CAIE P3 (Pure Mathematics 3) 2024 November

1 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 5 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(( x - 4 )\) the remainder is equal to 3 times the remainder when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\). Find the values of \(a\) and \(b\).

2 Find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln 3 x \mathrm {~d} x\). Give your answer in the form \(a \ln b + c\), where \(a\) and \(c\) are rational and \(b\) is an integer.
\includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-04_2720_38_105_2010}

3 The equation of a curve is \(\ln ( x + y ) = 3 x ^ { 2 } y\).
Find the gradient of the curve at the point \(( 1,0 )\).

4

1. Show that \(\sec ^ { 4 } \theta - \tan ^ { 4 } \theta \equiv 1 + 2 \tan ^ { 2 } \theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-07_2723_35_101_20}
2. Hence or otherwise solve the equation \(\sec ^ { 4 } 2 \alpha - \tan ^ { 4 } 2 \alpha = 2 \tan ^ { 2 } 2 \alpha \sec ^ { 2 } 2 \alpha\) for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\). [5]

5

1. By sketching a suitable pair of graphs, show that the equation \(2 + \mathrm { e } ^ { - 0.2 x } = \ln ( 1 + x )\) has only one root.
2. Show by calculation that this root lies between 7 and 9 .
   \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-08_2716_40_109_2009}
3. Use the iterative formula $$x _ { n + 1 } = \exp \left( 2 + \mathrm { e } ^ { - 0.2 x _ { n } } \right) - 1$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
   \(\left[ \exp ( x ) \right.\) is an alternative notation for \(\left. \mathrm { e } ^ { x } .\right]\)
   \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_481_789_262_639} The diagram shows the curve \(y = \sin 2 x ( 1 + \sin 2 x )\), for \(0 \leqslant x \leqslant \frac { 3 } { 4 } \pi\), and its minimum point \(M\). The shaded region bounded by the curve that lies above the \(x\)-axis and the \(x\)-axis itself is denoted by \(R\).

1. Given that the \(x\)-coordinate of \(M\) lies in the interval \(\frac { 1 } { 2 } \pi < x < \frac { 3 } { 4 } \pi\), find the exact coordinates of \(M\).
   \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_2718_35_107_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-11_2725_35_99_20}
2. Find the exact area of the region \(R\).

7
Let \(f ( x ) = \frac { 5 x ^ { 2 } + 8 x + 5 } { ( 1 + 2 x ) \left( 2 + x ^ { 2 } \right) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
   \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-13_2726_34_97_21}
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\mathrm { f } ( x )\).

8

1. Given that \(z = 1 + y \mathrm { i }\) and that \(y\) is a real number, express \(\frac { 1 } { z }\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are functions of \(y\).
2. Show that \(\left( a - \frac { 1 } { 2 } \right) ^ { 2 } + b ^ { 2 } = \frac { 1 } { 4 }\), where \(a\) and \(b\) are the functions of \(y\) found in part (a).
   \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-14_2716_35_108_2012}
3. On a single Argand diagram, sketch the loci given by the equations \(\operatorname { Re } ( z ) = 1\) and \(\left| z - \frac { 1 } { 2 } \right| = \frac { 1 } { 2 }\), where \(z\) is a complex number.
4. The complex number \(z\) is such that \(\operatorname { Re } ( z ) = 1\). Use your answer to part (b) to give a geometrical description of the locus of \(\frac { 1 } { z }\).

9 The position vector of point \(A\) relative to the origin \(O\) is \(\overrightarrow { O A } = 8 \mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is parallel to the vector \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\).

1. State a vector equation for \(l\).
2. The position vector of point \(B\) relative to the origin \(O\) is \(\overrightarrow { O B } = - t \mathbf { i } + 4 t \mathbf { j } + 3 t \mathbf { k }\), where \(t\) is a constant. The line \(l\) also passes through \(B\). Find the value of \(t\).
3. The line \(m\) has vector equation \(\mathbf { r } = 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\). The acute angle between the directions of \(l\) and \(m\) is \(\theta\), where \(\cos \theta = \frac { 1 } { \sqrt { 6 } }\).
   Find the possible values of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-18_542_559_251_753} A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time \(t\) minutes, the depth of the water in the tank is \(h\) metres. There is a tap at the bottom of the tank. When the tap is open, water flows out of the tank at a rate proportional to the square root of the volume of water in the tank.
4. Show that \(\frac { \mathrm { d } h } { \mathrm {~d} t } = - \lambda \sqrt { h }\), where \(\lambda\) is a positive constant.
   \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-18_2718_42_107_2007}
5. At time \(t = 0\) the tap is opened. It is given that \(h = 4\) when \(t = 0\) and that \(h = 2.25\) when \(t = 20\). Solve the differential equation to obtain an expression for \(t\) in terms of \(h\), and hence find the time taken to empty the tank.
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