CAIE P3 (Pure Mathematics 3) 2014 June

1 Find the set of values of \(x\) satisfying the inequality $$| x + 2 a | > 3 | x - a |$$ where \(a\) is a positive constant.

2 Solve the equation $$2 \ln \left( 5 - \mathrm { e } ^ { - 2 x } \right) = 1$$ giving your answer correct to 3 significant figures.

3 Solve the equation $$\cos \left( x + 30 ^ { \circ } \right) = 2 \cos x$$ giving all solutions in the interval \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).

4 The parametric equations of a curve are $$x = t - \tan t , \quad y = \ln ( \cos t )$$ for \(- \frac { 1 } { 2 } \pi < t < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot t\).
2. Hence find the \(x\)-coordinate of the point on the curve at which the gradient is equal to 2 . Give your answer correct to 3 significant figures.
3. The polynomial \(\mathrm { f } ( x )\) is of the form \(( x - 2 ) ^ { 2 } \mathrm {~g} ( x )\), where \(\mathrm { g } ( x )\) is another polynomial. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ^ { \prime } ( x )\).
4. The polynomial \(x ^ { 5 } + a x ^ { 4 } + 3 x ^ { 3 } + b x ^ { 2 } + a\), where \(a\) and \(b\) are constants, has a factor \(( x - 2 ) ^ { 2 }\). Using the factor theorem and the result of part (i), or otherwise, find the values of \(a\) and \(b\). [5]

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\includegraphics[max width=\textwidth, alt={}, center]{326d0ea0-8060-4439-8043-3301b281a30f-3_551_519_260_813} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(O A B\) is equal to \(x\) radians. The shaded region is bounded by \(A B , A C\) and the circular arc with centre \(A\) joining \(B\) and \(C\). The perimeter of the shaded region is equal to half the circumference of the circle.

1. Show that \(x = \cos ^ { - 1 } \left( \frac { \pi } { 4 + 4 x } \right)\).
2. Verify by calculation that \(x\) lies between 1 and 1.5.
3. Use the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { \pi } { 4 + 4 x _ { n } } \right)$$ to determine the value of \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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1. It is given that \(- 1 + ( \sqrt { } 5 ) \mathrm { i }\) is a root of the equation \(z ^ { 3 } + 2 z + a = 0\), where \(a\) is real. Showing your working, find the value of \(a\), and write down the other complex root of this equation.
2. The complex number \(w\) has modulus 1 and argument \(2 \theta\) radians. Show that \(\frac { w - 1 } { w + 1 } = \mathrm { i } \tan \theta\).

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\includegraphics[max width=\textwidth, alt={}, center]{326d0ea0-8060-4439-8043-3301b281a30f-3_391_826_1946_657} The diagram shows the curve \(y = x \cos \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y + 4 \sin \frac { 1 } { 2 } x = 0\).
2. Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis.

9 The population of a country at time \(t\) years is \(N\) millions. At any time, \(N\) is assumed to increase at a rate proportional to the product of \(N\) and \(( 1 - 0.01 N )\). When \(t = 0 , N = 20\) and \(\frac { \mathrm { d } N } { \mathrm {~d} t } = 0.32\).

1. Treating \(N\) and \(t\) as continuous variables, show that they satisfy the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = 0.02 N ( 1 - 0.01 N )$$
2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(N\).
3. Find the time at which the population will be double its value at \(t = 0\).

10 Referred to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } , \quad \overrightarrow { O B } = 2 \mathbf { i } + 4 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = 3 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }$$

1. Find the exact value of the cosine of angle \(B A C\).
2. Hence find the exact value of the area of triangle \(A B C\).
3. Find the equation of the plane which is parallel to the \(y\)-axis and contains the line through \(B\) and \(C\). Give your answer in the form \(a x + b y + c z = d\).