Bihar Secondary Teacher 2023: TRE 2.0, Class 9 to 10, Mathematics, re-examination of 18-12
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23/MAV–AC/M–2023–09 Booklet Series
Serial No.
Candidate’s Roll Number
Question Booklet
A
LANGUAGE, GENERAL STUDIES AND MATHEMATICS
Time Allowed : 2:30 Hours Maximum Marks : 150
Read the following instructions carefully before you begin to answer the questions.
IMPORTANT INSTRUCTIONS
1. This Question Booklet is divided into three Parts—Part–I, Part–II and Part–III. Part–I contains
questions of Language (Qualifying), Part–II contains questions of General Studies and Part–III
contains questions of Mathematics.
2. Part–I consists of Question Nos. 1 to 30, Part–II consists of Question Nos. 31 to 70 and Part–III
consists of Question Nos. 71 to 150. The questions and their responses are printed in English
and Hindi versions both of Part–II and Part–III.
3. All questions carry equal marks.
4. Immediately after commencement of the examination, you should check up your Question
Booklet and ensure that the Question Booklet Series is printed on the top right-hand corner
of the Booklet. Please check that the Booklet contains 56 printed pages including two pages
(Page Nos. 54 and 55) for Rough Work and no page or question is missing or unprinted or
torn or repeated. If you find any defect in this Booklet, get it replaced immediately by
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5. You must write your Roll Number in the space provided on the top of this page. Do not write
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PART—I
( LANGUAGE )
Direction (Q. Nos. 1 to 5) : Read the 4. What does the lion see in the
passage given below and choose the well?
correct answer for the questions that (A) Another lion
follow : (B) Its own reflection
A lion lived in a jungle and he killed (C) The rabbit
many animals everyday for his meal. (D) More than one of the above
All the animals of the jungle were (E) None of the above
worried about this and they go to the
lion to discuss the issue. The lion 5. What do you learn from this
agreed not to kill the animals if one story?
animal was sent to him everyday for his (A) To kill animals
meal. This goes on for many days. One (B) Wisdom is more strong
day, a rabbit goes to the lion. The (C) Small rabbit cannot win a
rabbit sees an old well and it makes a lion
plan to kill the lion. He tells the lion (D) More than one of the above
that there is another lion in the well
(E) None of the above
who is stronger and more powerful. The
lion then asks the rabbit to take him to Direction (Q. Nos. 6 and 7) : Choose
meet the lion in the well. the lion jumps the most appropriate prepositions in
into the well to see where the other lion the following sentences :
lives. The lion mistakes his own
6. Take the curtain _____ their
reflection in the water and takes it for
hooks.
the other lion and dies.
(A) from
1. Who were worried? (B) for
(A) Lion
(C) behind
(B) All the animals
(C) All the birds (D) More than one of the above
(D) More than one of the above (E) None of the above
(E) None of the above 7. In compliance _____ the office
2. Who makes a plan to kill the order.
lion? (A) with
(A) All the animals (B) of
(B) The lion in the well (C) for
(C) Rabbit (D) More than one of the above
(D) More than one of the above (E) None of the above
(E) None of the above
8. Paternal family belongs to _____
3. Who is more strong and side.
powerful?
(A) father
(A) All the animals
(B) The lion (B) mother
(C) The rabbit (C) grandson
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 2
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PART—II
( GENERAL STUDIES )
31. Given 168 ´ 32 = 5376, then 33. If 5( x + 3) = 25( 3x - 4), then the
5 × 376 ¸ 16 × 8 is equal to value of x is
5
(A) 0·032 (A)
11
(B) 0·32
11
(B)
5
(C) 3·2
11
(C)
(D) More than one of the above 3
(E) None of the above (D) More than one of the above
(E) None of the above
32. Which one of the following
numbers has rational square
root? 34. In a college, the ratio of the
number of boys to girls is 8 : 5.
If there are 160 girls, the total
(A) 0·4 number of students in the
college is
(B) 0·09 (A) 250
(B) 100
(C) 0·9
(C) 416
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 6
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35. How long will it take a sum 37. Which of the following
of money invested at 5% p.a. statements is true?
simple interest to increase its
value by 40%?
(A) Every rational number is
an integer.
(A) 8 years
(B) Every natural number is
(B) 5 years
a rational number.
(C) 7 years
(C) Every whole number is
a natural number.
(D) More than one of the above
(D) More than one of the above
(E) None of the above
(E) None of the above
36. The ratio of women and men
employees of an organization
is 5 : 7. If the number of women
employees is 50 less than 38. If the sum of the cubes of first
the men employees, then the n natural numbers is 2025,
number of women and men then what is the value of n?
employees, respectively, are
(A) 7
(A) 125, 175
(B) 12
(B) 150, 200
(C) 9
(C) 135, 185
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 8
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39. The speed of an electron in the 43. Endolymph is a fluid present in
orbit of the hydrogen atom in (A) Ascaris
the ground state is
(B) eye
C
(A)
1 × 37 (C) internal ear
C (D) More than one of the above
(B)
1370 (E) None of the above
C
(C) 44. At what temperature, a ferro-
137
magnetic material becomes
(D) More than one of the above paramagnetic?
(E) None of the above (A) Melting temperature
40. The aqueous solution whose pH (B) Curie temperature
is zero will be (C) Néel temperature
(A) acidic (D) More than one of the above
(B) amphoteric (E) None of the above
(C) alkaline
45. Which of the following is
(D) More than one of the above
a positively charged particle
(E) None of the above emitted by a radioactive
element?
41. Which of the following
interactions is weakest? (A) Beta ray
(A) Electromagnetic (B) Alpha ray
(B) Electrostatic (C) Cathode ray
(C) Gravitational (D) More than one of the above
(D) More than one of the above (E) None of the above
(E) None of the above
46. Stomata are tiny pores on the
42. Which of the following is an surface of leaves through which
insectivorous plant? plants take in
(A) Cuscuta (A) nitrogen
(B) Orobanche (B) carbon dioxide
(C) Drosera (C) water
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
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47. Where is the Headquarters of 51. Pyaar Ka Paudha Campaign was
the Integrated Defence Staff launched by Bihar Government
located in India? on
(A) New Delhi (A) 15th February, 2021
(B) Bengaluru (B) 26th January, 2022
(C) Chennai (C) 14th February, 2020
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
48. Where is the International Court 52. Which district of Bihar has the
of Justice (ICJ) located? lowest literacy rate?
(A) Hague (A) Araria
(B) New Delhi (B) Purnia
(C) Nawada
(C) Paris
(D) More than one of the above
(D) More than one of the above
(E) None of the above
(E) None of the above
53. What was the distance covered
49. Which city/district in Bihar
in Javelin throw by Neeraj
is famous for silk textile
Chopra in Tokyo Olympics?
production?
(A) 87·58 metres
(A) Bhagalpur
(B) 87·85 metres
(B) Patna
(C) 88·38 metres
(C) Champaran
(D) More than one of the above
(D) More than one of the above
(E) None of the above
(E) None of the above
54. Who among the following is
50. The World Earth Day is observed the Chairman of Cotton Council
every year on of India?
(A) 20th April (A) Shyam Makharia
(B) 22nd April (B) Suresh Amritlal Kotak
(C) 21st April (C) Kishore Mirani
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 12
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55. Ganga Ka Maidan is full of 57. Which of the following places
which type of soil? has the highest rainfall in
Bihar?
(A) Black soil
(A) Patna
(B) Red soil
(B) Kishangarh
(C) Alluvium soil
(C) Muzaffarpur
(D) More than one of the above
(D) More than one of the above
(E) None of the above
(E) None of the above
56. What is the height of
58. Where does the Chenab River
Someshwar peak of Bihar?
finally join the Indus River?
(A) 875 metres (A) Mithankot
(B) 850 metres (B) Tandi
(C) 825 metres (C) Ahmedpur Sial
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 14
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59. The Melghat Tiger Reserve is 61. Arrange the following districts
associated with of Bihar from North to South :
1. Sheohar
(A) Uttarakhand
2. Vaishali
(B) Maharashtra 3. Nalanda
4. Nawada
(C) West Bengal
Select the correct answer using
the codes given below.
(D) More than one of the above
(A) 3, 2, 1, 4
(E) None of the above (B) 1, 2, 3, 4
(C) 2, 3, 1, 4
60. Barabar Hills are situated in the (D) More than one of the above
vicinity of which city of Bihar?
(E) None of the above
(A) Makhdumpur
62. Tribeni Canal in West
Champaran obtains its water
supply from
(B) Munger
(A) Bhainsa Lotan
(C) Madhubani (B) Khaira Tola
(C) Thakrahan
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 16
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63. In which year was the ‘Madras 67. Who formed a Congress Ministry
Mahajan Sabha’ formed? in Bihar as Premier on 20 July,
1937?
(A) 1879
(A) Prof. Abdul Bari
(B) 1881
(B) Anugrah Narayan Sinha
(C) 1883
(C) Shri Krishna Sinha
(D) More than one of the above
(D) More than one of the above
(E) None of the above
(E) None of the above
64. Who began the ‘Self-Respect 68. With which of the following
Movement’ in India? places is Babu Kunwar Singh
(A) Jyotiba Phule associated?
(B) Narayan Swami (A) Arrah
(C) B. R. Ambedkar (B) Bhagalpur
(D) More than one of the above (C) Patna
(D) More than one of the above
(E) None of the above
(E) None of the above
65. Who formulated the powerful
slogan ‘India for Indians’ ? 69. Who established the Arya
Samaj?
(A) Raja Ram Mohan Roy
(A) Swami Vivekananda
(B) Swami Dayananda
(B) Swami Dayananda
Saraswati
Saraswati
(C) Swami Vivekananda
(C) Aurobindo Ghose
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
66. In which year did ‘All India 70. Where was the First Session of
Muslim League’ table the the Indian National Congress
demand of Separate Pakistan? held?
(A) 22 December, 1939 (A) Calcutta
(B) 23 March, 1940 (B) Bombay
(C) 8 August, 1940 (C) Madras
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 18
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PART—III
( MATHEMATICS )
71. Let D be a subset of real line. ¥ 1
74. Let a = ò dt
Consider the assertion : “Every 0 1+ t 2
infinite sequence in D has a Which of the following is true?
subsequence which converges in da 1
D.” This assertion is true if (A) =
dt 1 + t 2
(A) D = [0, ¥)
(B) a is a rational number
(B) D = [0,1] È [3, 4]
(C) sin(a ) = 1
(C) D = [ -1,1) È (1, 2]
(D) More than one of the above (D) More than one of the above
(E) None of the above
(E) None of the above
72. Evaluate :
1
n
n 75. Define f (x ) = for x > 0. Then
lim å x
2
n ®¥ k=0k +n2 f is uniformly continuous
p (A) on (0, ¥)
(A)
2 (B) on [r, ¥) for any r > 0
(B) p (C) on (0, r ] for any r > 0
p (D) More than one of the above
(C)
4
(E) None of the above
(D) More than one of the above
(E) None of the above 76. Let
nx
gn (x ) = , x Î [0, ¥)
73. If { x n } is a convergent sequence 1 + n 2x 2
in r and { y n } is a bounded
sequence in r, then we can Which of the following is true as
conclude that n ® ¥?
(A) { x n + y n } is convergent (A) gn ® 0 pointwise but not
uniformly
(B) { x n + y n } is bounded
(B) gn ® 0 uniformly
(C) { x n + y n } has no convergent
subsequence (C) gn (x ) ® x "x Î [0, ¥)
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 20
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77. Consider the vector space Pn of 79. Let A be a real matrix with
real polynomial in x of degree characteristic polynomial
less than or equal to n. Defined 3
(X - 1) . Pick the correct
T : P2 ® P3 by statement from below :
x
(Tf )(x ) = ò f (t)dt + f ¢(x )
0
(A) A is necessarily
Then the matrix representation diagonalizable
of T with respect to the bases
{ 1, x, x 2 } and { 1, x, x 2, x 3 } is
(B) If the minimal polynomial of
æ0 1 0 0 ö A is (X - 1) 3 , then A is
ç ÷
(A) ç 1 0 12 0 ÷ diagonalizable
ç0 2 0 1 ÷
è 3ø
(C) Characteristic polynomial of
æ0 1 0ö A 2 is (X - 1) 3
ç ÷
1 0 2÷
(B) ç 1 0÷
ç0 2
ç ÷ (D) More than one of the above
è0 0 13 ø
æ0 1 0 0 ö (E) None of the above
ç ÷
(C) ç 1 0 2 0 ÷
ç0 1 0 1 ÷
è 2 3ø
80. Let T : rn ® rn be a linear map
(D) More than one of the above that satisfies T 2 = T - I n . Then
(E) None of the above which of the following are true?
é 0 1ù
78. Let A=ê ú. Then the (A) T is invertible
ë -1 1û
smallest positive integer n such
that A n = I is (B) T - I n is not invertible
(A) 1
(C) T has a real eigenvalue
(B) 2
(C) 6
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 22
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81. The trace of the matrix 83. Let V be an inner product space
and let v1, v 2, v 3 Î V be an
20 orthogonal set of vectors. Which
æ 2 1 0ö
ç ÷ of the following statements is
ç 0 2 0÷ true?
ç ÷
è 0 0 3ø
(A) The vectors 5v1 + 6v 2 + 7v 3 ,
is 5v 2 + v 3 , v 2 + 3v 3 can be
extended to a basis of V .
(A) 7 20
(B) The vectors 4v1 + v 2 + 2v 3 ,
v 2 + 3v 3 , v 2 + 3v 3 can be
(B) 2 20 + 3 20 extended to an orthogonal
basis of V .
(C) 2 × 2 20 + 3 20
(C) The vectors v1 + v 2 + 2v 3 ,
v 2 + v 3 , 2v1 + v 2 + 3v 3 can be
(D) More than one of the above extended to a basis of V .
(D) More than one of the above
(E) None of the above
(E) None of the above
82. Which of the following real
quadratic forms on r 2 is positive
definite? 84. The number of group homo-
morphisms from z10 to z20 is
(A) Q (X ,Y ) = XY
(A) zero
(B) Q (X ,Y ) = X 2 + 2XY + Y 2 (B) one
(C) Q (X ,Y ) = X 2 - XY + Y 2 (C) ten
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 24
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85. Let S 7 denotes the group of 87. Let p ³ 5 be a prime. Then
permutations of the set
{ 1, 2, 3, 4,5, 6, 7} . Which of the (A) fp ´ fp has at least five
following is true? subgroups of order p
(B) every subgroup of fp ´ fp is
(A) There are no elements of of the form H1 ´ H 2 where
order 6 in S 7 H1, H 2 are subgroups of fp
(C) every subgroup of fp ´ fp is
(B) There are no elements of an ideal of the ring fp ´ fp
order 8 in S 7
(D) More than one of the above
(C) There are no elements of (E) None of the above
order 7 in S 7
88. Any group of order 25 is
(D) More than one of the above
(A) cyclic
(B) simple
(E) None of the above
(C) Abelian but not simple
(D) More than one of the above
86. Let p be a prime number. How
many distinct sub-rings (with (E) None of the above
unity) of cardinality p does the
field F p 2 have?
89. If R is a ring with unity a, b Î R
such that ab = 1, then
(A) 0
(A) ba is an idempotent in R but
(1 - ba) is not
(B) 1
(B) both ba and (1 - ba) are
idempotent elements in R
(C) p
(C) neither ba nor (1 - ba) is an
idempotent element in R
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 26
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90. Let G be a commutative group of 93. For the function
order 202. Then the number of
1 - e -z
element(s) of order 2 in G is f (z ) =
z
(A) 1 the point z = 0 is a/an
(A) essential singularity
(B) 2
(B) pole of order one
(C) 101 (C) removable singularity
(D) More than one of the above (D) More than one of the above
(E) None of the above
(E) None of the above
91. The class equation of a group of 94. Let g be any positively oriented
order 10 is circle whose centre is at the
origin. Then the value of the
(A) 1 + 1 + LLL+ (
110 times) = 10 cosz
integral ò dz is
g z2
(B) 1 + 1 + 3 + 5 = 10 (A) 2pi
(C) 1 + 2 + 2 + 5 = 10 (B) 0
(C) -2pi
(D) More than one of the above
(D) More than one of the above
(E) None of the above
(E) None of the above
92. The function f (z ) =|z|2 is 1
95. Let a-2 be the coefficient of 2 in
z
(A) continuous at the origin
the Laurent series of
only z +1
f (z ) = around z = 0 in
(B) continuous everywhere but z 3 (z 2 + 1)
nowhere differentiable the region 0 <|z|< 1. Then a-2 is
equal to
(C) continuous everywhere but (A) 1
nowhere differentiable,
except at the origin (B) 0
(C) -1
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 28
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96. Let f : c ® c be an entire 99. For any positive integers a and
function with f (0) = 1 and f (1) = 0. 3, there exist unique integers q
Then f maps and r such that a = 3q + r , where
r must satisfy
(A) open sets in c into open
sets in c (A) 1 < r < 3
(B) closed sets in c into closed (B) 0 < r < 3
sets in c
(C) 0 £ r < 3
(C) open sets in c into closed
sets in c (D) More than one of the above
(D) More than one of the above (E) None of the above
(E) None of the above
100. Consider the congruence
x n º 2(mod13). This congruence
97. A non-constant entire function
has a solution for x if
(A) has at least one zero in c
(A) n = 8
(B) cannot have finite number
of real zeros (B) n = 7
(C) cannot have uncountable (C) n = 6
number of zeros in c
(D) More than one of the above
(D) More than one of the above
(E) None of the above
(E) None of the above
98. f is an entire function. If f 101. Which of the following
satisfies the following two statements is true?
equations f (z + 1) = f (z ) and
(A) Every even integer n ³ 16
f (z + i) = f (z ) for every z in c then
divides (n - 1)!+ 3
(A) f º constant
(B) Every odd integer n ³ 16
(B) f (z ) Î r " z divides (n - 1)!
(C) f is a non-constant (C) Every even integer n ³ 16
polynomial divides (n - 1)!
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 30
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102. Let p be a prime number and 104. Let a, b be positive integers with
N p be the number of pairs of a > b and a + b = 24. Suppose
positive integers (x, y) such that that the following congruences
have a common integer
1 1 1 solution :
+ =
x y p4
2x º 3a(mod 5);x º 4b(mod 5)
Which among the following are
possible values of N p ? Which of the following
statements is true?
(A) 0
(A) a - b is divisible by 5
(B) 4
(B) 3b > a > 2b
(C) 9
(C) a > 3b
(D) More than one of the above
(D) More than one of the above
(E) None of the above
(E) None of the above
103. Consider the function
5 3
f (n ) = n - 2n + n , where n is a 105. For a Î r, let ëa û denotes the
greatest integer smaller than
positive integer. Which of the
or equal to a. Define
following statements is not
d : r ´ r ® [0, ¥) by
true?
d (x, y) = ë|x - y|û , x, y Î r
(A) For every positive integer k,
there exists a positive Then which of the following is
integer n such that f (n ) is true?
divisible by 2k .
(A) d (x, y) = 0 if and only if x = y,
(B) f (n ) is even for every integer x, y Î r
n ³ 20.
(B) d (x, y) = d (y, x ), x, y Î r
(C) For every integer n ³ 20,
either f (n ) is odd or f (n ) is (C) d (x, y) £ d (x, z ) + d (z, y),
divisible by 4. x, y, z Î r
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 32
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106. Let X be a countable set. Then 108. The equation
which of the following is not
(axy 3 + y cos x )dx +
true?
(x 2y 2 + b sin x )dy = 0
(A) There exists a metric d on X is exact for
such that (X , d ) is complete 2
(A) a = , b =1
3
(B) There exists a metric d on X
3
such that (X , d ) is not (B) a = 1, b =
complete 2
2
(C) a = 1, b =
(C) There exists a metric d on X 3
such that (X , d ) is compact
(D) More than one of the above
(D) More than one of the above (E) None of the above
109. An integrating factor of
(E) None of the above
dy
x + (3x + 1)y = xe -2x
dx
107. Let (X , d ) be a metric space and is
let A Í X . For x Î X , defined (A) xe 3x
d (x, A) = inf{ d (x, a):a Î A } (B) 3xe x
(C) xe x
If d (x, A) = 0 for all x Î X , then
which of the following assertions (D) More than one of the above
must be true?
(E) None of the above
(A) A is compact 110. The differential equation
representing the family of circles
(B) A is closed touching y-axis at the origin is
(A) linear and of first order
(C) A is dense in X
(B) linear and of second order
(C) non-linear and of first order
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 34
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111. Let, f , g :[ -1,1] ® r, f (x ) = x 3 , 113. The integral surface satisfying
the equation
g(x ) = x 2|x|. Then
¶z ¶z
y +x = x 2 + y2
(A) f and g are linearly ¶x ¶y
independent on [ -1,1] and passing through the curve
x = 1 - t, y = 1 + t, z = 1 + t 2 is
(B) f and g are linearly
dependent on [ -1,1] 1 2
(A) z = xy + (x - y 2 ) 2
2
(C) f (x ) g ¢(x ) - f ¢(x ) g(x ) is not
1 2
identically zero on [ -1,1] (B) z = xy + (x - y 2 ) 2
4
(D) More than one of the above
1 2
(C) z = xy + (x - y 2 ) 2
8
(E) None of the above
(D) More than one of the above
112. Let y be the solution of the (E) None of the above
initial value problem
114. Let the general integral of the
d 2y partial differential equation
+ y = 6 cos 2x, y(0) = 3, y ¢(0) = 1
dx 2 ¶z ¶z
(2xy - 1) + (z - 2x 2 ) = 2(x - yz )
¶x ¶y
Let the Laplace transform of y be
F (s). Then, the value of F (1) is be given by F (u, v) = 0, where
F : r 2 ® r is a continuously
17 differentiable function.
(A)
5 ( r is the set of all real numbers
and r 2 = {(x, y):x, y Î r})
13 Then which of the following is
(B) true?
5
(A) u = x 2 + y 2 + z, v = xz + y
11
(C) (B) u = x 2 + y 2 - z, v = xz - y
5
(C) u = x 2 - y 2 + z, v = yz + x
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 36
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115. The partial differential equation 117. Let f:[0,1] ® r be three times
continuously differentiable.
¶ 2u ¶ 2u ¶ 2u Suppose that the iterates
x + (x - y) -y +
¶x 2 ¶x¶y ¶y 2 defined by x n +1 = f(x n ), n ³ 0
1 æ ¶u ¶u ö converge to the fixed point x of f.
ç - ÷ =0 If the order of convergence is
4 è ¶y ¶x ø
three, then
is
(A) f ¢(x ) = 0, f ¢¢ = 0
(A) hyperbolic along the line
x +y = 0
(B) f ¢(x ) ¹ 0, f ¢¢ = 0
(B) elliptic along the line
x -y = 0
(C) f ¢(x ) ¹ 0, f ¢¢ ¹ 0
(C) parabolic along the line
x +y = 0
(D) More than one of the above
(D) More than one of the above
(E) None of the above
(E) None of the above
116. Consider the wave equation
118. While solving the equation
2
¶ u 2
¶ u x 2 - 3x + 1 = 0 using the
=4 , 0 < x < p, t > 0 Newton-Raphson method with
2
¶t ¶x 2
the initial guess of a root as 1,
with u(0, t) = u(p, t) = 0, the value of the root after one
¶u iteration is
u(x, 0) = sin x and = 0 at t = 0.
¶t
æp pö
Then u ç , ÷ is (A) 1·5
è2 2ø
(A) 2 (B) 0
(B) 1
(C) 1
(C) -1
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 38
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119. Consider the system of 121. The value of the integral
equations
1
I = ò exp(x 2 ) dx
é 5 2 1ù é x1 ù é 13ù -1
ê -2 5 2ú ê x ú = ê -22ú
ê ú ê 2ú ê ú using a rectangular rule is
êë -1 2 8úû êë x 3 úû êë 14úû
approximated as 2. Then, the
With the initial guess of the approximation error|I - 2| lies in
solution the interval
( 0) ( 0) ( 0)
[ x1 , x 2 , x 3 ]T = [1,1,1]T (A) (2e, 3e]
the approximate value of the
solution æ2 ù
(B) ç , 2eú
(1) (1) (1) è3 û
[ x1 , x 2 , x 3 ]T
after one iteration by the
æ eù
Gauss-Seidel method is (C) ç 0, ú
è 8û
(A) [2, - 4 × 4,1× 625]T
(D) More than one of the above
(B) [2, - 4, - 3]T
(C) [2, - 4, 3]T (E) None of the above
(D) More than one of the above
122. Which of the following is true?
(E) None of the above
(A) Every linear programming
120. For what values of a and b, the problem has a feasible
quadrature formula solution
1
ò-1 f (x )dx » af (-1) + f (b) (B) If a linear programming
problem has an optimal
is exact for all polynomials of solution then it is unique
degree £ 1?
(C) Extreme points of the disk
(A) a = 1, b = 1
x 2 + y 2 £ 1 are the points on
(B) a = -1, b = 1
the circles x 2 + y 2 = 1
(C) a = 1, b = -1
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 40
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123. The Linear Programming 125. A transportation problem is
Problem
Maximize z = x1 + x 2
(A) dual problem
subject to
x1 + 2x 2 £ 20 (B) non-linear programming
x1 + x 2 £ 15 problem
x2 £ 6
x1, x 2 ³ 0 (C) special case of linear
programming problem
(A) has exactly one optimum
solution (D) More than one of the above
(B) has more than one
(E) None of the above
optimum solution
(C) has no solution
126. Consider the map f : r2 ® r2
(D) More than one of the above
defined by
(E) None of the above
f (x, y) = (3x - 2y + x 2, 4x + 5y + y 2 )
124. The dual of the linear
programming problem : then
Minimize C T x subject to Ax ³ b
and x ³ 0 is (A) f is discontinuous at (0, 0)
(A) maximize b T w subject to
A T w ³ c and w ³ 0 (B) f is continuous at (0, 0) and
all directional derivatives
(B) maximize b T w subject to exists at (0, 0)
A T w £ c and w ³ 0
(C) f is differentiable at (0, 0) but
(C) maximize b T w subject to the derivative Df (0, 0) is not
AT w £ c and w is invertible
unrestricted
(D) More than one of the above
(D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 42
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127. The function 129. Let C be the circle
2 2
(x - 1) + y = 1, oriented counter-
f (x, y) = 3x 2y + 4y 3 - clockwise. Then the value of the
3x 2 - 12y 2 + 1 line integral
has a saddle point at 4 3 4
òC - 3 xy dx + x dy
(A) (0, 0)
is
(B) (0, 2)
(A) 6p
(C) (-2,1)
(B) 8p
(D) More than one of the above
(C) 12p
(E) None of the above
(D) More than one of the above
128. Let
(E) None of the above
2
x y
f (x, y) = for (x, y) ¹ (0, 0)
x + y2
2
130. The equation xy - z log y + e xz = 1
then can be solved in a neighbour-
hood of the point (0,1,1) as
¶f y = f (x, z ) for some continuously
(A) and f are bounded
¶x differentiable function f . Then
¶f (A) Ñf (0,1) = (2, 0)
(B) is bounded and f is
¶x
unbounded (B) Ñf (0,1) = (0, 2)
(C) Ñf (0,1) = (0,1)
(C) f is continuous
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 44
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131. Let V be the solid region in r 3 134. The number of surjective maps
bounded by the paraboloid from a set of 4 elements to a set
y = (x 2 + z 2 ) and the plane y = 4. of 3 elements is
Then the value of (A) 36
2 2
òòòV 15 x + z dV (B) 64
is (C) 69
(A) 128p (D) More than one of the above
(B) 64p (E) None of the above
(C) 28p
(D) More than one of the above 135. Which of the following sets is
countable?
(E) None of the above
(A) The set of all functions from
132. Which of the following is the r to { 0,1}
imaginary part of a possible
(B) The set of all functions from
value of ln( i )?
n to { 0,1}
(A) p
(C) The set of all finite subsets
(B)
p of n
2
(D) More than one of the above
p
(C) (E) None of the above
4
(D) More than one of the above 136. The maximum number of
(E) None of the above positive real roots of the
equation
133. The principal value of (-1)( -2i /p ) 2x 7 - x 4 + 4x 3 - 5 = 0
is
is
(A) e 2i
(A) 2
2
(B) e
(B) 3
-2
(C) e
(C) 6
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 46
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137. Find the shortest distance 139. Kepler’s law of planetary motion
between the lines are given below :
x - 8 y + 9 z - 10 I. The square of the periodic
= = and
3 -16 7 time of planets are
x - 15 y - 29 z - 5 proportional to the cubes of
= =
3 8 -5 the major axes of their
orbits.
(A) 16 II. The radius vector drawn
from the sun to the planet
(B) 14 describes equal arc in equal
time.
(C) 15 III. Each planet describes the
ellipse having the sun in
(D) More than one of the above one of its foci.
The order of these three laws are
(E) None of the above as follows :
(A) I, II, III
138. In polar coordinate system the
(B) III, II, I
component of velocity along
radius vector and perpendicular (C) III, I, II
to radius vector is
(D) More than one of the above
dr dq
(A) and r (E) None of the above
dt dt
140. In planetary motion
d 2r dq (A) the total angular momen-
(B) and r 2
2 dt
dt tum remains constant
(B) the angular speed remains
dr 1 dq constant
(C) and
dt r dt (C) the linear speed remains
constant
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 48
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141. A particle coming rest from 144. A fine uniform cycloidal tube is
infinity will reach the earth’s held with its axis vertical and
surface with a velocity vertex downwards and equal
weights of two liquids occupy
(A) gr lengths a and b. Then the
heights of the free surfaces of
(B) 2gr the liquids above the vertex are
in the ratio
(C) 2 gr (A) (3a + b) 2 : (3b + a) 2
(D) More than one of the above
(B) (3a + b):(3b + a)
(E) None of the above
(C) (3b + a) 2 : (3a + b) 2
142. The point in the immersed body
through which the resultant (D) More than one of the above
pressure of the liquid may be
taken to is known as
(E) None of the above
(A) centre of gravity
145. A box contains 10 mangoes out
(B) centre of depth
of which 4 are rotten. 2 mangoes
(C) centre of pressure are taken out together. If one of
them is found to be good, then
(D) More than one of the above probability that the other is also
good is
(E) None of the above
1
(A)
3
143. In an inclined plane submerged
in water, the centre of pressure
is located 8
(B)
15
(A) at the centroid
(B) below the centroid 5
(C)
13
(C) above the centroid
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 50
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146. Let X has probability density 148. What is the second moment
function about the origin for a random
variable X with probability
( - x 2 ) , x Î [ -1, 1]
ì0 × 751 density?
f (x ) = í
î 0 , otherwise
ìx , 0 < x < 2
f (x ) = í 2
What is the probability of î 0 , otherwise
æ 1 1ö
P ç- £ X £ ÷ ?
è 2 2ø (A) 2
(B) 3
(A) 0·68
(C) 4
(B) 0·60
(D) More than one of the above
(C) 0·65
(E) None of the above
(D) More than one of the above
149. If X is a real valued random
(E) None of the above variable, then which of the
following values cannot be
attained by E (X ) and E [ X 2 ]
147. For a given probability
distribution respectively?
(A) 0 and 1
1 æ 3ö
f (x ) = ç ÷, x = 0,1, 2, 3
8 èx ø (B) 2 and 5
for a random variable X , the (C) 2 and 3
moment generating function of (D) More than one of the above
this is
(E) None of the above
t 2
(A) (1 + e )
150. A and B are two independent
1 t events with equal probabilities
(B) e such that P (A È B ) = 0 × 75, then
4
(A) P (B ) = 0 × 25
1
(C) (1 + et ) 3 (B) P (A) = 0 × 75
8
(C) P (B ) = 0 × 5
(D) More than one of the above (D) More than one of the above
(E) None of the above (E) None of the above
23/MAV–AC/M–2023–09/116A-A 52
Text of page 1
Bihar Public Service Commission
School Teacher Written (Objective) Competitive Examination (Advt. No. 27/2023)
Examination Date : 18.12.2023
FINAL ANSWER KEY : LANGUAGE, GENERAL STUDIES AND MATHEMATICS
SERIES - A
1 B 31 B 61 B 91 D 121 B
2 C 32 B 62 A 92 C 122 C
3 E 33 B 63 E 93 C 123 A
4 B 34 C 64 E 94 B 124 B
5 B 35 A 65 B 95 A 125 C
6 E 36 A 66 B 96 A 126 B
7 A 37 B 67 C 97 C 127 C
8 A 38 C 68 D 98 A 128 D
9 D 39 C 69 B 99 C 129 B
10 B 40 A 70 B 100 B 130 A
11 D 41 C 71 B 101 C 131 A
12 E 42 C 72 C 102 C 132 C
13 A 43 C 73 B 103 C 133 B
14 E 44 B 74 C 104 A 134 A
15 A 45 B 75 B 105 B 135 C
16 B 46 B 76 A 106 E 136 B
17 B 47 A 77 B 107 C 137 B
18 A 48 A 78 C 108 A 138 A
19 E 49 A 79 C 109 A 139 B
20 D 50 B 80 A 110 C 140 A
21 B 51 C 81 C 111 A 141 B
22 A 52 B 82 C 112 B 142 C
23 E 53 A 83 A 113 C 143 B
24 D 54 B 84 C 114 A 144 A
25 E 55 C 85 B 115 C 145 C
26 A 56 A 86 B 116 C 146 A
27 C 57 A 87 A 117 A 147 C
28 A 58 A 88 C 118 B 148 A
29 A 59 B 89 B 119 C 149 C
30 B 60 A 90 C 120 A 150 CPage 1 of 56
Source: Bihar Public Service Commission. Question paper published by Bihar Public Service Commission.