PUBDET 2018 Question Paper Mathematics

PUBDET-2018 81240001
Subject: Mathematics (Booklet Number)

Duration: 90 minutes Full Marks: 100

Instructions
1. All questions are of objective type having four answer options for each. Only one option is
correct. Correct answer will carry full marks 2. In case of incorrect answer or any
combination of more than one answer, ½ marks will be deducted.
2. Questions must be answered on OMR sheet by darkening the appropriate bubble marked
A, B, C, or D.
3. Use only Black/Blue ball point pen to mark the answer by complete filling up of the
respective bubbles.
4. Do not make any stray mark on the OMR.
5. Write question booklet number and your roll number carefully in the specified locations of
the OMR. Also fill appropriate bubbles.
6. Write your name (in block letter), name of the examination centre and put your full
signature in appropriate boxes in the OMR.
7. The OMRs will be processed by electronic means. Hence it is liable to become invalid if
there is any mistake in the question booklet number or roll number entered or if there is
any mistake in filling corresponding bubbles. Also it may become invalid if there is any
discrepancy in the name of the candidate, name of the examination centre or signature of
the candidate vis-a-vis what is given in the candidate’s admit card. The OMR may also
become invalid due to folding or putting stray marks on it or any damage to it. The
consequence of such invalidation due to incorrect marking or careless handling by the
candidate will be sole responsibility of candidate.
8. Candidates are not allowed to carry any written or printed material, calculator, pen, docu-
pen, log table, any communication device like mobile phones etc. inside the examination
hall. Any candidate found with such items will be reported against & his/her candidature
will be summarily cancelled.
9. Rough work must be done on the question paper itself. Additional blank pages are given in
the question paper for rough work.
10. Hand over the OMR to the invigilator before leaving the Examination Hall.
11. This paper contains questions in both English and Bengali. Necessary care and precaution
were taken while framing the Bengali version. However, if any discrepancy(ies) is /are
found between the two versions, the information provided in the English version will stand
and will be treated as final

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1. 6.732 + 7.9 45 when divided by 4, the remainder is
6.732 + 7.9 45 -−L 4 à¡l¡ i¡N Ll−m i¡N−no q−h
(A) 2 (B) 3 (C) 1 (D) 0

2. 2+6 2+2 x 6 2+3 x 6 2 + 99 x 6 2 + 100 x 6
The value of 100
+ 99
+ 98
+ ⋯⋯ + + is equal to
4 4 4 42 4
2+6 2+2 x 6 2+3 x 6 2 + 99 x 6 2 + 100 x 6
100
+ 99
+ 98
+ ⋯⋯ + + -Hl j¡e q−h
4 4 4 42 4
1 1  1 1 
(A)  604 − 98  (B)  600 − 98 
2 4  3 4 
604
(C) (D) 200
3

3. The set of all real numbers in open interval ( - 2, 2) satisfying 2 x − 2x −1 − 1 = 2x −1 + 1 is

( - 2, 2) j¤š² A¿¹−ll −k pjÙ¹ h¡Ù¹h pwMÉ¡ 2 x − 2x −1 − 1 = 2x −1 + 1 pj£LlZ−L ¢pÜ L−l, −p…¢m
qm
(A) {−1,1} (B) {−1} ∪ [1,2 )
(C) ( −2, −1] ∪ [1,2 ) (D) ( −2, −1) ∪ {1}

4. 351 when divided by 8 leaves the remainder
351 -−L 8 à¡l¡ i¡N Ll−m i¡N−no q−h

(A) 6 (B) 5 (C) 7 (D) 3

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5. Let f,g : ℝ → ℝ be defined by f(x) = x + x and g(x) = x − x for all x ∈ ℝ .
f g be defined by ( f g )( x ) = f [ g(x)] for all x ∈ ℝ . Then

(A) f g ≡ g f on ℝ
(B) f g ≡ g f for all x ≥ 0 but not for x < 0.
(C) f g ≡ g f for all x < 0 but not for x ≥ 0.
(D) f g ≠ g f on ℝ

j−e Ll f,g : ℝ → ℝ ¢ejÀi¡−h pw‘¡a:
f(x) = x + x Hhw g(x) = x − x , pLm x ∈ ℝ -Hl SeÉ z
f g Hi¡−h pw‘¡a B−R −k ( f g )( x ) = f [ g(x)] ,pLm x ∈ ℝ -Hl SeÉz a−h

(A) ℝ -H f g ≡ g f
(B) pLm x ≥ 0 -Hl SeÉ f g ≡ g f , ¢L¿¹¥ x < 0 -Hl SeÉ eu
(C) pLm x < 0 -Hl SeÉ f g ≡ g f , ¢L¿¹¥ x ≥ 0 -Hl SeÉ eu
(D) ℝ -H f g ≠ g f
6. 3π
The principal value of argument of z (arg z) for z = 1 + i tan is
5
3π
z = 1 + i tan -Hl Bl…−j¾V-Hl j¤MÉj¡e (arg z) qm
5
3π 2π 2π 3π
(A) (B) (C) − (D) −
5 5 5 5

7. If log x ax, log x bx and log x cx are in A.P, where a, b, c, x belong to (1,∞ ) , then a, b, c are in
(A) A.P (B) G.P
(C) H.P (D) no specific relation exists between a, b and c

k¢c log x ax, log x bx J log x cx pj¡¿¹l −nËe£−a b¡−L,
−kM¡−e a, b, c, x ∈ (1,∞ ) , a−h a, b, c

(A) pj¡¿¹l −nËe£i¥š² q−h (B) …−Z¡šl −nËe£i¥š² q−h
(C) ¢hfl£a −nËe£i¥š² q−h (D) a, b J c-Hl j−dÉ −L¡e p¤¤¢e¢cÑø pÇfLÑ −eCz

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8. Choose the correct statement :
(A) Every non-singular matrix is orthogonal
(B) Every orthogonal matrix is non-singular
(C) Every symmetric matrix is orthogonal
(D) Every skew-symmetric matrix is non-singular

p¢WL E¢š²¢V h¡R¡C Ll :
(A) fË¢a¢V A¢h¢nø jÉ¡¢VÊ„ q−h mð jÉ¡¢VÊ„
(B) fË¢a¢V mð jÉ¡¢VÊ„ q−h A¢h¢nø jÉ¡¢VÊ„
(C) fË¢a¢V fË¢apj jÉ¡¢VÊ„ q−h mð jÉ¡¢VÊ„
(D) fË¢a¢V ¢h-fË¢apj q−h A¢h¢nø jÉ¡¢VÊ„z

9. a b aα + b
If the determinant b c bα + c is equal to zero, & b2 ≠ ac , then
aα + b bα + c 0

(A) a, b, c are in A. P
(B) a, b, c are in H.P
(C) α is a root of the equation ax 2 + 2bx + c = 0

(D) α is not a root of the equation ax 2 + 2bx + c = 0

a b aα + b
k¢c ¢eZÑ¡uL b c bα + c -Hl j¡e n§ZÉ qu J b2 ≠ ac qu, a−h
aα + b bα + c 0

(A) a, b, c pj¡¿¹l −nËe£−a b¡L−h
(B) a, b, c ¢hfl£a fËN¢a−a b¡L−h
(C) α , pj£LlZ ax 2 + 2bx + c = 0 -Hl HL¢V h£S q−h
(D) α , pj£LlZ ax 2 + 2bx + c = 0 -Hl HL¢V h£S q−h e¡

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10. The natural number x is chosen at random from the first 100 natural numbers. The
100
probability that x + > 50 is
x
fËbj 100 ¢V ü¡i¡¢hL pwMÉ¡ −b−L ü¡i¡¢hL pwMÉ¡ x kcªµRi¡−h −h−R −eJu¡ qmz −p−r−œ
100
x+ > 50 qJu¡l pñ¡he¡ qm
x

53 11 27 1
(A) (B) (C) (D)
100 20 50 2

If (1 + x ) = a0 + a1x + a2x 2 + ⋯ + anxn , ( n is positive integer) then the finite series
n
11.
a0 + a4 + a8 + a12 +⋯⋯ is equal to

k¢c (1 + x ) = a0 + a1x + a2x2 + ⋯ + anxn , qu ( n de¡aÈL f§ZÑpwMÉ¡), a−h pp£j −nËe£
n

a0 + a4 + a8 + a12 +⋯⋯ -Hl j¡e q−h

n
−1 nπ n
−2 nπ
(A) 2n−1 + 2 2 .cos (B) 2n−2 + 2 2 .cos
4 4

n− 2
n
−1 nπ n −1
n
−2 nπ
(C) 2 + 2 .cos
2
(D) 2 +2 2
.cos
4 4

12. k (k + 1)
Let n and k be positive integers such that n ≥ . The number of solutions
2
( x1 ,x 2 ,⋯ ,xk ) , x1 ≥ 1, x 2 ≥ 2,⋯xk ≥ k, all integers, satisfying the condition
x1 + x 2 + ⋯ + xk = n is

k (k + 1 )
n J k fËcš de¡aÈL f§ZÑpwMÉ¡ Hhw n ≥ −cJu¡ B−Rz x1 + x 2 + ⋯ + xk = n naÑ¢V−L ¢pÜ
2
L−l Hje pj¡d¡e {x1 ,x 2 ,⋯ ,xk } , −kM¡−e x1 ≥ 1, x2 ≥ 2,⋯xk ≥ k -Hl pwMÉ¡ qm

(m + k − 1)! (m − k − 1)!
(A) (B)
(k − 1)!m! (k + 1)!m!
(m + k + 1)! (m − k + 1)!
(C) (D)
(k + 1)!m! (k − 1)!m!
k (k + 1) k (k + 1)
Where m = n - H−r−œ m = n -
2 2

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13. At any point of a curve where the ordinate varies as the cube of the abscissa, a tangent is
drawn; where it cuts the curve again and another tangent is drawn & so on. Then
(A) There is no particular mathematical relation between the co-ordinates of the
points of contact.
(B) The ordinates of the points of contact form an A.P. where as the abscissas of the
points form a G.P.
(C) The abscissas of the points of contact form an A.P. where as the ordinates of the
points form a G.P.
(D) Both abscissas & ordinates form G.P.

HL¢V hœ²−lM¡l fË¢a¢V ¢h¾c¥l −L¡¢V, i¥−Sl ¢œO¡−al p−‰ plm−i−c B−Rz HL¢V ¢h¾c¥−a A¢ˆya
ØfnÑL hœ²−lM¡−L −k ¢h¾c¥−a −Rc L−l −pC ¢h¾c¥−a A¢ˆya ØfnÑL hœ²−lM¡¢V−L Bl HL¢V ¢h¾c¥−a
−Rc L−l Hhw HC fË¢œ²u¡ Qm−aC b¡−Lz −p−r−œ
(A) ØfnÑ ¢h¾c¥…¢ml ÙÛ¡e¡−ˆl j−dÉ −L¡e p¤¤¢e¢cÑø N¡¢Z¢aL pÇfLÑ −eC
(B) ØfnÑ ¢h¾c¥ pj§−ql −L¡¢V…¢m pj¡¿¹l −nËe£ NWe L−l Hhw i¥Spj§q …−Z¡šl −nËe£ NWe L−l
(C) ØfnÑ ¢h¾c¥ pj§−ql i¥S…¢m pj¡¿¹l −nËe£ NWe L−l Hhw −L¡¢Vpj§q …−Z¡šl −nËe£ NWe L−l
(D) i¥S J −L¡¢V ph…¢m …−Z¡šl −nËe£i¥š² q−hz

14. f(1 − h) − f(1)
Let f(x) = 3x10 − 7x 8 + 5x 6 − 21x 3 + 3x2 − 7 . Then lim
h→ 0 h3 + 3h
50 53 22
(A) does not exist (B) is (C) is (D) is
3 3 3
f(1 − h) − f(1)
j−e Ll f(x) = 3x10 − 7x 8 + 5x 6 − 21x 3 + 3x 2 − 7 z a−h lim -Hl
h→ 0 h3 + 3h

50 53 22
(A) A¢Ù¹aÆ b¡L−h e¡ (B) j¡e q−h (C) j¡e q−h (D) j¡e q−h
3 3 3

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15. Let f,g : ℝ → ℝ be such that f(x) g(x) is continuous at x 0 ∈ ℝ . Then
(A) f, g both must be continuous at x0.
(B) One of them must be continuous at x0, the other may not be
(C) both f, g may be discontinuous at x0.
(D) In case of discontinuity of any of them, the discontinuity must be removable
discontinuity.

j−e Ll f,g : ℝ → ℝ Hje −k f(x) g(x) A−frL x 0 ∈ ℝ ¢h¾c¥−a p¿¹az −p−r−œ
(A) f, g −L AhnÉC x0 ¢h¾c¥−a p¿¹a q−a q−h
(B) A−frLc¤¢Vl j−dÉ HL¢V−L AhnÉC x0 ¢h¾c¥−a p¿¹a q−a q−h
(C) f, g Ei−uC x0 ¢h¾c¥−a Ap¿¹a q−a f¡−l
(D) −L¡e HL¢V A−fr−Ll Ap¿¹¢al −r−œ Ap¿¹¢a¢V AhnÉC Afp¡lZ−k¡NÉ Ap¿¹¢a q−hz
16.  2 1 x
x sin + , x ≠ 0
Let f(x) =  x 2
0, x=0
Then, (A) f is increasing in any interval containing zero
(B) f is decreasing in any interval containing zero
(C) f is neither increasing nor decreasing in any interval containing zero
(D) Rolle’s theorem is applicable to f in [ - 1 , 1]

 2 1 x
x sin + , x ≠ 0
j−e Ll f(x) =  x 2
0, x=0
−p−r−œ (A) n§ZÉ d¡lZ L−l Hje fË¢a A¿¹l¡−m f œ²jhdÑj¡e
(B) n§ZÉ d¡lZ L−l Hje fË¢a A¿¹l¡−m f œ²jqÊÊ¡pj¡e
(C) n§ZÉ d¡lZ L−l Hje −k−L¡e A¿¹l¡−m f œ²jhdÑj¡e-J eu, œ²jqÊÊ¡pj¡e-J eu
(D) [ - 1 , 1]-H −l¡−ml Eff¡cÉ f-Hl −r−œ fËk¤š² euz

17. Let y = x be tangent to the curve y = ax , a > 0, a ≠ 1 . Then
y = ax , a > 0, a ≠ 1 -hœ²−lM¡l ØfnÑL y = x. a−h

1 1 1 1
(A) a = (B) a = (C) a = e e (D) a =
2 e ee

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18. 1
A man 5 ft. long walks away from the foot of a lamp-post 12 ft high at the rate of 3
2
m.p.h. Then,
(A) the shadow is decreasing at the rate of 1 m.p.h
(B) the shadow is increasing at the rate of 1 m.p.h

(C) the shadow is decreasing at the rate of 2 m.p.h

(D) the shadow is increasing at the rate of 2 m.p.h

1
12 g¥V EµQa¡ ¢h¢nø −L¡e h¡¢aÙ¹ñ-l f¡c¢h¾c¥ −b−L 5 g¥V EµQa¡pÇfæ −L¡e hÉ¢š² O¾V¡u 3
2
j¡Cm −h−N k¡œ¡ öl¦ L−lez −p−r−œ

(A) a¡yl R¡u¡ O¾V¡u 1 j¡Cm q¡−l qÊÊ¡p f¡u
(B) a¡yl R¡u¡ O¾V¡u 1 j¡Cm q¡−l c£OÑ qu
(C) a¡yl R¡u¡ O¾V¡u 2 j¡Cm q¡−l qÊÊ¡p f¡u
(D) a¡yl R¡u¡ O¾V¡u 2 j¡Cm q¡−l c£OÑ qu
19. P is the point of contact of the tangent from the origin to the curve y = log e x . The length
of the perpendicular drawn from the origin to the normal at P is
j§m ¢h¾c¥ −b−L hœ²−lM¡ y = log x -Hl A¢ˆa ØfnÑ−Ll ØfnÑ¢h¾c¥ qm Pz P ¢h¾c¥−a A¢ˆa
e

A¢imðl Efl j§m¢h¾c¥ −b−L A¢ˆa m−ðl °cOÑÉ q−h

1 1
(A) e2 + 1 (B) 2 e2 + 1 (C) (D)
e 2e
x
20.
The intercepts on x-axis made by the tangents to the curve, y = ∫ t dt, x ∈ ℝ which are
0

parallel to the line y = 2x are equal to
x
y = 2x Hl pj¡¿¹l¡m J y = ∫ t dt, x ∈ ℝ -Hl ØfnÑ−Ll x-A−r −R¢ca¡wn q−h
0

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

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π2 π4
21.
If Ι1 = ∫ f(sin2x).sinx dx, Ι2 = ∫ f(cos2x).cosx dx, then Ι1 : Ι2 =
0 0

π2 π4

¢c Ι1 = ∫ f(sin2x).sinx dx, Ι2 = ∫ f(cos2x).cosx dx, qu, a−h Ι1 : Ι2 =
0 0

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

22. Given P(x) = x 4 + ax 3 + bx 2 + cx + d such that x = 0 is the only real root of P′(x) = 0 . If
P(−1) < P(1) , then in the interval [ - 1, 1]
(A) P(−1) is the minimum but P(1) is not the maximum
(B) neither P(−1) is the minimum nor P(1) is the maximum
(C) P(−1) is the minimum and P(1) is the maximum
(D) P(−1) is not minimum but P(1) is the maximum

fËcš P(x) = x 4 + ax 3 + bx 2 + cx + d -Hhw P′(x) = 0 -Hl HLj¡œ h¡Ù¹h h£S qm x = 0z k¢c
P(−1) < P(1) qu, a−h A¿¹l¡m [ - 1, 1] -H

(A) P(−1) phÑ¢ejÀ ¢L¿¹¥ P(1) p−hÑ¡µQ eu
(B) P(−1) -J phÑ¢ejÀ eu Hhw P(1) J p−hÑ¡µQ eu
(C) P(−1) phÑ¢ejÀ J P(1) p−hÑ¡µQ j¡e
(D) P(−1) phÑ¢ejÀ eu ¢L¿¹¥ P(1) p−hÑ¡µQ
23. Suppose f(x) is differentiable for all x and f ′(x) ≤ 2 for all x. If f(1)=2 and f(4) =8, then f(2) =
j−e Ll pLm x-Hl SeÉ f(x) AhLme−k¡NÉ A−frL Hhw pLm x-Hl SeÉ f ′(x) ≤ 2 q−hz
k¢c f(1)=2 J f(4) =8 qu, a−h f(2) =
(A) 3 (B) 4 (C) 6 (D) 8

24. 3
2

The value of ∫
xsin πx dx is
−1

3
2

∫ xsin πx dx -Hl j¡e qm
−1

1  1  1 1  11 
(A) π  + 3  (B) π  − 3  (C)  + 3 (D)  − 3
 π   π  π π  π π 

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25. 3
The line y = mx bisects the area enclosed by the lines x = 0 , y = 0, x = and the curve
2
y = 1 + 4x − x 2 . Then m equals to

3
y = mx plm−lM¡ x = 0,y = 0,x = J hœ²−lM¡ y = 1 + 4x − x 2 -Hl à¡l¡ p£j¡hÜ −rœ−L ¢àd¡¢hiš²
2
L−lz −p−r−œ m-Hl j¡e q−h

15 14 13 16
(A) (B) (C) (D)
6 5 6 7
26.   π π  π
The area bounded by the curve y = tanx,  x ∈  − ,   the tangent to this curve at x =
  2 2  4
and x- axis in the first quadrant is
1 1 1
(A) log2 − square units (B) (A) log2 − 1 square units
2 4 2
1 1 1
(C) (A) log2 + square units (D) (A) log2 + 1 square units
2 4 2

 π π  π
hœ²−lM¡ y = tanx,  x ∈  − ,   Hhw x = ¢h¾c¥−a Eš² hœ²−lM¡l ØfnÑL J x-Ar à¡l¡ fËbj
  2 2  4
f¡−c p£j¡hÜ A’−ml −rœgm qm

1 1 1
(A) log2 − hNÑ HLL (B) (A) log2 − 1 hNÑ HLL
2 4 2

1 1 1
(C) (A) log2 + hNÑ HLL (D) (A) log2 + 1 hNÑ HLL
2 4 2

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n
27. x −[ x ]
Let [x] denote the greatest integer less than or equal to x. The value of ∫ [ x ] dx equals to
1

n x −[ x ]

[x], x-Hl −Q−u −R¡V h¡ pj¡e p−îÑ¡µQ f§ZÑpwMÉ¡ ¢e−cÑ¢na L−lz −p−r−œ ∫ [ x ] dx Hl j¡e q−h
1
n−1
(n − 1 ) (n − 1 )
n
23 22 34 33
(A) 1 + − + − +⋯ + −
loge 2 log e 2 log e 3 loge 3 log e (n − 1) loge (n − 1)
1 2 n−2
(B) 1 + + +⋯ +
log e 2 log e 3 log e (n − 1)

1 22 nn+1
(C) + +⋯ +
2 3 n+1
23 − 1 34 − 23 nn+1 − (n − 1)n
(D) + + ⋯⋯ +
3 4 n+1

28. Let Ι ( π ) = π3 , J( π ) = 3π . Then

j−e Ll Ι ( π ) = π3 , J( π ) = 3π z −p−r−œ
(A) Ι ( π ) > J( π ) (B) Ι ( π ) < J( π )
(C) Ι ( π ) = J( π ) (D) Ι ( π ) + J( π ) = 1
29. ax + b
Consider the function f(x) = , a,b,c,d ∈ ℝ & ad − bc ≠ 0 . Then
cx + d
(A) f(x) has maxima but no minima
(B) f(x) has minima but no maxima
(C) f(x) has both maxima & minima
(D) f(x) has neither maxima nor minima
ax + b
f(x) = , a,b,c,d ∈ ℝ J ad − bc ≠ 0 A−frL¢V ¢h−hQe¡ Ll : −p−r−œ
cx + d
(A) f(x) -Hl p−îÑ¡µQ j¡e B−R ¢L¿¹¥ phÑ¢ejÀ j¡e −eC
(B) f(x) -Hl phÑ¢ejÀ j¡e B−R ¢L¿¹¥ p−îÑ¡µQ j¡e −eC
(C) f(x) -Hl p−îÑ¡µQ J phÑ¢ejÀ Eiu j¡e-C B−R
(D) f(x) -Hl p−îÑ¡µQ j¡e-J −eC, phÑ¢ejÀ j¡e-J −eC

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30.  x2 
The domain of the function f ( x ) = sin  log2  is given by
−1

 2

 x2 
f ( x ) = sin−1  log2  A−fr−Ll p‘¡l A’m qm
 2
(A) [ 0, π] (B) [ −2, −1] ∪ [1,2] (C) [ −2, −1] (D) [1,2]

31. 1 dy
If 3f(x) + 2f   = x + 2 and y = xf(x) , then at x = - 1 equals to
x dx

k¢c 3f(x) + 2f   = x + 2 Hhw y = xf(x) qu, a−h x = - 1 ¢h¾c¥−a
1 dy
-Hl j¡e q−h
x dx
4 4 1 1
(A) − (B) (C) − (D)
5 5 5 5

32. Let f : ℝ → ℝ be such that f ( x ) − f ( y ) < x − y α where α is constant. Then

(A) f is derivable for all α (B) f is derivable for α > 1
(C) f is derivable for α < 1 (D) f is never differentiable.

j−e Ll f : ℝ → ℝ Hi¡−h pw”¡a B−R −k f ( x ) − f ( y ) < x − y α −kM¡−e α dË¥hLz −p−r−œ

(A) pLm α-Hl SeÉ f AhLme−k¡NÉ (B) pLm α > 1 -Hl SeÉ f AhLme−k¡NÉ
(C) pLm α < 1-Hl SeÉ f AhLme−k¡NÉ (D) f LM−e¡C AhLme−k¡NÉ eu

Let f : ℝ → ℝ not identically zero be such that f ( x + y 2n+1 ) = f(x) + [ f(y)]
2n +1
33. , n ∈ ℕ and x, y
are any two elements of domain of f and f ′ ( 0 ) ≥ 0 , then the value of f (5) is

j−e Ll f : ℝ → ℝ A−ic−k¡NÉ n§ZÉ eu Hhw Hje −k f ( x + y2n+1 ) = f(x) + [ f(y)]2n+1 , n ∈ ℕ J
f-Hl pw”¡l A’−ml A¿¹iѨš² −k−L¡e c¤¢V ¢h¾c¥ x J y Hhw f ′ ( 0 ) ≥ 0 z −p−r−œ f (5) -Hl j¡e
qm
(A) 0 (B) 1 (C) 3 (D) 5

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34. If α, β are roots of the quadratic equation ax 2 + bx + c = 0 (a,b,c ∈ ℝ)

1 − cos(ax 2 + bx + c)
then lim is
x →α (x − α)2
k¢c α, β ¢àO¡a pj£LlZ ax 2 + bx + c = 0 (a,b,c ∈ ℝ) -Hl h£S qu,
1 − cos(ax 2 + bx + c)
a−h lim q−h
x →α (x − α)2
a a2 a2
(B) ( α − β ) (C) − ( α − β ) (D) ( α − β )
2 2 2
(A) 0
2 2 2

35. Let f be twice differentiable function with f(0) = f(1) = f ′(0) = 0 , Then
(A) f ′′(x) is zero function. (B) f ′′(x) = 0
(C) f ′′(x) = 0 for some x ∈ ( 0,1 ) (D) f ′′(x) never vanishes

j−e Ll f A−frL¢V Hje −k f ′′ -Hl A¢Ù¹aÆ B−R Hhw f(0) = f(1) = f ′(0) = 0 quz a−h

(A) f ′′(x) n§ZÉ -A−frL (B) f ′′(x) = 0
(C) Hje ¢h¾c¥ x ∈ ( 0,1 ) f¡Ju¡ k¡−h k¡l SeÉ f ′′(x) = 0 q−h (D) f ′′(x) LM−e¡C n§ZÉ q−h e¡
36. Given log10 3 = 0.4771, log10 e = 0.4343 , the value of log10 30.5 , correct upto 3 places of
decimals is
−cJu¡ B−R log10 3 = 0.4771, log10 e = 0.4343 z −p−r−œ log10 30.5 -Hl j¡e ¢ae cn¢jL ÙÛ¡e
fkÑ¿¹ q−h

(A) 0.272 (B) 1.484 (C) 1.615 (D) 1.027

37. Consider the equation 2x + 5x = 3x + 4 x . Then
(A) the equation has only one solution in Z, the set of integers
(B) the equation has infinitely many solutions in Z
(C) the equation has exactly two solutions & both are in Z
(D) the equation has no solution in ℝ

2x + 5x = 3x + 4 x pj£LlZ¢Vl
(A) f§ZÑpwMÉ¡l −pV Z-l j¡œ HL¢V pj¡d¡e B−R
(B) Z-H Ap£j pwMÉL pj¡d¡e ¢hcÉj¡e
(C) pj£LlZ¢Vl j¡œ c¤¢V pj¡d¡e B−R Hhw Ei−uC Z-H A¿¹iѨš²
(D) ℝ -H pj£LlZ¢Vl −L¡e pj¡d¡e −eCz

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38. The equation 6x 2 + 4xy + 3y 2 + 8x − 2y + 4 = 0 is transformed into the form
Ax 2 + 2Hxy + By 2 = 1 by referring to parallel axes through a properly chosen point. The
point is
h¡R¡C Ll¡ ¢h¾c¥ ¢c−u pj¡¿¹l¡m Arà−ul p¡−f−r 6x2 + 4xy + 3y2 + 8x − 2y + 4 = 0 pj£LlZ¢V
Ax 2 + 2Hxy + By 2 = 1 BL¡−l f¢lh¢aÑa quz I ¢h¾c¥¢V qm

(A) ( −1,2 ) (B) ( −1,1 ) (C) (1, −2 ) (D) ( 2,0 )

39. A circle x 2 + y 2 + 2g1x + 2f1 y + c1 = 0 will bisect the circumference of the circle
2x 2 + 2y 2 + 4g 2x + 4f2 y + 2c2 = 0 is

hªš x2 + y2 + 2g1x + 2f1 y + c1 = 0 Bl HL¢V hªš 2x2 + 2y2 + 4g2x + 4f2 y + 2c2 = 0 -Hl f¢l¢d−L
¢àO¢äa L−lz −p−r−œ

(A) 2g1 ( g1 − g2 ) + 2f2 ( f1 − f2 ) = c1 − c2 (B) 2g2 ( g1 − g2 ) + 2f1 ( f1 − f2 ) = c1 − c2
(C) 2g1 ( g1 − g2 ) + 2f1 ( f1 − f2 ) = c2 − c1 (D) 2g2 ( g1 − g2 ) + 2f2 ( f1 − f2 ) = c1 − c2

40. MN is a chord of the parabola y 2 = 4ax, a > 0 , with vertex M. NP is drawn perpendicular to
MN meeting the axis at P. The projection of NP on the axis of the parabola is
A¢dhªš y 2 = 4ax, a > 0 -Hl MN HL¢V SÉ¡, M n£oÑ¢h¾c¥z NP , MN −lM¡l Efl mð J Ar−L P
¢h¾c¥−a −Rc L−lz A¢dhª−šl A−rl Efl NP-Hl fË−rf qm

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

41. Let ℓ be the length of focal chord of a parabola y 2 = 4ax, (a > 0) & let d be its distance
from the vertex. Then
A¢dhªš y 2 = 4ax, (a > 0) -Hl e¡¢iN¡j£ SÉ¡-Hl °cOÑÉ ℓ HLLz j−e Ll n£oÑ ¢h¾c¥ −b−L I SÉ¡-Hl
c§laÆ d HLLz −p−r−œ
1 1
(A) ℓ ∞ d2 (B) ℓ ∞ d (C) ℓ ∞ (D) ℓ ∞
d2 d

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42. A triangle ABC has a fixed base BC. If AB : AC = 1 : 2 , then locus of vertex A is
(A) a circle whose center is mid-point of BC
(B) a circle whose center is on BC but not the mid-point of BC
(C) a straight line
(D) a parabola

BC q’m ¢œi¥S ABC-Hl ¢e¢ŸÑø i¨¢jz k¢c AB : AC = 1 : 2 qu, a−h n£oÑ¢h¾c¥ A-Hl p’¡lfb q−h

(A) HL¢V hªš k¡l −L¾cÊ BC-Hl jdÉ¢h¾c¥
(B) HL¢V hªš k¡l −L¾cÊ BC-Hl Efl Ah¢ÙÛa ¢L¿¹¥ BC-Hl jdÉ¢h¾c¥ eu
(C) HL¢V plm−lM¡
(D) HL¢V A¢dhªš
43. The differential equation of the family of ellipse with the axes along x- axis and y-axis is
x-Ar J y-Ar àu−L Efhª−šl Aràu ¢h−hQe¡ L−l Efhªš-f¢lh¡−ll AhLm pj£LlZ q−h

2 2
d2 y  dy  dy d2 y  dy  dy
(A) xy 2 + x   − y = 0 (B) xy 2 + y   − x = 0
dx  dx  dx dx  dx  dx
2 2
d2 y  dy  dy d2 y  dy  dy
(C) xy 2 − x   + y = 0 (D) xy 2 − y   + x = 0
dx  dx  dx dx  dx  dx

44. PQRS is a rectangle whose sides are parallel to fixed directions. P lies on x-axis while Q and
S lie on the lines x = a and x = - a respectively. The locus of R is
(A) circle (B) straight line (C) pair of straight lines (D) parabola

PQRS HL¢V Bua−rœ k¡l h¡ý…¢m ¢e¢ŸÑø ¢c−Ll p−‰y pj¡¿¹l¡mz P, x-A−rl Efl Ah¢ÙÛa Hhw
Q J S kb¡œ²−j x = a J x = - a-Hl Efl Ah¢ÙÛaz R-Hl p’¡lfb q−h

(A) hªš (B) plm−lM¡ (C) plm−lM¡ k¤Nm (D) A¢dhªš
45. Let P be a point on the hyperbola x 2 − y 2 = a2 where a is a parameter such that P is nearest
to the line y = 2x. The locus of P is
j−e Ll P ¢h¾c¥¢V fl¡hªš x2 − y2 = a2 -Hl Efl Ah¢ÙÛa, −kM¡−e ‘a’HL¢V fËQmz P ¢h¾c¥¢V y = 2x
−lM¡l ¢eLVajz P-Hl p’¡lfb q−h

(A) 2x − y = 0 (B) 2y − x = 0 (C) 2x + y = 0 (D) x + 2y = 0

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46. Equation of the plane through the point (1,2, −3 ) & normal to the straight line joining the
points ( −1,3,4 ) & ( 5,2, −1 ) is

(1,2, −3 ) ¢h¾c¥N¡j£ Hhw ( −1,3,4 ) J ( 5,2, −1) ¢h¾c¥à−ul pw−k¡SL plm−lM¡l Efl A¢imð
a−ml pj£LlZ

(A) 3x − 2y − 5z − 19 = 0 (B) 6x − y − 5z − 19 = 0
(C) x − 2y − 3z − 17 = 0 (D) 6x − y + 5z − 19 = 0

47. Let A,B,C be three vectors in ℝ3 such that (B − A ) X ( C − A ) = O . Then

(A) A,B,C lie on a line (B) A,B,C do not lie on a line
(C) A,B,C are on a circle (D) Each is equidistant from the origin.

j−e Ll ℝ3 -−a A,B,C Hje −k (B − A ) X ( C − A ) = O z −p−r−œ
(A) A,B,C pj−lM q−h (B) A,B,C pj−lM eu
(C) A,B,C HL¢V hª−šl Ef−l Ah¢ÙÛa (D)fË¢a¢V j§m¢h¾c¥ −b−L pjc§lhaÑ£z
48. The unit vector which is orthogonal to the vector 2 ˆi − ˆj + 2 kˆ and is coplanar with the
vectors ˆi + ˆj − kˆ and 2 ˆi + ˆj − kˆ is

HL¢V HLL (unit) −iƒl 2 ˆi − ˆj + 2 kˆ −iƒ−ll Efl mð Hhw 2 ˆi + ˆj − kˆ J ˆi + ˆj − kˆ −iƒlà−ul
p−‰y HLC a−m B−Rz −iƒl¢V q−h

3 ˆi + 2 ˆj − 2 kˆ 3 ˆi − 2 ˆj − 2 kˆ 3 ˆi + 2 ˆj + 2 kˆ 3 ˆi − 2 ˆj + 2 kˆ
(A ) − (B) (C) (D)
17 17 17 17

49. A curve passing through the point (3, 2) for which the segment of any tangent line
contained between the co-ordinate axes is bisected at the point of tangency is given by
HL¢V hœ²−lM¡ (3, 2) ¢h¾c¥N¡j£ Hhw I hœ²−lM¡l −k−L¡e ØfnÑL Arà−ul j−dÉ Eš² ØfnÑ ¢h¾c¥−a
pj¢àM¢äa quz hœ²−lM¡¢Vl pj£LlZ qm
x 2 y2
(A) y 2 = 2cx (B) x 2 = 2cy (C) xy = 6 (D) + =1
2 3
( c is arbitrary constant) ( c HL¢V kcªµR dË¥hL )

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50.  13 
If cot  sin−1  = sin ( tan θ ) , then Ѳ is
−1

 17 
 13 
k¢c cot  sin−1  = sin ( tan θ ) qu, a−h Ѳ q−h
−1

 17 
2 13 2 2
(A) (B) (C) (D)
17 17 13 3

QP-PUBDET-2018-Math FINAL.docx Page: 18/19

PUBDET-2018
Subject: Mathematics

pju: 90 ¢j¢eV phÑ¡¢dL eðl: 100

¢e−cÑn¡hm£
1. HC fËnÀf−œl ph fËnÀC Ah−S¢ƒi fËnÀ Hhw fË¢a¢V fË−nÀl Q¡l¢V pñ¡hÉ Ešl −cJu¡
B−R k¡l HL¢V j¡œ p¢WLz p¢WL Ešl ¢c−m 2 eðl f¡−hz i¥m Ešl ¢c−m Abh¡
HL¡¢dL Ešl ¢c−m ½ eðl L¡V¡ k¡−hz
2. OMR f−œ A,B,C,D ¢Q¢q²a p¢WL Ol¢V il¡V L−l Ešl ¢c−a q−hz
3. OMR f−œ Ešl ¢c−a öd¤j¡œ L¡−m¡ h¡ e£m hm f−u¾V −fe hÉ¡hq¡l Ll−hz
4. OMR f−œ ¢e¢cÑø ÙÛ¡e R¡s¡ AeÉ −L¡b¡J −L¡e c¡N −c−h e¡z
5. OMR f−œ ¢e¢cÑø ÙÛ¡−e fËnÀf−œl eðl Hhw ¢e−Sl −l¡m eðl A¢a p¡hd¡ea¡l p¡−b
¢mM−a q−h Hhw fË−u¡Se£u Ol…¢m f§lZ Ll−a q−hz
6. OMR f−œ ¢e¢cÑø ÙÛ¡−e ¢e−Sl e¡j J fl£r¡ −L−¾cÊl e¡j ¢mM−a q−h Hhw ¢e−Sl pÇf¨ZÑ
p¡rl ¢c−a q−hz
7. OMR Ešlfœ¢V C−mLVÊ¢eL k−¿»l p¡q¡−kÉ fs¡ q−hz p¤a
¤ l¡w fËnÀ−fœl eðl h¡ −l¡m
eðl i¥m ¢mM−m Abh¡ i¥m Ol il¡V Ll−m Ešlfœ¢V A¢eh¡kÑ L¡l−Z h¡¢am q−a
f¡−lz HR¡s¡ fl£r¡bÑ£l e¡j, fl£r¡ −L−¾cÊl e¡j h¡ p¡r−l −L¡e i¥m b¡L−mJ Ešl fœ
h¡¢am q−u −k−a f¡−lz OMR Ešlfœ¢V i¡yS q−m h¡ a¡−a Ae¡hnÉL c¡N fs−mJ
h¡¢am q−u −k−a f¡−lz fl£r¡bÑ£l HC dl−el i¥m h¡ ApaÑLa¡l SeÉ Ešlfœ h¡¢am
q−m HLj¡œ fl£r¡bÑ£ ¢e−SC a¡l SeÉ c¡u£ b¡L−hz
8. −j¡h¡Cm−g¡e, LÉ¡mL¥−mVl, pÔ¡CXl¦m, mN−Vhm, −lM¡¢Qœ, NË¡g h¡ −L¡e dl−Zl a¡¢mL¡
fl£r¡ L−r Be¡ k¡−h e¡z Be−m −p¢V h¡−Su¡ç q−h Hhw fl£r¡bÑ£l JC fl£r¡
h¡¢am Ll¡ q−hz
9. fËnÀf−œl −n−o l¡g L¡S Ll¡l SeÉ gy¡L¡ S¡uN¡ −cJu¡ B−Rz AeÉ −L¡e L¡NS HC
L¡−S hÉhq¡l Ll−h e¡z
10. fl£r¡ Lr R¡s¡l B−N OMR fœ AhnÉ C f¢lcnÑL−L ¢c−u k¡−hz
11. HC fËnÀf−œ Cwl¡S£ J h¡wm¡ Eiu i¡o¡−aC fËnÀ −cJu¡ B−Rz h¡wm¡ j¡dÉ−j fËnÀ °al£l
pju fË−u¡Se£u p¡hd¡ea¡ J paLÑa¡ Ahmðe Ll¡ q−u−Rz a¡ p−šÄJ k¢c −L¡e Ap‰¢a
mr Ll¡ k¡u, −p−r−œ Cwl¡S£ j¡dÉ−j −cJu¡ fËnÀ ¢WL J Q¥s¡¿¹ h−m ¢h−h¢Qa q−hz

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