MATHEMATICS

SECTION – I
(SINGLE CORRECT ANSWER TYPE)
This section contains 20 multiple choice questions. Each question has 4 options (1), (2), (3) and (4) for its answer,
out of which ONLY ONE option can be correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -1 if not correct.

61. If 4i  7 j  8k , 2i  3 j  4k and 2i  5 j  7k are the position vectors of the vertices A, B and C

of triangle ABC, the position vector of the point where the bisector of A meets BC is
2 2 1
1) 6i  8 j  6k  2) 6i  8 j  6k  3) 6i  13 j  18k  4) 2  i  j  k 
3 3 3
 5a
62. The position vectors of A, B are a, b respectively. The position vector of C is  b then
3
1) C lies outside OAB but inside angle OBA
2) C is outside OAB but inside angle BOA
3) C is outside OAB but inside angle COA
4) Inside the OAB
63. ABCD is a parallelogram and P is a point on the segment AD dividing it internally in the
ratio 3 : 1 the line BP meets the diagonal AC in Q. Then the ratio AQ : AC is
1) 4 : 3 2) 3 : 4 3) 3 : 2 4) 2 : 3
64. Let ABCD be a parallelogram such that AB  q, AD  p and BAD be an acute angle. If
r is the vector that coincides with the altitude directed from the vertex B to the side AD,

then r is given by

1) r  3q 
 p
3 p.q
2) r  q 
p.q
p 3) r  q 
p.q
p 4) r  3q 
 p
3 p.q
p. p p. p p. p p. p
  
65. DA  a, AB  b and CB  k a when k>0 and X, Y are the midpoints of
  
DB and AC respectively, such that a  17 and XY  4 then k is

1) 18/17 2) 19/17 3) 4/17 4) 25/17

66. Let OABCDE is a regular hexagon, O is the origin and position vector of A and B are
a and b respectively. The position vector of point of intersection of OB and AC is

2 2 b b
1)
3
b 2)
3
ab  3)
3
4) a 
3
2 2 2
67. If a , b , c are unit vectors, then the value of a  2b  b  2c  c  2a does not exceed to

1) 9 2) 12 3) 18 4) 21
68. Let u  i  j , v  i  j and w  i  2 j  3k . If n is a unit vector such that u.n  0 and v.n  0 then
2 2
2 w.n  u  v 

1) 5 2) 7 3) 10 4) 8
69. If four points with position vectors a , b , c and d are coplanar and

 sin   a   2sin 2  b   3sin 3  c  d  O then the least value of sin 2   sin 2 2  sin 2 3 is

1) 1/14 2) 14 3) 6 4) 1/ 6
70. The locus of a point equidistant from two given points whose position vectors are
a and b is equal to

1 1
1)  r   a  b  . a  b   0 2)  r   a  b  . a  b   0
 2   2 
1
3)  r   a  b   .a  0 4)  r   a  b  .b  0
 2 

71. If a and b are any two vectors of magnitude 1 and 2 respectively and
2 2
1  3a.b  
 2a  b  3 a  b   47 then the angle between a and b is

1)  / 3 2)   cos1 1/ 4  3) 2 / 3 4) cos1 1/ 4 

72. Component of vector p  2i  4 j  3k in the direction perpendicular to the direction of

vector q  i  j  k is ai i  a2 j  a3 k then the value of a1  a2  a3 is
1) 14/3 2) 22/3 3) 8 4) 26/3
73. If a , b , c are unit vectors such that a.b  0  a.c and angle between b and c is  / 3 then
ab  ac 

1) 1/2 2) 1 3) 3 4) 1/3
74. Let a , b , c be three vectors such that a  0 and a  b  2a  c , a  c  1, b  4 and b  c  15 .
If b  2c   a then the value of  =
1) 4 2) 3 3) 2 4) 1
75. Let   4i  3 j  5k and   i  2 j  4k . Let 1 be parallel to  and  2 be perpendicular to  . If

  1   2 then the value of 5 2 . i  j  k is 
1) 6 2) 11 3) 7 4) 9
76. Let a  2i  7 j  5k , b  i  k and c  i  2 j  3k be three given vectors. If r is a vector such that

r  a  c  a and r.b  0 then r is equal to

11 11 914
1) 2 2) 11/7 2 3) 4)
7 5 7
  
77. Let ABC be a triangle such that BC  a, CA  b, AB  c, a  6 2, b  2 3 and b.c  12

consider the statements

 S1  :  a  b    c  b   c  6  2 2 1 , 
 S 2  :  A CB  cos 1  
2 / 3 then

1) Both S1 and S2 are true 2) Only S1 is true

3) Only (S2) is true 4) Both S1 and S2 are false
78. Let a   i  j   k and b  3i  5 j  4k be two vectors such that a  b  i  9 j  12k . Then the
projection of b  2a on b  a is
1) 2 2) 39/5 3) 9 4) 46/5
79. If Q   2,1, 2  and R   0, 5,1 then the perpendicular distance from P 1, 4, 2  to QR is
2 3 5 7
1) 26 2) 26 3) 26 4) 26
7 7 7 5

80. The vector c directed along the internal bisector of the angle between the vectors
a  7i  4 j  4k and b  2i  j  2k with c  5 6 is

5 5 5 5
1) 
3

7i  j  2 k  2) 
3

2i  7 j  k  3) 
3
i  7 j  2k  4) 
3
2i  7 j  k 
 SECTION-II
(NUMERICAL VALUE ANSWER TYPE)
This section contains 10 questions. The answer to each question is a Numerical value. If the Answer in the
decimals , Mark nearest Integer only. Have to Answer any 5 only out of 10 questions and question will be
evaluated according to the following marking scheme:
Marking scheme: +4 for correct answer, -1 in all other cases.

81. Let a  i  2 j   k , b  3i  5 j   k , a.c  7 , 2b.c  43  0, a  c  b  c . Then a.b is equal to

2
82. Let a and b be two vectors such that a  14, b  6 and a  b  48 then  a.b  

2 2 2 2 2
83. Let a and b be two vectors such that a  b  a  2 b , a.b  3 and a  b  75 then a is

equal to

84. a, b and c are unit vectors such that a  b  3c  4 . Angle between a and b is 1 , angle

 2 
between b and c is 2 and the angle between a and c lies in  ,  . Then the maximum
6 3 

value of cos 1  3cos 2 is

85. Let x and y be two real numbers such that 2sin x sin y  3cos y  6 cos x sin y  7 , then
tan 2 x  2 tan 2 y is equal to

86. In a quadrilateral ABCD, the point P divides DC internally such that DP : PC = 1 : 2 and
    
Q is the midpoint of the segment AC such that 2 DC  BC  2 AD  AB   PQ then  
87. In a tetrahedron OABC, the edges are of lengths OA  BC  a, OB  AC  b, OC  AB  c .
 
Let G1 and G2 be the centroids of the triangle ABC and AOC such that OG1  BG2 then
a2  c 2
the value of is
b2

88. l  2i  j  k , m  2i  j  k and n  i  j  k are three vectors and r is a vector satisfying

    2
r  m  n  m and r.l  0 then  r  where . denotes greatest integer function
 

89. Let a  i  2 j  k , b  i  j and c  i  j  k be three given vectors. If r is a vector such that

r  a  c  a and r.b  0 then r.a is equal to

90. Let x be a vector in the plane containing vectors a  2i  j  k , b  i  2 j  k . If the vector

17 6 2
 
x is perpendicular to 3i  2 j  k and its projection on a is
2
then the value of x is

equal to
MATHEMATICS
SECTION – 1 (Maximum Marks: 18)
This section contains SIX (06) questions.
Each question has FOUR options for correct answer(s). ONLY ONE of these four option is the correct answer.
For each question, choose the correct option corresponding to the correct answer.
Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +3 If only the correct option is chosen.
Zero Marks: 0 If none of the option is chosen.(i.e the question is un answered)
Negative Marks: -1 In all other cases.
1. STATEMENT 1 :Let OABC be a regular tetrahedron of side length unity, where O

is origin. If P is a point at a unit distance from origin such that OP is equally inclined
  
to OA, OB and OC at an angle  ,then p1  cos2  .
STATEMENT 2 :A,B,C,D are four points in the space and satisfy AB=3, BC=7,
CD=11 and DA=9, also if angle between AC and BD is  then p2  cos  .

Considering above two statements, the value of 3p1  p2  is,

(where [.] represent Greatest Integer Function)

A) 2 B) 3 C) 4 D) 5
 
2. Let a  f  x  iˆ  f '  x  ˆj and b  g  x  iˆ  g '  x  ˆj be any two non – zero vectors and

h  x  be antiderivative of f  x  g  x . If h 1  1, h  2  3, h  4  7, then vectors a

and b are linearly dependent for :
A) atleast one x 1,2 B) atleast one x 2,4

C) atleast two x 1,4 D) atleast one x 1,4

  
3. Let a, b and c be three non-coplanar unit vectors such that the angle between every
       
pair of them is . If a  b  b  c  pa  qb  rc , where p, q, r are scalars; then the value
3
of 20 p2  21q2  22r 2  is,

(where [.] is Greatest Integer Function)

A) 8 B) 9 C) 10 D) 6

4. Let A be the set of all integral solution of the inequality x2  x  2  0 and v1  iˆ  ˆj  kˆ ,
 
v2  aiˆ  bjˆ  ckˆ (where a, b, c  A ) be two non zero vectors such that v 2 is orthogonal to
 
v1 then number of such possible vector v2 is :

A) 12 B) 14 C) 16 D) 18
 2 3 6 
7 7 7 
 
 1  1
5.
7
 ˆ  7
 
ˆ
Let a  2 iˆ  3 ˆj  6k ; b  6 iˆ  2 ˆj  3k and ĉ be a unit vector and A    6
7
2
7
 
3
7
 ˆ
cˆ.iˆ cˆ. ˆj cˆ.k 
 

and AAT  I then 3 cˆ.iˆ  2 cˆ. ˆj  cˆ.kˆ  is __________ (where [.] represents greatest integer
function)
A) 1 B) 2 C) 3 D) 4
6. Let OABCD be a pentagon in which the sides OA and CB are parallel and the sides OD
and AB are parallel. Also OA:CB=2:1 and OD:AB=1:3. Let X be the point of
OX p
intersection of the diagonals OC and AD. If the ratio is given by , (where p, q are
XC q
positive integers which are relatively prime to each other), then p  q equals,
A) 6 B) 7 C) 8 D) 9
SECTION - 2 (Maximum Marks : 24)
This section contains SIX (06) questions.
Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is
(are) correct option(s).
For each question, choose the correct option(s) to answer the question.
Answer to each question will be evaluated according to the following marking scheme:
Full Marks : +4 If only (all) the correct option(s) is (are) chosen.
Partial Marks: +3 If all the four options are correct but ONLY three options are chosen.
Partial Marks: +2 If three or more options are correct but ONLY two options are chosen, both of which are
correct options.
Partial Marks : +1 If two or more options are correct but ONLY one option is chosen and it is a correct
option.
Zero Marks : 0 If none of the options is chosen (i.e. the question is unanswered).
Negative Marks: -2 In all other cases.
   
7. Let O be an interior point of ABC such that OA  2OB  3OC  0 and areas of
ABC, AOB, BOC and AOC are denoted by 1 ,  2 ,  3 and  4 respectively, then
1
is divisible by :
min.  2 , 3 ,  4 
A) 2 B) 3 C) 4 D) 6
8. Let f : R   0,1 be a continuous function, then which of the following pair of vectors
are
linearly dependent for some x 0,1 ?
   
A) a  f  
x iˆ  2 ˆ
j ; b  x 2ˆ
i  3 ˆ
j B) a  f  
x ˆ
i  3 ˆ
j ; b  x 2iˆ  2 ˆj
 
1 x
   
1 x
 
C) a    f  t  dt  iˆ  3 ˆj; b  xiˆ  2 ˆj D) a    f  t  dt  iˆ  2 ˆj; b  xiˆ  3 ˆj
 0   0 
9. An equilateral triangle OAB has side length 1. P is a point on the plane of the
triangle. If
   
OP   2  t  OA  tOB, t  R, then the possible value of AP can be:

1 1 3
A) B) C) D) 2
2 2 2
10. In triangle ABC, points D, E lie on sides AC and AB respectively. M and N are mid
points of side BD and CE respectively. If 1 is area of triangle AMN, 2 be the area of
triangle ADE,  be area of triangle ABC, 3 be area of quadrilateral BCDE, 4 be
area of quadrilateral MNDE. Which of the following is always true?
A) 41  3 B) 41  2   C) 1  4 D) 44  2  

11. Let V be the volume of the parallelopiped formed by the vectors a  a1 iˆ  a2 ˆj  a3kˆ,
 
b  b1 iˆ  b2 ˆj  b3kˆ , c  c1 iˆ  c2 ˆj  c3kˆ .If ar , br , cr (where r  1, 2,3 ) are non – negative real
3
numbers and   ar  br  cr   3L , then which of the following may hold CORRECT ?
r 1

A) L3  V B) L3  V C) L3  V D) 2L3  V
12. AB,AC,AD are adjacent edges of a parallelepiped. The vector along the diagonal AP is

given by a so that a  3 . The vector areas of the faces containing A,B,C and A,B, D
     
are respectively given by AB  AC  b ; AB  AD  c . The projections of the edges AB and
AC on the diagonal AP are both equal to 1. Then which of the following can be correct
 a 1     
 
A) AC   a  2b  c
3 9
 B) CD 
1 
3
a  b  c 
    a 1 
C) AB  a 
 1 
3
 
a  2b  c  D) AD  
3 9
 a  b  2c 
SECTION - 3 (Maximum Marks : 24)
This section contains SIX (06) questions. The answer to each question is a NUMERICAL VALUE
For each question, enter the correct numerical value of the answer using the mouse and the on-screen virtual
numeric keypad in the place designated to enter answer. If the numerical value has more than two decimal
places truncate/round- off the value to TWO decimal places.
Answer to each question will be evaluated according to the following marking scheme:
Full Marks: +4 If ONLY the correct numerical value is entered as answer.
Zero Marks: 0 In all other cases.
  
13. Let v1  b1iˆ  b2 ˆj  b3kˆ, v2  b4iˆ  b5 ˆj  b6kˆ and v3  b7iˆ  b8 ˆj  b9kˆ such that b .b .b  108,
1 2 3

b4  b5  b6  10 and b7  b8  b9  3 where bi  N i  1,2,.......9. If number of ordered

  
triplets  v1, v2 , v3  are P, then the sum of the digits in P is,
 2
aˆbˆ

14. Let g      2t  1 dt where  is the angle between â and bˆ . If volume of the

 
2
 aˆ.bˆ

parallelopiped whose co-terminous edges are represented by vectors aˆ , aˆ  bˆ and

  p
aˆ  aˆ  bˆ is (where 2 g    1  0 and ‘p’ and ‘q’ are coprime natural number), then the
q

least value of  p  q  is,

 
15. Let r1  ai  b j  ck  a  b  c where a, b, c are the solutions of 2 x  x  2{x} and r2  di

 1020000   
where d is the unit digit of  100  and 3r1  r2  m n then smallest value of   
m
100  3  n

(where [.] is Greatest Integer Function and {.} is Fractional part function and also m
  2  and n   2  are relatively prime Integers),
   
16. Let k , l , m, n are four distinct non-coplanar units vectors in space such that
          1   A
k  l  l  m  m  k  n  l  n  m  . The value of k  n can be expressed as , where A, B
11 B
are co- prime Natural Number, then the value of A  B is,

17. G is the centroid of triangle ABC and A1 and B1 are the midpoints of sides AB and AC,
respectively. If  1 be the area of quadrilateral GA1AB 1 and  2 be the area of triangle
2
ABC, then the value of is equal to
1

18. Given x, y, z, p, q, r, p1, q1, r1, l, m, n are scalars which satisfy the given relation
x  2 y  3z  4 p  5q  6r  7 p1  8q1  9r1  10l  11m  12n  12! , and given vectors,
   
u   xl  p1 p  i   xm  p1q  j   xn  p1r  k

   
v   yl  q1 p  i   ym  q1q  j   yn  q1r  k

   
w   zl  r1 p  i   zm  r1q  j   zn  r1r  k .
     
Then the value of u  v v  w w  u  is, (where [.] is Scaler Tripple Product of 3 Vectors)