1st Year Math Guess Paper

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• Note: Every real number is a complex number with 0 as its imaginary part.
• Note: A real number is self-conjugate.
• Note: The set C of complex numbers does not satisfy the order axioms. In fact there is no sense in saying that one complex number is greater or less than another.
• Note: (1) A = B if and only if they have the same elements means if A = B they have the same elements and if A and B have the same elements then A = B. (2) The phrase if and only if is shortly written as “if “.
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• Note: When we do not want to distinguish between proper and improper subsets, we may use the symbol ⊆ for the relationship. It is easy to see that: N ⊂ Z ⊂ Q ⊂ R.
• Note: (1) The power set of the empty set is not empty. (2) Let n (S) denoted the number of elements of a set S, then n{P(S)} denotes the number of elements of the power set of S. From examples 3 to 7 we get the following table of results:
• Notice that the elements common to A and B, namely the elements 2, 3 have been written only once in A U B because repetition of an element of a set is not allowed to keep the elements distinct.
• Note: In view of the deinition of complement and diference set it is evident that for any set A, A’ =U - A.
• Note: In all the above Venn diagrams only overlapping sets have been considered. Veriication in other cases can also be efected similarly. Detail of veriication may be written by yourselves.
• Note: (1) Evidently truth set of a tautology is the relevant universal set and that of an absurdity is the empty set Φ . (2) With the help of the above results we can express any logical formula in set- theoretic form and vice versa. We will illustrate this fact with the help of a solved example.
• Note: The elements of the set of this example are the fourth roots of unity.
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• Note: N is not a monoid w.r.t. addition because it has no identity w.r.t. addition.
• Note: The rule can obviously be extended to the product of three or more elements of a group.
• Note: Matrix multiplication is not commutative in general 
• Note: Matrices A and B are equivalent if B can be obtained by applying in turn a inite number of row operations on A.
• Note: Diagonal matrices are both upper triangular and lower triangular.
• Note: The solutions of an equation are also called its roots.
• Note: For simplicity, a conditional equation is called an equation.
• Note: An ininite sequence has no last term.
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• Note: Middle term of three consecutive terms in A.P. is the A.M. between the extreme terms.
• Note: The real values of r are usually taken but here other cases are considered to widen the out-look of the students.
• Note: In the above experiment, we have written the probability = 1/2(approx.) It may be remembered that the greater the number of trials, the more accurate is the estimate of the probability.
• Thus sum of odd coeicients of a binomial expansion equals to the sum of its even coeicients.