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```1.
2.
3.
4.
5.
 5c 
 3b 
 a
If log  , log  and log   are in
a
 5c 
 3b 
AP, where a, b, c are in GP, then a, b, c
are the length of sides of
(a) an isosceles triangle
(b) an equilateral triangle
(c) a scalene triangle
(d) None of the above
If the pth, qth and rth terms of an AP are
in GP then the common ratio of GP is
pq
(a)
rq
rq
(b)
q p
pr
(c)
pq
(d) None of the above
If a, b, c, d are four numbers such that
the first three are in AP while the last
three are in HP then
(b) ac = cd
(c) ab = cd
(d) None of the above
If Sr denotes the sum of the first r terms
S  Sr  1
of an AP then 3r
is equal to
S2r  S2r  1
(a) 2r-1
(b) 2r+1
(c) 4r+1
(d) 2r+3
If a, b, c are in AP then a  1 , b  1 , c  1
bc
6.
ca
ab
are in
(a) AP
(b) GP
(c) HP
(d) None of the above


If     then the minimum value
2
2
3
of cos  + sec 3  is
(a) 1
(b) 2
(c) 0
(d) None of the above
XIX
7.
FT12
Let Sn denote the sum of the cubes of
the first n natural numbers and Sn
denotes the sum of the first n natural
n S
numbers. Then  r is equal to
r 1 S r
n(n  1)(n  2)
6
n(n  1)
(b)
2
(a)
n 2  3n  2
(c)
6
(d) None of the above
8.
If a, b, c are in HP then
1
1

is
ba bc
equal to
2
(a)
b
2
(b)
ac
1 1

(c)
a c
(d) Both (a) and (b)
9. The number of real solutions of
1
1
x 2
 2 2
is
x 4
x 4
(a) 0
(b) 1
(c) 2
(d) infinite
10. The number of real solutions of the
equation ex  x is
(a) 1
(b) 2
(c) 0
(d) None of the above
11. The number of real solutions of the
equation log 0.5 x | x | is
(a)
(b)
(c)
(d)
1
2
0
None of the above
(1)
FT12
12. The equation
x  1  x 1  4x 1
has
No solution
One solutions
Two solutions
More than two solutions
The number of solutions of the equation
| x | = cos x is
(a) One
(b) Two
(c) Three
(d) Zero
If 3 x+1 = 6 log23 then x is
(a) 3
(b) 2
(c) log32
(d) log23
If ( 2 ) x  ( 3 ) x  ( 13 ) x / 2 then the
number of values of x is
(a) 2
(b) 4
(c) 1
(d) None of the above
The number of real solutions of the
6x
x
 2
equations 2
is
x2
x 4
(a) Two
(b) One
(c) Three
(d) None of the above
The number of real solutions of
(a)
(b)
(c)
(d)
13.
14.
15.
16.
17.
x 2  4x  2  x 2  9  4x 2 14x  6
is
(a) One
(b) Two
(c) Three
(d) None of the above
18. If [x]2 = [x+2], where [x] = the greatest
integer less than or equal to x, then x,
must be such that
(a) x = 2-1
(b) x [2,3)
(c) x[-1,0]
(d) None of the above
XIX
19. The number of solutions of | [x] – 2x| =
4, where [x] is the greatest integer ≤ x,
is
(a) 2
(b) 4
(c) 1
(d) Infinite
20. The set of real values of x satisfying | x
– 1 | ≤ 3 and | x – 1 | ≥ 1 is
(a) [2,4]
(b) [- (,2]  [4,)
(c) [-2,0][2,4]
(d) None of the above
21. The solution set of x 2  3x  4  1, xR, is
x 1
3,+)
-1,1)3,+)
[-1,1][3,+)
None of the above
22. The number of integral solutions of
x2 1
 is
x2 1 2
(a) 4
(b) 5
(c) 3
(d) None of the above
(a)
(b)
(c)
(d)
23. The value of the sum
13
 (i
n 1
n
 i n 1 )
where I = 1 is
(a) i
(b) i-1
(c) -i
(d) 0
24. If a+i=+i then b+i5 is equal to
(a)  +i
(b) -i
(c) -i
(d) --i
25. If  is a non real cube root of unity
then the expression 1-  )1-  2)1 4)1+  8) is equal to
(a) 0
(b) 3
(c) 1
(d) 2
(2)
FT12
26. If
x2-x+1=0
then
the
value
of
2
 n 1 
 is
n 1 
xn 
(a) 8
(b) 10
(c) 12
(d) None of the above
1 z
27. If | z | = 1 then
is equal to
1 z
(a) z
(b) z
(c) z+ z
(d) None of the above
z 1
28.
= 1 represents
z 1
(a) A circle
(b) An ellipse
(c) A straight line
(d) None of the above
ax a x
5
32. The value of
 x 
29. If a  x
a
ax a
x = 0 then x is
x
0
a
3
2a
0
pq pq
(a)
(b)
(c)
(d)
30.
qp
0
q  r is equal to
r p r q
0
(a) p + q + r
(b) 0
(c) p – q – r
(d) - p + q + r
31. The
value of the determinant
bc ca ab
p q r , where a, b, c are the pth,
1 1 1
qth and rth terms of a HP, is ap + bq + cr
(a) a + b +
(b) p + q + r)
(c) 0
(d) None of the above
XIX
a1x  b1y a 2 x  b 2 y a 3 x  b3 y
b1x  a1y b 2 x  a 2 y b3 x  a 3 y
b1x  a1
b2 x  a 2
b3 x  a 3
is equal to
(a) x 2  y 2
(b) 0
(c) a1a 2a 3x 2  b1b2b3 y2
(d) None of the above
33. If ,  are non-real numbers satisfying
x3 1  0
λ 1
α
α
λ β
1
β
1
λα
β
then
the
is equal to
value
of
(a) 0
(b) 3
(c) 3 + 1
(d) None of the above
 1  i, and  is a non-real cube root
of unity then the value of
34. If
1
2
1  i  2
i
1
1  i  
1  i 2  1
is equal to
1
(a) 1
(b) i
(c) 
(d) 0
35. The system of equations ax + 4y + z =0,
bx + 3y + z =0, cx + 2y + z = 0 has
nontrivial solutions if a, b, c are in
(a) AP
(b) GP
(c) HP
(d) None of the above
36. The equations x + y + z = 6, x + 2y + 3z
= 10, x + 2y + mz = n give infinite
number of values of the triplet x, y, z) if
(a) m = 3, n  R
(b) m = 3, n  10
(c) m = 3, n = 10
(d) None of the above
(3)
FT12
37. The system of equations 2x + 3y = 8, 7x
– 5y + 3 = 0, 4x – 6y +  = 0 is solvable
if  is
(a) 6
(b) 8
(c) –8
(d) –6
38. Indefinite Integra If (x )   cot 4 xdx  1 cot 3 x  cot x
3
39.
40.
41.
42.
 
and    then ( x ) is
2 2
(a)  -x
(b) x- 

(c)
-x
2
(d) None of the above
x
 x (1  log x )dx is equal to
(a) xxlogx+k
x
(b) e x  k
(c) x x  k
(d) None of the above
xdx
is equal to

1 x4
(a) tan 1 x 2  k
1
(b) tan 1 x 2  k
2
(c) log( 1  x 4 )  k
(d) None of the above
x5 / 2
dx is

1 x7
2
(a) log( x 7 / 2  x 7  1)  c
7
1
x7 1
log 7
c
(b)
2
x 1
(c) 2 1  x 7  c
(d) None of the above
The primitive of the function x | cos x |

when  x   is given by
2
(a) cos x  x sin x
(b)  cos x  x sin x
(c) xsinx-cosx
(d) None of the above
XIX
43.  e  x (1  tan x ) sec xdx is equal to
(a) e x sec c
(b) e x tan x  c
(c)  e x tan x  c
(d) None of the above
dx
is equal to
cos x  3 sin x
x Π
(a) log tan    +k
2 3
x 
(b) log tan    +k
2 3
1
x 
(c)
log tan    +k
2
2 3
(d) None of the above
44. 
45. In the expansion of
2  x 5 2 , the
coefficient of x 4 , if it exists, is
5  7  9 11  1 
 
25 2  4!  2 
4
5  7  9 11  1 
(b)
 
4!
 2
5  7  9 11
4
(a)
(c)
 2
21
(d) None of the above
46. The coefficient of x 5 in the expansion
1 x2
, x  1, is
1 x
(a) –1
(b) 2
(c) 0
(d) –2
47. The coefficient of x n in the expansion
of e2 x 3 is
of
2n
n!
e3  2 n
(b)
n!
(c) a p  a q
(a)
(d) None of the above
(4)
FT12
48. The coefficient of x10 in the expansion
of 10 x in ascending powers of x is
(a)
(b)
(c)
log e10 10
10 !
1
10 !
(log 10e)10
10 !
(d) None of the above
49. The constant term in the expansion of
3x  2 x
is
x2
(a) log e 3
(b) log e 6  log e
3
2
1
3
log e 6  log e
2
2
(d) None of the above
50. If x <1, the coefficient of x 3 in the
(a)
(b)
(c)
(d)
1
is
e  (1  x )
x
17
6
17

6
11

6
None of the above
log e 1  x 
1  x 2
3
(a)
2
(b)
2
C1
1
2
(d) None of the above
(c) 
XIX

3
4
x
x

 ...to   y
3
4
then y  y
2
2!

3
y
 ...to 
3!
is equal to
(a)  x
(b) x
(c) x + 1
(d) None of the above
53. The matrix λ 7  2 is a singular matrix
4 1

 2  1
3 
2 
if  is
2
(a)
5
5
(b)
2
(c)  5
(d) None of the above
0  4 1 
exists
A  2 λ  3 then A 1
1 2  1
i.e., A is
invertible) if
(a)   4
(b)   8
(c)  = 4
(d) None of the above
1
3
is
4
 2  3 1 
55. The rank of the matrix λ

2
2
18
11
18
(b)  =
11
18
(c)  = 
11
(d) None of the above
1
56. The value of
3 if
(a)  
51. If x <1, the coefficient of x 2 in the
expansion of
2
54. If
(c)
expansion of
52. If x  x
2
is

is equal to
40  20  10  80

1
3 10  2 5
70
3 10  2 5
(b)
70
3 10  2 5
(c)
50
(d) None of the above
(a)
(5)
FT12
57. The square root of 2 x  2 x 2  1 is
(a)
x  1  x 1
(b)
x  1  x 1
58. If x 
3 2
,y 
(b) 0
(c) –1
3 2
3 2
(d) None of the above
then the value
of x2+xy+y=is
(a) 5
(b) 99
(c) 98
(d) None of the above
59. If y =cos-1cosx) then
(a)
(b)
(c)
2 | x | x 2 1
1
2x x 2  1
1
2x x  1
(d) None of the above
XIX
2
1  x 2 1
with
x
1 x2 
x2
(a)
dy
5
at x =
is
dx
4
dy
x 1
then
is equal to
dx
x 1
1
63. The derivative of tan 1
respect to tan 1 x is
equal to
(a) 1
(b) -1
1
(c)
2
(d) None of the above
y  .....to
dy
ye
60. If x  e
then
is
dx
x
(a)
1 x
1
(b)
x
1 x
(c)
x
(d) None of the above
61. If y  tan 1
dy
at x =e is
dx
(a) 1
(c) x  1  x  1
(d) None of the above
3 2
62. If xy.yx = 16 then
(b) 1
1
1 x2
(d) None of the above
(c)
64. Let
x 3 sin x cos x
f (x)  6
1
0
p
p2
p3
, where p is a
d3
constant . Then
{f ( x)} at x  0 is
dx 3
(a) p
(b) p+p2
(c) p+p3
(d) Independent of p
65. If y =at2, x= 2at, where a is a constant,
d2y
1
then
at x  is
2
2
dx
1
(a)
2a
(b) 1
(c) 2a
(d) None of the above
66. Let A and B be two sets such that
A  B  A . Then A  B is equal to
(a) 
(b) B
(c) A
(d) None of the above
(6)
FT12
67. Of the members of three athletic teams
in a school 21 are in the cricket team,
26 are in the hockey team and 29 are in
the football team. Among them, 14 play
hockey and cricket, 15 play hockey and
football, and 12 play football and
cricket. Eight play all the three games.
The total number of members in the
three athletic team is
(a) 43
(b) 76
(c) 49
(d) None of the above
68. The relations “cingruence modulo m” is
(a) Reflexive only
(b) Transitive only
(c) Symmetric only
(d) An equivalence relation
XIX
69. Let f : R  R such that fx) =
1
,
1 x2
xR. Then f is
(a) Injective
(b) Surjective
(c) Bijective
(d) None of the above
70. Let R be the relation over the set of
integers such that m Rn if and only if m
is a multiple of n. Then R is
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) An Equivalence relation
(7)
```
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