AQA Further Paper 2 (Further Paper 2) 2019 June

Given that \(z\) is a complex number, and that \(z^*\) is the complex conjugate of \(z\), which of the following statements is not always true? Circle your answer. [1 mark] \((z^*)^* = z\) \quad\quad \(zz^* = |z|^2\) \quad\quad \((-z)^* = -(z^*)\) \quad\quad \(z - z^* = z^* - z\)

Which of the straight lines given below is an asymptote to the curve $$y = \frac{ax^2}{x-1}$$ where \(a\) is a non-zero constant? Circle your answer. [1 mark] \(y = ax + a\) \quad\quad \(y = ax\) \quad\quad \(y = ax - a\) \quad\quad \(y = a\)

The set \(A\) is defined by \(A = \{x : -\sqrt{2} < x < 0\} \cup \{x : 0 < x < \sqrt{2}\}\) Which of the inequalities given below has \(A\) as its solution? Circle your answer. [1 mark] \(|x^2 - 1| > 1\) \quad\quad \(|x^2 - 1| \geq 1\) \quad\quad \(|x^2 - 1| < 1\) \quad\quad \(|x^2 - 1| \leq 1\)

The positive integer \(k\) is such that $$\sum_{r=1}^{k} (3r - k) = 90$$ Find the value of \(k\). [3 marks]

A curve has equation \(y = \cosh x\) Show that the arc length of the curve from \(x = a\) to \(x = b\), where \(0 < a < b\), is equal to $$\sinh b - \sinh a$$ [4 marks]

A circle \(C\) in the complex plane has equation \(|z - 2 - 5\mathrm{i}| = a\) The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(\arg(z_1) = \frac{\pi}{4}\) Prove that \(a = \frac{3\sqrt{2}}{2}\) [6 marks]

The points \(A\), \(B\) and \(C\) have coordinates \(A(4, 5, 2)\), \(B(-3, 2, -4)\) and \(C(2, 6, 1)\)

1. Use a vector product to show that the area of triangle \(ABC\) is \(\frac{5\sqrt{11}}{2}\) [4 marks]
2. The points \(A\), \(B\) and \(C\) lie in a plane. Find a vector equation of the plane in the form \(\mathbf{r} \cdot \mathbf{n} = k\) [1 mark]
3. Hence find the exact distance of the plane from the origin. [1 mark]

A parabola \(P_1\) has equation \(y^2 = 4ax\) where \(a > 0\) \(P_1\) is translated by the vector \(\begin{bmatrix} b \\ 0 \end{bmatrix}\), where \(b > 0\), to give the parabola \(P_2\)

1. The line \(y = mx\) is a tangent to \(P_2\) Prove that \(m = \pm\sqrt{\frac{a}{b}}\) Solutions using differentiation will be given no marks. [4 marks]
2. The line \(y = \sqrt{\frac{a}{b}} x\) meets \(P_2\) at the point \(D\). The finite region \(R\) is bounded by the \(x\)-axis, \(P_2\) and a line through \(D\) perpendicular to the \(x\)-axis. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis to form a solid. Find, in terms of \(a\) and \(b\), the volume of this solid. Fully justify your answer. [5 marks]

1. Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 5 & 5 \\ -3 & 13 \\ 5 & 10 \end{bmatrix}$$ [5 marks]
2. Find matrices \(\mathbf{U}\) and \(\mathbf{D}\) such that \(\mathbf{D}\) is a diagonal matrix and \(\mathbf{M} = \mathbf{U}\mathbf{D}\mathbf{U}^{-1}\) [2 marks]
3. Given that \(\mathbf{M}^n \to \mathbf{L}\) as \(n \to \infty\), find the matrix \(\mathbf{L}\). [4 marks]
4. The transformation represented by \(\mathbf{L}\) maps all points onto a line. Find the equation of this line. [2 marks]

Prove by induction that \(f(n) = n^3 + 3n^2 + 8n\) is divisible by 6 for all integers \(n \geq 1\) [7 marks]

The line \(L_1\) has equation $$\frac{x-2}{3} = \frac{y+4}{8} = \frac{4z-5}{5}$$ The line \(L_2\) has equation $$\left(\mathbf{r} - \begin{bmatrix} -2 \\ 0 \\ 3 \end{bmatrix}\right) \times \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix} = \mathbf{0}$$ Find the shortest distance between the two lines, giving your answer to three significant figures. [8 marks]

Abel and Bonnie are trying to solve this mathematical problem: \(z = 2 - 3\mathrm{i}\) is a root of the equation \(2z^3 + mz^2 + pz + 91 = 0\) Find the value of \(m\) and the value of \(p\). Abel says he has solved the problem. Bonnie says there is not enough information to solve the problem.

1. Abel's solution begins as follows: Since \(z = 2 - 3\mathrm{i}\) is a root of the equation, \(z = 2 + 3\mathrm{i}\) is another root. State one extra piece of information about \(m\) and \(p\) which could be added to the problem to make the beginning of Abel's solution correct. [1 mark]
2. Prove that Bonnie is right. [4 marks]

1. Explain why \(\int_3^{\infty} x^2 e^{-2x} \, dx\) is an improper integral. [1 mark]
2. Evaluate \(\int_3^{\infty} x^2 e^{-2x} \, dx\) Show the limiting process. [9 marks]

Let $$S_n = \sum_{r=1}^{n} \frac{1}{(r+1)(r+3)}$$ where \(n \geq 1\)

1. Use the method of differences to show that $$S_n = \frac{5n^2 + an}{12(n+b)(n+c)}$$ where \(a\), \(b\) and \(c\) are integers. [6 marks]
2. Show that, for any number \(k\) greater than \(\frac{12}{5}\), if the difference between \(\frac{5}{12}\) and \(S_n\) is less than \(\frac{1}{k}\), then $$n > \frac{k-5+\sqrt{k^2+1}}{2}$$ [6 marks]

\includegraphics{figure_15} Two tanks, A and B, each have a capacity of 800 litres. At time \(t = 0\) both tanks are full of pure water. When \(t > 0\), water flows in the following ways: • Water with a salt concentration of \(\mu\) grams per litre flows into tank A at a constant rate • Water flows from tank A to tank B at a rate of 16 litres per minute • Water flows from tank B to tank A at a rate of \(r\) litres per minute • Water flows out of tank B through a waste pipe • The amount of water in each tank remains at 800 litres. At time \(t\) minutes (\(t \geq 0\)) there are \(x\) grams of salt in tank A and \(y\) grams of salt in tank B. This system is represented by the coupled differential equations \begin{align} \frac{dx}{dt} &= 36 - 0.02x + 0.005y \tag{1}
\frac{dy}{dt} &= 0.02x - 0.02y \tag{2} \end{align}

1. Find the value of \(r\). [2 marks]
2. Show that \(\mu = 3\) [3 marks]
3. Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\). [9 marks]