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I) Calculus

[a] Riemann Integration

(1) Riemann criterion for integrability  of a bounded function f defined on [a, b]

(2) Every monotonic function on [a, b] is Riemann integrable

(3) Every continuous function f defined on [a, b] is Riemann integrable

(iv) First and Second fundamental theorem of calculus

[b] Continuity

(1) Continuity of Functions : Example #1 - find the value of k

II) Linear Algebra

[a] Quotient Space

(1) Any two right cosets of V/W are either disjoint or identical

(2) Dimension theorem of a quotient space

(3) First isomorphism theorem for vector space, homomorphism, kernel and isomorphism of vector space

(4) Two finite dimensional vector spaces over the same field are isomorphic iff they are of the same dimensions

[b] Inner Product Space

(1) Theorem on Inner product space

[c] Linear Transformation

(1) Cayley Hamilton Theorem

(2) Linear Transformation Part I - Algebra of Linear Transformation

(3) Linear Transformation Part II - Inverse Linear Transformation and Isomorphism

(4) The Matrix of Linear Transformation and Linear Functionals

(5) Linear Functionals, Annihilator and Double Dual

III) Topology

[a] Connectedness

(1) Continuous image of connected set is connected

(2) A metric space (X, d) is connected iff every continuous function is constant

(3) Any two closed subset of metric space are connected iff they are disjoint

(4) (X, d) is disconnected iff there exists a non-empty proper open & closed subset of X

IV) General Aptitude

(1)  Three boys go to the same school. Every boy needs two cubes of the same color to play a game, but the cubes that different boys have are not necessarily differently colored. A teacher has red, blue and green cubes in one drawer. If the teacher draws the cubes without looking, what is the least number of cubes that need to be pulled out to be sure that every boy can get two cubes of the same color?

(2) There are 8 steps to the top of the stairs. In how many ways can we get to the top if we can climb one step or two steps?

(3) Calculate the largest integer n < 0 so that (1+frac{sqrt{3}}{3})^n is purely imaginary

(4) Two circles k1 and k2 intersect in such a way that outside their intersection is 10% of the area of the circle k1 and 60% the area of the circle k2. What is the ratio of the radii of the circles k1 and k2?What is the sum of the radii if the area of the figure is 94(pi)?

(5) What will be the sum of n terms of the series whose n^{th} term is 5.3^{n+1}+2n?

(6) The probability that a ticketless traveler is caught during a trip is 0.1. If the traveler makes 4 trips , the probability that he/she will be caught during at least one of the trips is:

(7) A woman starts shopping with Rs. X and Y paise, spends Rs. 3.50 and is left with Rs. 2Y and 2 X paise. The amount she started with is

(8) Female rabbits are either juvenile or adult. Each month, each juvenile grows to adulthood and each adult female gives birth (on average) to one juvenile female. Starting with one juvenile female rabbit, how many female rabbits will there be after 10 months?

(9) If mean and median of a frequency distribution are 20 and 22 respectively, then the mode of the distribution will be:

(10) If 6 men or 8 chidren complete one task in 24 days, then how many days will be required to complete the task for 7 men and 12 children?