CAIE P3 (Pure Mathematics 3) 2017 November

1
\includegraphics[max width=\textwidth, alt={}, center]{746d2c39-7d78-4478-bc36-15ea5e3ba72a-02_460_807_258_667} The diagram shows a sketch of the curve \(y = \frac { 3 } { \sqrt { } \left( 9 - x ^ { 3 } \right) }\) for values of \(x\) from - 1.2 to 1.2 .

1. Use the trapezium rule, with two intervals, to estimate the value of $$\int _ { - 1.2 } ^ { 1.2 } \frac { 3 } { \sqrt { \left( 9 - x ^ { 3 } \right) } } \mathrm { d } x$$ giving your answer correct to 2 decimal places.
2. Explain, with reference to the diagram, why the trapezium rule may be expected to give a good approximation to the true value of the integral in this case.

2 Showing all necessary working, solve the equation \(2 \log _ { 2 } x = 3 + \log _ { 2 } ( x + 1 )\), giving your answer correct to 3 significant figures.

3 By expressing the equation \(\tan \left( \theta + 60 ^ { \circ } \right) + \tan \left( \theta - 60 ^ { \circ } \right) = \cot \theta\) in terms of \(\tan \theta\) only, solve the equation for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

4 The curve with equation \(y = \frac { 2 - \sin x } { \cos x }\) has one stationary point in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).

1. Find the exact coordinates of this point.
2. Determine whether this point is a maximum or a minimum point.

5 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( x + 2 )$$ and it is given that \(y = 2\) when \(x = 1\). Solve the differential equation and obtain an expression for \(y\) in terms of \(x\).

6 The equation of a curve is \(x ^ { 3 } y - 3 x y ^ { 3 } = 2 a ^ { 4 }\), where \(a\) is a non-zero constant.

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x ^ { 2 } y - 3 y ^ { 3 } } { 9 x y ^ { 2 } - x ^ { 3 } }\).
2. Hence show that there are only two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points.

7 Throughout this question the use of a calculator is not permitted.
The complex number \(1 - ( \sqrt { } 3 ) \mathrm { i }\) is denoted by \(u\).

1. Find the modulus and argument of \(u\).
2. Show that \(u ^ { 3 } + 8 = 0\).
3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Re } z \geqslant 2\), where \(\operatorname { Re } z\) denotes the real part of \(z\).
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   \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 8 x ^ { 2 } + 9 x + 8 } { ( 1 - x ) ( 2 x + 3 ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

9 It is given that \(\int _ { 1 } ^ { a } x ^ { \frac { 1 } { 2 } } \ln x \mathrm {~d} x = 2\), where \(a > 1\).

1. Show that \(a ^ { \frac { 3 } { 2 } } = \frac { 7 + 2 a ^ { \frac { 3 } { 2 } } } { 3 \ln a }\).
2. Show by calculation that \(a\) lies between 2 and 4 .
3. Use the iterative formula $$a _ { n + 1 } = \left( \frac { 7 + 2 a _ { n } ^ { \frac { 3 } { 2 } } } { 3 \ln a _ { n } } \right) ^ { \frac { 2 } { 3 } }$$ to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

10 Two planes \(p\) and \(q\) have equations \(x + y + 3 z = 8\) and \(2 x - 2 y + z = 3\) respectively.

1. Calculate the acute angle between the planes \(p\) and \(q\).
2. The point \(A\) on the line of intersection of \(p\) and \(q\) has \(y\)-coordinate equal to 2 . Find the equation of the plane which contains the point \(A\) and is perpendicular to both the planes \(p\) and \(q\). Give your answer in the form \(a x + b y + c z = d\).