Download 2016 General Maths/ Math Methods Year 11

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2016
General Maths/
Math Methods
Year 11
Investigation Task
Time allowed: 2 hours
You are allowed: 1 bounded reference, 1 CAS, 1 scientific calculator
Working must be shown for questions worth 2 or more marks. Total: 60 marks
Theme: Finding Pythagorean triads containing a particular number
A Pythagorean triad is a set of 3 natural numbers, a, b and c, satisfying Pythagoras’ theorem, a 2 + b 2 = c 2 .
For examples, (3, 4, 5) , (6, 8, 10) and (56, 783, 785) are Pythagorean triads.
If the numbers in the triad have no common factor, the triad is called a primitive Pythagorean triad.
(3, 4, 5) and (56, 783, 785) are primitive Pythagorean triads.
(6, 8, 10) is not a primitive Pythagorean triad because the numbers 6, 8 and 10 have a common factor of 2.
In the following questions (a, b, c ) is a primitive Pythagorean triad, and c is the largest in the triad.
Question 1
Explain why a and b cannot be equal.
3 marks
Question 2
Explain why a and b cannot be both even or both odd.
4 marks
Question 3
a.
Express a in terms of b and c .
2 marks
b.
Hence explain why the smallest value of a is 3.
3 marks
2016 General Maths/Math Methods Year 11 Investigation Task
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How do you find a triad (or triads) containing a particular natural number as one of the two smaller
numbers?
Consider the following discussion:
Let p and q be two natural numbers, and p > q .
Question 4
pq and b =
p−q
, show that a 2 + b 2 is a perfect square.
2
a.
If a =
b.
Show that c = a 2 + b 2 =
p+q
.
2
3 marks
1 mark
c. Given (a, b, c ) is a Pythagorean triad, explain why both p and q must be even or both p and q must
be odd.
3 marks
d.
Let a =
pq be the particular number 5. State the possible values of p and q .
e.
Hence find the values of b and c .
2016 General Maths/Math Methods Year 11 Investigation Task
1 mark
2 marks
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How do you find a triad (or triads) containing a particular natural number as the largest number?
Question 5
a.
Let c =
p+q
be the particular number 5. State the possible values of p and q .
2
b. Hence find the values of a and b .
c.
Explain why only two sets of values of a and b exist.
2 marks
2 marks
2 marks
Question 6
Write down the two distinct Pythagorean triads containing the number 5.
2016 General Maths/Math Methods Year 11 Investigation Task
2 marks
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Question 7
Find all distinct Pythagorean triads consisting of 56 as one of the two smaller numbers in each triad.
Hint: Let a = 56 , find all possible values of p and q , and then find all possible values of b and c .
Set up a table of values to display your findings.
10 marks
2016 General Maths/Math Methods Year 11 Investigation Task
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Question 8
Find all Pythagorean triads (if they exist) consisting of 56 as the largest number in each triad.
Hint: Let c = 56 , find all possible values of p and q , and then find all possible values of a and b .
Set up a table of values to display your findings.
10 marks
2016 General Maths/Math Methods Year 11 Investigation Task
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6
Question 9
Find all primitive Pythagorean triads consisting of 1111111 as one of the smaller numbers in the triad.
Hint: Let a = 1111111 , find all possible values of p and q , and then find all possible values of b and c .
10 marks
End of task
2016 General Maths/Math Methods Year 11 Investigation Task
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