## Wednesday, May 20, 2009

### Answers/Solutions/Hints for DSE 2004-2008: Mathematics/Statistics Part

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DSE 2004 Mathematics/Statistics

ECOPOINT : MA Economics 2009 Entrance Coaching by DSE Alumni. 9999886629, 9873903314, 9899048237

Ans 1 (b) only proposition (i) is true (prove using contradiction: if possible \$ x Î BÇC, then x Î B and x Î CÞ ? )

Ans 2 (b) there exists a solar system in the milky way galaxy such that every planet in it has a moon revolving around it without any life forms.[negate the given statement]

Ans 26. (c) 2/3 [use baye’s theorem] Ans 27. (d) 3/32 Ans 28. (a) 2/3 Ans 29. (c ) 7.4

Ans 30. (d) 2p2

Part 2

Ans 2 A (a) np(1-p)(pn-2 + (1-p)n-2)

A (b) [1-np(1-p) (pn-2 + (1-p)n-2)]k-1 [np(1-p) (pn-2 + (1-p)n-2)]

Ans 2 B (a) a=1/48 [ use property of a pdf]

B (b) prob(3 <X <5) = 7/24 [ use 3ò5 a(x-3)dx ]

B (c ) prob (X³ 4) = ¾

DSE 2005 Mathematics/Statistics

ECOPOINT : MA Economics 2009 Entrance Coaching by DSE Alumni. 9999886629, 9873903314, 9899048237

Ans 5 (b) a tribe Ans 6 (d) a tribe

Ans 7 (b) there are atleast two sets of individuals that are both a Club and a Tribe

Ans 8 (a) a club

Ans 13 (d) every book by Bourbaki contains a chapter such that the validity of all the theorems in it is independent of the reader’s gender. [ negate the statement]

Ans 16 (a) P(E È F) = P ( Number of Heads > number of Tails or number of Tails > number of Heads) = 5/8

Ans 17 (b) 2/9 Ans 18 (b) ½ Ans 19 (c) 2p2 Ans 20 Do yourself

Ans 21 (b) [ i=1Sn (c-xi ) - =1Sn (cyi - xi )]/ c Ans 22 (c) 48/65 Ans 23 (c) 3

Part 2

Ans 1 A (a) required variance= 25/49 + 36/49 = 61/49

Ans 1 A (b) required variance= 50

Ans 1 A (c) Null hypothesis H0 : mA - mB =0

Z= [mean(XA) – mean(XB)] / Ö [25/36 + 36/49] = -25/1.2 ~ -20

Null hypothesis H0 is rejected

Ans 1 B (a) True

Ans 1 B (b) True

Ans 1 B (c) False

Ans 1 B (d) False

Ans 1 B (e) True

Ans 4 A Homogenous functions; Euler’s theorem.. prove!!!

Ans 4 B (a) a33 ¹ -3

Ans 4 B (b) Price = 61/6

DSE 2006 Mathematics/Statistics

ECOPOINT : MA Economics 2009 Entrance Coaching by DSE Alumni. 9999886629, 9873903314, 9899048237

Ans 9 (d) 3/8 Ans 12 Do yourself Ans 14 (d) any of the above three conditions is satisfied

Ans 15 (c) f is quasi convex Ans 16 (b) changes the sign of the determinant

Ans 17 should be 1/25 [ 5(1/5x 1/5x 1/5) = 5/125 =1/25 ] Ans 18 (d) 0.82

Ans 27 (c) Rs. 4000 Ans 28 (b) 0.56

Ans 31 (a) 5/14 Ans 32 (d) 2/7 Ans 33 (b) 0 Ans 34 (b) 128

Ans 37 (c) the line defined by y=x+2 and the vertical axis

Ans 46 (d) 7/15 Ans 47 (b) 121/256 Ans 48 (c) 4/13 Ans 49 (a) 28/30

Ans 52 (c) is greater than or equal to 0 at every xÎ Â

Ans 53 (d) in every election, every party has atleast one dishonest candidate. [ negate the statement]

Ans 54 (b) a gang Ans 54 (d) a gang

Ans 56 (b) there are atleast two sets of individuals that are both a family and a gang

Ans 57 (a) a family

Ans 58 (a) 45 Ans 59 (b) 3/8 Ans 60 (d) f(x) =|x| where xÎÂ and x ³0

DSE 2007 Mathematics/Statistics

ECOPOINT : MA Economics 2009 Entrance Coaching by DSE Alumni. 9999886629, 9873903314, 9899048237

Ans 11 (a) f(x) = x2

Ans 12 (c) det A = det B [ by operations on determinant]

Ans 13 (b) Df(x)=0 for some xÎ Â

Ans 14 (d) in every legislature and every party, there exists a legislature who doesnot pay taxes

Ans 15 (c) singular Ans 16 Do yourself Ans 17 (c) increases steadily from 0 to 1

Ans 18 (d) 4/7 Ans 19 (b) (i)28/30 (ii) 0.29 Ans 20 (c) 7.5 Ans 23 (a) (i) 1 km and (ii) Rs. 100

Ans 41 (a) { (x,r) Ân X Â| f(x)£ r } is convex Ans 42 (a) 2xe(x2) - ex

Ans 43 (d) converges neither to 1 nor to -1 Ans 44 (d) [-1,1] Ans 45 (c) (-1,1) Ans 46 (d) d(x,z) £ d(x,y) + d(y,z) Ans 47 (b) (A-B) X C = (AXC) – (BXC) Ans 48 (c) attains a saddle point

Ans 49 (c) 4.5 Ans 50 (d) during warm weather, my crops suffer from pests more than during cooler weather. Therefore, a warm environment must help pests to multiply.

Ans 51 (d) None of the above Ans 52 (3!3!)/6! (option not available) Ans 53 (b) 5/18

Ans 54 (b) (1/2, 3/4, 1/2) Ans 55 (b) 37.6 years Ans 56 (a) 0.025 Ans 57 (d) 5, 0.5, 39

Ans 58 (d) 0.79

Ans 59 (b) there is a significant difference in the performance of the classes at the 5% level but no significant difference at 1% level of significance

Ans 60 (b) there is no significant difference between the districts and the candidate is preferred in district A.

DSE 2008 Mathematics/Statistics

ECOPOINT : MA Economics 2009 Entrance Coaching by DSE Alumni. 9999886629, 9873903314, 9899048237

Ans 1 Do yourself Ans 2 (a) Ans 3 (d)

Ans 4 (c) III and IV [ consider any example based on the respective definitions of mutually exclusive sets and independent sets ]

Ans 5 (a) g(y) = f(Öy) " y= 1,4…..400 and g(y) = 0 otherwise

Ans 16 (a) both functions are linear

Ans 17 (b) -(yxy-1 + zx ln z)/ (xzx-1 + yz ln y) [ partially derivate, revise differentiation!! ]

Ans 18 (c) constant [ this is simple!!!] Ans 19 (a) will change the sign of det A (true for all nxn matrices ) Ans 20 (b) 3 ( think logically ) Ans 21 (b) Ans 22 (c) Ans 23 (a) Ans 24 (c) Ans 25 (c)

Ans 26 Do yourself Ans 27 Do yourself Ans 28 (d) Ans 29 (b) Ans 30 (c)

Ans 51 (d) the rows of A are linearly independent Ans 52 (a) strictly convex Ans 53 (c) exactly one solution Ans 54 (b) if player 1 plays appropriately, she can win regardless of how 2 actually plays

Ans 55 (a) Ans 56 do yourself!! Ans 57 (b) a set having a single point Ans 58 (a) ep

Ans 59 (c) is discontinuous at x=0 Ans 60 (c) linearly dependent