www.myengg.com BCECE 2016 First Stage Mathematics Question Paper
'he straight lines 4ay + 3by + c = 0, where (a + b + c) = 0 are concurrent at the point
101.
(A)
(B) 61
21
(C)
(D)
is equal to
1 1 1,+ 3"!+5T+
(A) (e + 1)
(D) (::
D
hannonic mean of a and b, then
g2 g2n- 1
(D)
(~, t)
(A)
+ .... +
an
gl g2n
gn gn + 1
2~'- 2~}D)
C;'p, -2;'p) (2~'2~)
(8)
(A) 0 (C) 2
. IS
(B)
108. The number of common tangents to the circles x 2 + y2 = 4 and x 2 + y2 - 6x - 8y - 24 = 0 is
a 1 + a2n
+ an + i
(~,~)
(C) (-
progression and a, gl' g2' g3' ..... g2n' b are in geometric progression, and h is the
a 2 + a 2n- i
(i, t)
The centre of the smallest circle touching the circle and the line x + y = 5-{2 has the coordinates
103. If a, ai' a 2, a 3, ... a 2n , b are in arithmetic
+
(t,~)
107. The equation of a circle is x 2 + y2 = 4.
(B) (e - I)
(:~ D
(C)
(B)
(C)
1 1 1 2T+4T+6!"+ ....... .
102.
(A) (4,3)
equal
(D) 3
to (A)
211 h
109. The locus of the vel1ices of the family of a\2 a2x
(B) 211h
parabolas y =
n
(C) I1h
(D) h
104. If A(cos a, sin a), B(sin a, - cos a), C(l, 2) are the vertices of a Ll ABC, then as
fJ.
varies, the locus of it centroid is
I x+4 x+8
x+6 x+l1
35 (C) xY=T6
(D) xy= 35
+ y - 2)2
(A) (0, 0) (C) (1,1)
I
16
(x _ y)2
+
16
I is (B) (0, 1) (D) (1,0)
111. The vertices of a triangle are (pq,
I is ( qr,
, (A)
2 (B) -2
~~) and (rp, ~), where p, q,
:J, r are the
roots of the equation y3 - 3y2 + 6y + 1 = 0. The coordinates of its centroid are (B) (2, - 1) (A) (1,2) (D) (2,3) (C) (1,-1)
(C) (x - 1)3 (D) x 3 + 3x2 + 3x + 1
Il
3 (B) xY=4"
9
(C) 3(x2 + y2) - 2x - 4y + 1 = 0 (D) 3(x2 + y2) - 2\' - 4y + 3 = 0
105. The value of
105 (A) xy= 64
(x
(B) x 2 + y2 - 2\' - 4y + 3 = 0
x+5 x+9 x+ 15
+ '2 - 2a is
110. The centre of ellipse
(A) x 2 + y2 - 2x- 4y + 1 = 0
I x+2 x+3
3
18
PBE
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www.myengg.com BCECE 2016 First Stage Mathematics Question Paper
101.
1
1
1
2!+4T+6!"+ ....... . 102.
1 1 + 3!
1 + 5!
cpf lfPf
+ ....... .
(A) (e + 1) (C) (:
~D
x
(B) (e-l)
~ern~~~ P\,hliC!) ~
(:~ D
(D)
(A)
103. ~ a, a j , a2, a 3, ... a2n , b 'fJl1RR ~ -q t
(C)
3lR a, gj' g2' g3' ..... g2n' b ~ ~-q t 3lR h, a ~ b cpf ~ lflUl t, Wl
(A)
G'~) (B) (2~' - 2~) (- 2~' - 2~}0) (2~' 2~)
108. <[ffi x 2 + y2 = 4 ~
x 2 + y2 - 6x - 8y - 24
a j + a2n + gj g2n cpflfPf
ern cpf fl'11C!),,{ul x 2 + y2 = 4 t 1 trn + Y = 5-J2 3lR ern Clil ~ m qffi
107. ~
t
~3lt~~t
t
i (D) 3
(A) 0 (C) 2
2n
(B)
(B) 2nh
h
(C) nh
n
h
(D)
105
104. ~ A (cos ex, sin ex), B(sin ex, - cos ex), q1, 2) ~ ~ABC ~ ~ ~,
Wl Wl
ex -q
(A) xy= 64
3 (B) xY=4"
35 (C) xY=16
16 (D) xy = 35
~iWnt, Wl~~~cpf~2l
110. ~
t
(x
(A) x 2 + y2 - 2x - 4y + 1 = 0 (B) x 2 + y2 - 2x - 4y + 3 = 0 .
111.
105.
x +4
v-l-'l
v-l-"
I
16
1
-+-= ~
cpf cr>'>! ~
(B) (0, 1) (D) (1,0)
~f5r~~~ (pg, plJ, (
1\
(
1\
y3 _ 3y2 + 6y + 1 = 0 ~ ~ ~ I ~ ~
x+15
~~ P\~~IiCfj ~
(A) (1,2) (C) (1, - 1)
(D) x 3 + 3x2 + 3x + 1
PBE
(x - y)2
lgr, ~~) ~ lrp, ~) t vrif p, g, r fl'1~ICfj"{UI
~ ~ ~ ~ ~ ~ I cpf lfPf t
x+8 x+l1 (A) ·2 (B) -2 (C) (x - 1)3
+
(A) (0, 0) (C) (1,1)
(D) 3(x2 + y2) - 2x - 4y + 3 = 0
x+2
+ y- 2)2 9
(C) 3(x2 + y2) - 2x - 4y + 1 = 0
I
= 0 ~ \j~
19
(B) (2, - 1) (D) (2, 3)
c
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www.myengg.com BCECE 2016 First Stage Mathematics Question Paper 112.
113. The range of function (
,.2 '\
f(x) = cos-I I~) is .
\1 +x-
(A)
(B)
[-¥,¥]
(C)
(D)
(0, ~]
120. If In =
f
(In x)n dx, then In + n In _ 1 is
equal to (In x )n (A) x
(B) x(lnx)n-l
(C) x (In x)n
(D)
(In x
114. If f(x) and g(x) are periodic functions with period 7 and 11 respectively, then the period ofF(x) = f(x)
g(~) -
(A) 1155 (C) 222
(B) 177 (D) 433
g(x)
{~) is
121. The value
t- I x
off (+x 1 )·~1'+x-'1) dx is o
(A)
41t
(B)
¥
(C) 1t
(D)
1t
"3
x
115. For x E
R,li~ (; ~ ;) .\
is equal to
122. The area enclosed between the curves y2 = x and y = Ixl is
~vv
(A) e
(B) e
(C) e-5
(D) e5
116. If f(l) = 2 and g'(-{2) = 4, then the derivative of f(tan x) with respect to g(sec x) at x =
1
(C)
-{i
(D) 2/3
(12 + 22 + ... + n2) (13 + 2 3 + ... + n 3) lun (1 6 + 26 + ... + n6) n-too
...[2
IS
12 (A) 19
(D) 2
(B) tan- 1
(~)
(C) tan- 1
(~)
(D) tan- 1
(i~D
8 (D) 19
+ c (B) xy = f(x) + cx (C) "y=f(x)+c+x (D) y(x+c)=f(x)
r_ .
e74)
7 (C) 12
(A) xy = f(x)
liS. The angie between the asymptotes of the x2 hyperbola 16 - 9 - 1 IS (A) tan- 1
8 (B) 21
124. The solution of differential equation ~ yf'(x) - y2 . I dx f(x) IS equa to
117. The value of nth derivative of xex is zero when (A) x = 0 (B) x=-1 (C) x= n (D) x=- n
C
(C) 113
.
41S (B)
(B)
123. The value of
1t.
(A) 1
6
(A)
125. A spherical balloon is being inflated so that its volume increases uniformly at the rate of 40 cm 3/min: At r = 8 cm, its surface area increases at the rate of (A) 8 cm 2/min (B) 10 cm2/min (C) 12 cm2/min 20
(D) 16 cm2/min
PBE
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www.myengg.com BCECE 2016 First Stage Mathematics Question Paper 112. fum t g(x) = 1 +
-rx 3fR
f(g(x» = 3 + 2~
=
(t + 3x - x 2) I (x - 4)
l1f1 iR ~ fu"
(B) 2 + x 2 (D) 2 + x
C:2x2) ~1fffi1t
~J [-~, 0]
t ~ l1f1 ~ ~ t (B) (0,00) (D) (4,00)
~
f(x)= eos- 1
t, "\if5T
+ x, G€[ f(x) CfiT
(A) 1 + 2x2 (C) 1 + x
113.
~ f(x)
120.
~ In =
(A) [0,
(B)
[-~: ~J
"liAt
(C)
(D)
(0,;]
(A)
(C)
J
(In x)n dx,
(In x
x
X
t
G€[
In + n In _ I
CfiT
(B) x(lnx)n-I (lnx
(lnx)n
(D)
t
-I
X
114. ~ t{x) 3fR g(x) ~: ~ qm:;r 7 3fR 11
~ ~ ~ -gi, G€[ F(x) = f(x) g(~) g(x)
<~) CfiT ~ qm:;r t
XE
_
'.p..,-rl.
K
q) 11.'1<-1,
(x - 3\\"
••
lun ~~ + x-t
2)
X
(1
')dxCfiTl1f1t
+ x)(1 +x-)
~
(B)
~
(C) It
(D) ~
(C) 1/3
(D) 2/3
3
.-\. cpl '11'1 Q
(A) 1
oo
1 (B) e
(A) e
Jo
(A)
(D) 433 _
(C) 222
(A) 1155 (B) 177
115.
121.
123.
lim (12 + 22 + ... + n 2) (13 + 23 + ... + n3)
116. ~ f/(I)
= 2 3fR g/(--J2) = 4, G€[ g(sec x)
~ ~aT f(tan x) CfiT ~ x =
(A) 1 ,~, _1 ~LJ
(B)
(1 6 + 2 6 + ... + n 6) CfiT"IiA t
4"It tR" t
(A)
-{2
(B)
8
21
(C)
7
12·
(D)
8
19
--J2
117. xeA' ~ n ii ~ CfiT l1f1 ~ t
(A) x=O (C) x = n 118. 3lre Rlrrrq=Xq=<'lr:lT1i
.Qy _ yf'(x) _ y2 + dx -
\ijq
,2
v2
16 - 9" =
f(x)
cp
~ 5R CfiT l1f1 Q
(A) xy = f(x) + e (B) xy = f(x) + ex (C) y = f(x) + e + x (D) y(x + c) = f(x)
(B) x=-l (D) x = - n
125. ~7J1c;r~cp'r~~~VfId1t ~ 1 ~ ~"1~fqlll
~~ 40 em 3/min. ~~~ C!Cfli(FlI"1
~Cf>TuTt
PBE
12
19
qq;a-rt Ir= 8 cmtR", ~~~iT~ 4 )
(A) tan- 1
C7
(B) tan- 1
(%)
~~~~
(C) tan- 1
(~)
(D) tan-I
(;4)
(A) 8 em2/min (C) 12 em2 /min
21
(B) 10 em 2/min (D) 16 em 2/min
c
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www.myengg.com BCECE 2016 First Stage Mathematics Question Paper 126. If cos (x - y), cos x, cos (x + y) , hannonic progression, then sec x is equal to (A)
±1
(C)
+-r=
I
-'1 3
I
(B)
±{i
(D)
±-J2
(B) 8115 (D) 2/3 134. The unit vectors orthogonal to the vector
127. Given f(a) = 5 cos a + 3 cos (a
+~) + 8,
- i + 2J + 2k and making equal angles with the x and y axes are
then the range of f(a) is (A) [0,16] (B) [6,16] (C) [10, 16] (D) [1, 15] 128. The number of solutions tan 8 tan 48 = 1 for 0 < (A) 5 (B) (C) 3 (D)
(C)
(D) ±"31 (Ai - 2jA- 2kA)
135. If(a x S)\ (a· S)2= 144 and lal = 4, then lSI is equal to (A) 16 (8) 8 (C) 3 136. If the vectors 2i - 3J + 4k,
D
I
(D) 12
i+
2J -; k
xi -
x
and J + 2k are coplanar, then is equal to 5 (D) I (A) (B) "8 (C)
a 2b2c 2 4R2 2 a b 2c 2 (D) 8R2
°
~
(B)
137. ( tan-I 1+ tan-I 1+ tan-I 1+ tan-I 1) is 357 8 equal to (A) 1t
(B)
~
1t
(C) "4
(0)
~
138. If the standard deviation of a variate X is a, then the standard deviation of aX + b is (8) aa (A) aa + b (C) a (D) lala
132. The number of values of lllE N for which y = e l1w is a solution of the differential . d 3y d2y ~ . equatIOn dx 3 - 3 dx 2 - 4 dx + 12y = IS
°
A A\
4 2
131. Each side of a square subtends an angle of 60° at the top of a tower h metres high standing in the centre of the square. If 'a' is the length of each side of the square, then (A) 2a 2 = h 2 (B) 2h2 = a 2 2 (C) 3a = 2h2 (D) 2a2 = 3h 2
(A) (C) 2
±3"1 (/\i + jA- kA) 1 (A
BO. If PI' P2' P3 are respectively the perpendicular from the vertices of a triangle to the opposite sides, then PI P2 P3 is equal to (R is circum-
4a 2b 2c 2 R2
(B)
(C) ±"3\2i-2j-k,
129. Ia 3 cos (B - C) is equal to (A) 3abc (B) 3(a+b+c) (C) abc (a + b + c) (D) zero
~
A A) ± 3"1 (A 2i + 2j -k
of the equation
e < 1t is
radius) a 2b 2c 2 (A)
(A)
139. 5 coins are tossed simultaneously. The probability that at least one head turning up is 5 7 31 1 (A) 32 (B) 32 (C) 16 (D) 32
°
(B) 1 (D) more than 2
22
PBE
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www.myengg.com BCECE 2016 First Stage Mathematics Question Paper 126.
(C) ±
127.
1
13
(A) 1/3
(B) [6, 16] (D) [1,15]
128. \(1"IICf)i(01 tan e, tan 4e = 1 ~ ~ ctt"ffis.m o
(A) 5
(B) 4
(C) 3
±31 (" 2i + 2j"") -k
(B)
±31 ("i + j" - kA)
(C)
"") ±31 (" 2i-2j-k
')
')
(a x br+ (a . br= 144 3lR
lal = 4, Ibl "Cfll11Rt (fq
(A) i6
130. ~ \fCf) f5r~ ~ ~ xl ~ :pITGit LR C¥
(A)
135. ~
(A) 3 abc (B) 3(a + b + c) (C) abc (a + b + c) (D) ~
m,
~ ~ ~ x ~2D y
1 ("i - 2j" - 2k") (D) ± 3
(D) 2
129. Ia 3 cos (B - C) "Cfll11R t
w+m: PI' P2' P3
2J + 2k
3f&fl xl "fI1'lR Cf>1ur q;:mf S'!" ~ ~ t
fee) ctt ~ t
(A) [0, 16] (C) [10,16]
(C) 5/13 (D) 2/3
134. ~ - j' +
~t t1e) = 5 cos e + 3 cos (e +~) + 8, (fq
(B) 8/15
(8) 8
(C) 3
(D) i2
PI P2 P3 "Cfll11R t
(Rcmrr~~
" xi1-\ "j + 2k
(A)
8
5"
"
fPldC'1ll1
(B)
5
"8
4
Q,
~
(fq x"Cfll11R Q
(C) 0
(D) 1
I tan- I -+ I tan- I -+ 1 tan- I -1) "Cfll 137. ( tan- I -+ 3 5 7 8 131. \fCf)qrf ctt ~ :pIT, qrf ~
R
LR ~ h
11Rt
~ ~ 1fAN ~ ~ LR, 60° "Cfll Cf>1ur q.mfi
t
l~qrfctt~:pITctt~ 'a'
(A) 2a2 = h 2 (C) 3a 2 = 2h2 132.
_
0
0
-:l-IQ'PC'1'1I
t
(A) 1t
(B) 2h2 = a2 (D) 2a2 = 3h2 d3y dx3
~ d 2y
-
12y = 0 "Cfll ~ Y = enu-
138. ~ -qcp T:R X
PBE
~
(C)
~
(D)
~
"Cfll lfRCP ~
cr t,
([q
aX + b "CflllfRCP ~ t
(A) (C)
. dy , dx +
~ dx2 ~ 4
m ~ fu<) mEN
aa + b a
(B) (D)
aa
lala
139. 5 ~ -qcp W~ ~ ~ ~ 1Cf>l'f xl Cf>l'f -qcp
LR m ~ lJHt ctt "ffis.m t (A) 0 (C) 2
(B)
(fq
'~' ~ 3iRctt ~1f!lCf)dl t
(B) 1 (D) 2xl~
1 (A) 32
23
5 (B) 32
7
(C)
16
31 (D) 32
c
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L
www.myengg.com BCECE 2016 First Stage Mathematics Question Paper 140. If f(x) IS defined on [0, 1] following ruic
.
fX
,
f(x) = I
.
i I-
X ,
=0
if X E Q . , If X Et Q
then fof(x) for all x E [0, I] is (A) x (8) I-x (e)
1
(D) zero
141. If the function f: e ~ C be defined by itx) = x 2 - I, then t'-l (-5) is (A) 24
(C)
:2i, - 2i}
147. In
the
1+
x)2 '
of ( 1 _ x
expansion
1 (8) 24
coefficient of xl1 will be
(D) {-2, 2}
(A) 4n
(8) 4n - 3
(e) 4n + 1
(D) 2 (211 + I)
the
142. For the equation zz + (-3 + 4i) z - (3 + 4i) z + k = 0 to represent a circle, the value of k is (A) 48
(8) 36
(C) 32
(D) ::; 25
148. The number of dit1erent words which can be formed from the letters of the word LUeKNOW· when the vowels always occupy even places is
143. If n is a positive integer, then value of
({3 + i)
n
_M + (" 3 -
(A) 211 cos
nn 1Ut
(D) 211 cos
I1rr -3-
144. The origin and the complex numbers represented by the roots of the equation z2 + az + b triangle if (A)
a = 2b
(e) b 2 = 3a
=
(e) 720
(D) 960
149. A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first 5 questions. The number of choices available to him is
nn
(C) 2"- I cos6
(8) 400
is
(8) 211 + I cos 6
6
._.
i)
n
(A) 120
(A) 196
(8) 280
(e) 346
(D) 140
0 form an equilateral
150. The sum of the series (8) a2 = 3b
13
1
(D) b = 2a
13 + 23 13 + 23 + 33 + 1 + 3 + 1 + 3 + 5 +. . . . .. upto
n temlS is
145. If A = {l, 2, 31, 8 = {4, 5, 6}, then which of the following is not a relation from A to 8? (A) RI = {(l, 4), (1, 5), (1, 6)} (8) R2
~
(8)
{(1, 5), (2, 4), (3, 6)}
.,
n
.,
(e) 24 (2n-+ 911 + 13)
(C) R3 = {(l, 4), (1, 5), (3, 6), (2,6), (3, 4)}
(D) R4 = {(4, 2), (2, 6), (5, I), (2, 4)}
a
1
4' (n-+ 2n + 1)
(D)
24
n
.,
12 (11- + 2) PBE
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www.myengg.com BCECE 2016 First Stage Mathematics Question Paper 140. ~ [0, 1]
lR f(x) f.i+«"1~fuJd
a\"2 + bx + c = 0 ~ ~ a, ~ ~
j .~ ,~xeQ
X
f(x) = { I - X ,
x 2 + qx + r = 0 -ij; ~ a + h, Cf>T lf1'1
~X ~ Q
~ + h ~, "Gq
t
(f
(B)
G~-;)
(A) x
(8) I-x
(D)
(e)
(D) ~
.!.(!?_Sl) 2 a p
1
141. ~ ~ f : C ~ cqft~
men
1 + X)2 147. (-1 -x
t
f(x) = x2 - 1 ~,"Gq {1(_5) t
1
(A) 24
(B) 24
(C) {2i,-2i}
(D) {-2,2}
3iR
-\-cp >RlTX ~ +-j
xl1
- \-,. .,.
Cf>T 7TlJl(J) ~ I'll
, . ! > '"-
(A) 411
(B) 4n - 3
(e) 4n + 1
(D) 2 (2n + 1)
148. LueKNOW ~ ~ 3l~ ~ ~ fqf~ ~ q;:n
~~~:
zZ + (-3 + 4-i) z - (3 + 4i) z + k = 0 cp1 ~ <[ff ~
(A) 48
143. ~
(A)
-ij; ~ k cpp:rH t
(8) 36
11 1;,J(f)
(-{3 --- i)
m
(C) 32
~ ~ t, "Gq
11
(A) 120
(D)
(-{3 + i)
~
25
(e) 720
(B) 400 (D) 960
+
149. \fCP fcmr2fi cp1 ~
wan 1f 13 w-n 1f ~
11
~ ~ >fCPR ~ ~ ~ >f2.fl1
qiPTH t
,..n ___ n1t
":'--l;US
1
(B)
6
(e) 211 - cos6 111t
,..n
['
q)l'f
+ I
n1t
'cose;
4~~
¥
10 ~
5 >rv.n 1f ~ q>ll x1
I ~ >TfC
f~ ~ ~
~t
n1t
(D) 211 cos-
(A) 196 (C) 346
3
(B) 280 (D) 140
144. ,,<\l1 z2 + az + b = 0 -ij; +rffi ~ ~ ~J3l ~ 'fCt ~ ~
'
150. J!,)uft
~
q.m'r~~
(A) a
= 2b
(e) b 2 = 3a
13
T+
(B) a 2 = 3b (D) b = 2a
13 + 23 13 + 23 + 33 1+3 + 1+3+5 + ......
-
n'lG1
qCf) Cf>T
145. ~ A = {t, 2, 3}, 8 = {4, 5, 6},"GqfTq 1f~~A~BqCf)~~t ?
(A)
'81 (n 3+ 40 + 3)
(B)
') '41 (0-+ 2n + 1)
(A) RI = {(l, 4), (1, 5), (1, 6)} n 2 (C) 24 (2n + 911 + 13)
(B) R2 = {(l, 5), (2, 4), (3, 6)} (e) R3= {(l,4), (1, 5}, (3, 6), (2, 6), (3, 4)}
n
,I
PBE
)
(D) 12(n~+2)
(D) R4 = {(4, 2), (2, 6), (5, 1), (2, 4)}
25
a
j
I
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