CAIE P3 (Pure Mathematics 3) 2021 November

1 Solve the equation \(4 \left| 5 ^ { x } - 1 \right| = 5 ^ { x }\), giving your answers correct to 3 decimal places.

2

1. Express \(5 \sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence state the greatest and least possible values of \(( 5 \sin x - 3 \cos x ) ^ { 2 }\).

3 The curve with equation \(y = x \mathrm { e } ^ { 1 - 2 x }\) has one stationary point.

1. Find the coordinates of this point.
2. Determine whether the stationary point is a maximum or a minimum.

4 Using the substitution \(u = \sqrt { x }\), find the exact value of $$\int _ { 3 } ^ { \infty } \frac { 1 } { ( x + 1 ) \sqrt { x } } \mathrm {~d} x$$

5

1. Show that the equation $$\cot 2 \theta + \cot \theta = 2$$ can be expressed as a quadratic equation in \(\tan \theta\).
2. Hence solve the equation \(\cot 2 \theta + \cot \theta = 2\), for \(0 < \theta < \pi\), giving your answers correct to 3 decimal places.

6 When \(( a + b x ) \sqrt { 1 + 4 x }\), where \(a\) and \(b\) are constants, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 2 }\) are 3 and - 6 respectively. Find the values of \(a\) and \(b\).

7

1. Given that \(y = \ln ( \ln x )\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \ln x }$$ The variables \(x\) and \(t\) satisfy the differential equation $$x \ln x + t \frac { \mathrm {~d} x } { \mathrm {~d} t } = 0$$ It is given that \(x = \mathrm { e }\) when \(t = 2\).
2. Solve the differential equation obtaining an expression for \(x\) in terms of \(t\), simplifying your answer.
3. Hence state what happens to the value of \(x\) as \(t\) tends to infinity.

8 The constant \(a\) is such that \(\int _ { 1 } ^ { a } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x = 6\).

1. Show that \(a = \exp \left( \frac { 1 } { \sqrt { a } } + 2 \right)\).
   \(\left[ \exp ( x ) \right.\) is an alternative notation for \(\left. \mathrm { e } ^ { x } .\right]\)
2. Verify by calculation that \(a\) lies between 9 and 11 .
3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

9 Two lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + s ( 4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + t ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) respectively.

1. Show that \(l\) and \(m\) are perpendicular.
2. Show that \(l\) and \(m\) intersect and state the position vector of the point of intersection.
3. Show that the length of the perpendicular from the origin to the line \(m\) is \(\frac { 1 } { 3 } \sqrt { 5 }\).

10 The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + 4 x + b\), where \(a\) and \(b\) are real constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\).

1. Find the values of \(a\) and \(b\).
2. State a second complex root of this equation.
3. Find the real factors of \(\mathrm { p } ( x )\).
4. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant \sqrt { 5 }\) and \(\arg z \leqslant \frac { 1 } { 4 } \pi\).
   2. Find the least value of \(\operatorname { Im } z\) for points in the shaded region. Give your answer in an exact form.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.