TS SET 2013: Paper II, Mathematical Sciences
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SUBJECT CODE SUBJECT PAPER
A-15-02 MATHEMATICAL SCIENCES II
HALL TICKET NUMBER
QUESTION BOOKLET NUMBER
OMR SHEET NUMBER
DURATION MAXIMUM MARKS NUMBER OF PAGES NUMBER OF QUESTIONS
1 HOUR 15 MINUTES 100 8 50
This is to certify that, the entries made in the above portion are correctly written and verified.
Candidates Signature Name and Signature of Invigilator
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1. Write your Hall Ticket Number in the space provided on the 1. D ç³#r ò³O ¿êVæü ÌZ CÐèþÓºyìþ¯èþ çÜ¦Ë ÌZ Ò$ çàÌý sìýMðüsý ¯èþ ºÆæÿ$ Æ>Äæý$ yìþ.
top of this page.
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2. This paper consists of fifty multiple-choice type of questions.
3. At the commencement of examination, the question booklet will 3. ç³Èæü {´ëÆæÿ ¿æýÐèþ$$¯èþ D {ç³Ôé²ç³{èþÐèþ$$ Ò$Mæü$ CÐèþÓºyæþ$èþ$ ¨. Ððþ$$§æþsìý I§æþ$
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discrepancy should be got replaced immediately by a {ç³Ôèý²Ë$ {MæüÐèþ$糧æþ®ÌZ ÌôýMæü´ùÐèþ#r Ìôý§é HÐðþO¯é óþyéË$ yæþ$r Ðèþ sìý §øçÙç³NÇèþÐðþ$O¯èþ
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where (C) is the correct response.
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and must not carry it with you outside the Examination Hall. The
candidate is allowed to take away the carbon copy of OMR Ðésìý° ç³Èæü Væü¨ ºÄæý$rMæü$ ¡çÜ$Mæü$ÐðþâæýÏMæü*yæþ§æþ$. ç³Èæü ç³NÆæÿ¢Æÿ$$¯èþ èþÆæÿ$Ðéèþ A¿æýÅÆæÿ$¦Ë$
Sheet and used Question paper booklet at the end of the {ç³Ôèý² ç³{é°², OMR ç³{èþ Äñý$$MæüP M>Ææÿ¾¯þ M>ï³° ¡çÜ$Mæü$ÐðþâæýÏÐèþ^èþ$a.
examination. 10. ±Í/¯èþËÏ Ææÿ Væü$ »êÌý ´ëÆÿ$$ sý ò³¯þ Ðèþ*{èþÐóþ$ Eç³Äñý*W ^éÍ.
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12. There is no negative marks for incorrect answers. 12. èþç³ çÜÐèþ*«§é¯éËMæü$ Ðèþ*Ææÿ$PË èþWY ç³# Ìôý§æþ$.
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MATHEMATICAL SCIENCES
Paper – II
3 3
1. If S is a set with 5 elements, then the 7. If T : IR → IR is a linear transformation
number of binary operations that can be defined by T(x, y, z) = (x + y, y + z, z – x)
defined on S which are not commutative is 3
(A) (1023) 2
15
(B) (1024) 2
15 and if B is the standard basis for IR , then
(C) 2
15 25
(D) 2 – 1 the matrix [T : B : B] =
2. The inverse of (1 3 5 7 9) o ( 5 7 9 2 4) o ⎡ 1 0 − 1⎤ ⎡ 1 1 0⎤
⎢ ⎥
(2 4 6 8) in the group ( S9, o ) is (A) ⎢ 1 1 0⎥ (B) ⎢ 0 1 1⎥
(A) (1 3 5)o(9 2 7 4 6 8) ⎢ ⎥
⎢⎣0 1 1⎥⎦ ⎢⎣− 1 0 1⎥⎦
(B) (1 3 5 9)o(2 7 4 6 8)
(C) (1 3 5 9 2)o(7 4 6 8)
(D) (1 3 5 9 2 7)o(4 6 8) ⎡ 1 − 1 0⎤ ⎡ 1 0 1⎤
⎢ ⎢ ⎥
(C) ⎢0 1 1⎥⎥ (D) ⎢− 1 1 1⎥
3. Let ω be the complex cube root of unity and
⎢⎣ 1 1 0⎥⎦ ⎢⎣ 0 1 0⎥⎦
G = {1, ω, ω2}. Let φ be a homomorphism
from the group ( , +) into the group (G, ⋅ )
8. In the vector space ( IR , +, .) over IR ,
3
defined by φ (n) = ωn for all n ∈ . Then
Ker φ = which one of the following is a linearly
(A) (B) 2 independent set ?
(C) 3 (D) 4 (A) {(1, 2, 3), (2, –1 , 4), (4, 3, 10)}
4. If X = {1, 2, 3, 4}, then which one of the (B) { (2, 1, 7), (3, – 2, 4), (–4, 1, –1)}
following is NOT a topology on X ?
(C) {(1, – 3, 2),(3, –1, 1), (1, 5, –3)}
(A) τ1 = { φ , X, {1,2}, {2, 3}, {1, 2, 3}}
(D) {(2, – 1, 4), (3, 1, 4), (1, –3, 4)}
(B) τ2 = { φ , X, {3}, {1,3}}
(C) τ3 = { φ , X, {1}, {1, 2}, {1, 2, 3}} 9. If V is the vector space of all 4 × 4 matrices
over IR , then the dimension of the vector
(D) τ4 = { φ , X, {1}}
space V over IR is
5. Which one of the following is NOT a dense
(A) 2 (B) 4
subset of IR in the standard topology on IR ?
(A) (B) IR – Q (C) 8 (D) 16
(C) (D) IR – 3
10. Let T be a linear transformation from IR
6. An example of a connected subset of IR
to IR 2 defined by T(x, y, z) = (x – y, y – z)
(in the standard topology) which is NOT
3
compact is for all (x, y, z) ∈ IR , then nullity of T =
(A) [0, 1] (B) (0, 1) (A) 3 (B) 2
(C) (0, 1) ∪ (1, 2) (D) [0, 1] ∪ [1, 2]
(C) 1 (D) 0
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11. The number of solutions of the 16. Which one of the following statements is
non-homogeneous system of equations FALSE ?
(A) The Lebesgue measure of an interval
given by is its length
x + 2y – z = 4 (B) The set of all Lebesgue measurable
2x – 3y +z = 3 sets is a σ - algebra
(C) Every Borel set is Lebesgue
5x – 4y + z = 5, is measurable
(A) 0 (B) 1 (D) Every measurable function is
Lebesgue integrable
(C) 2 (D) infinite
17. If z = 2 + 3i, the modulus of the complex
∞
( −1 )
n
7z −1
12. The series ∑ number is
n = 2 n ( log n )
5 3z + 1
(A) converges 1 (28 + 3i) 1 (28 + 3i)
(A) (B)
(B) diverges 13 130
(C) absolutely converges 1 (280 + 30i) 1 (280 + 3i)
(C) (D)
(D) conditionally converges 13 130
13. Define a function f : IR → IR by 1+i
f(x)= ⎨
⎧ x 2 sin x , if x ≠ 0 18. ∫ ( x + iy ) dz along y = x is
0
⎩0 , if x = 0 , then
(A) i (B) 0
(A) f is discontinuous at x = 0 (C) 1 (D) 1 + i
(B) f is continuous but not differentiable
19. A critical point of w = z2 is
at x = 0
(A) 1 (B) – 1
(C) lim f ( x ) does not exist (C) 0 (D) ∞
x→ 0
(D) f is differentiable at x = 0 20. If C is the circle x 2 + y 2 = 4, then
π 3 z 2 + 7 z + 1 dz =
2
∫ z −3
14. ∫ ln sin x dx = C
0 (A) 6i (B) 0
π ln 2 π ln 2 (C) – 6i (D) i
(A) (B) −
2 2 21. Given the following tabular data :
(C) ln 2 (D) does not exist
x : –1 0 2 3
15. The Lebesgue measure of the set of all y = f(x) : –8 3 1 2
integers The approximate value of f(1) correct to
(A) is infinite (B) is equal to 1 three decimal places is
(A) 3.447 (B) 3.538
(C) is equal to 0 (D) lies in (0, 1) (C) 3.666 (D) 3.764
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22. The approximate value of the integral 26. With the usual notation, the principle of
6 least action in the Jacobi’s form, is
dx
∫ 1 + x 2 by trapezoidal rule with six t2
0
(A) δ ∫ H − V dt = 0
intervals is t1
ρ2
(A) 1.3662 (B) 1.4108
(C) 1.1048 (D) 1.6332
(B) Δ ∫ H −V dρ = 0
ρ1
ρ
2
23. If φ1, φ2 are two solutions of (C) δ ∫ H + V dρ = 0
ρ
L(y) = y′′ + a1y′ + a 2 y = 0 (a1, a2 are 1
ρ
2
constants) on an interval I containing a (D) Δ ∫ T − V dρ = 0
ρ
point x0, then the Wronskian W (φ1, φ2 ) of 1
27. The kinetic energy of a particle of mass ‘m’
the solutions at x is moving in a plane, in polar coordinates r
and θ is
(A) W ( φ1 , φ2 ) ( x ) = ea1( x− x0 )W ( φ1 , φ2 )( x0 )
(B) m ( r + r θ )
− a1( x − x0 )
1 m ( r2 + r 2θ 2 ) 1 2 2 2
(B) W ( φ1 , φ2 ) ( x ) = e W ( φ1 , φ2 )( x0 ) (A)
2 2
(C) W ( φ1 , φ2 ) ( x ) = e−a1( x− x0 )W ( φ1 , φ2 )( 0 ) (C)
1 m ( r2 + 1 θ 2 )
(D) m ⎜⎛ r + θ ⎞⎟
1 1 2
2 r
2
2 ⎝ r ⎠
(D) W ( φ1 , φ2 ) ( x ) = ea1( x− x0 )W ( φ1 , φ2 )( 0 )
28. A solution of the integral equation
24. The Sturm Liouville operator L in the x
equation L(y) + λρy = 0 is y(x) = sin x + 2 ∫ cos(x − t )y(t ) dt is
0
d2 d (A) xe
x 2 x
(B) x e
(A) p 2 − q (B) q − p
dx dx x
(D) e
x
2
1 d p − 1 dq d ⎛ p d ⎞−q (C) e sin x
(C) (D) ⎜ ⎟ x
p dx 2 q dx dx ⎝ dx ⎠
29. The curve on which the functional
( p and q are some functions of x)
∫ [ (y′) + 12xy] dx; y(0) = 0 and y(1) = 2
1
2
25. An integrating factor for the equation
0
dy
( x + 1 ) − y = e3 x ( x + 1 )2 is can be extremized, is
dx
(A) y = x3
(A) (x + 1)e3x (B) − (x + 1) 3
(B) y = 2x – x
(C) y = x + x2 +1
3
⎛ 1 ⎞ 1
(C) − ⎜ 2 ⎟ 3
(D) (D) y = x + x
⎝ x + 1⎠ x +1
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30. The particular integral of 34. If A and B are independent non-null events
then which one of the following is NOT
(D2 − DD′)z = cos x cos 2y, with the usual
correct ?
notation, is (A) P(A ∩ B) = P(A ) ⋅ P(B)
C C
(A)
1 cos(x + 2y) − 1 cos(x − 2y) (B) P(A ∪ B) = 1− P(A )P(B )
2 6 (C) P(A ∩ B) = P(A ) ⋅P(B)
C C
1 1 (D) P(A ∪ B) = P(A ) + P(B)
(B) cos(x + 2y) + cos(x − 2y)
3 2
1 1 35. If p(n)
ii = 1 for all values of n, then the state
(C) cos(2x + y) − sin (x − 2y )
2 3 i is called ____________ state.
1 1 (A) Reflecting (B) Absorbing
(D) cos(x + 2y ) + cos (x − 2y)
3 6 (C) Communicating (D) Periodic
31. To understand the skewness of the 36. If T is the Transition Probability
variable which diagram is used ? Matrix (TPM) of a Markov Chain, then the
(A) Pictogram correct statement is
(B) Whisker diagram (A) T2 is again a TPM
(B) T2 + T is again a TPM
(C) Line
(C) α T is again a TPM for 0 < α < 1
(D) Range chart (D) log T is again a TPM
32. Let {Xn} be a sequence of iid r.v.s with 37. If X is a standard normal r.v. then the ratio
of quartile deviation, mean deviation and
mean μ < ∞ . Then Xn → μ in probability. standard deviation is
This statement relates to (A) 5 : 6 : 7 (B) 10 : 12 : 15
(A) Chebychev’s WLLN (C) 2 : 3 : 4 (D) 1 : 2 : 3
3
(B) Khintchine’s WLLN 38. If X is a Poisson r.v. with mean then its
2
variance is
(C) Kolmogorov’s SLLN
(D) Central limit theorem 2
(A) (B) 3
3
33. For a market research survey, a sample of
3
10 retail outlets are selected. The chances (C) (D) 2
2
for selecting a hyper market, super market,
39. Let X1, X2, …, Xn be a random sample form
departmental store and a restaurant are a normal population with mean μ and
0.10, 0.50, 0.20 and 0.20 respectively. What n
is the probability that at least three super ∑
variance σ . If s2 = 1 ( X i − X )2 , then
2
n i =1
markets are selected ?
the unbiased estimator for σ2 is
7 2
(A) 1− (B) 1 s2 1 s2
27 27 (A) (B)
n −1 n
7 1 n − 1s 2 n s2
(C) (D) (C) (D)
27 210 n n −1
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40. A sample of 50 observations drawn from ⎧8 xy , 0 < x < y < 1
a normal population with mean μ and 46. Let f ( x , y ) = ⎨ , then the
variance σ 2 . The sample mean and ⎩ 0 elsewhere
standard deviation are 11,795 and 14,054 marginal density of Y is
respectively. What is the 95% confidence 3 3
(A) 4y (B) y
interval for μ ?
(A) (11795, 15691) (B) (7855, 15291) (C) 4x2 (D) 4y
(C) (3896, 11795) (D) (12560, 13480) 47. A linear combination of treatments said to
41. Local control is the concept related to ___ be contrast if and only if
(A) The reduction of experimental limits (A) The sum of the treatment effects is
(B) The reduction of treatments zero
(C) The reduction of heterogeneity in the
experimental material (B) All the coefficients of the treatments
(D) The reduction of the cost of the are unity
experiment (C) The sum of coefficients of the
42. The length of 5% confidence interval is treatments is zero
_______ than the length of 1% confidence (D) The number of positive and negative
interval. coefficients is same
(A) Larger
(B) Less than or equal to 48. The total number of factorial effects in 2n
(C) Greater than or equal to experiment are
(D) Smaller n
(A) 2
43. The ratio estimator is more precise than (B) 1 – 2n
SRSWOR estimator for the populations n
(C) 2 – 1
total if n
(D) 2 + 1
1 Cx Cx
(A) p > 2 C (B) p > C 49. Which of the following statement is NOT
y y
C C true ?
1 y y
(C) p > (D) p > (A) degeneracy in LPP may arise at the
2 Cx Cx
initial stage
44. In sample surveys one can have the (B) a degenerate solution can never be
following error
optimum
(A) sampling error
(C) degeneracy may be a temporary
(B) non-sampling error
(C) both sampling and non-sampling phenomenon
errors (D) LPP can be used in solving a game
(D) no error 50. In an M/M/1 queuing model under
45. The X′AX and X′BX are independent equilibrium the mean arrival rate is 3 and
quadratic forms of normal vectors if and mean service rate is 4. What is the
only if probability that the server is busy ?
(A) AB = 0 (B) AB ≠ 0 (A) 0.25 (B) 0.75
–1
(C) A = B (D) A = B (C) 0.5 (D) 0
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II æ 7 A-15-02
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*A1502*
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Source: Telangana State Eligibility Test. Question paper published by Telangana State Eligibility Test.