IB Maths AA HL AA 1.5 Partial fractions and linear systems questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper
Paper 1, no calculatorPaper 2, calculator
Difficulty
EasyMediumHard

Question 1

Paper 1, no calculatorEasy6 marks

Let

R(x)=3x−2(x−4)(x+1).R(x)=\frac{3x-2}{(x-4)(x+1)}.
(a)

Express R(x)R(x) in partial fractions.

[3]
(b)

Hence determine the exact value of

∫01R(x) dx.\int_0^1 R(x)\,dx.
[3]

Question 2

Paper 1, no calculatorEasy6 marks

The rational function

R(x)=3x−1(x−3)(x+1)R(x)=\frac{3x-1}{(x-3)(x+1)}

is used as an integrand.

(a)

State the real values of xx for which R(x)R(x) is undefined.

[1]
(b)

Write R(x)R(x) in partial fractions.

[3]
(c)

Hence determine an antiderivative of R(x)R(x)

[2]

Question 3

Paper 1, no calculatorMedium12 marks

Let

f(x)=x+8x2+x−6,x∈R,f(x)=\frac{x+8}{x^2+x-6}, \qquad x\in\mathbb R,

where f(x)f(x) is defined only when its denominator is non-zero.

(a)

Write x2+x−6x^2+x-6 as a product of linear factors, and state the values of xx for which f(x)f(x) is undefined.

[2]
(b)

Write f(x)f(x) in partial fractions.

[4]
(c)

For parts (i) and (ii), consider −3<x<2-3<x<2

(i)

Hence determine an antiderivative FF of ff

[3]
(ii)

Hence evaluate ∫01f(x) dx\displaystyle\int_0^1 f(x)\,dx giving your answer exactly. Explain why the sign of your answer is consistent with the sign of f(x)f(x) on [0,1][0,1]

[3]

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