Vectors and 3D

FIITJEE
VECTOR AND 3D
CPP
Name:__________________________________ Batch: Date: _____________
Enrollment No.:__________________________ Faculty ID: MPJ Dept. of Mathematics
CPP. NO.–1
STRAIGHT OBJECTIVE TYPE
1. A (1, –1, –3) B (2, 1, –2) and C (–5, 2, –6) are the position vectors of the vertices of a triangle
ABC. The length of the bisector of its internal angle at A is
(A) 10 / 4 (B) 3 10 / 4
(C) 10 (D) none of these
2. In a triangle PQR, a = QR,b = RP and c = PQ . If
a = 3, b =4 and
( )= a ,
a. c − b
c . (a − b) a + b
2
then the value of a  b is
(A) 25 (B) 204
(C) 108 (D) 232
3. Consider the cube in the first octant with sides OP,OQ and OR of length 1 , along the x-axis,
 1 1 1
y-axis and z-axis, respectively, where O(0,0,0) is the origin. Let S  , ,  be the centre of
2 2 2
the cube and T be the vertex of the cube opposite to the origin O such that S lies on the
diagonal OT. If p = SP ,q = SQ , r = SR and t = ST , then the value of (p  q)  r  t ( ) is
1
(A) 1 (B) (C) 0 (D) 2
2
4. Let a = 2iˆ + ˆj − kˆ and b = î + 2ˆj + kˆ be two vectors. Consider a vector c =  a +  b,  ,   .If

( )
the projection of c on the vector a + b is 3 2 , then the minimum value of c − a  b .c ( ( ))
equals
(A) 2 (B) 8
(C) 12 (D) 18
5. Let a and b be vectors such that a. b = 0 . For some x, y  , let

( )
c = x a + y b + a  b . If c = 2 and the vector c is inclined at the same angle  to both
a and b , then the value of 8cos2  is
(A) 1 (B) 2 (C) 3 (D) 4
6. Let ,  and  be distinct real numbers. The points whose position vector’s are
 î + ˆj + kˆ ;  î + ˆj + kˆ and  î + ˆj +  kˆ
(A) are collinear (B) form an equilateral triangle
FIITJEE Ltd., Punjabi Bagh Centre, 31, 32, 33, Central Market, West Avenue Road, Punjabi Bagh, New Delhi - 26, Ph: 011-45634000.
(C) form a scalene triangle (D) form a right angled triangle
7. If the vectors a = 3iˆ + ˆj − 2k,

ˆ b = −î + 3ˆj + 4kˆ and c = 4iˆ − 2ˆj − 6kˆ constitute the sides of a
ABC, then the length of the median bisecting the vector c is
(A) 2 (B) 14
(C) 74 (D) 6
8. Let A(0, –1, 1), B(0, 0, 1), C(1, 0, 1) are the vertices of a ABC. If R and r denotes the
r
circumradius and inradius of ABC, then has value equal to
R
3 3
(A) tan (B) cot
8 8
 
(C) tan (D) cot
12 12
xy 2
9. If a = xiˆ − 2ˆj + 5kˆ and b = î + yjˆ − zkˆ are linearly dependent, then the value of equals
z
4 −3
(A) (B)
5 5
3 −4
(C) (D)
5 5
10. A vector of magnitude 10 along the normal to the curve 3x2 + 8xy + 2y2 – 3 = 0 at its point P(1,
0) can be
(A) 6iˆ + 8 ˆj (B) −6iˆ + 8 ˆj
(C) 6iˆ − 8ˆj (D) −6iˆ − 8 ˆj
11. If A(0, 1, 0), B(0, 0, 0), C(1, 0, 1) are the vertices of a ABC. Match the entries of column-I
with column-II.
Column-I Column-II
(A) Orthocentre of ABC (P) 2
2
(B) Circumcentre of ABC (Q) 3
2
(C) Area (ABC) (R) 3
3
(D) Distance between orthocenter (S) 3
and centroid
6
(E) Distance between orthocentre (T) (0, 0, 0)
and circumcentre.
(F) Distance between circumcentre (U)  1 1 1
and centroid  2, 2, 2
 
(G) Incentre of ABC (V)  1 1 1
3, 3, 3
 
(H) Centroid of ABC (W)  1 2 1 
 , , 
 1+ 2 + 3 1+ 2 + 3 1+ 2 + 3 
12. The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes
2x + y − 2z = 5 and 3x − 6y − 2z = 7 is,
(A) 14x + 2y – 15z = 1 (B)14x - 2y + 15z = 27
(C) 14x + 2y + 15z = 31 (D) -14x + 2y + 15z = 3
13. Let O be the origin and let PQR be an arbitrary triangle. The point S is such that
OP.OQ + OR.OS = OR.OP + OQ.OS = OQ.OR + OP.OS
Then the triangle PQR has S as its

(A) centroid (B) circumcentre
(C) incentre (D) orthocentre
PARAGRAPH (For Q.No.14,15)
Let O be the origin, and OX ,OY ,OZ be three unit vectors in the directions of the sides
QR ,RP ,PQ , respectively, of a triangle PQR.
14. OX  OY =
(A) sin (P + Q ) (B) sin 2R (C) sin (P + R ) (D) sin ( Q + R )
15. If the triangle PQR varies, then the minimum value of

cos (P + Q ) + cos ( Q + R ) + cos (R + P )
is
5 3 3 5
(A) − (B) − (C) (D)
3 2 2 3
16. Consider a pyramid OPQRS located in the first octant ( x  0, y  0, z  0 ) with O as origin,
and OP and OR along the x-axis and the y-axis respectively. The base OPQR of the pyramid
is a square with OP = 3 . The point S is directly above the mid-point T of diagonal OQ such
that TS = 3 . Then

(A) the acute angle between OQ and OS is
3
(B) the equation of the plane containing the triangle OQS is x − y = 0
3
(C) the length of the perpendicular from P to the plane containing the triangle OQS is
2
15
(D) the perpendicular distance from O to the straight line containing RS is
2
CPP. NO.–2
1. Consider the following 3 lines in space
L1 : r = 3iˆ − ˆj + 2kˆ + (2iˆ + 4 ˆj − k)
ˆ
L2 : r = î + ˆj − 3kˆ + (4iˆ + 2ˆj + 4k)
ˆ
L3 : r = 3iˆ + 2ˆj − 2kˆ + t(2iˆ + ˆj + 2k)
ˆ
Then which one of the following pair(s) are in the same plane.
(A) only L1L2 (B) only L2L3
(C) only L3L1 (D) L1L2 and L2L3
2. The vectors 3iˆ − 2ˆj + k,

ˆ î − 3ˆj + 5kˆ and 2iˆ + ˆj − 4kˆ form the sides of a triangle. Then triangle is
(A) an acute angled triangle (B) an obtuse angled triangle
(C) an equilateral triangle (D) a right angled triangle
3. Two vectors p, q on a plane satisfy p + q = 13, p − q = 1 and p = 3.

The angle between p and q , is equal to
 
(A) (B)
6 4
 
(C) (D)
3 2
4. ( )(
Consider the points A, B and C with position vectors −2iˆ + 3ˆj + 5kˆ , î + 2ˆj + 3kˆ and 7iˆ − kˆ )
respectively.
Statement-1: The vector sum, AB + BC + CA = 0
because
Statement-2: A, B and C form the vertices of a triangle.
(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for
statement-1.
(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation
for statement-1
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true.
5. The set of values of c for which the angle between the vectors cxiˆ − 6ˆj + 3kˆ and xiˆ − 2ˆj + 2cxkˆ
is acute for every x  R is
(A) (0, 4/3) (B) [0, 4/3]
(C) (11/9, 4/3) (D) [0, 4/3)
6. Let u = î + ˆj, v = î − ˆj and w = î + 2ˆj + 3kˆ . If n̂ is a unit vector such that u  nˆ = 0 and v  nˆ = 0,
then w  nˆ is equal to
(A) 1 (B) 2
(C) 3 (D) 0
7. If the vector 6iˆ − 3ˆj − 6kˆ is decomposed into vectors parallel and perpendicular to the vector
î + ˆj + kˆ then the vectors are:
( )
(A) − î + ˆj + kˆ & 7iˆ − 2ˆj − 5kˆ ( )
(B) −2 î + ˆj + kˆ & 8iˆ − ˆj − 4kˆ
( )
(C) +2 î + ˆj + kˆ & 4iˆ − 5ˆj − 8kˆ (D) none
8. Let r = a + l and r = b + m be two lines in space where a = 5iˆ + ˆj + 2k, ˆ b = − î + 7ˆj + 8k,
ˆ
l = −4iˆ + ˆj - kˆ and m = 2iˆ − 5j − 7kˆ then the p.v. of a point which lies on both of these lines, is
(A) î + 2ˆj + kˆ
(B) 2iˆ + ˆj + kˆ
(C) î + ˆj + 2kˆ
(D) non existent as the lines are skew
9. If the three points with position vectors (1, a, b) ; (a, 2, b) and (a, b, 3) are collinear in space,
then the value of (a + b) is
(A) 3 (B) 4
(C) 5 (D) none
10. If the vector v with magnitude 6 is along the internal bisector of the angle between
a = 7iˆ − 4ˆj − 4kˆ and b = −2iˆ − ˆj + 2kˆ then v  a equals
(A) 0 (B) 3
(C) 6 (D) 9
11. Let A(1, 2, 3), B(0, 0, 1), C(–1, 1, 1) are the vertices of a ABC.
(i) The equation of internal angle bisector through A to side BC is

(A) r = î + 2ˆj + 3kˆ + (3iˆ + 2ˆj + 3k)
ˆ (B) r = î + 2ˆj + 3kˆ + (3iˆ + 4ˆj + 3k)
ˆ
(C) r = î + 2ˆj + 3kˆ + (3iˆ + 3ˆj + 2k)ˆ (D) r = î + 2ˆj + 3kˆ + (3iˆ + 3ˆj + 4k)
ˆ
(ii) The equation of median through C to side AB is

(A) r = − î + ˆj + kˆ + p(3iˆ − 2k)
ˆ (B) r = − î + ˆj + kˆ + p(3iˆ + 2k)
ˆ
(C) r = − î + ˆj + kˆ + p( −3iˆ + 2k)
ˆ (D) r = −î + ˆj + kˆ + (3iˆ + 2ˆj)
(iii) The area (ABC) is equal to

9 17
(A) (B)
2 2
17 7
(C) (D)
2 2
12. Let OAB be a regular triangle with side length unity (O being the origin). Also, M, N are the
points of trisection of AB, M being closer to A and N closer to B. Position vectors of A, B, M
and N are a, b, m and n respectively. Which of the following hold(s) good?
2 1 5 1
(A) m = xa + yb  x = and y = (B) m = xa + yb  x = and y =
3 3 6 6
13 15
(C) m  n equals (D) m  n equals
18 18
13. In ABC, a point P is chosen on side AB so that AP : PB = 1 : 4 and a point Q is chosen on

MC
the side BC so that CQ : QB = 1 : 3. Segment CP and AQ intersect at M. If the ratio is
PC
a
expressed as a rational number in the lowest term as , find (a +b).
b
14. Let û = u1î + u2 ˆj + u3kˆ be a unit vector in 3
and ŵ =
6
(
1 ˆ ˆ
)
i + j + 2kˆ . Given that there exists
a vector v in 3
ˆ (uˆ  v ) = 1. Which of the following statement(s)
such that uˆ  v = 1 and w.
is(are) correct?
(A) There is exactly one choice for v
(B) There are infinitely many choices for such v
(C) If û lies in the xy-plane then u1 = u2
(D) If û lies in the xz-plane then 2 u1 = u3
15. In 3
, consider the planes P1 : y = 0 and P2 : x + z = 1 . Let P3 be a plane, different from P1
and P2 , which passes through the intersection of P1 and P2 If the distance of the point ( 0, 1, 0 )
from P3 is 1 and the distance of a point ( , ,  ) from P3 is 2, then which of the following
relations is (are) true?
(A) 2 +  + 2 + 2 = 0 (B) 2 −  + 2 + 4 = 0
(C) 2 +  − 2 − 10 = 0 (D) 2 −  + 2 − 8 = 0
16. In 3 , let L be a straight line passing through the origin. Suppose that all the points on L are
at a constant distance from the two planes P1 : x + 2y − z + 1 = 0 and P2 : 2x − y + z − 1 = 0 . Let
M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P1 .
Which of the following points lie(s) on M?
 5 2  1 1 1
(A)  0, − , −  (B)  − , − , 
 6 3  6 3 6
 5 1  1 2
(C)  − , 0,  (D)  − , 0, 
 6 6  3 3
CPP. NO.–3
1. If a + b + c = 0, a = 3, b = 5, c = 7, then the angle between a & b is

(A) /6 (B) 2/3
(C) 5/3 (D) /3
2. A line passes through the point A(iˆ + 2ˆj + 3k)

ˆ and is parallel to the vector V = (iˆ + ˆj + k).
ˆ The
shortest distance from the origin, of the line is
(A) 2 (B) 4
(C) 5 (D) 6
3. Let a, b, c be vectors of length 3, 4, 5 respectively. Let a be perpendicular to

(b + c ) , b ( )
to (c + a) and c to a + b . Then a + b + c is
(A) 2 5 (B) 2 2
(C) 10 5 (D) 5 2
→ →
4. Given a parallelogram ABCD. If AB = a, AD = b and AC = c, then DB  AB has the value
3a2 + b2 − c 2 a2 + 3b2 − c 2
(A) (B)
2 2
a − b2 + 3c 2
2
(C) (D) none of these
2
15 ˆ
5. If a vector a of magnitude 50 is collinear with vector b = 6iˆ − 8ˆj − k and makes an acute
2
angle with positive z-axis then
(A) a = 4b (B) a = −4b
(C) b = 4a (D) none of these
6. A, B, C and D are four points in a plane with pvs a, b,c & d respectively such that
(a − d)  (b − c ) = (b − d)  (c − a ) = 0. Then for the triangle ABC, D is its
(A) incentre (B) circumcentre
(C) orthocentre (D) centroid
7. a and b are unit vectors inclined to each other at an angle ,   (0, ) and a + b  1. Then

  2   2 
(A)  ,  (B)  ,  
3 3   3 
    3 
(C)  0,  (D)  , 
 3 4 4 
8. Image of the point P with position vector 7i − ˆj + 2kˆ in the line whose vector equation is,
r = 9iˆ + 5ˆj + 5kˆ + (iˆ + 3ˆj + 5k)
ˆ has the position vector
(A) (–9, 5, 2) (B) (9, 5, –2)
(C) (9, –5, –2) (D) none of these
9. ˆ c be three unit vectors such that aˆ + bˆ + cˆ is also a unit vector. If pairwise angles
Let â, b,
between a, ˆ cˆ are  ,  , and  respectively then cos 1 + cos 2 + cos 3 equals
ˆ b, 1 2 3
(A) 3 (B) –3
(C) 1 (D) –1
8
10. A tangent is drawn to the curve y = at a point A(x1, y1), where x1 = 2. The tangent cuts the
x2
→ →
x-axis at point B. Then the scalar product of the vectors AB and OB is
(A) 3 (B) –3
(C) 6 (D) –6
11. L1 and L2 are two lines whose vector equations are

( ) ( ) (
L1 : r =   cos  + 3 î + 2 sin  ˆj + cos  − 3 kˆ 
  )
ˆ( ˆ ˆ
L2 : r =  a i + bj + ck , )
where  and  are scalars and  is the acute angle between L1 and L2. If the angle ‘’ is
independent of  then the value of ‘’ is
 
(A) (B)
6 4
 
(C) (D)
3 2
2 2
a+b − a−b
12. Let a and b be two units vectors. The maximum value of 2 2
is equal to
a+b + a−b
(A) 1 (B) 2
(C) 4 (D) 6
13. Let a = î + ˆj + kˆ and r = xiˆ + yjˆ + zkˆ be a variable vector such that r  î, r  ˆj and r  kˆ be
positive integers. If r  ˆj  3 and r  a  12, then the total number of possible r is equal to
(A) 10C3 (B) 11C3
13
(C) C4 (D) 13C9
14. If the angle between the vectors

a = î + (cos x)jˆ + kˆ and b = (sin2 x − sin x)iˆ − (cos x)jˆ + (3 − 4 sin x)kˆ
 
is obtuse and x   0,  , then the exhaustive set of values of ‘x’ is equal to
 2
   
(A) x   0,  (B) x   , 
 6 6 2
   
(C) x   ,  (D) x   , 
6 3 3 2
15. Let PQR be a triangle. Let a = QR, b = RP and c = PQ . If a = 12, b = 4 3 and

b.c = 24, then which of the following is (are) true?
2 2
c c
(A) − a = 12 (B) + a = 30
2 2
(C) a  b + c  a = 48 3 (D) a.b = −72
Paragraph for 16-17
16. Let
a = i + 2j + 3k
b = 2i + 3j + 4k
c = 4i + j
d = 5i + 2j + k
Consider the sequence of vectors,
x1 = a,
x2 = c + 
1 (
 x − c .d 
d,) x3 = a + 
2(
 x − a .b 
)
b ,
 2   2 
 d   b 
   
x4 = c + 
3 (
 x − c .d 
d,) x5 = a + 
4(
 x − a .b 
)
b ,
 2   2 
 d   b 
   
x6 = c + 
5 (
 x − c .d 
d,) and so on
 2 
 d 
 
16. lim x 2n =
n→
k 7 4 1
(A) 10i + 6j − (B) −i − j − k (C) i + 18j + k (D) 5i + 12j − k
3 2 3 3
17. lim x 2n − x 2n+1 =

n→
49 98 118
(A) 0 (B) (C) (D)
3 3 3
CPP. NO.–4
1. ( ) ( )
Cosine of angle between the vectors a + b and a − b if a = 2, b = 1 and a ^ b = 60o is
(A) 3/7 (B) 9 / 21
(C) 3 / 7 (D) none of these
2. An arc AC of a circle subtends a right angle at the centre O. The point B divides the arc in the
ratio 1 : 2. If OA = a & OB = b, then the vector OC in terms of a & b , is
(A) 3 a − 2b (B) − 3 a + 2b
(C) 2a − 3 b (D) −2a + 3 b
3. Given three vectors a, b & c each two of which are non collinear. Further if a + b is collinear ( )
( )
with c , b + c is collinear with a and a = b = c = 2. Then the value of a  b + b  c + c  a
(A) is 3 (B) is –3
(C) is 0 (D) cannot be evaluated
4. The vector equations of two lines L1 and L2 are respectively

r = 17iˆ − 9ˆj + 9kˆ + (3iˆ + ˆj + 5k)
ˆ and r = 15iˆ − 8 ˆj − kˆ + (4iˆ + 3 ˆj)
I. L1 and L2 are skew lines
II. (11, –11, –1) is the point of intersection of L1 and L2
III. (–11, 11, 1) is the point of intersection of L1 and L2
( )
IV. cos–1 3 / 35 is the acute angle between L1 and L2
then, which of the following is true?
(A) II and IV (B) I and IV
(C) IV only (D) III and IV
5. For some non zero vector V , if the sum of V and the vector obtained from V by rotating
it by an angle 2 equals to the vector obtained from V by rotating it by  then the value
of , is
 
(A) 2n  (B) n 
3 3
2 2
(C) 2n  (D) n 
3 3
where n is an integer.
6. Let u, v, w be such that u = 1, v = 2, w = 3. If the projection of v along u is equal to that

of w along u and vectors v, w are perpendicular to each other then u − v + w equals
(A) 2 (B) 7
(C) 14 (D) 14
7. If a and b are non zero, non collinear vectors, and the linear combination
(2x − y)a + 4b = 5a + (x − 2y)b holds for real x and y then x + y has the value equal to
(A) –3 (B) 1
(C) 17 (D) 3
→ →
8. In the isosceles triangle ABC, AB = BC = 8, a point E divides AB internally in the ratio 1 : 3,
→ →
 →

then the cosine of the angle between CE and CA is  where CA = 12 
 
3 7 3 8
(A) − (B)
8 17
3 7 −3 8
(C) (D)
8 17
9. If p = 3a − 5b ; q = 2a + b ; r = a + 4b ; s = −a + b are four vectors such that sin(p ^ q) = 1 and

sin(r ^ s) = 1 then cos(a ^ b) is
19
(A) − (B) 0
5 43
19
(C) 1 (D)
5 43
10. Given an equilateral triangle ABC with side length equal to ‘a’. Let M and N two points
AB
respectively on the side AB and AC such that AN = KAC and AM = . If BN and CM are
3
orthogonal then the value of K is equal to
1 1
(A) (B)
5 4
1 1
(C) (D)
3 2
MATRIX MATCH TYPE
11.
Column-I Column-II
(A) P is point in the plane of the triangle ABAC pv’s of A, B and (P) centroid
C are a, b and c respectively with respect to P as the
origin.
( )( )
If b + c  b − c = 0 and (c + a)  (c − a) = 0 , then w.r.t. the
triangle ABC, P is its
(B) If a, b, c are the position vectors of the three non collinear (Q) orthocentre
points A, B and C respectively such that the vector
V = PA + PB + PC is a null vector then w.r.t. the ABC, P
is its
(C) If P is a point inside the ABC such that the vector (R) Incentre
R = (BC)(PA) + (CA) (PB) + (AB) (PC) is a null vector then
w.r.t. the ABC, P is its
(D) If P is a point in the plane of the triangle ABC such that the (S) circumcentre
scalar product PA  CB and PB  AC vanishes, then w.r.t
the ABC, P is its
INTEGER TYPE
  2
12. In a triangle ABC, if BC = − and AC = where    then find the value of
  
(1 + cos 2A + cos 2B + cos 2C)
CPP. NO.–5
1. Let V1 = 3ax 2 î − 2(x − 1)jˆ and V2 = b(x − 1)iˆ + x 2 ˆj , where ab < 0. The vector V1 and V2 are
linearly dependent for
(A) atleast one x in (0, 1) (B) atleast one x in (–1, 0)
(C) atleast one x in (1, 2) (D) no value of x
2. If p & s are not perpendicular to each other and r  p = q  p & r  s = 0, then r =

 qs
(A) p  s (B) q +  p
ps
 qs
(C) q −  p (D) q + p for all scalars 
ps
→ → →
3. Given a parallelogram OACB. The lengths of the vectors OA, OB & AB are a, b, and c
→ →
respectively. The scalar product of the vectors OC & OB is
a2 − 3b2 + c 2 3a2 + b2 − c 2
(A) (B)
2 2
3a2 − b2 + c 2 a2 + 3b2 − c 2
(C) (D)
2 2
2
(a + 3b)  (3a − b)
2
4. Vectors a and b make an angle  = . If a = 1, b = 2 then =
3
(A) 225 (B) 250
(C) 275 (D) 300
→
 → →
 2
5. In a quadrilateral ABCD, AC is the bisector of the  AB ^ CD  which is ,
  3
→ → →
 → ^ → 
15 AC = 3 AB = 5 AD then cos  BA CD  is
 
14 21
(A) − (B) −
7 2 7 3
2 2 7
(C) (D)
7 14
6. If the two adjacent sides of two rectangles are represented by the vectors
p = 5a − 3b ; q = −a − 2b and r = −4a − b ; s = −a + b respectively, then the angle between the
1 1
vectors x = ( p + r + s ) and y = ( r + s )
3 5
 19   19 
(A) is –cos–1   (B) is cos–1  
 5 43   5 43 
 19 
(C) is  –cos–1   (D) can not be evaluated
 5 43 
→ →
7. If the vector product of a constant vector OA with a variable vector OB in a fixed plane OAB
be a constant vector, then locus of B is
→ →
(A) a straight line perpendicular to OA (B) a circle with centre O radius equal to OA
→
(C) a straight line parallel to OA (D) none of these
ONE OR MORE MAY BE CORRECT
8. Which of the following conclusion(s) hold(s) true for a non-zero vector a .

(A) a  b = a  c  b = c (B) a  b = a  c  b = c
(C) a  b = a  c and a  b = a  c  b = c (D) aˆ + bˆ = aˆ − bˆ  aˆ  bˆ = 0
INTEGER TYPE
9. ( ) ( )
Let O be an interior point of ABC such that 2 OA + 5 OB + 10 OC = 0. If the ratio of the ( )
area of ABC to the area of AOC is t, where ‘O’ is the origin. Find [t]. (where [ ] denotes
greatest integer function.)
10. If the distance from the point P(1, 1, 1) to the line passing through the points Q(0, 6, 8) and
R(–1, 4, 7) is expressed in the form p / q where p and q are coprime, then find the value of
(p + q)(p + q − 1)
2
11. Let S(t) be the area of the OAB with O (0, 0, 0), A (2, 2, 1) and B (t, 1, t+ 1). The value of the
e
 e3 + a 
definite integral  (S(t))2 ln t dt, is equal to   where a, b  N, find (a + b).
1  b 
12. Given f2(x) + g2(x) + h2(x)  9 and U(x) = 3f(x) + 4g(x) + 10h(x), where f(x), g(x) and h(x) are
continuous  x  R. If maximum value of U(x) is N, then find N.
13. In 2
, if the magnitude of the projection vector of the vector iˆ + ˆj on 3iˆ + ˆj is 3 and if 
=2+ 3  , then possible value(s) of |  | is (are)
CPP. NO.–6
1. For non-zero vectors a, b, c, a  b  c = a b c holds if and only if

(A) a  b = 0, b  c = 0 (B) c  a = 0, a  b = 0
(C) a  c = 0, b  c = 0 (D) a  b = b  c = c  a = 0
2. The vectors a = î + 2ˆj + 3k ; b = 2iˆ − ˆj + kˆ & c = 3iˆ + ˆj + 4kˆ are so placed that the end point of
one vector is the starting point of the next vector. Then the vectors are
(A) not coplanar (B) coplanar but cannot form a triangle
(C) coplanar but can form a triangle (D) coplanar & can form a right angled triangle
3. Given the vectors

u = 2iˆ − ˆj − k
v = î − ˆj − 2kˆ
w = î − kˆ
If the volume of the parallelopiped having −cu, v and cw as concurrent edges, is 8 them ‘c’
can be equal to
(A) 2 (B) 4
(C) 8 (D) 21
4. Given a = xiˆ + yjˆ + 2k,

ˆ b = î − ˆj + k, ( )
ˆ c = î + 2ˆj ; a ^ b =  / 2, a  c = 4 then
(A) [a b c]2 = a (B) [a b c] = a

2
(C) [a b c] = 0 (D) [a b c] = a
5. Let a = a1î + a2 ˆj + a3kˆ ; b = b1î + b2 ˆj + b3kˆ ; c = c1î + c 2 ˆj + c 3kˆ be three non-zero vectors such
that c is a unit vector perpendicular to both a and b . If the angle between a and b is
2
a1 b1 c1

then a2 b2 c2 =
6
a3 b3 c3
(A) 0 (B) 1
1 3
(C) (a12 + a22 + a32 ) (b12 + b22 + b32 ) (D) (a12 + a22 + a32 ) (b12 + b22 + b32 ) (c12 + c 22 + c 32 )
4 4
6. For three vectors u, v, w which of the following expressions is not equal to any of the remaining
three?
(A) u  (v  w) (B) (v  w)  u
(C) v  (u  w) (D) (u  v)  w
7. Let a = î + ˆj, b = ˆj + kˆ & c = a + b. If the vectors, î − 2ˆj + k,

ˆ 3iˆ + 2ˆj − kˆ & c are coplanar then

is:

(A) 1 (B) 2
(C) 3 (D) –3
8. A rigid body rotates about an axis through the origin with an angular velocity 10 3
radians/sec. If  points in the direction of î + ˆj + kˆ then the equation to the locus of the points
having tangential speed 20 m/sec is
(A) x2 + y2 + z2 – xy – yz – zx –1 = 0 (B) x2 + y2 + z2 –2xy – 2yz – 2zx – 1 = 0
(C)x2 + y2 + z2 –xy – yz – zx –2 = 0 (D) x2 + y2 + z2 – 2xy – 2yz – 2zx = 2
9. A rigid body rotates with constant angular velocity  about the line whose vector equation is,
r = (iˆ + 2ˆj + 2k).
ˆ The speed of the particle at the instant it passes through the point with p.v.
2iˆ + 3ˆj + 5kˆ is
(A)  2 (B) 2
(C)  / 2 (D) none of these
Let AB = 3iˆ − ˆj, AC = 2iˆ + 3ˆj and DE = 4iˆ − 2ˆj. The

A
10.
area of the shaded region in the adjacent figure, is D
(A) 5 (B) 6
(C) 7 (D) 8
E
B C
11. Equation of the line which passes through the point with p.v. (2, 1, 0) and perpendicular to the
plane containing the vectors î + ˆj and ˆj + kˆ is
(A) r = (2, 1,0) + t(1, − 1, 1) (B) r = (2, 1,0) + t( −1, 1, 1)
(C) r = (2, 1,0) + t(1, 1, − 1) (D) r = (2, 1,0) + t(1, 1, 1)
12. If a, b, c be three non zero vectors satisfying the condition a  b = c and b  c = a then which
of the following always hold(s) good?
(A) a, b, c are orthogonal in pairs (B) a, b, c  = b
(C) a, b, c  = c
2
(D) b = c
INTEGER TYPE
13. Vector V is perpendicular to the plane of vectors a = 2iˆ − 3ˆj + kˆ and b = î − 2ˆj + 3kˆ and
2
satisfies to condition V  (iˆ + 2ˆj − 7k)
ˆ = 10. Find V .
2
14. Let two non-collinear vectors a and b inclined at an angle be such that
3
a = 3 and b = 4. A point P moves so that at any time the position vector OP (where O is
the origin) is given as OP = (et + e–t) a + (et – e–t) b . If the least distance of P from origin is
2 a − b where a, b, N then find the value of (a + b).
15. If the three vectors V1 = î − ajˆ − ak,

ˆ V = biˆ − ˆj + bk,
2
ˆ V = ciˆ + cjˆ − kˆ are linearly dependent then
3
–1 –1 –1
find the value of (1 + a) + (1 + b) + (1 + c) .
CPP. NO.–7
1. The altitude of a parallelopiped whose three coterminous edges are the vectors,
A = î + ˆj + kˆ ; B = 2iˆ + 4ˆj − kˆ & C = î + ˆj + 3kˆ with A and B as the sides of the base of the
parallelopiped, is
(A) 2 / 19 (B) 4 / 19
(C) 2 38 / 19 (D) none of these
2. Consider ABC with () ()

A  a ; B  b and C  c . () If ( )
b  a + c = b  b + a  c ; b − a = 3;
→ →
c − b = 4 then the angle between the medians AM & BD is
 1   1 
(A)  − cos−1   (B)  − cos−1  
 5 13   13 5 
 1   1 
(C) cos−1   (D) cos−1  
 5 13   13 5 
3. If A(–4, 0, 3) ; B (14, 2, –5) then which one of the following points lie on the bisector of the
angle between OA and OB (‘O’ is the origin of reference)
(A) (2, 1, –1) (B) (2, 11, 5)
(C) (10, 2, –2) (D) (1, 1, 2)
4. Position vectors of the four angular points of a tetrahedron ABCD are A(3, –2, 1); B(3, 1, 5);
C(4, 0, 3) and D(1, 0, 0). Acute angle between the plane faces ADC and ABC is
(A) tan–1(5/2) (B) cos–1(2/5)
(C) cosec–1(5/2) (D) cot–1(3/2)
5. The volume of the tetrahedron formed by the coterminus edges a, b, c is 3. Then the volume
of the parallelepiped formed by the coterminus edges a + b, b + c, c + a is
(A) 6 (B) 18
(C) 36 (D) 9
6. If a = î + ˆj + kˆ & b = î − 2ˆj + k,
ˆ then the vector c such that a  c = 2 & a  c = b is
(A)
1 ˆ
3
(3i − 2ˆj + 5kˆ ) (B)
1 ˆ
3
(
− i + 2ˆj + 5kˆ )
(C)
1 ˆ
3
( i + 2ˆj − 5kˆ) (D)
1 ˆ
3
(
3i + 2ˆj + kˆ )
7. a, b and c be three vectors having magnitudes 1, 1 and 2 respectively. If a  (a  c) + b = 0,
then the acute angle between a & c is
(A) /6 (B) /4
(C) /3 (D) 5/12
8. If a = î + ˆj + k,
ˆ b = 4iˆ + 3ˆj + 4kˆ and c = î + ˆj + kˆ are linearly dependent vectors and c = 3,
then
(A)  = 1,  = –1 (B)  = 1,  = 1
(C)  = –1,  = 1 (D)  = 1,  = 1
9. A vector of magnitude 5 5 coplanar with vectors î + 2ˆj & j + 2kˆ and the perpendicular vector
2iˆ + ˆj + 2kˆ is
(A) 5 (5iˆ + 6ˆj − 8k)
ˆ (B)  5 (5iˆ + 6ˆj − 8k)
ˆ
(C) 5 5 (5iˆ + 6ˆj − 8k)
ˆ (D)  (5iˆ + 6ˆj − 8k)
ˆ
10. Let  = 2iˆ + 3ˆj − kˆ and  = i + j is a unit vector, then the maximum value of          
is equal to
(A) 2 (B) 3
(C) 4 (D) 9
Paragraph for questions nos. 11 to 13
Consider three vectors p = î + ˆj + k,

ˆ q = 2iˆ + 4ˆj − kˆ and r = î + ˆj + 3kˆ and let s be a unit vector,
then
11. p, q and r are

(A) linearly dependent
(B) can form the sides of a possible triangle
(C) such that the vectors ( q − r ) is orthogonal to p
(D) such that each one of these can be expressed as a linear combination of the other two
12. If (p  q)  r = up + vq + wr, then (u + v + w) equals to

(A) 8 (B) 2
(C) –2 (D) 4
13. The magnitude of the vector (p  s ) ( q  r ) + ( q  s ) ( r  p ) + ( r  s ) (p  q) is

(A) 4 (B) 8
(C) 18 (D) 2
14. Given the following information about the non zero vectors A, B and C
(i) (A  B)  A = 0 (ii) B  B = 4
(iii) A  B = −6 (iv) B  C = 6
Which one of the following holds good?
(A) A  B = 0 (B) A  (B  C) = 0
(C) A  A = 8 (D) A  C = −9
15. If A, B, C and D are four non zero vectors in the same plane no two of which are collinear
then which of the following hold(s) good?
(A) (A  B)  (C  D) = 0 (B) (A  C)  (B  D)  0
(C) (A  B)  (C  D) = 0 (D) (A  C)  (B  D)  0
16. Let first and second row vectors of matrix A be r1 = [1 1 3] and r2 = [2 1 1] and let the third
row vector be in the plane of r1 and r2 and perpendicular to r2 with magnitude 5 , then which
of the following is/are true? [Note: Tr (P) denotes trace of matrix P]
(A) Tr (A) = 3
(B) Volume of parallelopiped formed by r2 , r3 and r2  r3 equals 30.
(C) Row vectors are linearly dependent
(D)  r1  r2 r2  r3 r3  r1  = 0
INTEGER TYPE
17. If u, v, w are non-zero and non-coplanar vectors then find the number of ordered pairs (p, q)
so that 3u pv pw  − pv w qu − 2w qv qu  = 0  p, q  R.
CPP. NO.–8
1. Consider three vectors p = î + ˆj + k,

ˆ q = 2iˆ + 4ˆj − kˆ and r = î + ˆj + 3k.
ˆ If p, q and r denotes
the position vector of three non-collinear points then the equation of the plane containing these
points is
(A) 2x – 3y + 1 = 0 (B) x – 3y + 2z = 0
(C) 3x – y + z – 3 = 0 (D) 3x – y – 2 = 0
2. The intercept made by the plane r  n = q on the x-axis is

q î  n
(A) (B)
î  n q
( )
(C) î  n q (D)
q
n
3. If the distance between the planes

8x + 12y – 14z = 2
and 4x + 6y – 7z = 2
1 N(N + 1)
can be expressed in the form where N is natural then the value of is
N 2
(A) 4950 (B) 5050
(C) 5150 (D) 5151
4. A plane passes through the point P(4, 0, 0) and Q(0, 0, 4) and is parallel to the y-axis. The
distance of the plane from the origin is
(A) 2 (B) 4
(C) 2 (D) 2 2
5. If from the point P(f, g, h) perpendiculars PL, PM be drawn to yz and zx planes then the
equation to the plane OLM is
x y z x y z
(A) + − = 0 (B) + + = 0
d g h f g h
x y z x y z
(C) − + = 0 (D) − + + = 0
f g h f g h
6. IF the plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1 (k) with x-axis, then k is equal to
(A) 3 / 2 (B) 2/7
(C) 2/3 (D) 1
7. The plane XOZ divides the join of (1, –1, 5) and (2, 3, 4) in the ratio  : 1, then  is
(A) –3 (B) –1/3
(C) 3 (D) 1/3
8. A variable plane forms a tetrahedron of constant volume 64 K3 with the coordinate planes and
the origin, then locus of the centroid of the tetrahedron is
(A) x3 + y3 + z3 = 6K3 (B) xyz = 6k3
(C) x2 + y2 + z2 = 4K2 (D) x–2 + y–2 + z–2 = 4K–2
9. Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular.
Let the area of triangles ABC, ACD and ADB be 3, 4 and 5 sq. units respectively. Then the
area of the triangle BCD, is
(A) 5 2 (B) 5
(C) 5 / 2 (D) 5/2
10. Which of the following planes are parallel but not identical?
P1 : 4x – 2y + 6z = 3
P2 : 4x – 2y – 2z = 6
P3 : –6x + 3y – 9z = 5
P4 : 2x – y – z = 3
(A) P2 & P3 (B) P2 & P4
(C) P1 & P3 (D) P1 & P4
11. A parallelopiped is formed by planes drawn through the points (1, 2, 3) and (9, 8, 5) parallel to
the coordinate planes then which of the following is not the length of an edge of this rectangular
parallelopiped
(A) 2 (B) 4
(C) 6 (D) 8
12. Vector equation of the plane r = î − ˆj + (iˆ + ˆj + k)

ˆ + (i − 2ˆj + 3k)ˆ in the scalar dot product form
is
(A) r  (5iˆ − 2ˆj + 3k)
ˆ =7 (B) r  (5iˆ + 2ˆj − 3k)
ˆ =7
(C) r  (5iˆ − 2ˆj − 3k)
ˆ =7 (D) r  (5iˆ + 2ˆj + 3k)
ˆ =7
13. The vector equations of the two lines L1 and L2 are given by
L1 : r = 2iˆ + 9ˆj + 13kˆ + (iˆ + 2ˆj + 3k) ; L 2 : r = −3iˆ + 7 ˆj + pkˆ + ( − î + 2ˆj − 3k)
ˆ
Then the lines L1 and L2 are
(A) skew lines for all pR
(B) intersecting for all pR and the point of intersection is (–1, 3, 4)
(C) intersecting lines for p = –2
(D) intersecting for all real pR
14. Consider the plane (x, y, z) = (0, 1, 1) + (1, –1, 1) + (2, –1, 0). The distance of this plane
from the origin is
(A) 1/3 (B) 3 / 2
(C) 3/2 (D) 2 / 3
15. The equation of the plane which has the property that the point Q (5, 4, 5) is the reflection of
point P(1, 2, 3) through that plane, is ax + by + cz = d where a, b, c, d  N. Find the least value
of (a + b + c + d).
16. Let P be the image of the point (3, 1, 7) with respect to the plane x − y + z = 3 . Then the
x y z
equation of the plane passing through P and containing the straight line = = is
1 2 1
(A) x + y − 3z = 0 (B) 3x + z = 0
(C) x − 4y + 7z = 0 (D) 2x − y = 0
CPP. NO.–9
x − 2 y − 9 z − 13 x−a y−7 z+2
1. The value of ‘a’ for which the lines = = and = = intersect, is
1 2 3 −1 2 −3
(A) –5 (B) –2
(C) 5 (D) –3
2. Given A(1, –1, 0) ; B(3, 1, 2) ; C(2, –2, 4) and D(–1, 1, –1) which of the following points neither
lie on AB nor on CD?
(A) (2, 2, 4) (B) (2, –2, 4)
(C) (2, 0, 1) (D) (0, –2, –1)
x −1 y − 2 z − 3
3. For the line = = , which one of the following is incorrect?
1 2 3
x y z
(A) it lies in the plane x – 2y + z = 0 (B) it is same as line = =
1 2 3
(C) it passes through (2, 3, 5) (D) it is parallel to the plane x – 2y + z – 6 = 0
4. Given planes
P1 : cy + bz = x
P2 ; az + cx = y
P3 ; bx + ay = z
P1, P2 and P3 pass through one line, if
(A) a2 + b2 + c2 = ab + bc + ca (B) a2 + b2 + c2 + 2abc = 1
(C) a2 + b2 + c2 = 1 (D) a2 + b2 + c2 + 2ab + 2bc + 2ca + 2abc = 1
x − x1 y − y1 z − z1
5. The line = = is
0 1 2
(A) parallel to x-axis (B) perpendicular to x-axis
(C) perpendicular to YOZ plane (D) parallel to y-axis
x−2 y−3 z−4 x −1 y − 4 z − 5

6. The lines = = and = = are coplanar if
1 1 −k k 2 1
(A) k = 0 or –1 (B) k = 1 or –1
(C) k = 0 or –3 (D) k = 3 or –3
7. The line which contains all points (x, y, z) which are of the form (x, y, z) = (2, –2, 5) +
(1, –3, 2)intersects the plane 2x – 3y + 4z = 163 at P and intersects the YZ plane at Q.
If the distance PQ is a b where a, b  N and a > 3 then (a + b) equals
(A) 23 (B) 95
8. Let L1 be the line r1 = 2iˆ + ˆj − kˆ + (i + 2k)

ˆ and let L2 be the line r = 3iˆ + ˆj + (iˆ + ˆj − k).
2
ˆ Let 
be the plane which contains the line L1 and parallel to L2. The distance of the plane  from the
origin is
(A) 1/7 (B) 2 / 7
9. The value of m for which straight line 3x – 2y + z + 3 = 0 = 4x – 3y + 4z + 1 is parallel to the

plane 2x – y + mz – 2 = 0 is
(A) –2 (B) 8
(C) –18 (D) 11
10. The distance of the point (–1, –5, –10) from the point of intersection of the line
x − 2 y +1 z − 2
= = and the plane x – y + z = 5 is
2 4 12
(A) 2 11 (B) 126
(C) 13 (D) 14
11. P(p) and Q(q) are the position vectors of two fixed points and R(r ) is the position vector of
a variable point. If R moves such that (r − p)  (r − q) = 0, then the locus of R is
(A) a plane containing the origin ‘O’ and parallel to two non collinear vectors OP and OQ
(B) the surface of a sphere described on PQ as its diameter.
(C) a line passing through the points P and Q
(D) a set of lines parallel to the line PQ
Let P and Q be points on the lines

L1 : r = 6iˆ + 7ˆj + 4k + (3iˆ − ˆj + k) and L 2 : r = −9 ˆj + 2kˆ + ( −3iˆ + 2ˆj + 4k)
ˆ respectively which
are nearest to each other.
12. The distance between P and Q equals

(A) 5 10 (B) 3 30
(C) 4 8 (D) 6 7
13. The equation of plane OPQ (where O is origin), is

(A) 23x + 9y – z = 0 (B) 2x – 3y + 10z = 0
(C) 2x + 3y – 10z = 0 (D) 23x – 9y + z = 0
14. The volume of parallelopiped formed by OP, OQ and OR where O is origin and R(1, –1, 0)
is equal to
(A) 64 (B) 78
(C) 86 (D) 96
15. Consider a plane P passing through A(, 3, ), B(–1, 3, 2) and C(7, 5, 10) and a straight line
L with positive direction cosines passing through A, bisecting BC and makes equal angles with
the coordinate axes. Let L1 be a line parallel to L and passing through origin. Which of the
following is(are) correct?
(A) The value of ( + ) is equal to 5.
x −1 y −1 z −1
(B*) Equation of straight line L1 is = = .
1 1 1
(C*) Equation of the plane perpendicular to the plane P and containing line L1 is x – 2y + z = 0.
(D*) Area of triangle ABC is equal to 3 2.
INTEGER TYPE
16. A line L passing through the point P(1, 4, 3), is perpendicular to both the lines
x −1 y + 3 z − 2 x + 2 y − 4 z +1
= = and = =
2 1 4 3 2 −2
If the position vector of point Q on L is (a1, a2, a3) such that (PQ)2 = 357, then find the sum of
all possible values of (a1 +a2 + a3).
17. Two intersecting lines lying in a plane P1 have equations
x −1 y − 3 z − 4 x −1 y − 3 z − 4
= = and = = .
2 1 −3 −1 2 4
If the equation of plane P2 is 2x – y + z = 21 and distance between he planes P1 and P2 is d,
then find the value of d2.
18. Consider the lines L1; r = 2iˆ + kˆ + (iˆ − ˆj) and L 2 : r = (iˆ − ˆj + 2k)
ˆ where , R. Find the
distance of point M ( )
2, 8, 0 from the plane passing through the point N(1, –1, 5) and whose
normal is perpendicular to both the lines L1 and L2.
19. If planes r  (iˆ + ˆj + k)

ˆ = 1, r  (iˆ + 2ajˆ + k)
ˆ = 2 and r  (aiˆ + a 2 ˆj + k)
ˆ = 3 intersect in a line, then find
the number of real values of a.
MATCH THE COLUMN
20. Consider the following four pairs of lines in column-I and match them with one or more entries
in column-II.
Column-I Column-II
L1 : x = 1 + t, y = t, z = 2 – 5t
(A) (P) non coplanar lines
L2 : r = (2, 1, –3) + (2, 2, –10)
x −1 y − 3 z − 2
L1 : = =
2 2 −1
(B) (Q) lines lie in a unique plane
x−2 y−6 z+2
L2 : = =
1 −1 3
L1 : x = –6t, y = 1 + 9t, z = –3t infinite planes containing both the
(C) (R)
L2 : x = 1 + 2s, y = 4 – 3s, z = s lines
x y −1 z − 2
L1 : = =
1 2 3
(D) (S) lines do not intersect
x − 3 y − 2 z −1
L2 : = =
−4 −3 2
21. P(0, 3, –2); Q(3, 7, –1) and R(1, –3, –1) are 3 given points. Let L1 be the line passing through
P and Q and L2 be the line through R and parallel to the vector V = î + k.
Column-I Column-II
(A) perpendicular distance of P from L2 (P) 7 3
(B) shortest distance between L1 and L2 (Q) 2
(C) area of the triangle PQR (R) 6
19
(D) distance from (0, 0, 0) to the plane PQR (S)
147
CPP. NO.–10
1. If a, b, c be three non-coplanar and p, q, r are reciprocal vectors to a, b & c respectively,
then ( a + mb + nc )  ( p + mq + nr ) is equal to: (where , m, n are scalars)
(A) 2 + m2 + n2 (B)  m + m n + n
2. The three vectors î + ˆj, ˆj + k,

ˆ kˆ + î taken two at a time form three planes. The three unit vectors
drawn perpendicular to these three planes form a parallelopiped of volume
(A) 1/3 (B) 4
(C) 3 3 / 4 (D) 4 / 3 3
3. If x & y are two non collinear vectors and a, b, c represent the sides of a ABC satisfying
(a − b)x + (b − c)y + (c − a)(x  y) = 0 then ABC is
(A) an acute angle triangle (B) an obtuse angle triangle
(C) a right angle triangle (D) a scalene triangle
4. Given three non-zero, non-coplanar vectors a, b, c and r1 = pa + qb + c and r2 = a + pb + qc

if the vectors r1 + 2r2 and 2r1 + r2 are collinear then (p, q) is
(A) (0, 0) (B) (1, –1)
(C) (–1, 1) (D) (1, 1)
5. If the vectors a, b, c are non-coplanar and , m, n are distinct scalars, then
(
 a + mb + nc
 ) ( b + mc + ma) ( c + ma + nb) = 0 implies
(A) m + m n + n = 0 (B)  + m + n = 0
(C) 2 + m2 + n2 = 0 (D) 3 + m3 + n3 = 0
6. Let r1, r2 , r3 .......rn be the position vectors of points P1, P2, P3.......Pn relative to the origin O.
If the vector equation a1r1 + a2 r2 + ......... + an rn = 0 holds, then a similar equation will also hold
w.r.t. to any other origin provided
(A) a1 + a2 + …. + an = n (B) a1 + a2 + … +an = 1
(C) a1 + a2 + …. + an = 0 (D) none of these
7. The orthogonal projection A of the point A with position vector (1, 2, 3) on the plane
3x – y + 4z = 0 is
 1 5 
(A) (–1, 3, –1) (B)  − , , 1
 2 2 
1 5 
(C)  , − , − 1 (D) (6, –7, –5)
2 2 
8. ˆ  (aˆ + b)
If â and bˆ are unit vectors then the vector defined as V = (aˆ  b) ˆ is collinear to the
vector
(A) â + bˆ (B) bˆ − aˆ
(C) 2aˆ − b (D) â + 2bˆ
9. If â and bˆ are orthogonal unit vectors then for any non zero vector r , the vector (r  a)
ˆ
equals
(A)  r aˆ bˆ  (aˆ + b)
ˆ (B)  r aˆ bˆ  aˆ + (rˆ  a)(a ˆ
ˆ ˆ  b)
(C)  r aˆ bˆ  bˆ + (rˆ  b)(b
ˆ ˆ  a)
ˆ (D)  r aˆ bˆ  bˆ + (rˆ  a)(a ˆ
ˆ ˆ  b)
Consider a plane
x + y – z = 1 and the point A(1, 2, –3)
A line L has the equation
x = 1 + 3r
y=2–r
z = 3 + 4r
10. The co-ordinate of a point B of line L, such that AB is parallel to the plane, is
(A) 10, –1, 15 (B) –5, 4, –5
(C) 4, 1, 7 (D) –8, 5, –9
11. Equation of the plane containing the line L and the point A has the equation
(A) x – 3y + 5 = 0 (B) x + 3y – 7 = 0
(C) 3x – y – 1 =0 (D) 3x + y – 5 = 0
Let a = b = 2 and c = 1. Also (a − c)  (b − c) = 0
2
12. a − b + 2c  (a + b) has the value equal to
(A) 12 (B) 10
(C) 8 (D) 6
2
13. a + b − c equals
(A) 5 (B) 6
(C) 7 (D) 8
14. Difference between of the maximum and minimum value of a + b is equal to

(A) 2 (B) 3
(C) 4 (D) 1
Consider the thee vectors p, q and r such that

p = î + ˆj + k ; q = î − ˆj + k
p  r = q + cp and p  r = 2
15. The value of p q r  is

5 2c 8
(A) − (B) −
r 3
(C) 0 (D) greater then zero
16. If x is a vector such that then x is

(A) c(iˆ − 2ˆj + k)
ˆ (B) a unit vector
1
(C) indeterminate, as p q r  (D) − (iˆ − 2ˆj + k)
ˆ
2
17. If y is a vector satisfying (1 + c)y = p  (q  r ) then the vector x, y, r
(A) are collinear
(B) are coplanar
(C) represent the coterminus edges of a tetrahedron whose volume is c cubic units
(D) represent the coterminus edges of a parallelepiped whose volume is c cubic units
REASONING TYPE
x − 4 y + 5 z −1 x − 2 y +1 z
18. Given = = and = =
2 4 −3 1 3 2
Statement-1: The lines intersect.
Statement-2: They are not parallel.
statement-1.
for statement-1.
19. Consider three vectors a, b & c

( ) ( ) (
Statement-1: a  b = (iˆ  a)  b î + ( ˆj  a)  b ˆj + (kˆ  a)  b kˆ )
Statement-2: c = (iˆ  c)iˆ + ( ˆj  c)jˆ + (kˆ  c)kˆ
statement-1.
for statement-1.
20. Statement-1: Let a = î + 2ˆj − 3kˆ and b = 2iˆ + ˆj − k then the vector x satisfying a  x = a  b
and a  x = 0 is of length 10 .
Statement-2: If p, q, r are non-zero distinct vectors such that p  q = p  r, then p is parallel
to (q − r ).
statement-1.
for statement-1.
21. Statement-1: Let the vector a = î + ˆj + kˆ be vertical. The line of greatest slope on a plane with
normal b = 2iˆ − ˆj + kˆ is along the vector î − 4ˆj + 2k.
ˆ
Statement-2: If a is vertical, then the line of greatest slope on a plane with normal b is along
the vector (a  b)  b.
statement-1.
for statement-1.
MULTIPLE OBJECTIVE TYPE
22. () () () ()
If A a ; B b ; C c and D d are four points such that
a = −2iˆ + 4ˆj + 3kˆ ; b = 2iˆ − 8ˆj ; c = î − 3 ˆj + 5kˆ ; d = 4iˆ + ˆj − 7k
d is the shortest distance between the lines AB and CD, then which of the following is True?
(A) d = 0, hence AB and CD intersect
 AB CD BD 
(B) d =  
AB  CD
23
(C) AB and CD are skew lines and d =
13
 AB CD AC 
(D) d =  
AB  CD
23. Which of the following statement(s) is(are) incorrect?

(A) The relation (u  v) = u  v is only possible if atleast one of the vectors u and v is
null vector.
(B) Every vector contained in the line r(t) = 1 + 2t, 1 + 3t, 1 + 4t is parallel to the vector 1, 1, 1 .
(C) If scalar triple product of three vectors u, v, w is larger than u  v then w  1.
(D) The distance between the x-axis and the line x = y = 1 is 2.
24. Given the equations of the line 3x – y +z + 1 = 0 = 5x + y + 3z . Then which of the following is
correct.
1 5
y− z+
x 8 = 8
(A) symmetrical form of the equations of lines is =
2 −1 1
1 5
x+ y−
(B) symmetrical form of the equations of line is 8 = 8 = z
1 1 −2
(C) equation of the plane through (2, 1, 4) and perpendicular to the given lines is
2x – y + z – 7 = 0
(D) equation of the plane through (2, 1, 4) and perpendicular to the given lines is
x + y – 2z + 5 = 0
25. Given three vectors

U = 2iˆ + 3ˆj − 6k ; V = 6iˆ + 2ˆj + 3kˆ ; W = 3iˆ − 6 ˆj − 2kˆ
Which of the following hold good for the vectors U, V and W ?
(A) U, V and W are linearly dependent (B) (U  V)  W = 0
(C) U, V and W form a triplet of mutually perpendicular vectors
(D) U  (V  W ) = 0
26. Consider the family of planes x + y + z = c where c is a parameter intersecting the coordinate
axis at P, Q, R and , ,  are the angles made by each member of this family with positive x,
y and z axis. Which of the following interpretations hold good for this family?
(A) each member of this family is equally inclined with the coordinate axis.
(B) sin2 + sin2 + sin2 = 1.
(C) cos2 + cos2 + cos2 = 2.
(D) for c = 3 area of the triangle PQR is 3 3 sq. units.
27. Let a and b be two non-zero and non-collinear vectors then which of the following is/are
always correct?
(A) a  b = a b î  î + a b ˆj  ˆj + a b kˆ  kˆ
(B) a  b = (a  î) (b  î) + (a  ˆj) (b  ˆj) + (a  k)ˆ (b  k)
ˆ
ˆ ˆ and v = aˆ  bˆ then u = v
(C) if u = aˆ − (aˆ  b)b
(D) if c = a  (a  b) and d = b  (a  b) then c + d = 0.
MATCH THE COLUMN
28.
Column-I Column-II
Centre of the parallelopiped whose 3 coterminous
(A) edges OA, OB and OC have position vectors (P) a+b+c
a, b and c respectively where O is the origin, is
OABC is a tetrahedron where O is the origin.

Positions vectors of its angular points A, B and C
a+b+c
(B) are a, b and c respectively. Segments joining (Q)
2
each vertex with the centroid of the opposite face
are concurrent at a point P whose p.v.’s are
Let ABC be a triangle the position vectors of its
angular points are a, b and c respectively. If a+b+c
(C) (R)
a−b = b−c = c −a then the p.v. of the 3
orthocentre of the triangle is
Let a, b, c be 3 mutually perpendicular vectors of
the same magnitude. If an unknown vectors x
satisfies the equation (S) a+b+c
(D)
( )
a  (x − b)  a + b  ((x − c) ) + c  ((x − a)  c ) = 0. 4
Then x is given by
ABC is a triangle whose centroid is G, orthocentre
is H and circumcentre is the origin. If position
(E) vectors of A, B, C , G and H are a, b, c, g and h
respectively, then h in term of a, b and c is equal
to
29.
Column-I Column-II
Let O be an interior point of AABC such that
(A) OA + 2OB + 3OC = 0, then the ratio of the area of ABC (P) 0
to the area of AOC, is with O as the origin
(B) Let ABC be a triangle whose centroid is G, orthocentre is (Q) 1

H and circumcentre is the origin ‘O’. If D is any point in
the plane of the triangle such that no three of O, A, B, C
and D are collinear satisfying the relation
AD + BD + CH + 3HG = HD then the value of the scalar
‘’ is
If a, b, c and d are non zero vectors such that no three
of them are in the same plane and no two are orthogonal
(C) (b  c)  (a  d) + (c  a)  (b  d) (R) 2
then the value of the scalar
(a  b)  (d  c)
is
(S) 3
SUBJECTIVE TYPE
30. Given a tetrahedron D-ABC with AB = 12, CD = 6. If the shortest distance between the skew

line AB and CD is 8 and the angle between them is , then find the volume of tetrahedron.
6
31. Given A = 2iˆ + 3ˆj + 6k,

ˆ B = î + ˆj − 2kˆ and C = î + 2ˆj + k.
ˆ Compute the value of
A  (A  (A  B)  C .
32. A vector V = v1î + v 2 ˆj + v 3kˆ satisfies the following conditions:

(i) magnitude of V is 7 2
(ii) V is parallel to the plane x – 2y + z = 6
(iii) V is orthogonal to the vector 2iˆ − 3ˆj + 6kˆ
and (iv) V  î  0.
Find the value of (v1 + v2 + v3).
33. Let (p  q)  r + (q  r )q = (x 2 + y 2 )q + (14 − 4x − 6y)p and (r  r )p = r where p and q are two

non-zero non-collinear vectors and x and y are scalars. Find the value of (x + y).
1
34. Let a and b be two vectors such that a = = b . If  denotes the minimum value of
2
1 1
2
+ 2
, then find he number of solution of equation sec2  =  in interval (–4, 8].
a+b a−b
35. Let
a = î + ˆj − 2k,
ˆ
b = 3i − 4ˆj + k,ˆ
c = î + ˆj + k,
ˆ
( )(
Then find the value of a + 2b  ( c + 2a )  b + 2c . ( ))
36. Let P be a point in the first octant, whose image Q in the plane x + y = 3 lies on the z-axis. Let
the distance of P from the x-axis be 5. If R is the image of P in the xy-plane, then the length of
PR is _______.
ANSWER KEY
CPP-1
1. B 2. C 3. B 4. D
5. C 6. B 7. D 8. B
9. D 10. A, D
11. (A) → T ; (B) → U ; (C) → P ; (D) → R ; (E) → Q ; (F) → S ; (G) → W ; (H) → V
12. C 13. D 14. A 15. B
16. BCD
CPP-2
1. D 2. D 3. A 4. C
5. D 6. C 7. A 8. A
9. B 10. D 11. (i) D, (ii) B, (iii) B
12. A, C 13. 13 14. BC 15. BD
16. AB
CPP-3
1. D 2. A 3. D 4. A
5. B 6. C 7. B 8. B
9. D 10. A 11. A 12. A
13. A 14. B 15. ACD 16. B
17. A
CPP-4
1. A 2. B 3. B 4. A
5. A 6. C 7. B 8. C
9. D 10. A
11. (A) → S ; (B) → P ; (C) → R ; (D) → Q 12. 0
CPP-5
1. A 2. C 3. D 4. D
5. C 6. B 7. C 8. C, D
9. 3 10. 4950 11. 7 12. 1125
13. 1 or 2
CPP-6
1. D 2. B 3. A 4. D
5. C 6. C 7. D 8. C
9. A 10. C 11. A 12. A, C
13. 75 14. 0488 15. 2
CPP-7
1. C 2. A 3. D 4. A
5. C 6. B 7. A 8. D
9. D 10. B 11. C 12. B
13. A 14. A, B, D 15. B, C
16. B, C, D 17. 1
CPP-8
1. D 2. A 3. D 4. D
5. A 6. B 7. D 8. B
9. A 10. C 11. B 12. C
13. C 14. C 15. 17 16. C
CPP-9
1. D 2. A 3. C 4. B
5. B 6. C 7. A 8. B
9. A 10. C 11. C 12. B
13. D 14. D 15. B, C, D 16. 16
17. 54 18. 3 19. 0
20. (A) → R ; (B) → Q ; (C) → Q, S ; (D) → P, S
21. (A) → R ; (B) → Q ; (C) → P ; (D) → S
CPP-10
1. A 2. D 3. A 4. D
5. B 6. C 7. B 8. B
9. C 10. D 11. B 12. B
13. C 14. A 15. B 16. D
17. C 18. D 19. A 20. D
21. D 22. B, C, D 23. A, B, D 24. B, D
25. B, C, D 26. A, B, C 27. A, B, C
28. (A) → Q ; (B) → S ; (C) → R ; (D) → Q ; (E) → P
29. (A) → S ; (B) → R ; (C) → Q 30. 48 31. 343
32. 12 33. 5 34. 24 35. 189
36. 8

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FIITJEE

2. In a triangle PQR, a = QR,b = RP and c = PQ . If

diagonal OT. If p = SP ,q = SQ , r = SR and t = ST , then the value of (p  q)  r  t ( ) is

4. Let a = 2iˆ + ˆj − kˆ and b = ˆi + 2ˆj + kˆ be two vectors. Consider a vector c =  a +  b,  ,   .If

5. Let a and b be vectors such that a. b = 0 . For some x, y  , let

(A) 1 (B) 2 (C) 3 (D) 4

7. If the vectors a = 3iˆ + ˆj − 2k,

(C) 14x + 2y + 15z = 31 (D) -14x + 2y + 15z = 3

OP.OQ + OR.OS = OR.OP + OQ.OS = OQ.OR + OP.OS

Then the triangle PQR has S as its

(C) incentre (D) orthocentre

PARAGRAPH (For Q.No.14,15)

(A) sin (P + Q ) (B) sin 2R (C) sin (P + R ) (D) sin ( Q + R )

15. If the triangle PQR varies, then the minimum value of

2. The vectors 3iˆ − 2ˆj + k,

3. Two vectors p, q on a plane satisfy p + q = 13, p − q = 1 and p = 3.

(i) The equation of internal angle bisector through A to side BC is

(ii) The equation of median through C to side AB is

(iii) The area (ABC) is equal to

13. In ABC, a point P is chosen on side AB so that AP : PB = 1 : 4 and a point Q is chosen on

(A) There is exactly one choice for v

(B) There are infinitely many choices for such v

(C) If û lies in the xy-plane then u1 = u2

(D) If û lies in the xz-plane then 2 u1 = u3

1. If a + b + c = 0, a = 3, b = 5, c = 7, then the angle between a & b is

2. A line passes through the point A(iˆ + 2ˆj + 3k)

3. Let a, b, c be vectors of length 3, 4, 5 respectively. Let a be perpendicular to

11. L1 and L2 are two lines whose vector equations are

14. If the angle between the vectors

15. Let PQR be a triangle. Let a = QR, b = RP and c = PQ . If a = 12, b = 4 3 and

Paragraph for 16-17

17. lim x 2n − x 2n+1 =

4. The vector equations of two lines L1 and L2 are respectively

6. Let u, v, w be such that u = 1, v = 2, w = 3. If the projection of v along u is equal to that

9. If p = 3a − 5b ; q = 2a + b ; r = a + 4b ; s = −a + b are four vectors such that sin(p ^ q) = 1 and

2. If p & s are not perpendicular to each other and r  p = q  p & r  s = 0, then r =

ONE OR MORE MAY BE CORRECT

8. Which of the following conclusion(s) hold(s) true for a non-zero vector a .

1. For non-zero vectors a, b, c, a  b  c = a b c holds if and only if

3. Given the vectors

4. Given a = xiˆ + yjˆ + 2k,

(A) [a b c]2 = a (B) [a b c] = a

7. Let a = ˆi + ˆj, b = ˆj + kˆ & c = a + b. If the vectors, ˆi − 2ˆj + k,

Let AB = 3iˆ − ˆj, AC = 2iˆ + 3ˆj and DE = 4iˆ − 2ˆj. The

ONE OR MORE MAY BE CORRECT

15. If the three vectors V1 = ˆi − ajˆ − ak,

2. Consider ABC with () ()

Paragraph for questions nos. 11 to 13

Consider three vectors p = ˆi + ˆj + k,

11. p, q and r are

12. If (p  q)  r = up + vq + wr, then (u + v + w) equals to

13. The magnitude of the vector (p  s ) ( q  r ) + ( q  s ) ( r  p ) + ( r  s ) (p  q) is

ONE OR MORE MAY BE CORRECT

1. Consider three vectors p = ˆi + ˆj + k,

2. The intercept made by the plane r  n = q on the x-axis is

3. If the distance between the planes

12. Vector equation of the plane r = ˆi − ˆj + (iˆ + ˆj + k)

x−2 y−3 z−4 x −1 y − 4 z − 5

8. Let L1 be the line r1 = 2iˆ + ˆj − kˆ + (i + 2k)