4PM1/01R Jun 2019, Question 10: Simultaneous Equations with a Curve

Part (a) · 3 marks

Express 6x − x² as D(x + E)² + F with integer D, E, F; state the three values.

1. 16x − x² = −(x² − 6x)
2. 2= −[(x − 3)² − 9]
3. 3= −(x − 3)² + 9: D = −1, E = −3, F = 9

AnswerD = −1, E = −3, F = 9

Part (b) · 2 marks

From the completed square, give the largest value of f(x) and the x value where it happens.

1. 1Maximum value = F = 9
2. 2It occurs when x − 3 = 0, so x = 3

Answer(i) 9 (ii) x = 3

Part (c) · 4 marks

Solve f(x) = x² − 4x + 8 to find the coordinates of both intersection points of C and S.

1. 16x − x² = x² − 4x + 8
2. 22x² − 10x + 8 = 0
3. 3(2x − 2)(x − 4) = 0 → x = 1, x = 4
4. 4y = 5 and y = 8

Answer(1, 5) and (4, 8)

Part (d) · 4 marks

Set up and evaluate a definite integral of the difference between the two curves to find the area of R.

1. 1Area = ∫ from 1 to 4 of [(6x − x²) − (x² − 4x + 8)] dx
2. 2= ∫ from 1 to 4 of (−2x² + 10x − 8) dx
3. 3= [−2x³/3 + 5x² − 8x] from 1 to 4
4. 4= 16/3 − (−11/3) = 9

Answer9

More past-paper questions on simultaneous equations with a curve

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