TS SET 2012: Paper II, Mathematical Sciences
Text of page 1
Test Paper : II Test Booklet Serial No. : ______________________
OMR Sheet No. : ________________________________
Test Subject : MATHEMATICAL
SCIENCES Hall Ticket No.
Test Subject Code : A-15-02 (Figures as per admission card)
Name & Signature of Invigilator
Name : ____________________________________ Signature : ___________________________________
Paper : II
Subject : MATHEMATICAL SCIENCES
Time : 1 Hour 15 Minutes Maximum Marks : 100
Number of Pages in this Booklet : 8 Number of Questions in this Booklet : 50
Instructions for the Candidates A¿æýÅÆæÿ$¦ËMæü$ çÜ*^èþ¯èþË$
1. Write your Hall Ticket Number in the space provided on the top 1. D ç³#r ò³O ¿êVæü ÌZ CÐèþÓºyìþ¯èþ çÜ¦Ë ÌZ Ò$ çàÌý sìýMðüsý ¯èþ ºÆæÿ$ Æ>Äæý$ yìþ.
of this page. 2. D {ç³Ôèý² ç³{èþÐèþ$$ Äæý*¿ñýO ºçßý$âñýO_eMæü {ç³Ôèý²Ë¯èþ$ MæüÍW E ¨.
2. This paper consists of fifty multiple-choice type of questions. 3. ç³Èæü {´ëÆæÿ ¿æýÐèþ$$¯èþ D {ç³Ôé²ç³{èþÐèþ$$ Ò$Mæü$ CÐèþÓºyæþ$èþ$ ¨. Ððþ$$§æþsìý I§æþ$
3. At the commencement of examination, the question booklet will °Ñ$çÙÐèþ$$ËÌZ D {ç³Ôé²ç³{èþÐèþ$$¯èþ$ ðþÇ_ Mìü §æþ ðþÍí³¯èþ A Ôé˯èþ$ èþç³µ°çÜÇV>
be given to you. In the first 5 minutes, you are requested to open çÜÇ^èþ*çÜ$Mø yìþ.
the booklet and compulsorily examine it as below : (i) D {ç³Ôèý² ç³{èþÐèþ$$¯èþ$ ^èþ*yæþyé°Mìü MæüÐèþÆÿõ³h A ^èþ$¯èþ E¯èþ² M>Wèþç³# ïÜË$¯èþ$
(i) To have access to the Question Booklet, tear off the paper _ ^èþ yìþ. íÜtMæüPÆÿ ïÜË$Ìôý° Ðèþ$ÇÄæý$$ C¨ÐèþÆæÿMóü ðþÇ_ E¯èþ² {ç³Ôé²ç³{èþÐèþ$$¯èþ$
seal on the edge of this cover page. Do not accept a booklet Ò$Ææÿ$ A XMæüÇ ^èþÐèþ§æþ$ª.
without sticker-seal and do not accept an open booklet.
(ii) MæüÐèþÆæÿ$ õ³h ò³O Ðèþ$${¨ _¯èþ çÜÐèþ*^éÆæÿ {ç³M>Ææÿ D {ç³Ôèý²ç³{èþÐèþ$$ÌZ° õ³iË
(ii) Tally the number of pages and number of questions in
the booklet with the information printed on the cover
çÜ Qůèþ$ Ðèþ$ÇÄæý$$ {ç³Ôèý²Ë çÜ Qůèþ$ çÜÇ^èþ*çÜ$Mø yìþ. õ³iË çÜ QÅMæü$
page. Faulty booklets due to pages/questions missing
çÜ º «¨ _ V>± Ìôý§é çÜ*_ _¯èþ çÜ QÅÌZ {ç³Ôèý²Ë$ ÌôýMæü´ùÐèþ#r Ìôý§é °f{ç³
or duplicate or not in serial order or any other
M>Mæü´ùÐèþ#r Ìôý§é {ç³Ôèý²Ë$ {MæüÐèþ$糧æþ®ÌZ ÌôýMæü´ùÐèþ#r Ìôý§é HÐðþO¯é óþyéË$ yæþ$r
discrepancy should be got replaced immediately by a
Ðèþ sìý §øçÙç³NÇèþÐðþ$O¯èþ {ç³Ôèý² ç³{é°² Ððþ r¯óþ Ððþ$$§æþsìý I§æþ$ °Ñ$ÚëÌZÏ ç³È>
correct booklet from the invigilator within the period
ç³ÆæÿÅÐóþæüMæü$°Mìü ÇW C_aÐóþíÜ §é°Mìü º§æþ$Ë$V> çÜÇV>Y E¯èþ² {ç³Ôèý²ç³{é°² ¡çÜ$Mø yìþ.
of 5 minutes. Afterwards, neither the Question Booklet
èþ§æþ¯èþ èþÆæÿ {ç³Ôèý²ç³{èþÐèþ$$ Ðèþ*Ææÿaºyæþ§æþ$ A§æþ¯èþç³# çÜÐèþ$Äæý$ CÐèþÓºyæþ§æþ$.
will be replaced nor any extra time will be given. (iii) ò³O Ñ«§æþ V> çÜÇ^èþ*çÜ$Mö¯èþ² èþÆ>Óèþ {ç³Ôé²ç³{èþ çÜ Qůèþ$ OMR ç³{èþÐèþ$$ ò³O
(iii) After this verification is over, the Test Booklet Number A§óþÑ«§æþ V> OMR ç³{èþÐèþ$$ çÜ Qůèþ$ D {ç³Ôé²ç³{èþÐèþ$$ ò³O °ÇªçÙtt çÜ¦Ë ÌZ Æ>Äæý$ÐèþÌñý¯èþ$.
should be entered in the OMR Sheet and the OMR Sheet 4. {ç³ {ç³Ôèý²Mæü$ ¯éË$Væü$ {ç³éÅÐèþ*²Äæý$ {ç³çܵ §æþ¯èþË$ (A), (B), (C) Ðèþ$ÇÄæý$$ (D)
Number should be entered on this Test Booklet. Ë$V> CÐèþÓºyézÆÿ$$. {ç³{ç³Ôèý²Mæü$ çÜÆðÿO¯èþ {ç³çܵ §æþ¯èþ¯èþ$ G¯èþ$²Mö° Mìü §æþ ðþÍí³¯èþ Ñ«§æþ V>
4. Each item has four alternative responses marked (A), (B), (C) OMR ç³{èþÐèþ$$ÌZ {ç³ {ç³Ôé² çÜ QÅMæü$ CÐèþÓºyìþ¯èþ ¯éË$Væü$ Ðèþé¢ÌZÏ çÜÆðÿO¯èþ
and (D). You have to darken the circle as indicated below on the {ç³çܵ §æþ¯èþ¯èþ$ çÜ*_ ^óþ Ðèþ颰² »êÌý ´ëÆÿ$$ sý ò³¯þø Mìü §æþ ðþÍí³¯èþ Ñ«§æþ V>
correct response against each item. ç³NÇ ^éÍ.
Example : A B C D E§éçßýÆæÿ×ý : A B C D
(C) çÜÆðÿO¯èþ {ç³çܵ §æþ¯èþ AÆÿ$$óþ
where (C) is the correct response.
5. {ç³Ôèý²ËMæü$ {ç³çܵ §æþ¯èþ˯èþ$ D {ç³Ôèý²ç³{èþÐèþ$$ø CÐèþÓºyìþ¯èþ OMR ç³{èþÐèþ$$ ò³O¯èþ
5. Your responses to the items are to be indicated in the OMR Sheet
CÐèþÓºyìþ¯èþ Ðèþé¢ÌZϯóþ ç³NÇ _ Væü$Ç¢ ^éÍ. AÌêM>Mæü çÜÐèþ*«§é¯èþ ç³{èþ ò³O ÐóþÆöMæü ^ør
given to you. If you mark at any place other than in the circle in
the Answer Sheet, it will not be evaluated.
Væü$Ç¢õÜ¢ Ò$ {ç³çܵ §æþ¯èþ Ðèþ$*ÌêÅ Mæü¯èþ ^óþÄæý$ºyæþ§æþ$.
6. Read instructions given inside carefully. 6. {ç ³Ôèý² ç³{èþÐèþ$$ ÌZç³Ë C_a¯èþ çÜ*^èþ¯èþ˯èþ$ gê{Væüèþ¢V> ^èþ§æþÐèþ yìþ.
7. Rough Work is to be done in the end of this booklet. 7. _èþ$¢ç³°° {ç³Ôèý²ç³{èþÐèþ$$ _ÐèþÆæÿ C_a¯èþ RêäçܦËÐèþ$$ÌZ ^óþÄæý*Í.
8. If you write your name or put any mark on any part of the OMR 8. OMR ç³{èþÐèþ$$ ò³O °È~èþ çÜ¦Ë ÌZ çÜ*_ ^èþÐèþËíܯèþ ÑÐèþÆ>Ë$ èþí³µ _ CèþÆæÿ çÜ¦Ë ÌZ
Answer Sheet, except for the space allotted for the relevant Ò$ Væü$Ç¢ ç³#¯èþ$ ðþÍõ³ Ñ«§æþ V> Ò$ õ³Ææÿ$ Æ>Äæý$yæþ V>± Ìôý§é CèþÆæÿ _à²Ë¯èþ$ ò³rtyæþ
entries, which may disclose your identity, you will render yourself V>± ^óþíܯèþrÏÆÿ$$óþ Ò$ A¯èþÆæÿáèþMæü$ Ò$Æóÿ »ê«§æþ$ÅËÐèþ#éÆæÿ$.
liable to disqualification. 9. çç³Èæü ç³NÆæÿ¢Æÿ$$¯èþ èþÆ>Óèþ Ò$ {ç³Ôèý²ç³{é°² Ðèþ$ÇÄæý$$ OMR ç³{é°² èþç³µ°çÜÇV>
9. You have to return the test question booklet and OMR Answer ç³Èæüç³ÆæÿÅÐóþæüMæü$yìþMìü CÐéÓÍ. Ðésìý° ç³Èæü Væü¨ ºÄæý$rMæü$ ¡çÜ$Mæü$ÐðþâæýÏMæü*yæþ§æþ$.
Sheet to the invigilators at the end of the examination compulsorily
10. ±Í/¯èþËÏ Ææÿ Væü$ »êÌý ´ëÆÿ$$ sý ò³¯þ Ðèþ*{èþÐóþ$ Eç³Äñý*W ^éÍ.
and must not carry it with you outside the Examination Hall.
10. Use only Blue/Black Ball point pen. 11. ÌêVæüÇ£æþÐþ$ sôýº$ÌýÞ, M>ÅÍMæü$ÅÌôýrÆÿË$, GË[M>t°Mü ç³ÇMæüÆ>Ë$ Ððþ$$§æþËVæü$¯èþÑ ç³ÈæüVæü¨ÌZ
11. Use of any calculator or log table etc., is prohibited. Eç³Äñý*W ^èþyæþ °õÙ«§æþ .
12. There is no negative marks for incorrect answers. 12. èþç³ çÜÐèþ*«§é¯éËMæü$ Ðèþ*Ææÿ$PË èþWY ç³# Ìôý§æþ$.
IIæ 1 A-15-02
Text of page 2
*A1303* IIæ 2 A-15-02
Text of page 3
*A1303*
MATHEMATICAL SCIENCES
Paper – II
x2 7. The largest eigen value of the matrix
1. Define f n = , x is real, n = 0, 1, 2, … ,
(1 + x 2 )n
∞ ⎡ 3 10 5 ⎤
then for x ≠ 0, ∑ n = 0 f n ( x) =
A = ⎢⎢ − 2 − 3 − 4⎥⎥
is
(A) 0 (B) 1 ⎢⎣ 3 5 7 ⎥⎦
(C) 1 + x (D) 1 + x2
(A) 2 (B) 3 (C) 4 (D) 6
2. The function f(x) = sin(1/x), x > 0 is
(A) Uniformly continuous on R+ 8. Let T: R3 → R be a linear transformation
(B) Uniformly continuous on R
defined by T (1,1, 1) = 3, T (0, 1,− 2) = 1
(C) Continuous on R+
and T (0, 0, 1) = − 2
(D) Not continuous on R+
Then T (x, y, z) =
1
3. ∫ x (1 − x 2 ) n dx = (A) 8x + 3y − 2z (B) 3x − 8y − 2z
0
(C) 8x − 3y − 2z (D) 2x − 3y + 8z
(A) 1 (B) 12
1 1 9. If f (z) = u (x, y) + iv (x, y) and
(C) (D) g (z) = u (x, y) − iv (x, y) are analytic functions
2( n + 1) ( 2n + 1)
defined on the same domain and
nα
4. If p > 0 and α ε R, then lim n→∞ = f (1 + i) = 2 + 3i, then g ( 4 + 3i) =
(1 + p ) n
(A) 4 + 3i (B) 4 − 3i
(A) 0 (B) 1
(C) 2 + 3i (D) 2 − 3i
(C) ∞ (D) not defined
10. If an entire function f (z) is bounded in
5. If T: R3 → R2 is a linear transformation
absolute value for all z, then f (z) =
defined by T (x, y, z) = ( x + 2y − z, y + z,
(A) Constant (B) e z
x + y − 2z), then the rank of T is equal to
(C) 1/z (D) z
(A) 0 (B) 1 (C) 2 (D) 3
6. Let A and B be n×n matrices with real ∂f ( z )
11. If f (z) = sinz + e3z, then =
numbers as elements. Then which one of ∂z
the following statements is NOT true ? (A) 0 (B) 3e3 z
(A) Rank of A = Rank of A′ (C) cos z (D) cos z + 3e3 z
(B) Rank of (AB) = Rank of (BA)
(C) Rank of (A + B) < Rank of A + Rank of B z 2 + 3z + 2
12. The residue of f (z) = at z = 0 is
(D) If C is obtained after a finite number z 2 (z − 1)
of elementary operations on A, then
Rank of A = Rank of C (A) 0 (B) − 1 (C) 2 (D) − 5
IIæ 3 A-15-02
Text of page 4
*A1303*
13. If G is a finite abelian group, then G is 18. Consider the linear space X = C [0, 1]
isomorphic to the direct product of its with norm f = sup{ f(t) : 0 ≤ t ≤ 1}. Let
(A) normal subgroups F = {f ε X : f (½) = 0} and G = {gε X : g (½) ≠ 0}.
(B) abelian subgroups
Then
(C) sylow subgroups
(A) F is not closed and G is open
(D) cyclic subgroups (B) F is closed but G is not open
14. Which one of the following is NOT true? (C) F is not closed and G is not open
(D) F is closed and G is open
(A) There are φ (n) premature n-th roots
of unity where φ (n) is the Euler 19. An integrating factor for the differential
φ - function equation ydx + (x2y − x) dy = 0 is
(A) x (B) 1/x (C) 1/x2 (D) ex
(B) If ω is a primitive n-th root of unity,
then F0 (ω) is the splitting field of xn − 1 20. For the differential equation y′′− x 2 y′− xy = 0
over F0 an equivalent system of first order
(C) If ω1 , ω2 , ..., ωφ( n ) are the φ (n) equations is
primitive n-th roots of unity, then any (A) dy/dx = y+z; dz/dx = x2z + y
2
automorphism of F0 (ω1 ) takes ω1 (B) dy/dx = z; dz/dx = xy + x z
into some ωi (C) dy/dx = xz; dz/dx = y + xz
(D) dy/dx = x +z; dz/dx = y + x2z
(D) [F0 (ω1 ) : F0 ] > φ(n)
21. The eigen functions of the Sturm − Liouville
15. If R is a commutative ring with unit element
problem x′′ + α x = 0, x(0) = 0 and x(1) = 0,
and M is an ideal of R, then M is a maximal for 0 < t < 1, n = 0, 1, 2, … are
ideal of R if and only if R/M is
(A) xn(t) = sin(2n+1) π t
(A) a quotient ring (B) a field
(B) xn(t) = cos(2n+1) π t/2
(C) a division ring (D) an ideal
(C) xn(t) = sin(2n+3) π t/2
16. The solution of x 2 ⊕ 1 = 0 in the field (D) xn(t) = cos(2n+3) π t/2
(Z5 ,⊕,•) is 4
2
(
22. The stationary function of ∫ xy′ − y ′ dx )
(A) 0, 1 (B) 1, 2 (C) 2, 3 (D) 3, 4 0
subject to y(0) = 0 and y(4) = 3 is
17. The number of units in the domain of
Gaussian integers is (A) y= ½ (x2 − 1) (B) y= ½ (x2+1)
(A) 1 (B) 2 (C) 3 (D) 4 (C) y = ¼ (2x2 + x) (D) y = ¼ (x2 − x )
IIæ 4 A-15-02
Text of page 5
*A1303*
23. The general form of Hamiltonian equations 27. The solution of the partial differential
of motion for a conservative holonomic ∂u ∂u
equation 3 + 2 = 0, u(x,0) = 4e − x
dynamical system for k = 1, 2, 3, …, n are ∂x ∂y
is u =
∂H ∂H
(A) qk = ∂t , pk = − ∂q (A) 2e5x + 3y+ 2e- 3x + 2y
k k
(B) 3e- 5x + 3y+ e3x - 2y
∂H ∂H
(B) qk = − ∂t , p k = − ∂q (C) 4e- (2x - 3y)/2
(D) 4e- (3x - 2y)/2
k k
∂H ∂H
(C) qk = ∂t , pk = − ∂t 28. The Newton-Raphson method of finding
k k
a root of a polynomial or transcendental
∂H ∂H
(D) qk = ∂t , pk = − ∂p equation has the rate of convergence
k k
(A) 3 (B) 2
24. If the kinetic energy of a body of mass m (C) 1.67 (D) 1.4
moving in one-dimension is ½ mx•2 and its
potential energy is ½ kx2 where k is a 29. An approximate solution for the system of
constant then its Lagrange’s equations of linear equations
motion is 10x + 2y + z = 9, 2x + 20y − 2z = − 44 ;
− 2x + 3y + 10z = 22 using Gauss-Seidal
(A) kx + mx = 0 (B) mx + kx = 0 method is
(C) mx + kx = 0 (D) kx + mx = 0 (A) x = 1.094, y = − 1.908, z = 2.946
(B) x = 1.082, y = − 1.914, z = 2.965
25. The resolvent kernel R (x, t; λ) for the
(C) x = 1.108, y= − 1.932, z = 2.981
K (x, t) = x2t2, a = −1, b = 1 is
(D) x = 1.013, y = − 1.996, z = 3.001
3x t
2 2
4x t 2 2
6 dx
(A) , < 1 2 (B) , <3 2
1 − 2 3 − 2 30. The value of the integral I = ∫0 by
1 + x2
5 x 2t 2 5 x 2t 2 using Simpson’s 1/3 rule, is
(C) , < 5 2 (D) , <2 5
5 − 2 5 + 2 (A) 1.2886 (B) 1.3268
26. The solution of the partial differential (C) 1.3662 (D) 1.3874
equation (y − z)p + (z − x)q = x − y is
31. For any discrete distribution
(A) f (x + y + z) = x2+ y2+ z2
(A) S.D > M.D from mean
(B) f (x + y + z) = xy + yz + zx
(B) S.D = M.D
(C) f (x2 + y2 + z2) = xyz
(C) S.D < M.D from mean
(D) f (x2 + y2 + z2) = x + y + z
(D) No relation between S.D and M.D
IIæ 5 A-15-02
Text of page 6
*A1303*
32. For a random variable x moments of all 36. The Poisson process satisfies the
order exit and μ j denote jth central moment,
following conditions
then
2 (A) regularity
(A) μ 2 j+1 > μ2 j μ2 j+2
(B) homogeneity
2
(B) μ 2 j+1 < μ 2 j μ 2 j+ 2
(C) independent increments
(C) μ 2 j+1 = μ 2 j μ 2 j+ 2
(D) all the above three (A), (B) and (C)
(D) either B or C is correct
37. For a Binomial distribution with p = 0.5 has
33. The probability distribution of a random
variable X is f(x) = k sin( π x/5), 0 < x < 5, maximum probability when
then the ratio between median of X and k (A) X = n/2 if n is even
is
(B) x = ½ (n − 1) and x = ½ (n + 1) if n is odd
(A) 25: π (B) 5:2
(C) 1:2 (D) 10: π (C) x = n
(D) both A) and B) are correct
34. If φ (t) denotes the characteristic function
of a random variable, then 38. The probability mass function of a Poisson
i) φ(0) = 1, variate with parameter λ truncated at 0 is
ii) φ(t) < 1,
e − λ λx e − λ λx
iii) φ(t) is continuous every where (A)
x!
(B)
x!
(1 − e −λ )
(A) only i) is correct but not ii) and iii)
e − λ λx
(B) only ii) is correct but not iii) (C) (D) 1 − e − λ
(C) all i), ii) and iii) are correct
(
x! 1 − e −λ )
(D) none of i), ii) and iii) are correct
39. Let x1, x2, … , xn be a random sample
35. If you wish to estimate the proportion of drawn from an exponential distribution with
engineers and scientists who have studied parameter λ , then the distribution function
statistics and you wish your estimate to be of Y = min(x1, x2, … , xn) is
correct within 2% with probability 0.95 or
more, how large sample would you take if (
(A) n(n − 1) 1 − e−nλy )
you have no idea what the true proportion
(
(B) 1 − e −nλy )
is ?
(A) 1200 (B) 800
(C) ½ n(n − 1) 1 − e −nλy( )
(C) any number (D) 95 (
(D) 1− e−λy )
IIæ 6 A-15-02
Text of page 7
*A1303*
40. The minimum variance bound estimator 46. The arrival rate and service rates of a
for θ in a Cauchy population with M/M/1 queueing system are 3 and 4, then
parameter θ is the probability of system emptiness is
(A) sample mean (A) 0.25 (B) 0.75
(B) sample median (C) 0.3 (D) 0.5
(C) reciprocal of sample median
(D) does not exist 47. A simplex is a
(A) half plane
41. In a Poisson distribution with unit mean
(B) convex polyhedran
the mean deviation about mean is
(C) hull
(A) 2/e (B) e/2 (C) e (D) 1
(D) envelop
42. With normal notations the variance of
48. If A = [aij] is the payoff matrix then saddle
M.L.E. is
point exist when
(A) I (θ ) (B) 1 / I (θ )
(A) Minj Maxi aij = Max. Min aij
(C) √ I (θ) (D) − 1 / I (θ) (B) Minj Maxi aij < Max. Min aij
(C) Minj Maxi aij > Max. Min aij
43. Neyman-Pearson lemma provides
(D) All the above three are correct
(A) An unbiased test
(B) a most powerful test 49. If f(x) is the p.d.f of a life time distribution
(C) an admissible test then the hazard rate of distribution is
(D) sufficient test (A) f(x)/F(x)
44. With the usual notation the sum of squares (B) f(x)/(1 − F(x))
due to any factorial effect is (C) F(x)/f(x)
(D) log f(x)/(1 − F(x))
( )2 ( )2
(A) (B)
∑ γ i ci2 ∑ γ i2 ci 50. The variance covariance matrix of a
( )2 2
⎡4 0 0⎤
(C) (D) ( ) ⎢ ⎥
random vector x is ∑ = ⎢0 8 2⎥ , then
∑ γ i2 ∑ γ i ci
⎢⎣0 2 8⎥⎦
45. The information on auxiliary variate is used
the first principal component explains
in the following sampling scheme
______ percentage of variations on the
(A) double sampling
(B) two stage sampling dependent variable.
(C) cluster sampling (A) 90 (B) 45
(D) quota sampling (C) 50 (D) 60
IIæ 7 A-15-02
Text of page 8
*A1303*
Space for Rough Work
IIæ 8 A-15-02Official key not published for this paper.
Page 1 of 8
Source: Telangana State Eligibility Test. Question paper published by Telangana State Eligibility Test.