EVALUATION STRATEGY AND COMMON MISTAKES
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***  Q1 HAS A MISTAKE, AND IS ABANDONED. EVALUATION WILL BE IN 80 - 8 = 72.  ***
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                       Q2,3,4 are checked by AH
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Q2
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        Indicate Catalan Number                                                 [2 marks]
        Justify and Derive Catalan number                                       [8 marks]
                (i) Reason out from first princples
                (ii) Map to another problem having Catalan number of options
        [ Derivation NOT given: deducted 6-8 marks (depending on the severity) ]

Q3(a)
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        Derive the l_n expression                                               [2 marks]
        Calculations for L(x)                                                   [4 marks]
        [ Derivation NOT given: deducted 3-4 marks (depending on the severity)
          Derivation has error: deducted 1-2 marks (depending on the severity) ]

Q3(b)
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        Generate Roots of 1-3x+x^2                                              [1 mark]
        Calculate coefficient of X^n in the L(x)                                [4 marks]
        Derive coefficients for the two roots                                   [1 mark]
        [ Derivation has error: deducted 1-2 marks (depending on the severity) ]

Q4
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        Properly calculated and solved                                          [10 marks]
        [ Derivation NOT given: deducted 6-9 marks (depending on the severity)
          Derivation has error: deducted 3-6 marks (depending on the severity) ]

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                      Q5,6,7,8 are checked by AD
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Q5(a)
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	One mark each for the following:
		  (i) A is closed under addition
		 (ii) A is closed under additive inverse
		(iii) A is closed under multiplication
		 (iv) Multiplication in A is commutative
		  (v) A contains the identity matrix
	No credit/discredit for showing other obvious properties of matrices
        (like commutativity of addition, associativity, distributivity).

Q5(b)
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	One mark each for the following:
		  (i) Define an explicit function f : A -> C (or f : C -> A)
		 (ii) Show that f(A+B) = f(A)+f(B)
		(iii) Show that f(AB) = f(A)f(B)
		 (iv) Show that f is injective.
                  (v) Show that f is bijective.

Q6(a)
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	The following needs to be shown.
	 (i) If a,b in rad(I), then a - b is in rad(I)				[3 marks]
	(ii) If a is in rad(I) and r is in R, then ra (= ar) is also in rad(I)	[2 marks]

	Common mistakes in (i)
		Take n the same for both a and b				[-1]
		Take a^n and b^m in I, and show (a - b)^lcm(m,n) is in I.
		This is incorrect since we may have m = n. Then the
		binomial expansion has terms a^ib^j with both i,j<m=n. Also,
		taking (a - b)^(mn) requires much care to complete the proof.	[-2]

Q6(b)
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	An explicit example is asked for, not a justification where the proof
	of Part (a) fails.							[-5]

Q7(a)
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	Show closure under multplication modulo p				[2 marks]
	Correctly mention the order of QR_p					[1 mark]
	Justify why this is the order of QR_p					[1 mark]

Q7(b)
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	Let s = sum(Z_p^*), s1 = sum(QR_p), and s2 = sum(QNR_p).

	  (i) Identification that s = s1 + s2 (mod p)				[1 mark]
	 (ii) Show that s is a multiple of p					[1 mark]
	(iii) Find the formula for s2 = p(p-1)(p+1)/24 (mod p)			[2 marks]
	 (iv) Argue why (p-1)(p+1)/24 is an integer (or s = 0 (mod p))		[2 marks]
	      (Note: This is true for all primes p >= 5, but it does not
	      matter even if (p-1)(p+1)/24 is not an integer, because
	      s2 is NOT EQUAL (but CONGRUENT modulo p) to p(p-1)(p+1)/24,
	      so it suffices to argue that 1/24 makes sense modulo p.

Q8(a)
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	Given the condition, prove that G is commutative. No credit for proving
	the converse of the statement.

Q8(b)
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	Arguing that <a> is finite (this is totally wrong).			[-5]
	Handling only the special case: <a> is finite.				[-3]