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Specialist Maths Trial Exam 2 2005 Solutions

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Q1a

rij, ,

v = dr/dtij,

At , r = 100i – 60j, v = 30i – 50j,

kmh-1.

Q1b Let and

, , ,

, ,

Q1c Initial position (i.e. at ), r = 80i. At ,

r = 170i – 150j.

0 80 170 x

Crossroad

Highway

– 150

y = – x

Q1d At , v = 30i – 50j, a unit vector in the direction of v is (3i – 5j). Let a unit vector parallel to the crossroad be ai + bj. and (ai + bj) (3i – 5j) = ,

. Solve for a, b, , .

Q1e The straight line representing the crossroad passes through (100, – 60) and has a gradient . Equation of the crossroad is

, .

Q2a Let , ,

, square both sides and then expand, , simplify,

, square both sides and simplify,

, complete the square,

Q2b

1.5

–3 0 1 Re

–1.5

Q2c Let ,

z

r y

 x

, ,

Q2d ,

The centre of the ellipse is (– 1, 0), area enclosed by the ellipse

Q2e Let , , when , , when , .

Q2f The ellipse is formed by dilating the unit circle at the origin horizontally by a factor of 2 and vertically by a factor of , and then translating it to the left by a unit. Translation does not affect the area. The area of the unit circle is , and after dilations the area becomes .

Q3a for ,

Hence .

Q3b y

–0.1 0 0.4 0.8 0.9 x

Q3c Total distance

= 40 km

Q3d , .

Max acceleration at (see graph).

kmh-2

ms-2.

Q3e , .

When , , (use graphics calculator to sketch , find at ). kmh-2.

Q4a Normal reaction

Friction

Air resistance

30o

Weight

Q4b Friction ,

air resistance .

Resultant force

When , , .

When , ,

Q4c Given ,

, .

Q4d When the object reaches terminal speed, ,

, hence ms-1.

Q4e Use . Given

Estimated time = 4.0 s

Q4f

7.37

0 t

Q5a

0 2 4 x

– 1

Q5b , , .

Volume

Q5c ,

, hence

, .

Q5d , ,

, At , ,

and .

Q5e Use graphics calculator to sketch and . Find the intersection, m

Please inform re conceptual,

mathematical and/or typing errors