CAIE P3 (Pure Mathematics 3) 2013 June

1 Find the quotient and remainder when \(2 x ^ { 2 }\) is divided by \(x + 2\).

2 Expand \(\frac { 1 + 3 x } { \sqrt { } ( 1 + 2 x ) }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

3 Express \(\frac { 7 x ^ { 2 } - 3 x + 2 } { x \left( x ^ { 2 } + 1 \right) }\) in partial fractions.

4

1. Solve the equation \(| 4 x - 1 | = | x - 3 |\).
2. Hence solve the equation \(\left| 4 ^ { y + 1 } - 1 \right| = \left| 4 ^ { y } - 3 \right|\) correct to 3 significant figures.

5 For each of the following curves, find the gradient at the point where the curve crosses the \(y\)-axis:

1. \(y = \frac { 1 + x ^ { 2 } } { 1 + \mathrm { e } ^ { 2 x } }\);
2. \(2 x ^ { 3 } + 5 x y + y ^ { 3 } = 8\).

6 The points \(P\) and \(Q\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O P } = 7 \mathbf { i } + 7 \mathbf { j } - 5 \mathbf { k } \quad \text { and } \quad \overrightarrow { O Q } = - 5 \mathbf { i } + \mathbf { j } + \mathbf { k }$$ The mid-point of \(P Q\) is the point \(A\). The plane \(\Pi\) is perpendicular to the line \(P Q\) and passes through \(A\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
2. The straight line through \(P\) parallel to the \(x\)-axis meets \(\Pi\) at the point \(B\). Find the distance \(A B\), correct to 3 significant figures.

7

1. Without using a calculator, solve the equation $$3 w + 2 \mathrm { i } w ^ { * } = 17 + 8 \mathrm { i }$$ where \(w ^ { * }\) denotes the complex conjugate of \(w\). Give your answer in the form \(a + b \mathrm { i }\).
2. In an Argand diagram, the loci $$\arg ( z - 2 \mathrm { i } ) = \frac { 1 } { 6 } \pi \quad \text { and } \quad | z - 3 | = | z - 3 \mathrm { i } |$$ intersect at the point \(P\). Express the complex number represented by \(P\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), giving the exact value of \(\theta\) and the value of \(r\) correct to 3 significant figures.

8

1. Show that \(\int _ { 2 } ^ { 4 } 4 x \ln x \mathrm {~d} x = 56 \ln 2 - 12\).
2. Use the substitution \(u = \sin 4 x\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 24 } \pi } \cos ^ { 3 } 4 x \mathrm {~d} x\).

9

1. Express \(4 \cos \theta + 3 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
2. Hence
   (a) solve the equation \(4 \cos \theta + 3 \sin \theta = 2\) for \(0 < \theta < 2 \pi\),
   (b) find \(\int \frac { 50 } { ( 4 \cos \theta + 3 \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).

10 Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, \(t\) minutes later, the volume of liquid in the tank is \(V \mathrm {~cm} ^ { 3 }\). The liquid is flowing into the tank at a constant rate of \(80 \mathrm {~cm} ^ { 3 }\) per minute. Because of the leak, liquid is being lost from the tank at a rate which, at any instant, is equal to \(k V \mathrm {~cm} ^ { 3 }\) per minute where \(k\) is a positive constant.

1. Write down a differential equation describing this situation and solve it to show that $$V = \frac { 1 } { k } \left( 80 - 80 \mathrm { e } ^ { - k t } \right)$$
2. It is observed that \(V = 500\) when \(t = 15\), so that \(k\) satisfies the equation $$k = \frac { 4 - 4 e ^ { - 15 k } } { 25 }$$ Use an iterative formula, based on this equation, to find the value of \(k\) correct to 2 significant figures. Use an initial value of \(k = 0.1\) and show the result of each iteration to 4 significant figures.
3. Determine how much liquid there is in the tank 20 minutes after the liquid started flowing, and state what happens to the volume of liquid in the tank after a long time.