.». Marking Scheme: Term 2 2009, 2U Question 3

MARKS

» Part (a) (i)

ܲ = ‫ ݁ܣ‬௞௧ ∴

݀ܲ = ݇‫ ݁ܣ‬௞௧ ݀‫ݐ‬

1 for differentiation

= ݇ܲ

(ii)

‫ = ܣ‬1000 But when ‫ = ݐ‬10, ܲ = 1400. ∴ 1400 = 1000݁ ଵ଴௞ ଻ ହ

=݁

ଵ଴௞

1 for substitution and simplification

଻

ln ቀହቁ = 10݇ ଵ

݇ = ଵ଴ ln 1.4 (iii)

1 for exact value of ݇

When ‫ = ݐ‬20, ܲ = 1000݁ ଶ଴௞

(iv)

= 1000݁ ଶ௟௡ଵ.ସ

1 for substitution

= 1960

1 for final answer

When ܲ = 3000, 3000 = 1000݁ ௞௧ 3 = ݁ ௞௧ ln 3 = ݇‫ݐ‬

1 for substitution

ଵ

‫ = ݐ‬୩ ln 3 ≈ 32.65 … ∴ Population of 3000 is reached during year 1998.

1 for correct year

Woo | 2009

» Part (b)

MARKS

(i) Parabola and line intersect at ܲሺ‫݌‬, ‫ݍ‬ሻ. ݊‫ ݔ‬+ ݉‫ = ݕ‬1 ‫= ݔ‬

ଵି௠௬

1 for changing subject and solving simultaneously

௡

Substitute into equation of parabola (‫ݍ = ݕ‬ሻ: ‫ ݍ‬ଶ = 4ܽ ቀ

ଵି௠௤ ௡

ቁ

݊‫ ݍ‬ଶ = 4ܽ − 4ܽ݉‫ݍ‬

1 for substituting ‫ݔ‬ and ‫ ݕ‬for ‫ ݌‬and ‫ݍ‬

݊‫ ݍ‬ଶ + 4ܽ݉‫ ݍ‬− 4ܽ = 0 (ii)

Line is tangent to parabola if discriminant of ݊‫ ݍ‬ଶ + 4ܽ݉‫ ݍ‬− 4ܽ is 0. ܾ ଶ − 4ܽܿ = 0 ሺ4ܽ݉ሻଶ − 4ሺ−4ܽ݊ሻ = 0 16ܽଶ ݉ଶ + 16ܽ݊ = 0 ܽ݉ଶ + ݊ = 0

1 for substitution into discriminant

1 for simplification

Woo | 2009

» Part (c)

MARKS

(i) Katie wins by rolling a 2, 4 or 6. Ming wins by rolling a 1, 3, 5 or 6.

1 2

1 2

win

win

2 3

lose

lose

1 3

Katie

win

1 2

lose

1 2

Ming

Katie ଵ

∴ ܲሺKatie wins on throw 1ሻ = ଶ (ii)

1 for probability

ܲሺKatie wins on throw 2ሻ = ܲሺKatie losesሻ × ܲሺMing losesሻ × ܲሺKatie winsሻ ଵ

ଵ

½ for reasoning

ଵ

=ଶ×ଷ×ଶ ଵ

= ଵଶ (iii)

ଵ

ଵ

½ for probability

ଵ

ܲሺKatie winsሻ = ଶ + ଵଶ + ଻ଶ … ௔

ଵ

= ଵି௥ ଵ

ଵ

where ܽ = ଶ , ‫଺ = ݎ‬

1 for limiting sum

ହ

=ଶ÷଺ ଷ

=ହ

1 for probability

Woo | 2009