alternatives : (More than one are correct)
Q.46 B,C,D
Q.45 A,C,D
Q.42 A,D
Q.41 A,B,C,D
Q.49
A,B,C,D
Q.53 B,C,D
Q.43
Q.47
Q.50 B,D
Q.51
Q.54 B,C
Q.55
A,B
Q.44
A,B
Q.48
A,D
Q.52
A,B,C
B,C,D
A,D
B,D
SUBJECTIVE:
Q.1
Q.2
Q.5
Q.8
p=
q (q 3  3)
4
,decreasing in q ∈ (–1, 1), q ≠ 0
Q.9

8
Q.6
3 units
Q.3
–2
Q.10 (a)
Q.11
C–
Q.7
1/2
Q.4
2 i  2 j  k  . r = 3
4
2n  1
(−8, 4, 2, −11) or (8, 4, 2, 5)
1 4 4 4
, , , 
3 3 3 3
1 t2 1
t 2  1 1
 ln
where t = cot2x;
2
4
t2 1 1
1 t2 1
t 2 1 1
 ln
(b)
+ C, where t = tan2x
2
4
t2 1 1
  
   (a  b ) . n  
 
r  a    (a  b) 
  c 
c .n


Q.12 2 [tan–1(sin x + cos x) +
Q.13
Q.15
1
ln
2 2
2  sin x  cos x
+C
2  sin x  cos x
x 1 y  4 z  3


, ; (–9, 20, 4) ; (11, –12, 2)
 10
16
1
x  2 y 1 z


8
12
13
Q.16
Q.14
e–1
1  b
1  a
 b 
 b 

 a ln   
 b ln    , 2

2
a 2  b2  2
 a 
 a  a  b  2
ANSWER KEY