1. Question bank on Vector, 3D and Complex Number & Misc. Subjective Problem
Select the correct alternative : (Only one is correct)
Q.1 If

a = 11
1 ,

b = 23 ,
 
a b
 = 30 , then
 
a b
 is :
(A) 10 (B) 20 (C) 30 (D) 40
Q.2 ThepositionvectorofapointPmovinginspaceisgivenby k̂
)
t
sin
5
(
ĵ
)
t
cos
4
(
î
)
t
cos
3
(
R
OP 




.
The time 't' when the point P crosses the plane 4x – 3y + 2z = 5 is
(A)
2

sec (B)
6

sec (C)
3

sec (D)
4

sec
Q.3 Indicate the correct order sequence in respect of the following :
I. The lines
3
4
x


=
1
6
y


=
1
6
y


and
1
1
x


=
2
2
y


=
2
3
z 
are orthogonal.
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
Q.4* If
2
1
z
z
is purelyimaginarythen
2
1
2
1
z
z
z
z


is equal to :
(A) 1 (B) 2 (C) 3 (D) 0
Q.5 In a regulartetrahedron, the centres ofthe four faces are thevertices of a smallertetrahedron. The ratio
of the volume ofthe smaller tetrahedron to that of the larger is
n
m
,where m and n arerelativelyprime
positive integers. The value of (m + n)is
(A) 3 (B) 4 (C) 27 (D) 28
Q.6 If
  
a b c
, , arenon-coplanarunitvectorssuchthat
    
a x bxc b c
( ) ( )
 
1
2
thentheanglebetween b
&
a

 is
(A) 3 /4 (B) /4 (C) /2 (D) 
Q.7 The sineof angle formed bythe lateral faceADC and plane of thebaseABC of thetetrahedronABCD
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 i j k
  
2 3 4
   ,

b i j k
  
3 2
   ,

c i j k
  
  
2 is :
(A) 7 (B) 5 (C) 4 (D) none

2. Q.10 Let w
,
v
,
u



bethevectors suchthat 0
w
v
u 





,if 5
w
&
4
v
,
3
u 





then the value of
u
.
w
w
.
v
v
.
u







 is :
(A) 47 (B) 25 (C) 0 (D) 25
Q.11 Let

a = j
ˆ
î  ,

b= k̂
j
ˆ  ,

c= î
k̂  . If

d is a unit vector such that
    
a d b c d
. [ , , ]
 
0 then

d
(A) )
k̂
2
j
ˆ
î
(
6
1


 (B) )
k̂
j
ˆ
î
(
3
1


 (C) )
k̂
j
ˆ
î
(
3
1


 (D) ± k̂
Q.12* If z be a complex number for which z
z
 
1
2 , then the greatest value of z is :
(A) 2 1
 (B) 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 :
(A)  
 
 
 
r xa
a a
a b
  
1
.
(B)  
 
 
 
r xb
b b
a b
  
1
.
(C)
  
r xa b
  (D)
  
r xb a
 
where x is anyscalar.
Q.14 Thelines
k
4
z
1
3
y
1
2
x






and
1
5
z
2
4
y
k
1
x 




arecoplanar if
(A) k = 1 or – 1 (B) k = 0 or – 3 (C) k = 3 or – 3 (D) k = 0 or – 1
Q.15 Whichoneof thefollowingstatementis 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  = 30º &  = 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)InaregulartetrahedronOABCwhere'O'istheorigin,thevector OA OB OC
  
  isperpendicular
to the planeABC.
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 ofthe point (3, 4, 5) from x-axis is
(A) 3 (B) 5 (C) 34 (D) 41
Q.18 Givennonzerovectors C
and
B
,
A



,thenwhich oneofthefollowingis false?
(A)Avector orthogonal to C
and
B
A



 is ± C
)
B
A
(





(B)Avector orthogonal to B
A


 and B
A


 is ± B
A



(C)Volume of the parallelopiped determined by C
and
B
,
A



is |
C
·
B
A
|




(D)Vector projection of B
onto
A


is
|
B
|
B
·
A




3. Q.19 Given three vectors
  
a b c
, , such that they are nonzero, noncoplanar vectors, then which of the
followingarenoncoplanar.
(A)
     
a b b c c a
  
, , (B)
     
a b b c c a
  
, ,
(C)
     
a b b c c a
  
, , (D)
     
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  =
1
2
( 'O' is the origin) is :
(A)Acircle parallel to yz plane withcentre on the xaxis
(B) a coneconcentric with positive xaxis havingvertex at the originand the slant height equal to the
magnitudeofthevector
(C)a rayemanatingfrom the origin and making an angleof60ºwith xaxis
(D) a disc parallel to yz plane withcentre on xaxis & radius equal to| |
OP

sin 60º
Q.22 Alinewithdirectionratios(2,1,2)intersectsthelines )
k̂
j
ˆ
î
(
j
ˆ
r 






and )
k̂
ĵ
î
2
(
î
r 






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 areA(1, 1, 2), B(4, 3, 1) & C(2, 3, 5).The vector representing the internal
bisector of theangleAis :
(A)   
i j k
  2 (B) 2 2
  
i j k
  (C) 2 2
  
i j k
  (D) 2 2
  
i j k
 
Q.24* Lowest degree ofa polynomial with rational coefficients if one of its rootis, i

2 is
(A) 2 (B) 4 (C) 6 (D) 8
Q.25 Aplane vector has components 3 & 4 w.r.t. the rectangular cartesian system. This system is rotated
through an angle

4
in anticlockwisesense.Then w.r.t. thenew system the vectorhascomponents :
(A) 4, 3 (B)
7
2
1
2
, (C)
1
2
7
2
, (D) none
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 k̂
j
ˆ
î
cos 

 , k̂
j
ˆ
cos
î 

 & k̂
cos
j
ˆ
î 

 (      2 n ) are coplanar then the value of





 




2
ec
cos
2
ec
cos
2
ec
cos 2
2
2
equal to
(A) 1 (B)  (C) 3 (D) none of these
Q.28* The straight line (1 + 2i)z + (2i – 1) z = 10i on the complex plane, has intercept onthe imaginaryaxis
equal to
(A) 5 (B)
2
5
(C) –
2
5
(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 thatangularpoint, is
(A) a
3 (B) a
2 (C) a
2
3
(D) a
3
2
Q.30 Which one of the following does not hold for the vector V

=  
  
a x b x a ?
(A) perpendicular to

a (B) perpendicular to

b
(C) coplanar with

a &

b (D) perpendicular to

a x

b .
Q.31* Let z1,z2 &z3 bethecomplex numbersrepresentingthevertices ofatriangleABC 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 triangleABC, the point Pis its :
(A) centroid (B) orthocentre (C) circumcentre (D) incentre
Q.32 Given the position vectors of the vertices of a triangleABC, )
a
(
A  ; )
b
(
B ; )
c
(
C .Avector r

is
parallel to the altitude drawn from the vertexA, making an obtuse angle with the positiveY-axis. If
|
r
|

= 34
2 ; k̂
3
j
ˆ
î
2
a 



; k̂
4
j
ˆ
2
î
b 



; k̂
2
j
ˆ
î
3
c 



, then r

is
(A) k̂
6
j
ˆ
8
î
6 

 (B) k̂
6
j
ˆ
8
î
6 
 (C) k̂
6
j
ˆ
8
î
6 

 (D) k̂
6
j
ˆ
8
î
6 

Q.33* The complex number, z = 5 + i has anargument which is nearlyequal to :
(A) /32 (B) /16 (C) /12 (D) /8
Q.34 If k̂
j
ˆ
î
a 



, k̂
j
ˆ
î
b 



, k̂
j
ˆ
2
î
c 



, then the value of
c
.
c
b
.
c
a
.
c
c
.
b
b
.
b
a
.
b
c
.
a
b
.
a
a
.
a


















equal to
(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 k̂
3
j
ˆ
2
î
A 



and k̂
5
j
ˆ
4
î
3
B 



Q.36 The value of the scalar 2
2
)
B
·
A
(
|
B
A
|





 is equal to
(A) 8 (B) 10
7 (C) 7
10 (D) 64
Q.37 Equation of a line passing through the point with position vector j
ˆ
3
î
2  and orthogonal to the plane
containingthevectors A

and B

,is
(A) k̂
j
ˆ
)
3
2
(
î
)
2
(
r 








(B) k̂
j
ˆ
)
3
2
(
î
)
2
(
r 








(C) k̂
j
ˆ
)
3
2
(
î
r 







(D) none

5. Q.38 Equation of a plane containing the point with position vector )
k̂
j
ˆ
î
( 
 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 k̂
4
j
ˆ
î
2
C 



is
(A) 1 (B) 4/3 (C) 8/3 (D) 8
Q.40 Vector component of A

perpendicular to thevector B

is given by
(A) 2
B
)
B
A
(
B






(B) 2
B
)
B
A
(
A






(C) 2
A
)
B
A
(
B






(D) 2
A
)
B
A
(
A






Select the correct alternatives : (More than one are correct)
Q.41 If a, b, c are different real numbers and a i b j ck
  
  ; bi c j a k
  
  & ci a j bk
  
  are position
vectors of threenon-collinear pointsA, B &C then:
(A) centroid oftriangleABC is  
a b c
i j k
 
 
3
  
(B)  
i j k
  is equallyinclinedto the threevectors
(C)perpendicular from the originto theplane of triangleABC meet at centroid
(D)triangleABCis an equilateraltriangle.
Q.42 The vectors
  
a b c
, , are of thesame length & pairwise form equal angles. If

a i j
 
  &

b j k
 
  , the
pv's of

c can be :
(A) (1, 0, 1) (B)  






4
3
1
3
4
3
, , (C)
1
3
4
3
1
3
, ,






 (D)  






1
3
4
3
1
3
, ,
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 linearlyindependent set of vectors &K a K b K c K d
1 2 3 4
   
   = 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 pointsA, B, C & D respectivelyin three dimensional space & satisfy
the relation3 2 2
   
a b c d
   =0, then :
(A)A, B, C & D are coplanar
(B) the line joiningthe points B& D divides the line joining the pointA& C in the ratio 2 : 1.
(C) theline joiningthe pointsA& C divides the line joiningthepoints B & D in the ratio 1: 1
(D) the four vectors
  
a b c
, , &

d arelinearlydependent.
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) x =  1 ; y = 1 & 

a +

b  = 2 cos
 
a b











2
(B) x =  1 ; y = 1 & cos
 
a b






 + 

a +

b  cos  
  
a a b
,  







=  1
(C) 

a +

b  =  2 cot
 
a b











2
cos
 
a b











2
& x =  1 , y = 1 (D) none
Q.48 The lines with vector equations are ;

r1 =  
     
3 6 4 3 2
   ~ 
i j i j k
 and

r2 =  
     
2 7 4
   ~ 
i j i j k
 are such that :
(A) theyare coplanar (B) theydo not intersect
(C) theyare skew (D) the angle between them is tan1 (3/7)
Q.49* Given a, b,x, yR thenwhich of the followingstatement(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 2
  
i j k
  makes with the plane contained by the two vectors
2 3
  
i j k
  and   
i j k
  2 is given by:
(A) cos–1
1
3





 (B) sin–1
1
3





 (C) tan–1
 
2 (D) cot–1
 
2
Q.51 ThevolumeofarighttriangularprismABCA1B1C1 isequalto3.Ifthepositionvectorsoftheverticesof
the baseABC areA(1, 0, 1) ; B(2,0, 0) and C(0, 1, 0) the position vectors of the vertexA1 can be:
(A) (2, 2, 2) (B) (0, 2, 0) (C) (0,  2, 2) (D) (0,  2, 0)
Q.52* If xr = CiS

2r





 for 1  r  n r, n  N then :
(A) Limit
n   Re xr
r
n








1
=  1 (B) Limit
n   Re xr
r
n








1
= 0
(C) Limit
n   Im xr
r
n








1
= 1 (D) Limit
n   Im xr
r
n








1
= 0

7. Q.53 If a line has a vector equation,  

r i j i j
   
2 6 3
   
 then which of the following statements holds
good ?
(A)the lineis parallel to 2 6
 
i j
 (B) the line passes through thepoint 3 3
 
i j

(C) the line passes through the point  
i j
 9 (D)thelineis parallel to xyplane
Q.54 If
  
a b c
, , are non-zero, non-collinear vectors such that a vector

p = a b cos  
 
2  
  
a b c and a vector

q = a c cos  
 
  
  
a c b then

p +

q is
(A) parallel to

a (B) perpendicular to

a
(C) coplanarwith
 
b c
& (D) coplanar with a

and c

Q.55* The greatest valueof the modulus ofof the complex number 'z' satisfyingtheequality z
z

1
= 1 is
(A)
 
1 5
2
(B)
3 5
2

(C)
3 5
2

(D)
5 1
2

SUBJECTIVE:
Q.1 Let j
ˆ
î
3
a 


and ĵ
2
3
î
2
1
b 


and b
)
3
q
(
a
x 2





 , b
q
a
p
y





 . If y
x


 , then express p
as a function of q, say p = f (q), (p  0 & q  0) and find the intervals of monotonicity of f (q).
Q.2 Using onlythe limit theorems
1
x
x
n
Lim
1
x 

l
= 1 and
x
1
e
Lim
x
0
x


= 1. Evaluate
1
x
x
n
x
x
Lim
x
1
x 


 l
.
Q.3 The three vectors

a i j k
  
4 2
   ;

b i j k
  
2   and

c i k
 
2  are all drawn from the point with
p.v. j
ˆ
î . Findthe equationof theplanecontainingtheir endpoint in scalar dot product form.
Q.4 (cos cos )
2 1 2 1
2
2
n n
x x dx
 


z


where n N

Q.5 Let points P, Q & R have position vectors, 1
r

= 3i  2j  k; 2
r

= i + 3j + 4k & 3
r

= 2i+j2k
respectively, relative toan origin O. Find the distance of Pfrom the plane OQR.
Q.6 Evaluate:  


3
1
dx
)
3
x
)(
2
x
)(
1
x
(
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 is5 6 where

A = ai + bj + ck;

B= di + 3j + 4k &

C =3i + j  2k.

8. Q.8  




















1
n
0
k
n
1
k
n
k
n
dx
x
n
1
k
n
k
x
n
Lim
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 thepvofthefoot of the perpendicularfrom Pon the
plane L.
Q.10 Evaluate: (a) 

dx
x
cos
x
sin
x
cos
x
sin
3
4
4
, x  




 
2
,
0 ; (b) 

dx
x
cos
x
sin
x
cos
x
sin
3
4
4
Q.11 Findtheequationofthestraightlinewhichpassesthroughthepointwithpositionvector a

,meetstheline
c
t
b
r




 and is parallel to the plane 1
n
.
r 


.
Q.12 Integrate:   x
sin
x
cos
dx
3
3
.
Q.13 Findtheequationofthelinepassingthroughthepoint(1, 4,3)whichis perpendiculartobothofthelines
2
1
x 
=
1
3
y 
=
4
2
z 
and
3
2
x 
=
2
4
y 
=
2
1
z


Also find all points on this linethe square of whose distance from (1, 4,3) is 357.
Q.14
1
n
n
2
2
n
2
n
1
n
n
Lim



 






 

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.
Q.16 Let I = 


2
0
dx
x
sin
b
x
cos
a
x
cos
and J = 


2
0
dx
x
sin
b
x
cos
a
x
sin
, where a > 0 and b > 0.
Compute the values of Iand J.

9. ANSWER KEY
Select
the
correct
alternative
:
(Only
one
is
correct)
Q.1
B
Q.2
B
Q.3
C
Q.4
A
Q.5
D
Q.6
A
Q.7
B
Q.8
A
Q.9
A
Q.10
B
Q.11
A
Q.12
A
Q.13
A
Q.14
B
Q.15
B
Q.16
D
Q.17
D
Q.18
D
Q.19
A
Q.20
D
Q.21
B
Q.22
A
Q.23
D
Q.24
B
Q.25
B
Q.26
C
Q.27
B
Q.28
A
Q.29
D
Q.30
B
Q.31
D
Q.32
A
Q.33
B
Q.34
C
Q.35
A
Q.36
C
Q.37
A
Q.38
C
Q.39
B
Q.40
A
Select
the
correct
alternatives
:
(More
than
one
are
correct)
Q.41
A,B,C,D
Q.42
A,D
Q.43
A,B
Q.44
A,B,C
Q.45
A,C,D
Q.46
B,C,D
Q.47
A,B
Q.48
B,C,D
Q
.
49
A,B,C,D
Q.50
B,D
Q.51
A,D
Q
.
52
A,D
Q.53
B,C,D
Q.54
B,C
Q.55
B,D
SUBJECTIVE:
Q.1
p
=
,decreasing
in
q

(–1,
1),
q

0
Q.2
–
2
Q.3
=
3
Q.4
Q.5
3
units
Q.6
1/2
Q.7
(8,
4,
2,
11)
or
(8,
4,
2,
5)
Q.8
Q.9
Q.10
(a)
C
–
where
t
=
cot
2
x;
(b)
+
C,
where
t
=
tan
2
x
Q.11
Q.12
2
[tan
–1
(sin
x
+
cos
x)
+
ln
+
C
Q.13
,
;
(–9,
20,
4)
;
(11,
–12,
2)
Q.14
e
–1
Q.15
Q.16
,