CAIE P3 (Pure Mathematics 3) 2024 November

Expand \((9 - 3x)^{\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients. [4]

1. By sketching a suitable pair of graphs, show that the equation \(\cot 2x = \sec x\) has exactly one root in the interval \(0 < x < \frac{1}{2}\pi\). [2]
2. Show that if a sequence of real values given by the iterative formula $$x_{n+1} = \frac{1}{2}\tan^{-1}(\cos x_n)$$ converges, then it converges to the root in part (a). [1]

The square roots of \(6 - 8i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(6 - 8i\) in exact Cartesian form. [5]

Solve the equation \(5^x = 5^{x+2} - 10\). Give your answer correct to 3 decimal places. [3]

1. The complex number \(u\) is given by $$u = \frac{(\cos \frac{1}{4}\pi + i \sin \frac{1}{4}\pi)^4}{\cos \frac{1}{2}\pi - i \sin \frac{1}{2}\pi}$$ Find the exact value of \(\arg u\). [2]
2. The complex numbers \(u\) and \(u^*\) are plotted on an Argand diagram. Describe the single geometrical transformation that maps \(u\) onto \(u^*\) and state the exact value of \(\arg u^*\). [2]

\includegraphics{figure_6} The variables \(x\) and \(y\) satisfy the equation \(ay = b^x\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((0.50, 2.24)\) and \((3.40, 8.27)\), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to 1 significant figure. [4]

1. Show that the equation \(\tan^3 x + 2 \tan 2x - \tan x = 0\) may be expressed as $$\tan^3 x - 2 \tan^2 x - 3 = 0$$ for \(\tan x \neq 0\). [3]
2. Hence solve the equation \(\tan^3 2\theta + 2 \tan 4\theta - \tan 2\theta = 0\) for \(0 < \theta < \pi\). Give your answers in exact form. [3]

The parametric equations of a curve are $$x = \tan^2 2t, \quad y = \cos 2t,$$ for \(0 < t < \frac{1}{4}\pi\).

1. Show that \(\frac{dy}{dx} = -\frac{1}{2}\cos^3 2t\). [4]
2. Hence find the equation of the normal to the curve at the point where \(t = \frac{1}{8}\pi\). Give your answer in the form \(y = mx + c\). [4]

With respect to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} -3 \\ -2 \\ 2 \end{pmatrix}.$$

1. The point \(D\) is such that \(ABCD\) is a trapezium with \(\overrightarrow{DC} = 3\overrightarrow{AB}\). Find the position vector of \(D\). [2]
2. The diagonals of the trapezium intersect at the point \(P\). Find the position vector of \(P\). [5]
3. Using a scalar product, calculate angle \(ABC\). [4]

A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40\pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8\pi r\). The balloon remains a sphere at all times.

1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac{dr}{dt} = \frac{50 - r}{5r^2}.$$ [3]
2. Find the quotient and remainder when \(5r^2\) is divided by \(50 - r\). [3]
3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\). [6]
4. Find the value of \(t\) when the radius of the balloon is 12. [1]

Let \(f(x) = \frac{2e^{2x}}{e^{2x} - 3e^x + 2}\).

1. Find \(f'(x)\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = f(x)\). [5]
2. Use the substitution \(u = e^x\) and partial fractions to find the exact value of \(\int_{\ln 5} f(x) dx\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form. [9]