CAIE P3 (Pure Mathematics 3) 2011 June

1 Solve the inequality \(| x | < | 5 + 2 x |\).

2

1. Show that the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$ can be written as a quadratic equation in \(x\).
2. Hence solve the equation $$\log _ { 2 } ( x + 5 ) = 5 - \log _ { 2 } x$$

3 Solve the equation $$\cos \theta + 4 \cos 2 \theta = 3$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

4
\includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-2_339_1395_1089_374} The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The tangent at \(C\) meets \(A B\) produced at \(T\). The angle \(B O C\) is \(x\) radians. The area of the shaded region is equal to the area of the semicircle.

1. Show that \(x\) satisfies the equation $$\tan x = x + \pi$$
2. Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( x _ { n } + \pi \right)\) to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 The parametric equations of a curve are $$x = \ln ( \tan t ) , \quad y = \sin ^ { 2 } t$$ where \(0 < t < \frac { 1 } { 2 } \pi\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the equation of the tangent to the curve at the point where \(x = 0\).

6 A certain curve is such that its gradient at a point \(( x , y )\) is proportional to \(x y\). At the point \(( 1,2 )\) the gradient is 4 .

1. By setting up and solving a differential equation, show that the equation of the curve is \(y = 2 \mathrm { e } ^ { x ^ { 2 } - 1 }\).
2. State the gradient of the curve at the point \(( - 1,2 )\) and sketch the curve.

7

1. The complex number \(u\) is defined by \(u = \frac { 5 } { a + 2 \mathrm { i } }\), where the constant \(a\) is real.
   1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
   2. Find the value of \(a\) for which \(\arg \left( u ^ { * } \right) = \frac { 3 } { 4 } \pi\), where \(u ^ { * }\) denotes the complex conjugate of \(u\).
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z | < 2\) and \(| z | < | z - 2 - 2 \mathrm { i } |\).

8

1. Express \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in partial fractions.
2. Hence obtain the expansion of \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

9 Two planes have equations \(x + 2 y - 2 z = 7\) and \(2 x + y + 3 z = 5\).

1. Calculate the acute angle between the planes.
2. Find a vector equation for the line of intersection of the planes.

10
\includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-3_451_933_1777_605} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).

1. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\) is equal to \(2 - \frac { 17 } { \mathrm { e } ^ { 3 } }\).
2. Find the \(x\)-coordinate of the maximum point \(M\) on the curve.
3. Find the \(x\)-coordinate of the point \(P\) at which the tangent to the curve passes through the origin.