1. Let m and n be two positive integers such that m + n + mn=186. Then the value of m + n is
a) 17
b) 11
c) 26
d) 186
2. How many digits are there in 350 when it is expressed in the decimal form?
a) 25
b) 50
c) 24
d) 26
3. What is the last digit of 72016?
a) 7
b) 1
c) 8
d) 4
4. A polynomial of odd degree with real coefficients must have
a) At least one real root
b) No real root
c) Only real roots
d) At least one root which is not real
5. |z+i| = 2∫z2 dz/4-z2 =
a) 0
b) 2Πi
c) -2Πi
d) 1
6. The number of diagonals of a convex dodecagon is
a) 66
b) 54
c) 55
d) 60
7. At what time after 4 O'clock the hour and the minute hands will lie opposite to each other?
a) 4-50'-31"
b) 4-52'-51"
c) 4-53'-23"
d) 4-54'-33"
8. What is the next number in the Sequence 24,30,33,39,51,...
a) 57
b) 69
c) 54
d) 81
9. Let X be a metric space and A be a subset of X. If A is a connected set with at least two distinct elements, then the number of distinct points in A is
a) uncountable
b) 2
c) More than 2, but finite
d) countably infinite
10. A single cell spherical organism contains 70% of water by volume. If it loses 10% of its water content, how much would its surface area change by approximately?
a) 3%
b) 5%
c) 6%
d) 7%
11. What is the value of the series k=1Σ∞ (1+2+.... +k)/k!
a) e
b) e/2
c) 3e/2
d) 1+(e/2)
12. A row space of a 40 x 70 matrix A has dimension 17. What is the dimension of solutions of Ax=0 ?
a) 17
b) 53
c) 23
d) 57
13. What is the area of the triangle bounded by the lines y=2x, y=-2x and y=6?
a) 36
b) 18
c) 12
d) 24
14. The sum of first n natural numbers with one number missed is 40. What is the number that was missed?
a) 5
b) 3
c) 9
d) 2
15. How many non-negative integers between 1000 and 10,000 are there such that the sum of the digits of the number is divisible by three?
a) 3000
b) 2999
c) 3334
d) 3001
16. Up to Isomorphism, the number of abelian groups of order 108 is:
a) 12
b) 9
c) 6
d) 5
17. Ten balls are put in 6 slots at random. Then the expected total number of balls in the two extreme slots is
a) 10/6
b) 10/3
c) 6/10
d) 1/6
18. Let G be simple group of order 60. Then G has
a) Six Sylow-5-subgroups
b) Four Sylow-3-subgroupls
c) A Cyclic subgroup of order 6
d) A unique element of order 2
19. What is the maximum sum of the numbers of Saturdays and Sundays in a leap year?
a) 105
b) 106
c) 107
d) 104
20. Students of a school are divided into 4 groups. What is the probability that three friends get into the same group?
a) ¾
b) 1/64
c) 1/16
d) 1/3
21. Let A be a subset of R with more than one element. Let a∈A. If A\{a} is compact, then
a) A is compact
b) Every subset of A must be compact
c) A must be finite set
d) A is disconnected
22. Let A be a 10 x 10 invertible matrix. Which of the following is not necessarily true?
a) The rows of A form a basis of R10
b) Null space of A contains only the 0 vector
c) A has 10 distinct eigen values
d) Image of the linear transformation x→Ax on R10 is R10
23. If 1Σ∞(-1)n+1/n = log2, then 1Σ∞(-1)n/n (n+1) =
a) 1-2log2
b) 1+log2
c) (log2)2
d) -(log2)2
24. The number of elements in the set {m:1 £m£1000, m and 1000 are relative prime} is
a) 100
b) 250
c) 300
d) 400
25. What is the smallest positive integer in the set {24x+60y+2000z | x, y, z are integers}
a) 4
b) 12
c) 6
d) 24
26. Suppose you expand the product (x1+y1)(x2+y2)...(x20+y20). How many terms will have only one x and rest y?
a) 1
b) 5
c) 10
d) 20
27. What is the angle between the minute and hour hands of a clock at 7:35?
a) 0°
b) 17.5°
c) 19.5°
d) 20°
28. If A is a 5 X 5 real matrix with trace 15 and if 2 and 3 are eigenvalues of A, each with algebraic multiplicity 2, then the determinant of A is equal to
a) 0
b) 24
c) 120
d) 180
29. What is the total number of positive integer solutions to the equation (x1+x2+x3)(y1+y2+y3+y4)=15?
a) 1
b) 2
c) 3
d) 4
30. A group G is generated by the elements x, y with the relations x3=y2=(xy)2=1. The order of G is
a) 6
b) 4
c) 12
d) 8.