4PM1/02 Jan 2023, Question 3: Simultaneous Equations with a Curve

Part (a) · 3 marks

Express f(x) as A(x + B)² + C and state the constants A, B and C.

1. 1f(x) = 8(x² + (5/4)x) − 3
2. 2= 8[(x + 5/8)² − 25/64] − 3
3. 3= 8(x + 5/8)² − 25/8 − 3
4. 4= 8(x + 5/8)² − 49/8

AnswerA = 8, B = 5/8, C = −49/8

Part (b) · 2 marks

Using your completed square, find the x value giving the minimum and state that minimum value of f(x).

1. 1The minimum is where x + 5/8 = 0
2. 2x = −5/8
3. 3Minimum value = C = −49/8

Answer(i) x = −5/8 (ii) −49/8

Part (c) · 2 marks

Find the x coordinates where the curve y = f(x) crosses the x-axis.

1. 18x² + 10x − 3 = 0
2. 2(4x − 1)(2x + 3) = 0
3. 3x = 1/4 or x = −3/2

Answerx = −3/2 and x = 1/4

Part (d) · 4 marks

Solve algebraically for the coordinates of the two intersection points of C and the line y = 2x + 13.

1. 18x² + 10x − 3 = 2x + 13
2. 28x² + 8x − 16 = 0, so x² + x − 2 = 0
3. 3(x + 2)(x − 1) = 0, x = −2 or x = 1
4. 4y = 2(−2) + 13 = 9 and y = 2(1) + 13 = 15

Answer(−2, 9) and (1, 15)

Part (e) · 2 marks

On one set of axes, sketch curve C and line l using results from parts b, c and d.

1. 1Curve: positive quadratic with minimum below the x-axis, one root either side of the y-axis
2. 2Line: positive gradient, y-intercept 13
3. 3They meet at (−2, 9) and (1, 15)

AnswerU-shaped parabola with minimum (−5/8, −49/8) below the x-axis, crossing the x-axis at −3/2 and 1/4; straight line with positive gradient through (0, 13), crossing the curve at (−2, 9) and (1, 15)

More past-paper questions on simultaneous equations with a curve

• 4PM1/01 Jun 2023 Q11 · 14 marks
• 4PM1/02 Jun 2023 Q4 · 8 marks
• 4PM1/02 Jun 2021 Q6 · 13 marks
• 4PM1/01 Jan 2020 Q5 · 11 marks
• 4PM1/02 Jan 2020 Q6 · 8 marks
• 4PM1/01 Jun 2020 Q7 · 13 marks