KEAM 2017 Question Paper Maths

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WARtr{ING: in the Examination will DISQUALIFY THE CAI\DIDATE.
PAPER _II MATHEMATICS-2ol7
Version Question Booklet ?,182728
code B1 Serial Number :
Time: 150 Minutes Number of Questions: 120 Maximum Marks: 480

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Maths-II-Bli20l7 2

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PRINTED PAGES 32

1l
I qr is equal to
li q r+l
(A) q-p (B) q+p (C) q (D) p (E) 0

2. LetA= 50 andB: l-o I .If4A+58-C:0,thenCrs
10 l-r 0
(A)
Is 25 (B) 20 sl (c) Is I (D)
5 25
(E)
05
L-i 0 -1 ol Lo 25 I 5 5 25

I -1
3. IfU: l1 T6 , then U-l 1S

.12 ''12
(A) ur (B) u (C) I (D) 0 G) U2

0 -1 0
4, IfA= 1 0 0 , then A-l is
0 0 I

(A) AT (B) A2 (c) A (D) I (E) 0

Maths-II-Bl/2017 J lP.T.O.

5. x+y x-y
If 2x+ 0
z x+z , then the values of-r, y and z are respectively
)=(? I )
(A) 0,0, I (B) l, 1,0 (c) -r,0,0 (D) 0,0,0 (E) l, l, I

6
I
(; 0 a[i].'ral is equa,,o
(A)
ON
(B) (x) (c) (1:) (D) (19 (E) 6
I6

I 2 4
7. If I 3 5 is singular, then the value ofa is
I 4 a
(A) a:-6 (B) a:s (C) (D) a:6 (E) a=0
":-5
(t 2
8. rrlo 4
Io 0 ':]tr=til 'ihen(x,v,z) is equa,,o
(A) (1, 6, 6) (B) (1, _6, l) (c) (1, l, 6) (D) (6, _1, 1)(E) (_1,6, l)

Maths-II-Bl/2017 4

9 rfA: (l 25 , then
(A) A2-2A+2I=0 (B) A2-3A+2I=0
(C) A2-5A+2I=0 (D) 2e2-A+t=o
(E) A2+3A+2I=0

x+y I then(x,y,p,q) equals
10. ,t( 2x+y
P-q p+q )=(l 0
,

(A) 0, l, o, 0 (B) 0, -1, 0, 0 (c) 1,0, 0, 0 (D) 0, l, 0, t (E) l, 0, l, 0

ll. The value of t'
|.vq;tEl-l.,rrq
(A) I (B) 2 (c) 4 (D) 3 (E) 5

3 4 2
12. The value of 82t -l6tt -gtl is
(A) -1 (B) -2 (c) -3 (D) -4 (E) -5

13. Letx=2bearoot of y-4x2-l4x+q=0. Thenyis equal to
(A) (x-2) (ax-6) (B) (x-2) (ax+6)
(C) (x-2) (4x-6) (D) (.r-2) (-4x+6)
(E) (x-2)(4x+3)

Maths-II-B1/2017 5 lP.T.O.

14. If x, and ,2 zne the roots of 3x2 -2x-6-0, then xl +xl is equal to
50 40
(A)
9
(B)
9
(c)
30
9
(D)
20
9
(Dli
15. Let x, and x, be the roots of the equation x2+px-3=0. lf
t+x3=tO, then the
value ofp is equal to
(A) -4 or 4 (B) -3 or 3 (C) --2 or 2 (D) -1 or I (E) O

16. If the product of roots of the equation mx2 +6x+(2m_l)=0 is _1, then the
value of rn is

(A) (B) (c) 3 (D) -1 (E) -3
+

17. tff(x):"- 4 4
x'+4x+4 TEW +
x3 +2r2 '
,(+) equar to
then
',
(A) I (B) 2 (c) -l (D) 3 (E) 4

Maths-II-BI/2017 6

r8. If x and y are the roots of the equation x2 +bx+l-}, then the value of
ll rs
x+b y+b
-+-
(A) (B) Q)* (D) 2b (E) l
; '
r9. The equations x5 + ax+l=0 and x6 +m2 +l=0 have a common root. Then a is
equal to
(A) r (B) -2 (C) -3 (D) -l (E) 0

20. The roots of axz +x+l-}, where a*0, are in the ratio I :1. Then a is equal to

(A)
+
(B)
+
(c) i (D) l (E) 0

21. lf z2 +z+l-O where z is a complex number, then the value of
2 2 2
I I
1
z+- + (,' +-z +
1
+T equals
z z z
(A) 4 (B) s (c) 6 (D) 7 (E) 8

Maths-II-Bl/2017 1 lP.T.O

I 11
22. Let A= I -1-w2 .2 , where w*l is a complex number such that w3=1.
l
1 ww'
Then A equals

(1t) 3w+u? 1n) 3w2 (c) 3(w-w2) (D) -3w2 (E) 3#+1
5t
23. If 2 -9i
I
9i I =r+0/, then
10 9 i
(A) x-1, y=7 (B) x=0, y=l
(c) x-1, y=0 (D) x-0, y=0
(E) x--1, y=0

24. Ifz: cos ;1t -rsln lt , thet z2 - z+1 is equal to
J J

(A) 0 (B) I (c) -l (D)
1t
(E) r
z

Maths-II-Bl/2017 8

TE 7t

25.
l+cos
n a
+, stn
is equal to
1t 7t
l+cos
n -, sln a
I
(A) 0 (B) I (c) I (D)
1
(D-+

4k k
26. rfA: 0kk ard det (A) : 256, then l/<l equals
00 k
(A) 4 (B) s (c) 6 (D) 7 (E) 8

27. rfA: l0
1l , then A" +zI is equal to

(A) I (B) nA (C) I+zA (D) I-zA (E) zA-I

28 rr fzf : s *a.=fi,thenRe(w) is equal to

(A)o (B)+ (q2s (D) I (E) -1

9 lP.T.O.
Maths-II-Bl/2017

29. IfA: I11I , then A2o1? is equal to
1a) z20r5e (B) 22016A (c) 220t4A (D) 22or7A (E) 22o2oA

30 rf a=eia, trt"n is equal to
{}
(A) cot
0
z (B) tan o (C) icot| (D)
.e
tlan, (E) 2 tan e

31. Three numbers x, y, and z are in arithmetic progression. lf x+y+z_-3 and
xyz -8, then *2 + y2 + ,2 is equal to
(A) e (B) 10 (C) 2t (D) 20 (E) I

32. The 30th term of the arithmetic progression 10, 7, 4 is
(A) -e7 (B) -87 (C) _77 (D) 47 (E) _57

33. The arithmetic mean of two numbers x arrd y is 3 and geometric mean is 1
Then x2 +y2 is equal to
(A) 30 (B) 3l (C) 32 (D) 33 (E) 34

Maths-II-Bl/2017 10

34. The solution of 32r-1=811-r is
(A) 23 (B)
I
6
G)I rn); to+
35. The sixth term in the sequence is 3, l, JI ls

1 I I I
(D+
(A) (B) (c) (D)
27 9 8l t7

36. Three numbers are in arithmetic progression. Their sum is 21 and the product
of the first number and the third number is 45. Then the product of these three
numbers is
(A) 3rs (B) e0 (c) 180 (D) 270 (E) 450

37. lf a+1,2a+1,4a-I are in arithmetic progression, then the value ofa is
(A) I (B) 2 (c) 3 (D) 4 (E) 5

Maths-II-Bl/2017 lt lP.T.O.

38. Two numbers x and y have arithmetic mean 9 and geometric mean 4' Then x
and y are the roots of

(A) x2-lEx-16=0 (B) x2-l8x+16=0
(C) x2+l8x-16=0 (D) x2+18x+16=0
(E) x2-17x+16=O

39. Three unbiased coins are tossed. The probability of getting at least 2 tails is
I I
(A)
i (B)
I
4
(c)
2
(D)
3
@)1

40 A single letter is selected from the word TRICKS. The probability that it is
eitherTorRis
I I 2
(A) (B) (c) (D) (E) +
* 4 1 J J

41. From 4 red balls, 2 white balls and 4 black balls, four balls are selected' The
probability of getting 2 red balls is
8 9 10
@)+ (B)
2I
(c)
2I
(D)
2l
(D
;i

MathsJI-Bl/2017 t2

42. In a class, 60% of the students know lesson I,40% know lesson II and 20o/o
know lesson I and lesson II. A student is selected at random. The probability
that the student does not know lesson I and lesson II is

(A)o (B);431.) (o; (D)5 (E);

43. Two distinct numbers x and y are chosen from 1,2,3, 4,5' The probability that
the arithmetic mean of x andy is an integer is

(A)o re)+ (c); to)? (E)1
44 The number of 3 x 3 matrices with entries -l or +1 is
(A) 24 (B) 25 G) 26 (D) 21 (E) 2e

45. Let S be the set of all 2 x 2 symmetric matrices whose entries are either zero or
one. A matrix X is chosen from S. The probability that the determinant of X is
not zero 1s

(A)
I
5
(B)
I
,
(q; to)+ @)?

Maths-II-Bl/2017 l3 lP.T.O.

46. The number of words that can be formed by using all the letters of the word
PROBLEM onJy once is
(A) s! (B) 6! (C) 7t (D) 8! (E) et

47. The number ofdiagonals in a hexagon is
(A) 8 (B) e (c) 10 (D) 1l (E) 12

48. The sum ofodd integers from I to 2001 is

(A) 10012 1n) 10002 G) rc022 1n) 10032 (E) 9ss2
49. Two balls are selected from two black and two red balls. The probability that
the two balls will have no black ball is

(A)+ <e)+ (c)+ I
to) JO ral
50. If z -i9 + i19 , then z is equal to
(A) 0+0i (B) 1+0, (c) 0+l (D) t+2i (E) t+3i

Maths-II-B 1/20 17 t4

51. The mean for the data 6, 7, 10, 12, 13, 4, 8, 12 is
(A) e (B) E (c) 7 (D) 6 (E) 5

52, The set of all real numbers satisffing the inequality x-2<l is
(A) (3,.") (B) t3,-) (C) [-3,"") (D) (--'-3) (E) (*,3)

53 rrH,o, *,*
(A) xe(-3,."1 @) xe(3,.") (C) .xe(2,-; (D) xe(1,-) (E) re (-1"")

54. The mode ofthe data 8, I l, 9, 8, I l, 9, 7,8,7,3,2,8 is
(A) ll (B) e (c) 8 (D) 3 (E) 7

55 If the mean of six numbers is 41, then the sum of these numbers is
(A) 246 (B) 236 (C) 226 (D) 216 (E) 206

56. It f;f <Dat=xz +e'1.x>0;, then/(l) is equal to
(A) l+e (B) 2+e (C) 3+e (D) e (E) 0

Maths-II-Bl/2017 l5 lP.T.O.

57. !*r.:
x1 -

(A) -x3t2 +x[/a +c (B) ,"'
1c1
lf '' +2x,', +c (D) *''' +x l2 +c (E) *t''

58. In a flight 50 people speak Hindi,20 speak English and l0 speak both English
and Hindi. The number of people who speak at least one of the two languages
is
(A) 40 (B) s0 (c) 20 (D) 80 (E) 60

59. If f (i =4,then the value of f (-f (*)) is equal to
(A) , (B) 0 (C) -, (D) 1 (E) 2

60. Two dice are thrown simultaneously. what is the probability of getting two
numbers whose product is even ?

I I 2 I
(A) (B) (c) (D)
; 4 , J
(E)
16

Maths-II-B I /20 I 7 t6

61.
nm ''lr+;-',lz-, is equal to
x+0 x

(A) +
'!2
@ J, (q o (D) Does not exist (E) 2J' 1

dx
62. is equal to
e' + e-' +2

<oti,*" rrl**. Q#*" @)l--+c re>!,+"
63. tan
(n 0\
tan
(" e) is equal to
[a*zJ* l4-1)
e
(A) sec 0 (B) 2 sec 0 (C) secT (D) sin 0 (E) cos 0

.o
-,---;-
dx
64 | rs equal to
'-t x' +x+2
fi (C) r (D) 0 (E) -r
(A)
4
@;

Maths-II-B1/2017 t7 [P.T.O.

65.
,: 6i",
li-#dr isequalto
" vsln r + vcos )r

(A) 0 (B) -r G)+ @; (u);
66. If (,r,y) is equidistant fiom (a+b,b-a) and (a-b,a+b),then
(A) x+y=g (B) bx-ay-A
(C) ax-by--O (D) bx+cy-g
(E) ax+by-O

67. If the points (1, 0), (0, l) and (x, 8) are collinear, then the value ofx is equal to
(A) 5 (B) -6 (C) 6 (D) 7 (E) -7

68. The minimum value of the function maxlx, x2) is equal to

(A) 0 (B) l (c) 2 <o) + @)l
69 Let f (x + y) = f (r) f (y) for all x and y. Ifl(O) = t, f (3) -3 and /'(0) = I l, then
/'(3) is equal to
(A) r r (B) 22 (C) 33 (D) 44 (E) s5

Maths-II-B1/2017 18

f -3 (,)
70. If /(e)=/(9)=0, then lim
x-$ x-3 is equal to
(A) 0 (B) .f(0) (c) f'(3) (D) ,f(e) (E) I

1l 1l
71. The value of cos 4*' + cos
4-x ls
(A) J2sin2x @) JTsinx (C).Ecos2x (D) ^6cos;r (E) .Dcosx

72 Area of the triangle with vertices (-2,2),(1,5) and (6, -1) is

(A)rs te)? e)+ @)+ rnl?
73. The equation of the line passing through (-3, 5) and perpendicular to the line
throughthepoints (1,0) and (-4, 1) is
(A) 5x+y+10=0 (B) 5x-Y+20=0
(C) 5x-y-10=0 (D) 5x+Y+20=0
(E) 5y-:r-10=0

74. The coefficient of x5 in the expansion of 11+ x2;511+ x;a is
(A) 30 (B) 60 (c) 40 (D) l0 (E) 4s

Maths-II-B1/2017 t9 lP.T.O.

75. The coefficient of ,a in the expansion of (l-2x)5 is equal to
(A) 40 (B) 320 (C) -320 (D) -32 (E) 80

76. The equation 5x2 +yz +y=g represents
(A) an ellipse (B) a parabola
(C) a hyperbola (D) a circle
(E) a straight line

77. The center of the ellipse 4x? + y2-8;+4y-8 =0 is
(A) (0,2) (B) (2,-1) (c) (2,1) (D) (1,2) (E) (1,_2)

78. The area bounded by the curves y=-y2a3 and y=Q ig

(A) .6+t (s) JJ (c) 4\6 (D) s$ (D 6\6
2 2 5
d3v dy
79, The order ofthe differential equation +
(*) + =0 is
dx
3 dx

(A) 3 (B) 4 (c) I (D) 5 (E) 6

Maths-II-B1/2017 20

80. rf f()=J/aa 4 , +hen f'(2) is equal to
J2,
(A) 0 (B) -l (c) I (D) 2 (E) -2

E1. The area of the circle ,2 -2*+ yz -l}y+k=O is 25n. The value of t is equal
to
(A) -l (B) I (c) 0 (D) 2 (E) 3

2017 G
82 I 2016 Ji* 4013-x dx is equal to

(A)
I
4
(B)
J
,
(c)
2017
2
(,)+ (E) s08

83 The solution ot
fl* y t^, -sec x,y(0) =Q i5
(A) ysecx=tanx (B) Ytanx=secx
(C) tanx=y tanr (D) xsecx=tanY
(E) / cot.x = sec r

84 If the vectors zi+zi+et,2i+)'i+6i, zi-li+iare coplanar, then the value
of L is
(A) -10 (B) I (c) 0 (D) lo (E) 2

Maths-II-B1/2017 2l lP.T.O.

85. The distanceberween (2, l, 0) nd 2x+ y+22+5_0 is
(A) l0 10 l0
(B)
3
(c) (D) s (E) I
9

E6. The equation of the hyperbola with vertices (0, tl5) and foci (0, t20) is

(A)
xy
2 2
x2 v
2
1 (B) I
175 225 625 t25
2
v x2 2
x2
(c) (D) v
22s t25 I 65 65 I

(E)
u2 ,2
--__-l
225 17s

I 53 + 63 + 3.6. 15.2 1
87. The value of
I + 4(6) + 6(3 6) + 4(2 t 6) + 129 6 is equal to
,o 7 6 2t
(); (B) (c)
19 7
(D)
19
@)+

Maths-II-Bl/2017 22

88. The equation of the plane that passes through the points (1, 0, 2), (-1, 1,2)
(5,0, 3) is
(A) x+2y-42+7 -0 (B) x+2y-32+7 -0
(C) x-2y+42+7-0 (D) 2y-42-7+x=0
(E) x+2y+32+7 -0

89. The vertex of the parabola y2 -4y-x+3-0 is
(A) (-1,3) (B) (-1,2) (c) (2,-l) (D) (3,-l) (E) (1,2)

90 If a,b,e are vectors such that d+6+d-0 ana lal=Z,l5l=S,lal=1, then the
angle between d and 5 is
fi T' TL
(D) r
(A) (B) (c) (E) 0
J 6 Z

91. Let f (x)-)v3 -9ax2 +l2a2x+L where a > 0. The minimum of/is attained at

a point q and the maximum is attained at apointp. lf p3 -q,thena is equal to
(A) I (B) 3 (c) 2 (D) J' (D+

Maths-II-Bl/2017 23 lP.T.O

92 For all real numbers x and y, it is known that the real valued function /
satisfies/(.x)+/1fl-f (x+y).rf f(1)=7, tt'"" r11?/(') is equal to
(A) 7x51x102 (B) 6xs0x102
(C) 7x50x102 (D) 6x25x102
(E) 7x50x101
2
The eccentricity ofthe ellipse t*;')' *( Y*4
3 1
93 ls
t6

(A)
'12
I
(B)
2J'
1
(c)
I
,
(D)
4
I
(E) +
4.,t2

94. l_r**1x,*31d, is equal to
3 I I
(A)
4
(B) (c) (D) I (E) 0
4 ,

Maths-II-Bl/2017 24

95. If xe 0 T1t ,ve 0 11t and sin x + cos y:2, then the value ofx +y is equal to
n fi
(A) 2n (B) r (c) (D) (E) 0
4 z
96 Let a, a + r and a + 2r be positive real numbers such that their product is 64
Then the minimum value of a t 2r is equal to

(A)4 (B)3 (c)2 (D)+ (E)r

11111
97. The sum S :--L--L--L-_L_
9l' 317',! ' 5!5! ' 7!3! ' 9! is equal to
)10 2e 27 26 )5
(A)
? (B)
l0!
(c)
10!
(D)
l0!
(E)
iI
lx x2 x3
98. lt f(x)= Ir 2x ^2
5X , then /'(x) is equal to
2 6x
l0

(1t) x3 +6x2 (B) 6,x3 (C) 3x (D) 6x2 (E) 0

Maths-II-B1/2017 25 lP.T.O.

2
99. dx is equal to
' l + (x')'

2
(A) taril x2 + c (B) ;J tan -l x 3 +c
I I
(c) ; tan -11r3;+,
,t^ x+c
(D) -i,
J
tan ' ,3 +"
,l
(E)

100. Let f,(x) bethe zft derivative otf(x).Theleastvalueofn sothat f,=fn*,
where /(x) =v2 +
"' is
(A) 4 (B) 5 (c) 2 (D) 3 (E) 6

l0l. sin 765'is equal to
(A) I (B) 0 (c) 6 (D)
I
(E)+
2 , ,J2

MathsJI-Bl/2017 26

102. The distance ofthe point (3, -5) from the line 3x - 4y - 26 : 0 is
(A)+ ru)? (q; r,); t,),
103. The difference between the maximum and minimum value of the function
7111- $1t2 +t+t)dt on[2, 3] is

(A)
39
6
(B)
49
6
(c)
59
6
tn) ? @)?

104. lf a a:nd, b Ne the non zero distinct roots of x2 +ax+b=0, then the minimum
value of x2 +ax+b is
.)
9
(c) -9
2
(A) (B) (D) (E) I
I 4 4 3

Maths-II-Bl/2017 27 lP.r.o.

10s. If the straight line y -.!ya
" touches the ellipse ** ,, =rthen c is equal to
(A) 0 (B) tJ65 G) 4a (D) tJ' (E) r13

106. The equations M-y=2,2x-3y=-)y 2Iird3x-2y--l are consistent for
(A) l,=-4 (B) l,=1,4 (C) ?y=1,-4 (D) I=_1,4 (E) ),=_l

lrl * lyl = t} in the ry plane represents
107. The set {(;r, l) :

(A) a square
(B) a circle
(C) an ellipse
(D) a rectangle which is not a square
(E) a rhombus which is not a square

Maths-II-81/2017 28

108. The value ofcos tan
lr)).
[AJJ,'
(A)
4
5
(B)i (q; t,)? (E) 0

109. Let A(6, -l), B(1, 3) and C(x, 8) be three points such that AB : BC. The values
ofx are
(A) 3,s (B) -3,s (c) 3,-5 (D) 4,s (E) -3,-5

110. In an experiment with I 5 observations on x, the following results were
available

Zx2 -zl3o
Ir =170
One observation that was 20, was found to be wrong and was replaced by the
correct value 30. Then the corrected variance is
(A) e.3 (B) 8.3 (C) 188.6 (D) t77.3 (E) 78

x+2 y-4 z-5
I 11. The angle between the pair of lines 'r2 = *--* *O
184
IS

(A) cos ,( 21 )
(
(B) cos-r t_t zt\
sJ:n ) IsJ38./
-l zq\ ( zs.t
(c) cos -,( @) cos-l t-t
s$8 ) Is*8.,1
-t 26\
,(
@) cos
s$B )
-t

Maths-II-B1/2017 29 [P.T.O.

ll2. Let d be aunitvector. \f @-Q.@+A)=l2,thenthemagnitudeof i is
(A) .6 (B) .6 (q Jio (D) J3 (pl J,

113. The area of the triangular region whose sides are y=2x+1,y-3x+l and x=4
ls
(A) s (B) 6 (c) 7 (D) 8 (E) e

ll4. lf nC,-1=36, nC, =84 and nC,*1=l)$, then the value of r is
(A) e (B) 3 (c) 4 (D) s (E) 6

115. Let f(x+y)-f(*)f(y) and /(x)=l+5in(3x)g(x), where g is differentiable.
Then f'(x) is equal to
(A) 3f(x) (B) s(0) (c) /(x)s(o) (D) 3g(x) (E) 3/(x)g(0)

x-l 1 I
116. The roots ofthe equation I x-l I =0 are
I I x-l
(A) 1,2 (B) -1,2 (c) -r,1 (D) t,-2 (E) l, 1

Maths-II-B 1/2017 30

tt1. If the 7th and 8d term of the binomial expansion (2a -3b)n are equal, then
2a+3b .
,;=T5 rs equal to
6)# @)# G)* @)# (E)
2n-l
13-"
II 8. Standard deviation of first z odd natural numbers is
(A) ..6
(n+2)(n+t) ,2 -l
@) (c) (D) (E\ 2n
J J "
f19. Let S : {1,2,3,..., l0}. The number of subsets of S containing only odd
numbers is
(A) ls (B) 3l (c) 63 (D) 7 (E) 5

120. Theareaoftheparallelogramwithvertices(0,0),(7,2)(5,9)and(12, ll)is
(A) 50 (B) 54 (c) sl (D) 52 (E) 53

Maths-II-Bl/2017 31 lP.T.O.

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Maths-II-Bl/2017 32