The rate of increase of the temperature of the water at any time t is modelled by the differential equationdθ/dt = λ(120 - θ), θ ≤ 100 where λ is a positive constant.
Given that θ = 20 when t = 0,
(a) solve this differential equation to show that θ = 120 – 100e–λt
When the temperature of the water reaches 100 °C, the kettle switches off.
(b) Given that λ = 0.01, find the time, to the nearest second, when the kettle switches off.
(a) Find dy/dx in terms of x and y.
A point Q lies on the curve.
The tangent to the curve at Q is parallel to the y-axis.
Given that the x coordinate of Q is negative,
(b) use your answer to part (a) to find the coordinates of Q.
r , where λ is a scalar parameter.
The point A lies on l and has coordinates (3, – 2, 6).
The point P has position vector (–p i + 2p k) relative to O, where p is a constant.
Given that vector PA is perpendicular to l,
(a) find the value of p.
Given also that B is a point on l such that ∠BPA = 45°,
(b) find the coordinates of the two possible positions of B.
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