CAIE P3 (Pure Mathematics 3) 2019 November

1 Solve the inequality \(2 | x + 2 | > | 3 x - 1 |\).

2 The polynomial \(6 x ^ { 3 } + a x ^ { 2 } + b x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 24 . Find the values of \(a\) and \(b\).

3 Showing all necessary working, solve the equation \(\frac { 3 ^ { 2 x } + 3 ^ { - x } } { 3 ^ { 2 x } - 3 ^ { - x } } = 4\). Give your answer correct to 3 decimal places.

4

1. By first expanding \(\tan ( 2 x + x )\), show that the equation \(\tan 3 x = 3 \cot x\) can be written in the form \(\tan ^ { 4 } x - 12 \tan ^ { 2 } x + 3 = 0\).
2. Hence solve the equation \(\tan 3 x = 3 \cot x\) for \(0 ^ { \circ } < x < 90 ^ { \circ }\).

5

1. By sketching a suitable pair of graphs, show that the equation \(\ln ( x + 2 ) = 4 \mathrm { e } ^ { - x }\) has exactly one real root.
2. Show by calculation that this root lies between \(x = 1\) and \(x = 1.5\).
3. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 4 } { \ln \left( x _ { n } + 2 \right) } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

1. Express \(w\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
   The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The complex number \(v\) is such that \(| v | = 2 | u |\) and \(\arg v = \arg u + \frac { 1 } { 3 } \pi\).
2. Sketch an Argand diagram showing the points representing \(u\) and \(v\).
3. Explain why \(v\) can be expressed as \(2 u w\). Hence find \(v\), giving your answer in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.

7 The plane \(m\) has equation \(x + 4 y - 8 z = 2\). The plane \(n\) is parallel to \(m\) and passes through the point \(P\) with coordinates \(( 5,2 , - 2 )\).

1. Find the equation of \(n\), giving your answer in the form \(a x + b y + c z = d\).
2. Calculate the perpendicular distance between \(m\) and \(n\).
3. The line \(l\) lies in the plane \(n\), passes through the point \(P\) and is perpendicular to \(O P\), where \(O\) is the origin. Find a vector equation for \(l\).

8
\includegraphics[max width=\textwidth, alt={}, center]{5b5ed7d1-028e-4f9a-ae9e-26071d0df678-14_604_497_262_822} The diagram shows the graph of \(y = \sec x\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).

1. Use the trapezium rule with 2 intervals to estimate the value of \(\int _ { 0 } ^ { 1.2 } \sec x \mathrm {~d} x\), giving your answer correct to 2 decimal places.
2. Explain, with reference to the diagram, whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part (i).
3. \(P\) is the point on the part of the curve \(y = \sec x\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\) at which the gradient is 2 . By first differentiating \(\frac { 1 } { \cos x }\), find the \(x\)-coordinate of \(P\), giving your answer correct to 3 decimal places.

9 The variables \(x\) and \(t\) satisfy the differential equation \(5 \frac { \mathrm {~d} x } { \mathrm {~d} t } = ( 20 - x ) ( 40 - x )\). It is given that \(x = 10\) when \(t = 0\).

1. Using partial fractions, solve the differential equation, obtaining an expression for \(x\) in terms of \(t\). [9]
2. State what happens to the value of \(x\) when \(t\) becomes large.

10
\includegraphics[max width=\textwidth, alt={}, center]{5b5ed7d1-028e-4f9a-ae9e-26071d0df678-18_449_787_262_678} The diagram shows the graph of \(y = \mathrm { e } ^ { \cos x } \sin ^ { 3 } x\) for \(0 \leqslant x \leqslant \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. Find the \(x\)-coordinate of \(M\). Show all necessary working and give your answer correct to 2 decimal places.
2. By first using the substitution \(u = \cos x\), find the exact value of the area of \(R\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.