TS SET 2014: Paper III, Mathematical Sciences
Text of page 1
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
SUBJECT CODE SUBJECT PAPER
A-15-03 MATHEMATICAL SCIENCES III
HALL TICKET NUMBER QUESTION BOOKLET
NUMBER
OMR SHEET NUMBER
DURATION MAXIMUM MARKS NUMBER OF PAGES NUMBER OF QUESTIONS
2 HOUR 30 MINUTES 150 16 75
This is to certify that, the entries made in the above portion are correctly written and verified.
Candidate’s Signature Name and Signature of Invigilator
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$$ yðþ¿ñýOI§æþ$ ºçßý$âñýO_eMæü {ç³Ôèý²Ë¯èþ$ MæüÍW E ¨.
2. This paper consists of seventy five multiple-choice type of
questions. 3. ç³Èæü {´ëÆæÿ ¿æýÐèþ$$¯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$$¯èþ$
seal on the edge of this cover page. Do not accept a booklet Ò$Ææÿ$ A XMæüÇ ^èþÐèþ§æþ$ª.
without sticker-seal and do not accept an open booklet.
(ii) Tally the number of pages and number of questions in (ii) MæüÐèþÆæÿ$ õ³h ò³O Ðèþ$${¨ _¯èþ çÜÐèþ*^éÆæÿ {ç³M>Ææÿ D {ç³Ôèý²ç³{èþÐèþ$$ÌZ° õ³iË
the booklet with the information printed on the cover çÜ Qůèþ$ Ðèþ$ÇÄæý$$ {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$.
will be replaced nor any extra time will be given.
(iii) After this verification is over, the Test Booklet Number (iii) ò³O Ñ«§æþ V> çÜÇ^èþ*çÜ$Mö¯èþ² èþÆ>Óèþ {ç³Ôé²ç³{èþ çÜ Qůèþ$ OMR ç³{èþÐèþ$$ ò³O
should be entered in the OMR Sheet and the OMR Sheet A§óþÑ«§æþ V> OMR ç³{èþÐèþ$$ çÜ Qůèþ$ D {ç³Ôé²ç³{èþÐèþ$$ ò³O °ÇªçÙtt çÜ¦Ë ÌZ Æ>Äæý$ÐèþÌñý¯èþ$.
Number should be entered on this Test Booklet. 4. {ç ³ {ç ³ Ôèý²Mæü$ ¯éË$Væü$ {ç³éÅÐèþ*²Äæý$ {ç³çܵ §æþ¯èþË$ (A), (B), (C) Ðèþ$ÇÄæý$$ (D)
4. Each item has four alternative responses marked (A), (B), (C) Ë$V> CÐèþÓºyézÆÿ$$. {ç³{ç³Ôèý²Mæü$ çÜÆðÿO¯èþ {ç³çܵ §æþ¯èþ¯èþ$ G¯èþ$²Mö° Mìü §æþ ðþÍí³¯èþ Ñ«§æþ V>
and (D). You have to darken the circle as indicated below on the OMR ç³{èþÐèþ$$ÌZ {ç³ {ç³Ôé² çÜ QÅMæü$ CÐèþÓºyìþ¯èþ ¯éË$Væü$ Ðèþé¢ÌZÏ çÜÆðÿO¯èþ
correct response against each item. {ç³çܵ §æþ¯èþ¯èþ$ çÜ*_ ^óþ Ðèþ颰² »êÌý ´ëÆÿ$$ sý ò³¯þø Mìü §æþ ðþÍí³¯èþ Ñ«§æþ V>
Example : A B C D ç³NÇ ^éÍ.
where (C) is the correct response. E§éçßýÆæÿ×ý : A B C D
(C) çÜÆðÿO¯èþ {ç³çܵ §æþ¯èþ AÆÿ$$óþ
5. Your responses to the items are to be indicated in theOMR Answer
Sheet given to you. If you mark at any place other than in the5. {ç³Ôèý²ËMæü$ {ç³çܵ §æþ¯èþ˯èþ$ D {ç³Ôèý²ç³{èþÐèþ$$ø CÐèþÓºyìþ¯èþ OMR ç³{èþÐèþ$$ ò³O¯èþ
circle in the Answer Sheet, it will not be evaluated. CÐèþÓºyìþ¯èþ Ðèþé¢ÌZϯóþ ç³NÇ _ Væü$Ç¢ ^éÍ. AÌêM>Mæü çÜÐèþ*«§é¯èþ ç³{èþ ò³O ÐóþÆöMæü ^ør
6. Read instructions given inside carefully. Væü$Ç¢õÜ¢ Ò$ {ç³çܵ §æþ¯èþ Ðèþ$*ÌêÅ Mæü¯èþ ^óþÄæý$ºyæþ§æþ$.
7. Rough Work is to be done in the end of this booklet. 6. {ç³Ôèý² ç³{èþÐèþ$$ ÌZç³Ë C_a¯èþ çÜ*^èþ¯èþ˯èþ$ gê{Væüèþ¢V> ^èþ§æþÐèþ yìþ.
8. If you write your name or put any mark on any part of the OMR 7. _èþ$¢ç³°° {ç³Ôèý²ç³{èþÐèþ$$ _ÐèþÆæÿ C_a¯èþ RêäçܦËÐèþ$$ÌZ ^óþÄæý*Í.
Answer Sheet, except for the space allotted for the relevant
8. OMR ç³{èþÐèþ$$ ò³O °È~èþ çÜ¦Ë ÌZ çÜ*_ ^èþÐèþËíܯèþ ÑÐèþÆ>Ë$ èþí³µ _ CèþÆæÿ çÜ¦Ë ÌZ
entries, which may disclose your identity, you will render yourself
liable to disqualification. Ò$ Væü$Ç¢ ç³#¯èþ$ ðþÍõ³ Ñ«§æþ V> Ò$ õ³Ææÿ$ Æ>Äæý$yæþ V>± Ìôý§é CèþÆæÿ _à²Ë¯èþ$ ò³rtyæþ
9. The candidate must handover the OMR Answer Sheet to the V>± ^óþíܯèþrÏÆÿ$$óþ Ò$ A¯èþÆæÿáèþMæü$ Ò$Æóÿ »ê«§æþ$ÅËÐèþ#éÆæÿ$.
invigilators at the end of the examination compulsorily and 9. ç ³Èæü ç³NÆæÿ¢Æÿ$$¯èþ èþÆ>Óèþ Ò$ OMR ç³{é°² èþç³µ°çÜÇV> ç³Èæü ç³ÆæÿÅÐóþæüMæü$yìþMìü CÐéÓÍ.
must not carry it with you outside the Examination Hall. The Ðésìý° ç³Èæü Væü¨ ºÄæý$rMæü$ ¡çÜ$Mæü$ÐðþâæýÏMæü*yæþ§æþ$. ç³Èæü ç³NÆæÿ¢Æÿ$$¯èþ èþÆæÿ$Ðéèþ A¿æýÅÆæÿ$¦Ë$
candidate is allowed to take away the carbon copy of OMR {ç³Ôèý² ç³{é°², OMR ç³{èþ Äñý$$MæüP M>Ææÿ¾¯þ M>ï³° ¡çÜ$Mæü$ÐðþâæýÏÐèþ^èþ$a.
Sheet and used Question paper booklet at the end of the
10. ±Í/¯èþËÏ Ææÿ Væü$ »êÌý ´ëÆÿ$$ sý ò³¯þ Ðèþ*{èþÐóþ$ Eç³Äñý*W ^éÍ.
examination.
10. Use only Blue/Black Ball point pen. 11. ÌêVæüÇ£æþÐþ$ sôýº$ÌýÞ, M>ÅÍMæü$ÅÌôýrÆÿË$, GË[M>t°Mü ç³ÇMæüÆ>Ë$ Ððþ$$§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$PË èþWY ç³# Ìôý§æþ$.
IIIÀ 1 A-15-03
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
Text of page 2
*A1503*
HE
RE
WR
ITE
NO
T
DO
IIIÀ 2 A-15-03
Text of page 3
*A1503*
MATHEMATICAL SCIENCES
Paper – III
∞
m
3. The improper integral ∫ x n dx converges
1. If f : [ 0, 1] → is defined by 0
1+ x
only if
⎧⎪ 0 if x = 0 (A) n > m + 1 (B) n = m
f ( x) = ⎨ 1 if 1 < x ≤ 1 for r = 1, 2, 3, ...
⎪⎩2r −1 2r 2r−1 (C) n ≤ m + 1 (D) n ≠ m
1 4. If f : ( a, b ) → is monotonic increasing
then ∫ f ( x ) dx = function and a < c < b then f (c – 0), the
0
left hand limit of f at c is equal to
2 3
(A) (B) (A) inf {f (t) : c < t < b}
3 4
(B) inf {f (t) : a < t < c}
4 5
(C) (D) (C) sup {f (t) : a < t < c}
5 6
(D) sup {f (t) : c < t < b}
2. The correct statement among the
5. If is the collection of all Lebesgue
following is
measurable sets in then the incorrect
(I) fn ( x ) = xn converges uniformly to 0 on
statement among the following is :
[0, a) for any a with 0 < a < 1 (A) is σ - algebra of subsets of
nx
(II) fn ( x ) = converges uniformly (B) Every open set in is a member of
1+ n2 x 2
(C) Every closed set in is a member of
to 0 on
(D) Every member of is a Borel set
(III) fn ( x ) = x n−1
(1− x ) converges only
6. The function
pointwise on (0, 1)
(IV) fn ( x ) = xn converges uniformly on [0, 1) ( )
f ( x, y) = x3 + y3 − 6 x2 + y2 +12xy − 75( x + y)
has maximum value at the point
(A) I (B) II
(A) (5, 5) (B) (–5, –5)
(C) III (D) IV (C) (1, 7) (D) (7, 1)
IIIÀ 3 A-15-03
Text of page 4
*A1503*
7. In the metric space 2, the Euclidean 8. Assertion (A) : In the normed linear space
plane, consider the lists given below : C [a, b] of continuous real-valued
functions on [a, b] with the supremum
List - I List - II
norm, the set P of all polynomials is a
dense subset
(a) A closed (1) {( 1, 0 ∈
n ) 2
: n = 1, 2, 3, ... }
unbounded set Reason (R): Weierstrauss approximation
(b) A bounded set (2) {( x, y) ∈ 2
: x2 + y2 ≤ 7 } theorem holds
which is not closed (A) (A) is true but (R) is false
(c) An unbounded (3) φ
(B) Both (A) and (R) are false
set which is both
open and closed (C) Both (A) and (R) are true but (R) is not
(d) A non-empty (4) {( 0, n) ∈ 2
: n = 0, ± 1, ± 2, ... } a correct explanation for (A)
compact set
(D) Both (A) and (R) are true; and (R) is
(5) {(x, y) ∈ 2 2
: ( x + 1) + y2 > 4 } the correct explanation for (A)
2
9. The function f ( x, y) = x + y − 3x − 12y + 20
(6) 3 3
has minimum at the point
The correct matching of List-I from List-II is
(A) (1, 2) (B) (–1, 0)
(a) (b) (c) (d)
(C) (1, 0) (D) (–1, –2)
(A) 3 2 6 1
∞
dx
(B) 4 3 5 2 10. The value of the integral ∫ is
−∞ 1+ x2
(C) 4 1 6 2 π
(A) (B) 2 π
2
(D) 3 2 4 1 3π
(C) π (D)
2
IIIÀ 4 A-15-03
Text of page 5
*A1503*
14. The solution of the system of equations
11. The determinant of the matrix x1 + 2x2 – x3 = –1
⎛ 1 3 − 3⎞ 2x1 + 2x2 + x3 = 1
⎜ ⎟
A = ⎜ −3 −5 2⎟ 3x1 + 5x2 – 2x3 = –1 is
is
⎜ ⎟
⎝ −4 4 −6 ⎠
⎡4 ⎤ ⎡ 4⎤
(A) 48 (B) 40 (A) ⎢2 ⎥ (B) ⎢ −2⎥
⎢1 ⎥ ⎢1⎥
⎣ ⎦ ⎣ ⎦
(C) 38 (D) 50
⎡ −4⎤ ⎡ 4⎤
12. Let T : 2→ 3 be the linear ⎢ ⎥ ⎢ ⎥
(C) ⎢ −2⎥ (D) ⎢ −3⎥
transformation defined by ⎣1⎦ ⎣ −1⎦
T ( a1, a2 ) = ( a1 + 3a2 , 0, 2a1 − 4a2 )
⎛ 1 2 1⎞
⎜ ⎟
Let β and γ be the standard ordered basis 15. The rank of the matrix A = ⎜ 1 3 4 ⎟ is
2 3
for and respectively. Then the ⎜ ⎟
⎝ 2 3 − 1⎠
γ
matrix corresponding [ T]β is
(A) 1 (B) 3
⎡1 3 ⎤ ⎡3 1⎤
(A) ⎢0 0 ⎥ (B) ⎢0 1⎥ (C) 0 (D) 2
⎢2 − 4 ⎥ ⎢2 − 4 ⎥
⎣ ⎦ ⎣ ⎦
16. Let f (z) be analytic in z ≤ 1 and f ( z ) ≤ 1
⎡ 1 3⎤ ⎡ −1 3 ⎤
(C) ⎢ (D) ⎢ 0 0 ⎥
0 0⎥ with f ( 0 ) = 1+ i . Then f (i) – f (1) =
⎢ −2 − 4 ⎥ ⎢ ⎥
⎣ ⎦ ⎣ −2 − 4 ⎦ 2
(A) 0 (B) i − 1
(C) i (D) 1
⎛ 1 2 ⎞ and ⎛1 0 ⎞
13. Let the matrix A = ⎜ ⎟ I= ⎜ ⎟
⎝ −2 1 ⎠ ⎝ 0 1⎠
then A2 – 2A + 5I is equal to π
17. ∫ ecos θ cos ( sin θ ) dθ =
⎛ 1 1⎞ 0
1 0⎞
(A) ⎛⎜ ⎟ (B) ⎜ 1 1⎟
⎝ 0 1⎠ ⎝ ⎠ (A) 2 π (B) π
⎛ 0 0⎞ ⎛ 0 −1⎞ π π
(C) ⎜ 0 0 ⎟ (D) ⎜ −1 0 ⎟ (C) (D)
⎝ ⎠ ⎝ ⎠ 2 4
IIIÀ 5 A-15-03
Text of page 6
*A1503*
18. Suppose C is the circle |z| = 2 positively 22. The number of groups G such that
1
oriented. Then ∫ z2 + 2iz −1 dz = G =119, where z (G) is the centre of G,
z( G)
C
is
(A) 2 π i (B) − 2π (A) 1 (B) 0
(C) 2 (D) infinite
(C) 2π (D) 0
2π
23. Let α = cos 25π + isin 25π and β = sin .
5
19. Suppose f (z) is analytic on z ≤ 1 such
Then the degree of the field Q ( α ) over
that f ( z ) − z < z on |z| = 1. Then the Q (β ) is
number of zeros of f (z) in |z| < 1 is (A) 1 (B) 2
(C) 4 (D) infinite
(A) 1 (B) 2
24. The order of Galois group of x4 + x2 + 1
over Q is
(C) 3 (D) 5 (A) 1 (B) 3
(C) 2 (D) 5
∞
sin5x dx =
20. ∫ x 25. Let F, K be fields such that K ⊆ F and
0
(A) π (B) 2π ( )
u ∈ F. If ⎡⎣K ( u) : K ⎤⎦ = 5 then ⎡⎣K u : K ⎤⎦ =
2
(A) 25 (B) 5
π π
(C) (D) (C) 10 (D) 15
2 4
21. Suppose C is the circle |z| = 1 positively 26. Consider [i] the ring of Gaussian
oriented, and f ( z ) = ez .
z( z −2 )( z −3 )...( z −9 ) integers and the maximal ideal
Then ∫ f ( z ) dz = M = {a + bi : 3 a, 3 b} in [i] . Then the
c
R
order of the quotient ring is
−πi πi M
(A) (B)
9! 9! (A) 3 (B) 5
−2πi 2πi (C) 7 (D) 9
(C) (D)
9! 9!
IIIÀ 6 A-15-03
Text of page 7
*A1503*
27. A maximal ideal in [ x] among the 31. If the complementary function of the
following is differential equation y ′′ − y ′ − 6y = 0 is
y c = Aeαx + Beβx , then α2β2 =
(A) x4 + 4 (B) x3 + 1
(A) 4 (B) 16
5 2
(C) x +1 (D) x +2
(C) 64 (D) 36
28. Let T be the Cantor’s set in . Then which
32. If the solution of the diferential equation
of the following is incorrect ?
(D3 – D2 – 4D + 4) y = e3x is
(A) T is closed y = c1ex + c 2e2x + c3e−2x + Ke3x , then K =
1
(B) T is compact (A)
10
(C) T is bounded
(B) 1
5
(D) T is connected
(C) 5
29. Consider the topology
(D) 10
2
T = {φ, X, {x} , {z, w} , {x, z, w} , {y, z, w, u}} 33. If y ( 0 ) = 2, y′ ( 0 ) = −1 and d y + y = 0,
dx2
then y =
on X = {x, y, z, w, u}. Then the number
(A) 2sinx + cosx (B) 2cosx + sinx
of components of X is
(C) 2cosx – sinx (D) 2sinx – cosx
(A) 1 (B) 2
34. The general solution of the differential
(C) 3 (D) 4
equation
30. Suppose X is a compact metric space. ( x2z − y3 ) dx + 3xy2dy + x3dz = 0 is
Then which of the following statements is
not true ? (A) x2z2 + y3 = c
(A) X is separable 2 3
(B) x z + y = cx
(B) X is closed
(C) x2z + y3 = c
(C) X is sequentially compact
2 2 3
(D) x z + y = cx
(D) X is not separable
IIIÀ 7 A-15-03
Text of page 8
A *A1503*
35. If x 2 + y 2 + lz3 + mz2 + 2 = 0 is the surface 40. By dividing [0, 1] into 4 equal sub intervals,
that intersects the system of surfaces 1
dx
the value of ∫ 1+ x (using trapezoidal rule)
z (x + y) = C (3z + 1) orthogonally and 0
passes through the circle x2 + y2 = 1, correct to 3 decimals is
z = 1 then l + m = (A) 0.693 (B) 0.694
(A) 3 (B) –3
(C) 0.697 (D) 0.699
(C) 2 (D) –2
1⎛ dy ⎞ ⎞
2
∂z 2 ⎛
z = ey , = 1 when x = 0, then the 41. ∫ ⎜ y + ⎜ ⎟ ⎟ dx along the path y = x2 is
36. If 0⎝
⎜ ⎝ dx ⎠ ⎠⎟
∂x
∂2z 23 18
solution of + z = 0 is z = (A) (B)
∂x2 15 15
y
(A) sin x + e cos x 28 33
(C) (D)
y
(B) e sin x + cos x 15 15
y
(C) sin x cos x + e 42. If the solution of
y
(D) sin x cos x – e
1
u′ ( x ) + ∫ exp ( x − y ) u( y ) dy = f ( x ), u( 0) = 0
37. The partial differential equation obtained 0
by eliminating a, b from the equation 1
z = xy + y x2 − a2 − b2 is
( )
is u ( x ) = g ( x ) + λ e − 1 ∫ e g ( y ) dy
x −y
0
(A) px – qy = pq (B) px + qy + pq = 0
x
(C) px + qy = pq (D) px – qy + pq = 0
where g ( x ) = ∫ f ( t ) dt, then λ =
0
38. If a cubic polynomial f (x) is such that
f (0) = 1, f (1) = 0, f (2) = 1 and f (3) = 10,
e
then f (4) = (A)
1+ e
(A) 43 (B) 33
(B) 1 – e
(C) 23 (D) 13
−e
(C)
1+ e
39. With the standard notation 4μ2 − δ2 =
1
(A) 4 (B) 3 (D)
1− e
(C) 2 (D) 1
IIIÀ 8 A-15-03
Text of page 9
*A1503*
43. The resolvent kernel of 47. Which of the following is not a property of
a distribution function F (x) of a random
x
ϕ ( x ) = f ( x ) + ∫ ( x − t ) ϕ ( t ) dt ( t < x ) is variable X ?
0 (A) P ( a < X ≤ b ) = F ( b ) − F ( a )
(A) sinh (x – y) (B) F ( x ) ≤ F ( y ) if x < y
(B) sin (x – y)
(C) F ( −∞ ) = 0 and F ( ∞ ) = 1
(C) cos (x – y) 1
(D) F ( 0 ) =
(D) cosh (x – y) 2
48. Let X be a random variable with mean μ
44. Assume that a piston executes a simple 2
and variance σ > 0. Let k > 0 be fixed.
harmonic motion with an amplitude 0.15 m.
If it passes through the centre of its motion Then which of the following is not correct ?
with a speed of 0.3 m/s then the period of 1
oscillation (in per seconds) is (A) P ( X − μ ≥ kσ ) ≤
k2
(A) π (B) 2π 1
(B) P ( X − μ ≤ kσ ) ≥ 1−
π 2π k2
(C) (D)
2 3 σ2
(C) P ( X − μ ≥ k ) ≤
k2
(2
)
45. If F = ax + by i + c xy i is conservative, 1
(D) P ( X − μ ≥ k ) ≤ 2
k
then
(A) a + b + c = 0 49. Let {xn} be a sequence of i.i.d. random
2
(B) a = bc variables with finite mean μ and finite
variance. Consider the following two
(C) c = – 2b
statements :
2 2
(D) b + c = 2 ab n
2
P: ∑
n ( n + 1) i =1
p
i ⋅ xi ⎯⎯ →μ ;
46. An event A is independent of itself if and
n
6 p
only if Q: ∑
n (n + 1)( 2n + 1) i =1
i2 xi ⎯⎯ →μ
(A) P (A) = 0.5
then which of the following is true ?
( )
(B) P ( A ) + P A = 1 (A) Both P and Q are false
(B) Both P and Q are true
(C) P (A) = 0 or P (A) = 1
(C) P is true but Q is false
(D) P (A) = 0.25 (D) P is false but Q is true
IIIÀ 9 A-15-03
Text of page 10
*A1503*
50. Let {xn} be a sequence of i.i.d. random
variables with mean μ and finite variance. ( 2
)
53. Let X be normal N μ, σ . Let I be the set
Let Xn be the mean of the first n random of all integers. Then P ( X ∈ I) is equal to
variables. Now, consider the statement
(A) 1 (B) 0
P:
(
n Xn − μ ) ⎯⎯
d
→ N ( 0, 1)
(C)
1
2
(D)
1
2π
Xn
Then which of the following condition is 54. Let X be the mean of n i.i.d. standard
necessary for P to be true ?
Cauchy random variables. Then X is
(A) X1 follows Poisson P ( μ )
(B) X follows Binomial B (k, μ ) distributed as
1
(C) X follows Normal N μ, σ
1
( 2
) (A) Standard normal
(D) X1 follows Normal N (μ, 1)
(B) Laplace
51. If Transition Probability Matrix (TPM) of a
⎡ 21 21 ⎤ (C) Standard Cauchy
Markov chain (MC) is P = ⎢ ⎥ , then its 1
⎢⎣ 34 41 ⎥⎦ (D) Cauchy with scale parameter
n
stationary distribution is given by
⎛ 1 1⎞ ⎛ 3 1⎞ 55. The procedure of improving the efficiency
(A) ⎜ , ⎟ (B) ⎜ , ⎟
⎝2 2⎠ ⎝4 4⎠
of an unbiased estimator with the use of a
⎛3 2⎞ ⎛ 4 1⎞
(C) ⎜ , ⎟ (D) ⎜ , ⎟ sufficient statistic was invented by
⎝5 5⎠ ⎝5 5⎠
⎡ 0 ⋅ 1 0 ⋅ 2 0 ⋅ 7⎤ (A) Lehmann
⎢ ⎥
52. If TPM of a MC is P = ⎢ 0 ⋅ 2 0 ⋅ 2 0 ⋅ 6 ⎥ ,
⎢⎣ 0 ⋅ 6 0 ⋅ 1 0 ⋅ 3⎥⎦ (B) Scheffe
then P ( X 3 = 1 X1 = 0 ) is
(C) Cramer
(A) 0.26 (B) 0.17
(C) 0.14 (D) 0.13 (D) C.R. Rao
IIIÀ 10 A-15-03
Text of page 11
*A1503*
56. Let X1, X2 , , Xn be a random sample
( 2
)
59. Let X N θ, σ . Which of the following
of size n drawn from uniform U ( 0, θ )
distribution. If X (n) is the largest is a simple hypothesis ?
observation in the sample, then which of
(A) H : θ = θ0 (B) H : σ = σ0
the following is an unbiased estimator of θ ?
(A) Sample mean
(C) H : θ = θ0, σ = σ0 (D) H : θ ≠ θ0
(B) Sample median
(C) X(n) 60. The mean of R in Runs test under H0 with
n+1
(D) X
n ( n) usual notations is given by
57. The efficiency of sample mean as 2m 2n
(A) +1 (B) +1
compared to sample median as estimator m+n m+n
of the mean of a normal population in 2mn 2mn
(C) (D) +1
percentage is m+n m+n
(A) 64 (B) 157
61. The Gauss Markov theorem
(C) 317 (D) 31.5 establishes that the G.L.S. estimator
∧
( ) X V Y is
−1
58. A random sample of size 1 is taken from of β, β = X1 V −1X 1 −1
a p.d.f. (A) Unbiased estimator only
2 (θ − x ) (B) Error free estimator
f ( x, θ ) = 2
, 0 < x < θ ; f ( 0, θ ) = 0,
θ
(C) Both (A) and (B)
elsewhere. The most powerful test of
H0 : θ = θ0 Vs H1 : θ = θ1, θ1 < θ0 at (D) Best linear unbiased estimator
level α is given by 62. The log linear models are analogous to
(A) φ ( x ) = 1 if x > θ0 ⎣⎡1− 1− α ⎦⎤ ANOVA models with
(A) Unequal number of observations
(B) φ ( x ) = 1 if x < θ0 ⎡⎣1− 1− α ⎤⎦
(B) Equal number of observations
(C) φ ( x ) = 1 if x > θα0
(C) Multiple observations
(D) φ ( x ) = 1 if x < θ 0 α
(D) Both (A) and (B)
IIIÀ 11 A-15-03
Text of page 12
*A1503*
63. If R1.23 = 1, then 67. If A1, A 2 , , A q are independently
(A) At least one regression residual is distributed with Ai distributed according to
non-zero
Wishart distribution W ( Σ, ni ) then
(B) The multiple linear regression
q
equation of X on X and X is
1 2 3 A = ∑ Ai is distributed according to
considered as perfect for predictions i=1
(C) All total correlations involving X are
1
zero ⎛ q ⎞
(A) W ⎜ Σ, ∑ ni ⎟
(D) All partial correlations involving X are ⎝ i =1 ⎠
1
zero
(B) W ( Σ, ni )
64. In a logistic regression the S-shaped curve
is not symmetrical about its ⎛ n ⎞
(C) W ⎜ Σ−1, ∑ ni ⎟
(A) Increase in point of inflection ⎝ i =1 ⎠
(B) The curve increases rapidly
⎛1 ⎞
(D) W ⎜ , ni ⎟
(C) The curve decreases ⎝Σ ⎠
(D) Point of inflection
68. Let A and ∑ be partitioned into q and p – q
65. The independent variables in logistic rows and columns
regression are called as
(A) Variates ⎛A A ⎞ ⎛ Σ11 Σ12 ⎞
A = ⎜ 11 12 ⎟ ; Σ = ⎜ ⎟.
(B) Covariates A A
⎝ 21 22 ⎠ Σ Σ
⎝ 21 22 ⎠
(C) Both (A) and (B)
(D) Logit If A is distributed according to Wishart
66. A square symmetric matrix A and its distribution W ( Σ, n) then A 11 is
associated quadratic form is called
positive definite if distributed according to
(A) x′Ax ≤ 0 for every ‘x’ not equal to the
(A) W ( Σ, n)
null vector
(B) x′Ax > 0 for every ‘x’ not equal to the
(B) W ( Σ11, n)
null vector
(C) x′Ax < 0 for every ‘x’ not equal to the (C) W ( Σ12 , n)
null vector
(D) x′Ax ≥ 0 for every ‘x’ not equal to the
null vector
−1
(
(D) W Σ11, n )
IIIÀ 12 A-15-03
Text of page 13
*A1503*
69. The function a′x is known as 73. Suppose f(x) is a density on ( 0, ∞ ) with
(A) Multiple Discriminant Analysis distribution function f (x) which is DFR then
(B) Linear Logistic Regression the function g (x) = log f(x) is
(C) Linear Discriminant Function
(A) Convex
(D) Logistic Discrimination
70. If the entries in Rows of a Latin square are (B) Concave
same as its columns, the Latin square is
called (C) Constant
(A) Conjugate
(D) The function value is 1
(B) Self conjugate
(C) Orthogonal 74. In which method we use the formula
(D) Symmetric
{ }
Min Xb : Xb < 0 to obtain the learning
i i
71. The method of confounding is a device to
variable
reduce the size of
(A) Experiments (A) Dual Simplex
(B) Replications
(B) Duality Problem
(C) Blocks
(D) All the above (C) Big M-method
72. The systematic sampling and S.R.S. shall (D) Simplex Method
give estimates of equal precisions if the
inter class correlation between the units
75. If the arrival rate is 3 per hour and service
of the same systematic sample, from a
population of size N and a sample of size rate is 6 per hour then the traffic intensity
n, is equal to
ρ equals to
1 −1
(A) (B)
N−n n−1 (A) 6 (B) 3
−1 −1
(C) (D)
N−n N−1 (C) 2 (D) 0.5
IIIÀ 13 A-15-03
Text of page 14
*A1503*
Space for Rough Work
IIIÀ 14 A-15-03
Text of page 15
*A1503*
Space for Rough Work
IIIÀ 15 A-15-03
Text of page 16
*A1503*
Space for Rough Work
IIIÀ 16 A-15-03Official key not published for this paper.
Page 1 of 16
Source: Telangana State Eligibility Test. Question paper published by Telangana State Eligibility Test.