Edexcel GCE Core Maths C4 Advanced June 2013

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C4 Edexcel Core Mathematics June 2013 Question 6

Differential Equation

6. Water is being heated in a kettle. At time t seconds, the temperature of the water is θ°C.

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.

6(b) C4 Edexcel Core Mathematics June 2013 Question 7

Implicit Differentiation

7. A curve is described by the equation x2 + 4xy + y2 + 27 = 0

(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.

7(b)Tangent parallel to the y-axis.

C4 Edexcel Core Mathematics June 2013 Question 8

Vector Equation of a Line

8. With respect to a fixed origin O, the line l has equation

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.

8(b) Working with vector equation of a line