SPS SPS FM Pure (SPS FM Pure) 2021 June

A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$ Find the gradient of the curve at the point where \(t = -2\) [2]

The equation \(x^3 - 3x + 1 = 0\) has three real roots.

1. Show that one of the roots lies between \(-2\) and \(-1\) [2 marks]
2. Taking \(x_1 = -2\) as the first approximation to one of the roots, use the Newton-Raphson method to find \(x_2\), the second approximation. [3 marks]
3. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x_1 = -1\) [1 mark]

Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).

1. Find the position vector of \(P\). [2]
2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [2]

Solve the quadratic equation \(x^2 - 4x - 1 - 12i = 0\) writing your solutions in the form \(a + bi\). [8]

\(\int_1^2 x^3 \ln(2x) dx\) can be written in the form \(p \ln 2 + q\), where \(p\) and \(q\) are rational numbers. Find \(p\) and \(q\). [5 marks]

1. Use the binomial expansion, in ascending powers of \(x\), to show that $$\sqrt{4-x} = 2 - \frac{1}{4}x + kx^2 + ...$$ where \(k\) is a rational constant to be found. [4] A student attempts to substitute \(x = 1\) into both sides of this equation to find an approximate value for \(\sqrt{3}\).
2. State, giving a reason, if the expansion is valid for this value of \(x\). [1]

1. Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]

Prove by induction that, for \(n \in \mathbb{Z}^+\) $$f(n) = 2^{n+2} + 3^{2n+1}$$ is divisible by 7 [6]

\includegraphics{figure_2} Figure 2 shows a sketch of part of the graph \(y = f(x)\), where $$f(x) = 2|3 - x| + 5, \quad x \geq 0$$

1. State the range of \(f\) [1]
2. Solve the equation $$f(x) = \frac{1}{2}x + 30$$ [3] Given that the equation \(f(x) = k\), where \(k\) is a constant, has two distinct roots,
3. state the set of possible values for \(k\). [2]

A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation \(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics{figure_3} Find the maximum vertical height above the platform of the sculpture. [8 marks]

1. Given that \(u = 2^x\), write down an expression for \(\frac{du}{dx}\) [1 mark]
2. Find the exact value of \(\int_0^1 2^x \sqrt{3 + 2^x} dx\) Fully justify your answer. [6 marks]

$$\mathbf{A} = \begin{pmatrix} 2 & a \\ a-4 & b \end{pmatrix}$$ where \(a\) and \(b\) are non-zero constants. Given that the matrix \(\mathbf{A}\) is self-inverse,

1. determine the value of \(b\) and the possible values for \(a\). [5] The matrix \(\mathbf{A}\) represents a linear transformation \(M\). Using the smaller value of \(a\) from part (a),
2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line. [3]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{4\sin 2x}{e^{\sqrt{2}x-1}}, \quad 0 \leq x \leq \pi$$ The curve has a maximum turning point at \(P\) and a minimum turning point at \(Q\) as shown in Figure 5.

1. Show that the \(x\) coordinates of point \(P\) and point \(Q\) are solutions of the equation $$\tan 2x = \sqrt{2}$$ [4]
2. Using your answer to part (a), find the \(x\)-coordinate of the minimum turning point on the curve with equation $$y = 3 - 2f(x)$$ [2]

The height \(x\) metres, of a column of water in a fountain display satisfies the differential equation \(\frac{dx}{dt} = -\frac{8\sin 2t}{3\sqrt{x}}\), where \(t\) is the time in seconds after the display begins. Solve the differential equation, given that initially the column of water has zero height. Express your answer in the form \(x = f(t)\) [7 marks]

Given that there are two distinct complex numbers \(z\) that satisfy $$\{z: |z - 3 - 5i| = 2r\} \cap \left\{z: \arg(z - 2) = \frac{3\pi}{4}\right\}$$ determine the exact range of values for the real constant \(r\). [7]