CAIE P3 (Pure Mathematics 3) 2017 March

1 Solve the equation \(\ln \left( 1 + 2 ^ { x } \right) = 2\), giving your answer correct to 3 decimal places.

2 Solve the inequality \(| x - 4 | < 2 | 3 x + 1 |\).

3

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

4

1. Express \(8 \cos \theta - 15 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$8 \cos 2 x - 15 \sin 2 x = 4$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

5 The curve with equation \(y = \mathrm { e } ^ { - a x } \tan x\), where \(a\) is a positive constant, has only one point in the interval \(0 < x < \frac { 1 } { 2 } \pi\) at which the tangent is parallel to the \(x\)-axis. Find the value of \(a\) and state the exact value of the \(x\)-coordinate of this point.

6 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\). The plane \(p\) has equation \(3 x + y - 5 z = 20\).

1. Show that the line \(l\) lies in the plane \(p\).
2. A second plane is parallel to \(l\), perpendicular to \(p\) and contains the point with position vector \(3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). Find the equation of this plane, giving your answer in the form \(a x + b y + c z = d\). [5]

7
\includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-12_444_382_258_886} A water tank has vertical sides and a horizontal rectangular base, as shown in the diagram. The area of the base is \(2 \mathrm {~m} ^ { 2 }\). At time \(t = 0\) the tank is empty and water begins to flow into it at a rate of \(1 \mathrm {~m} ^ { 3 }\) per hour. At the same time water begins to flow out from the base at a rate of \(0.2 \sqrt { } h \mathrm {~m} ^ { 3 }\) per hour, where \(h \mathrm {~m}\) is the depth of water in the tank at time \(t\) hours.

1. Form a differential equation satisfied by \(h\) and \(t\), and show that the time \(T\) hours taken for the depth of water to reach 4 m is given by $$T = \int _ { 0 } ^ { 4 } \frac { 10 } { 5 - \sqrt { } h } \mathrm {~d} h$$
2. Using the substitution \(u = 5 - \sqrt { } h\), find the value of \(T\).

1. Showing all your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( z ) = 0\).
2. Find the other three roots of the equation \(\mathrm { p } ( z ) = 0\).

9 Let \(\mathrm { f } ( x ) = \frac { x ( 6 - x ) } { ( 2 + x ) \left( 4 + x ^ { 2 } \right) }\).

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 \(x ^ { 2 }\).

10
\includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-18_337_529_260_808} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\). The \(x\)-coordinate of the point \(P\) is equal to e, and the normal to the curve at \(P\) meets the \(x\)-axis at \(Q\).

1. Find the \(x\)-coordinate of \(Q\).
2. Show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is a constant.
3. Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the \(x\)-axis and the normal \(P Q\).