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Unformatted text preview: n integrating factor, solve this differential equation.
Give your answer in the form y = f(x). (ii) Calculate, correct to two decimal places, the value of y when x = 1.
(10) 548 (c) Sketch the graph of y = f(x) for 0 ≤ x ≤ 1. Use your sketch to explain why your
approximate value of y is greater than the true value of y.
(4)
(Total 24 marks) 803. (a) Write the number 10 201 in base 8.
(4) (b) Prove that if a number is divisible by 7 that the sum of its base 8 digits is also divisible
by 7.
(5) (c) Using the result of part (b), show that the number 10 201 is not divisible by 7.
(2)
(Total 11 marks) 804. Let a and b be two positive integers.
(a) Show that gcd(a, b) × lcm(a, b) = ab
(6) (b) Show that gcd(a, a + b) = gcd(a, b)
(7)
(Total 13 marks) 805. Find the remainder when 67101 is divided by 65.
(Total 6 marks) 806. Solve the system of linear congruences
x ≡ 1(mod3); x ≡ 2(mod5); x ≡ 3(mod7).
(Total 6 marks) 549 807. The matrix below is the adjacency matrix of a graph H with 6 vertices A, B, C, D, E, F.
A
A 0 B 1
C 0 D 1 E 1
F 0 (a) B
1
0
1 C
0
1
0 D
1
0
1 E
1
0
1 F
0 1
0 0 1 0 0 1 0 1 0 0 1
1 0 1 1 0 Show that H is not planar.
(3) (b) Find a planar subgraph of H by deleting one edge from it.
(3) (c) Show that any subgraph of H (excluding H itself) is planar.
(4)
(Total 10 marks) 808. Let G be the graph below.
A
6 7
8 B E
12 10 C (a) 8 7 5 9 9 D Find the total number of Hamiltonian cycles in G, starting at vertex A.
Explain your answer.
(3) 550 (b) (i) Find a minimum spanning tree for the subgraph obtained by deleting A from G.
(3) (ii) Hence, find a lower bound for the travelling salesman problem for G.
(3) (c) Give an upper bound for the travelling salesman problem for the graph above.
(2) (d) Show that the lower bound you have obtained is not the best possible for the solution to
the travelling salesman problem for G.
(3)
(Total 14 marks) 809. Let Sn be the sum of the first n terms of an arithmetic sequence, whose first three terms are u1,
u2 and u3. It is known that S1 = 7, and S2 = 18.
(a) Write down u1. (b) Calculate the common difference of the sequence. (c) Calculate u4. Working: Answers:
(a) …………………………………………..
(b) ..................................................................
(c) ……………………………………..........
(Total 6 marks) 551 810. Consider the line L with equation y + 2x = 3. The line L1 is parallel to L and passes through the
point (6, –4).
(a) Find the gradient of L1. (b) Find the equation of L1 in the form y = + mx + b. (c) Find the xcoordinate of the point where line L1 crosses the xaxis. Working: Answers:
(a) …………………………………………..
(b) ..................................................................
(c) ……………………………………..........
(Total 6 marks) 552 811. Consider the expansion of (x2 – 2)5.
(a) Write down the number of terms in this expansion. (b) The first four terms of the expansion in descending powers of x are
x10 – 10x8 + 40x6 + Ax4 + ...
Find the value of A. Working: Answers:
(a) ..................................................................
(b) ...........................................
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This note was uploaded on 06/24/2013 for the course HISTORY exam taught by Professor Apple during the Fall '13 term at Berlin Brothersvalley Shs.
 Fall '13
 Apple
 The Land

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