CAIE P3 (Pure Mathematics 3) 2019 June

1 Use logarithms to solve the equation \(5 ^ { 3 - 2 x } = 4 \left( 7 ^ { x } \right)\), giving your answer correct to 3 decimal places.

2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x ^ { 2 } \cos 2 x \mathrm {~d} x = \frac { 1 } { 32 } \left( \pi ^ { 2 } - 8 \right)\).

3 Let \(f ( \theta ) = \frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta }\).

1. Show that \(\mathrm { f } ( \theta ) = \tan \theta\).
2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \mathrm { f } ( \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).

4 The equation of a curve is \(y = \frac { 1 + \mathrm { e } ^ { - x } } { 1 - \mathrm { e } ^ { - x } }\), for \(x > 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is always negative.
2. The gradient of the curve is equal to - 1 when \(x = a\). Show that \(a\) satisfies the equation \(\mathrm { e } ^ { 2 a } - 4 \mathrm { e } ^ { a } + 1 = 0\). Hence find the exact value of \(a\).

5 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) y \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ^ { 2 } + 5$$ It is given that \(y = 2\) when \(x = 0\). Solve the differential equation obtaining an expression for \(y ^ { 2 }\) in terms of \(x\).

6
\includegraphics[max width=\textwidth, alt={}, center]{87392b1c-3683-45b4-8d55-36760b5f0cc1-10_547_531_260_806} The diagram shows the curve \(y = x ^ { 4 } - 2 x ^ { 3 } - 7 x - 6\). The curve intersects the \(x\)-axis at the points \(( a , 0 )\) and \(( b , 0 )\), where \(a < b\). It is given that \(b\) is an integer.

1. Find the value of \(b\).
2. Hence show that \(a\) satisfies the equation \(a = - \frac { 1 } { 3 } \left( 2 + a ^ { 2 } + a ^ { 3 } \right)\).
3. Use an iterative formula based on the equation in part (ii) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

7 The curve \(y = \sin \left( x + \frac { 1 } { 3 } \pi \right) \cos x\) has two stationary points in the interval \(0 \leqslant x \leqslant \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. By considering the formula for \(\cos ( A + B )\), show that, at the stationary points on the curve, \(\cos \left( 2 x + \frac { 1 } { 3 } \pi \right) = 0\).
3. Hence find the exact \(x\)-coordinates of the stationary points.

8 Throughout this question the use of a calculator is not permitted.
The complex number \(u\) is defined by $$u = \frac { 4 \mathrm { i } } { 1 - ( \sqrt { } 3 ) \mathrm { i } }$$

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. Find the exact modulus and argument of \(u\).
3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | < 2\) and \(| z - u | < | z |\).

9 Let \(\mathrm { f } ( x ) = \frac { 2 x ( 5 - x ) } { ( 3 + x ) ( 1 - x ) ^ { 2 } }\).

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 \(x ^ { 3 }\).

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )\).

1. The point \(P\) has position vector \(4 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\). Find the length of the perpendicular from \(P\) to \(l\).
2. It is given that \(l\) lies in the plane with equation \(a x + b y + 2 z = 13\), where \(a\) and \(b\) are constants. Find the values of \(a\) and \(b\).
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