Exam-style Questions: Advanced Algebra

1. The function f is given by:

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Given that:

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where A, B and C are constants,

(i) find the values of B and C,

(ii) show that A = 0.

(Marks available: 4 marks)

Answer

Answer outline and marking scheme for question: 1

Give yourself marks for mentioning any of the points below:

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(max 3 marks)

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(max 1 mark)

(Marks available: 4)

2. The function f is defined for the domain by:

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a) Deduce that f is a decreasing function for all.

(Marks available: 2)

Answer

Answer outline and marking scheme for question: 2

a) The function f(x) is decreasing if f '(x) is negative. Therefore we need to show that f '(x) is less than zero, shown below:

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rearranging this and changing back to square roots gives:

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rearranging further gives:

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which is always true for all.

(2 marks)

(Marks available: 2)

3. A function is given by:

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find the value of the constants A and B.

(Marks available: 3)

Answer

Answer outline and marking scheme for question: 3

Give yourself marks for mentioning any of the points below:

Adding the two fractions on the right hand side gives:

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As the denominators are now the same the numerators must match as well.

Therefore:

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Selecting x = -1, will eliminate the constant B.

This gives:

-2 -1 = A (-3 + 2 )

-3 = -A

A = 3

Selecting x = 2/3 to this will eliminate the constant A.

This gives:

-2/3 x 2 -1 = B (-2/3 +1)

-4/3 -1 = B (1/3)

-7/3 = 1/3 B

-7 = B

B = -7

Therefore:

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(Marks available: 3)