TS SET 2012: Paper III, Mathematical Sciences
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Test Paper : III Test Booklet Serial No. : ______________________
OMR Sheet No. : ________________________________
Test Subject : MATHEMATICAL SCIENCES
Hall Ticket No.
Test Subject Code : A-15-03
(Figures as per admission card)
Name & Signature of Invigilator
Name : ____________________________________ Signature : ___________________________________
Paper : III
Subject : MATHEMATICAL SCIENCES
Time : 2 Hours 30 Minutes Maximum Marks : 150
Number of Pages in this Booklet : 16 Number of Questions in this Booklet : 75
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IIIÀ 1 A-15-03
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*A1503* IIIÀ 2 A-15 – 03
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*A1503*
MATHEMATICAL SCIENCES
Paper – III
1. The series 1 + 1p + 1p + ... converges if ⎧1, if x is rational
2 3 4. If f(x) = ⎨ , then
(A) P < 1 (B) p > 1 ⎩0, if x is irrational
(C) p < 1 (D) p = 0
(A) f has discontinuities of the second
2. If {fn} be a sequence of integrable functions
kind at every point x
defined on [a, b], then
(A) If fn → 0 a.e and each fn is a bounded (B) f has discontinuities of the first kind
b at every point x
function, then ∫a f n (x)dx → 0
(C) f(x+) exists
(B) If fn → 0 a.e and fn are uniformly
bounded functions, then (D) f(x−) exists
b
∫a f n (x)dx → 0 5. Compact subsets of metric spaces are
b
(C) If ∫a f n (x)dx → 0 , then fn → 0 a.e (A) convex
b (B) open
(D) If fn → 0 a.e, then ∫a f n (x)dx → 0
(C) closed
3. Suppose a and c are real numbers, c>0,
(D) connected
and f is defined on [ −1, 1] by
fn(x) = xa sin(x– c ), if x ≠ 0 6. Let E0 denote the set of all interior points
= 0, if x = 0 of a set E. Then which one of the
statements is NOT true?
Then which one of the following
statements is true? (A) E0 is always open
(A) f ′ is bounded ⇔ a <1 + c
(B) E is open if and only if E0 = E
(B) f′ is continuous ⇔ a < 1+ c
(C) f′ (0) exists ⇔ a < 1 (C) E is closed
(D) f is continuous ⇔ a > 0 (D) If G ⊂ E and G is open, then G ⊂ E0
IIIÀ 3 A-15 – 03
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*A1503*
7. The Cayley Hamilton theorem states that 10. Which of the following map is NOT a linear
transform?
(A) The characteristic roots of a non-
(A) T : V 3 ( R ) → V 2 ( R ) defined by
singular matrix are distinct
T(x,y,z) =( x – y, x – z)
(B) The characteristic equation of a
(B) T : V 3 ( R ) → V 1 ( R ) defined by
matrix admits a non-zero solution
T(x,y,z) =x2 + y 2 + z 2
(C) Every matrix satisfies its own
(C) D: V → V defined by D(f(x)) = df(x) / dx,
characteristic equation
where V is a vector space of
(D) The eigen values of any matrix are polynomials f(x) in x over R
linearly independent
x
(D) I : V → V defined by I(f(x)) = ∫0 f(x) dx ,
8. If A is an upper triangular matrix, then
where V is a vector space of
which one of the following statements is
polynomials f(x) in x over R.
NOT true?
(A) A′ is a lower triangular matrix 11. If w is a subspace of V4(R) generated by
the vectors (1,–2, 5, –3 ), ( 2, 3, 1, –4),
(B) |A| is the product of its diagonal
( 3, 8, –3, –5), then dim(w) =
elements
(C) Rank of A is equal to the order of A (A) 1 (B) 2
(D) Eigen values of A are the diagonal (C) 3 (D) 4
elements of A
12. A l i n e a r t r a n s f o r m w h i c h
9. If α and β are two eigen values of a matrix reduces the quadratic form
A, then the corresponding eigen vectors x2 + 4y2 + 4z2 + 4xy + 4xz + 16yz to
of A are the canonical form is
(A) x = u + 2v, y= v – 2w, z = v + 1 2 w
(A) linearly dependent
(B) orthogonal (B) x = u – 2v, y= v+ 1 2 w, z = v – 1 4 w
(C) orthonormal (C) x = u – 4v, y= v − 1 2 w, z = v + 1 2 w
(D) linearly independent (D) x = u + 4v, y= v + 1 w, z = v – 1 w
2 2
IIIÀ 4 A-15 – 03
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*A1503*
13. If u(x,y) = 2x – x2 + ky2 is to be harmonic 16. The Taylor series expansion of
function, then k should be equal to z −1
f (z) = about the point z =1 is
z +1
(A) 0
(A) 1
2 (z − 1) − ( 1 2 )2 (z −1)2 + ( 1 2 )3 (z − 1)3 − ...
(B) 1
(B) 1
2 ( z − 1) + ( 1 2 )2 (z − 1) 2 + ( 1 2 )3 (z − 1)3 + ...
(C) 2
(C) ( z − 1) − 1 2 ( z − 1) 2 + ( 1 2 )2 ( z − 1) 3 − ...
(D) 3
(D) ( z − 1) + 1 2 ( z − 1) 2 + ( 1 2 )2 ( z − 1) 3 + ...
14. The bilinear transformation which maps
1, i, –1 to 2, i, –2 respectively is sin z
17. The residue of f (z) = at the pole
z cos z
2i − 6z z = π /2 is
(A) w =
iz − 3
(A) 1 (B) – 2/ π
zi − 3
(B) w = (C) π /2 (D) 0
2i − 6
2i + 6z 18. If f(z) has a pole of order 3 at z = a, then
(C) w =
iz + 3 Res {f(z)} =
2i + 6 z
{ }
2
(D) w = (A)
1 d 3
( z − a ) f (z )
iz − 2 6 dz 2
z =a
{ }
2
z 1 d 3
15. The value of I = ∫ 2
dz over (B) 2 2 ( z − a ) f ( z )
c z − 3z + 2 dz z=a
C: z − 2 = 1
2 is
{ }
2
1 d 2
(C) (z − a) f (z)
(A) 2πi 6 dz2
z =a
(B) 4πi
{ }
2
(C) 6πi 1 d 2
(D) − 2
(z − a ) f ( z)
(D) 8πi 2 dz z=a
IIIÀ 5 A-15 – 03
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*A1503*
19. If w = logz, then w is not analytic 22. Let O(G) = pq with p, q primes, p > q where
G is a group. Then which one of the
(A) On real axis
following statements is NOT true?
(B) on negative real axis (A) If q | p – 1 , then there exists a non-
abelian group of order pq
(C) at z=0 only
(B) If q | p – 1 , then G is cyclic
(D) for complex z
(C) G has a subgroup of order p and a
20. If H and K are two subgroups of a group G, subgroup of order q
then HK is a subgroup of G if and only if (D) Any two non-abelian groups of order
pq are isomorphic
(A) O(HK) = O(KH)
23. Let G be a group of order 112.132. Then
(B) HK =KH
the number of 13 – Sylow subgroups in G
(C) O(H) = O(K) are
(D) either O(H) or O(K) is a prime number (A) 1 (B) 2
(C) 3 (D) 4
21. If H is a subgroup of G and N(H) = { a ε G |
24. If p is a prime number, then the splitting
a Ha–1 =H} then N(H) is
field over F, the field of rational numbers,
(A) a normal subgroup of G of the polynomials xp – 1 is of degree
(B) a subgroup of G (A) 2p+1
(B) p +1
(C) an abelian group
(C) p – 1
(D) a cyclic group
(D) 2p – 1
IIIÀ 6 A-15 – 03
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*A1503*
25. Let I1 be the ideal generated by x4+ 3x2+ 2 29. Let y1(x) and y2(x) be a set of fundamental
and I2 be the ideal generated by x3 + 1 in
solutions of the equation
Q[x]. If F1 = Q[x]/I1 and F2 = Q[x]/I2, then
(A) F1 is not a field while F2 is a field y′′ + p( x )y′ + q(x)y = 0 a < x < b, where
(B) neither F1 nor F2 is a field p(x) and q(x) are continuous in [a, b] and
(C) F1 and F2 are fields x0 ε (a,b). Then which of the following
(D) F1 is a field but F2 is not a field
statements is NOT true?
26. The particular integral of the equation
2 2 (A) y1(x) and y2(x) both can not have a
yy′ = y y′ + ( y′) ; y(0) = − 1 2 and
y ′(0) = 1 is local minimum at x
0
2 x/2
(A) 2y – 1 = 3y e
(B) y1(x) and y2(x) both can not have a
(B) 2y – 1 = ye5x/2
local maximum at x0
(C) 2y – 3 = 8ye3x/2
(D) 2y + 3 = 6y2ex/2 (C) y1(x) can have a local maximum at
x 0 and y 2 (x) can have a local
27. The singular solution of the equation
xp 2 – 2yp + 4x = 0 is minimum at x0
(A) x (B) 2x
(D) y 1 (x) and y2 (x) both can vanish
(C) 3x (D) 4x
simultaneously at x0
28. Consider the boundary value problem:
y′′ + λy = 0, y (− π) = y ( π) , y′(− π) = y′(π)
30. A complete solution of the partial
then to each eigen value λ there
corresponds differential equation √p + √q = 2x is
(A) a unique eigen function
(A) z= 1 4 (zx+a)2 + a2y +b
(B) two eigen functions
(C) two linearly independent eigen (B) z= 1 6 (zx+a)3 + a2y +b
functions
(C) z= 1 6 (zx+a)2 + ay +b
(D) two mutually dependent eigen
functions (D) z= 1 4 (zx+a) + ay +b
IIIÀ 7 A-15 – 03
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*A1503*
31. A particular integral of the equation 35. The approximate value of y(0.2) obtained
(D 2 – D′ 2 )z = cos(x + y) is
upto 3 decimal places by solving y′ = x+y,
(A) x cos(x+y) y(0) =1 with h = 0.2 using the fourth order
(B) ½ x cos(x+y) Runge - Kutta method, is
(C) x2sin(x+y) (A) 1.216
(D) ½ x sin(x+y)
(B) 1.232
32. A solution of the equation
(C) 1.243
∂u ∂u
= 4 , u(0, y) = 8e–3y is
∂x ∂y (D) 1.263
(A) u= 8 e–12x−3y
36. The shortest curve joining two fixed points
–4x–3y
(B) u= 6 e
on a given surface and lying entirely on
(C) u= 4 e–4x– y
that surface is called a
– 8x– 3y
(D) u= 2 e
(A) cycloid
33. A real root of the equation x log10x = 1.2
(B) geodesic
correct to 3 decimal places obtained by
(C) catenary
Newton’s iteration method , is
(D) helix
(A) 2.740 (B) 2.752
37. A triangle enclosing the greatest area for
(C) 2.758 (D) 2.762
a given perimeter, is
34. Given that y′ = – x2y, y(0) =2, then the
(A) an isosceles triangle
value of y (0.2) obtained by modified
(B) an equilateral triangle
Eulers method with h = 0.1 is
(C) a right angled triangle
(A) 1.912 (B) 1.918
(D) a scalene triangle
(C) 1.923 (D) 1.932
IIIÀ 8 A-15 – 03
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*A1503*
40. A body of mass m is moving in two
d ⎛ ∂ f ⎞ ∂f
38. The condition that ⎜ ⎟− = 0 for dimensions with G as centre of mass. If v
d x ⎜⎝ ∂ y ′ ⎠⎟ ∂ y
is the velocity of G and k is the radius of
x gyration of the body about a line passing
the integral I = ∫ 2 f (x, y, y′)dx to have a
x1 through G, perpendicular to the plane of
motion and θ is the angle between a line
stationary value, is
fixed in the body with a line fixed in space,
then the kinetic energy of that body is
(A) only a necessary condition
+ k )
2 2 2
(A) 1
2 m(v
(B) a necessary and sufficient condition
(B) m ( v 2 + k 2θ 2 )
(C) only a sufficient condition
2 2 2
(C) 1 m(2 v
2 +k θ )
(D) neither necessary nor sufficient 2 2 2
(D) 1 m( v
2 + 2k θ )
condition
41. The number of degrees of freedom of a
particle moving in space, is
39. A particle of mass m with polar coordinates
(A) 4 (B) 2
r, θ is subjected to a central force (C) 1 (D) 6
described by the potential v(r). Then its 42. Let L be the Lagrangian of a conservative
system under no constraints and qk be a
Lagrangian is
∂L
generalised coordinate. If = 0 , then
∂q k
1 2 2
(A) mr + − v( r ) its generalised momentum is
2
d ⎛ ∂L ⎞
(A) ⎜ ⎟
1
(B) m( r + r ) − v (r )
2 2 2
dt ⎝⎜ ∂q k ⎟⎠
2
d ⎛ ∂L ⎞
1 (B) ⎜ ⎟
(C) m(r + ) − v (r ) dt ⎝⎜ ∂q k ⎟⎠
2 2
2
(C) constant of motion
1 2 2
(D) m(r + ) − v ( r ) (D) undefined
2
IIIÀ 9 A-15 – 03
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*A1503*
43. The eigen values of the integral equation 47. For a negatively skewed distribution, the
1 2
y(x) = λ ∫ ( 2xt − 4 x ) y(t)dt , are correct relation between mean, median
0
and mode is
(A) 2, 3 (B) 2, – 3
(A) mean=median= mode
(C) 3, 3 (D) – 3, 3
(B) mean < median < mode
44. The Voltera’s equation of the second kind
obtained by reducing the integral equation (C) median< mean < mode
x 2 2 2 (D) mode < mean < median
∫0 ( 2 + x − t ) y(t)dt = x , is y(x) =
x 48. With the usual notation Bowley’s
(A) 2 x − 2 ∫0 xy ( t ) dt
coefficient of skewness
x
(B) x + 2 ∫0 xy ( t ) dt Q3 + Q1 − 2M
(A)
Q3 − Q1
x
(C) 2 x − 3 ∫0 xy ( t ) dt Q3 − Q1 − 2M
(B)
Q3 + Q1
x
(D) x − 2 ∫0 xy ( t ) dt
Q3 − Q1 + 2M
(C)
Q3 − Q1
45. The extremals of ∫
1
−1
{( y′′) + y}dx subject
2
Q3 − Q1 + 2M
to y(+1) =0 and y′ (+1) =0, is (D)
Q3 + Q1
1 2 1
(A) ( x − 1) (B) − ( x − 1)
24 24
49. If a random variable U is uniformly
1 2 2 1 2 distributed over the interval (0, 1), then the
(C) − ( x − 1) (D) ( x − 1)
24 24
distribution of Y = tan( π (U – 0.2)) is
46. The mean absolute deviation is minimum
(A) Normal
when it is measured from
(A) mean (B) Laplace
(B) median
(C) Cauchy
(C) mode
(D) standard deviation (D) Exponential
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50. Which of the following is NOT the 53. Identify in the following, a stochastic
distribution function of a random variable? process that is an example of discrete time,
⎧ 0 if x < 0 discrete state space stochastic process
(A) F ( x ) = ⎨ −x
⎩1 − e if x ≥ 0 (A) Yule process
⎧ 0 if x < 0 (B) renewal process
(B) F ( x ) = ⎨ − x
⎩e if x ≥ 0
(C) Wiener process
⎧0 if x < 0 (D) Branching process
⎪
(C) F ( x ) = ⎨ 12 if 0 ≤ x < 1
⎪1 if x ≥ 0
⎩ 54. If the mean of a binomial distribution is 3,
then its variance could be
(D) F ( x) = ⎧
0 if x < 1
⎨
⎩1 if x ≥ 1 (A) 7.5
51. The domain of a probability measure is (B) 5.5
(A) the sample space
(C) 3.5
(B) the interval [0,1]
(C) the sigma field of subsets of the (D) 1.5
sample space
(D) the real line 55. Let {x n } be a sequence of i.i.d non-
52. Let {xn} be a Markovian chain on S= {1, 2, 3} negative r.v’s with common mean θ
with the following transition probability and N be a Poisson r.v with mean λ ,
⎡0 23 31 ⎤ which is independent of xi’s. Then
⎢ ⎥
matrix : P = 21 0 21 , then which of the
⎢ ⎥ E(x 1 + x 2 +… + x n ) =
⎢⎣ 21 21 0⎥⎦
(A) λ / θ
following properties hold good for {xn} ?
(A) {xn} is irreducible (B) λ θ
(B) all states are aperiodic
(C) λ
(C) all states are persistent
(D) all the above three (D) θ
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56. If X and Y are two i.i.d standard normal 59. Let x1, x2, … , xn be a random sample from
variates then E(X/Y)=
( n n 2
N( θ , θ 2 ) and T = Σi =1 x i , Σi =1 x i . )
Then T is
(A) 1
(A) complete and sufficient for θ
(B) 0
(B) complete but not sufficient for θ
(C) does not exist
(C) not complete but sufficient for θ
(D) ½
(D) neither complete nor sufficient for θ
57. With the usual notations the standard error
60. An example of a distribution which does
of sample variance is
not possess monotone likelihood ratio
2
(A) σ √(2/n) property is
2
(B) σ (A) Normal
2
(C) σ √(n/2) (B) Cauchy
2 (C) exponential
(D) σ /2n
(D) Gamma
58. Let (x1, x2, … , xn) be a random sample
61. Neyman - Pearson most powerful test with
from U( θ –1 , θ +1). Then the M.L.E of θ is
power 1 – β and size α satisfies
(A) x(1)+1
(A) α < 1 – β
(B) x(n) – 1
(B) β < 1– α
(C) ½ (x(1) + x (n) )
(C) 1 – β < α
1 2
(D)
3
(x( n ) − 1) + (x( n ) + 1)
3 (D) α < β
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62. If the sample size in Wald - Wolfowitz runs 64. The non-parametric test used for testing
test is large, the number of runs in the equality of more than two means of the
population is
pooled ordered sample is distributed with
mean (A) Kruskal Wallis
2m (B) Median test
(A) +1
m+n
(C) Run test
2n
(B) +1
m+n (D) Rank test
2mn
(C) 65. With the usual notations 1– R1.232 equals
m+n
to
2mn
(D) +1
m+n (A) (1 − γ12 )(1 − γ13.2 2 )
63. If two regression lines are perpendicular
(B) (1 − γ 2 )(1 − γ
13.2 )
12
to each other then the relation between the
(C) (1− γ12 )(1− γ 13.2 )
two regression coefficients is
2 2
(D) (1 − γ12 )(1 − γ13.2 )
(A) β xy = β yx
66. Confounding is a technique used for
(B) β xy β yx =1
(A) reducing the block size
(C) – β xy = β yx (B) increasing the block size
(C) reducing the number of factors
(D) β xy > β yx
(D) reducing the number of blocks
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67. A sample of size n is drawn from a 70. For a BIBD with b blocks such that each
dichotomous population. If the sample has
proportion p of items of category I and treatment occurs in r blocks, the rank of
proportion q of category II then the
variance of proportion is the incidence matrix N satisfies
2
(A) s p = pq /( n − 1)
(A) rank(N) = b – (r –1)
2
(B) s p = pq / n
2
(C) s p = npq /( n − 1) (B) rank(N) > b – (r – 1)
2 2
(D) s p = p q /( n − 1)
(C) rank(N) < b – (r – 1)
68. For the linear programming problem,
(D) rank(N) = b+ (r – 1)
maximize z = x + 2y
subject to : x + y >1, x + 2y < 10, y< 4, 71. In the context of two group classification
x > 0, y > 0
problem, the maximum likelihood
(A) Optimum solution is unique
allocation rule for assigning x0 in terms of
(B) optimum solution exists but not
unique usual notation is
(C) optimum solution is unbounded
f1( x 0 )
(D) optimum solution does not exist (A) assign x0 to π I if >1
f2 (x 0 )
69. The hazard function corresponding to the f1 ( x 0 ) p2
(B) assign x0 to π I if >
f 2 ( x 0 ) p1
−1 − x
pdf: f(x ; θ ) = θx e , x > 0, θ >0 is
(A) always increasing in x f1 ( x 0 ) c(1 / 2) p2
(C) assign x0 to π I if >
f 2 ( x 0 ) c( 2 / 1) p1
(B) always decreasing in x
(C) decreasing in x if θ <1 f1 ( x 0 ) c(1 / 2) p1
(D) assign x0 to π I if >
f 2 ( x 0 ) c( 2 / 1) p2
(D) increasing in x if θ <1
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72. In a two factor fixed -effects model with 74. If X and Y are two i.i.d r.v’s with mean 0
interaction, where A has ‘a’ levels and B and variance σ 2, then P[|x + y |> 2]<
has ‘b’ levels, we assume that
(A) σ 2
a b b
(A) Σi =1α i = 0 , Σ j=1 βi = 0 , Σ j=1 γ ij = 0 ,
(B) σ 2/2
a
Σi=1 γ ij = 0
(C) σ 2/4
a b a
(B) Σi =1α i = 0 , Σ j=1 βi = 0 , Σi =1 γ ij = 0
(D) 2 σ – 2
a b b
(C) Σi =1α i = 0 , Σ j=1 βi = 0 , Σ j=1 γ ij = 0
75. For economic order quantity models
a b
(D) Σi =1 γ ij = 0 , Σ j=1 γ ij = 0 i) the demand rate is constant always
ii) the replenishment rate is either finite
73. The bath-tup curve type of hazard function or infinite
is associated with iii) the items are non-deteriorating only
Then
(A) Weibull distribution
(A) i) is correct
(B) exponential distribution
(B) i) is not correct but ii) and iii) are
(C) two component mixture of Weibull correct
distributions (C) ii) is correct
(D) Pareto distribution (D) i), ii) and iii) are correct
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Space for Rough Work
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Source: Telangana State Eligibility Test. Question paper published by Telangana State Eligibility Test.