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Nagoya University Entrance Examination (Year 2012) :Mathematics

*Time limit: 150 min.

1. Let a be a positive constant and y = x3a2x be the equation of the curve C in the xy-plane.

(1) Let l be a tangential line of C that passes through the point A(t, t3 – a2t) on C. Find the area S(t) of a figure bounded by l and C. Assume t ≠ 0.

(2) Let b be a real number. Find the number of tangent lines of C that pass through the point B(2a, b) in the xy-plane.

(3) Assume that only two tangent lines of C, l1 and l2, pass through the point B(2a, b). Neither l1 nor l2 passes through the origin (0, 0). The area of the figure bounded by l1 and C is S1. The area of the figure bounded by l2 and C is S2. Assuming S1S2, find the value of S1/S2.

2. For a positive integer n, a function fn(x) of a real number x is defined by:

where f0(x) = xex.

(1) Find f1(x).

When , find a definite integral assuming that real numbers a, b, and c are constants.

(3) Find f2n(x) for a positive integer n.

3. Let n be an integer (n ≥ 2). A single number, between 1 to n, is written on n pieces of card. A different number is written on each card. Randomly select one card from these n pieces of card, record the number written on the card, and then return it to the stack of cards. Repeat this trial three times and assign the minimum and the maximum integer values as X and Y, respectively. Answer the following questions. Note: j and k are positive integers and j + kn. s is a positive integer and sn - 1.

(1) Find the probability that satisfies both conditions: Xj and Yj + k.

(2) Find the probability that satisfies both conditions: X = j and Y = j + k.

(3) Find the probability P(s) that satisfies the condition YX = s.

(4) When n is an even number, find s that maximizes P(s).

4. Let m and p be odd numbers (m, p ≥ 3). m is not divisible by p.

(1) Find the coefficient of the x2 term in the polynomial expansion of (x – 1)101.

(2) Show that (p – 1)m + l is divisible by p.

(3) Show that (p – 1)m + l is not divisible by p2.

(4) Let r be a positive integer and s = 3r–1m. Show that 2s + 1 is divisible by 3r.