AQA Further Paper 2 (Further Paper 2) 2024 June

It is given that $$\begin{bmatrix} 2 \\ 1 \\ 9 \end{bmatrix} \times \begin{bmatrix} 5 \\ \lambda \\ -6 \end{bmatrix} = 0$$ where \(\lambda\) is a constant. Find the value of \(\lambda\) Circle your answer. [1 mark] \(-28\) \quad\quad \(-8\) \quad\quad \(8\) \quad\quad \(28\)

The movement of a particle is described by the simple harmonic equation $$\ddot{x} = -25x$$ where \(x\) metres is the displacement of the particle at time \(t\) seconds, and \(\ddot{x}\) m s\(^{-2}\) is the acceleration of the particle. The maximum displacement of the particle is 9 metres. Find the maximum speed of the particle. Circle your answer. [1 mark] \(15\) m s\(^{-1}\) \quad\quad \(45\) m s\(^{-1}\) \quad\quad \(75\) m s\(^{-1}\) \quad\quad \(135\) m s\(^{-1}\)

The function g is defined by $$g(x) = \text{sech } x \quad\quad (x \in \mathbb{R})$$ Which one of the following is the range of g? Tick (\(\checkmark\)) one box. [1 mark] \(-\infty < g(x) \leq -1\) \quad \(\square\) \(-1 \leq g(x) < 0\) \quad \(\square\) \(0 < g(x) \leq 1\) \quad \(\square\) \(1 \leq g(x) \leq \infty\) \quad \(\square\)

The function f is a quartic function with real coefficients. The complex number \(5i\) is a root of the equation \(f(x) = 0\) Which one of the following must be a factor of \(f(x)\)? Circle your answer. [1 mark] \((x^2 - 25)\) \quad\quad \((x^2 - 5)\) \quad\quad \((x^2 + 5)\) \quad\quad \((x^2 + 25)\)

The first four terms of the series \(S\) can be written as $$S = (1 \times 2) + (2 \times 3) + (3 \times 4) + (4 \times 5) + ...$$

1. Write an expression, using \(\sum\) notation, for the sum of the first \(n\) terms of \(S\) [1 mark]
2. Show that the sum of the first \(n\) terms of \(S\) is equal to $$\frac{1}{3}n(n + 1)(n + 2)$$ [2 marks]

The cubic equation $$x^3 + 5x^2 - 4x + 2 = 0$$ has roots \(\alpha\), \(\beta\) and \(\gamma\) Find a cubic equation, with integer coefficients, whose roots are \(3\alpha\), \(3\beta\) and \(3\gamma\) [3 marks]

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are defined as follows. $$\mathbf{A} = \begin{bmatrix} p - 2 & p - 1 \\ 0 & 1 \end{bmatrix} \quad\quad \mathbf{B} = \begin{bmatrix} 1 & 2p - 1 \\ 0 & 4 - p \end{bmatrix}$$ Find the values of \(p\) such that \(\mathbf{A}\) and \(\mathbf{B}\) are commutative under matrix multiplication. Fully justify your answer. [4 marks]

The vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are such that \(\mathbf{a} \times \mathbf{b} = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\) and \(\mathbf{a} \times \mathbf{c} = \begin{bmatrix} 0 \\ 0 \\ 3 \end{bmatrix}\) Work out \((\mathbf{a} - 4\mathbf{b} + 3\mathbf{c}) \times (2\mathbf{a})\) [4 marks]

A curve passes through the point \((-2, 4.73)\) and satisfies the differential equation $$\frac{dy}{dx} = \frac{y^2 - x^2}{2x + 3y}$$ Use Euler's step by step method once, and then the midpoint formula $$y_{r+1} = y_{r-1} + 2hf(x_r, y_r), \quad x_{r+1} = x_r + h$$ once, each with a step length of \(0.02\), to estimate the value of \(y\) when \(x = -1.96\) Give your answer to five significant figures. [4 marks]

The matrix \(\mathbf{C}\) is defined by $$\mathbf{C} = \begin{bmatrix} 3 & 2 \\ -4 & 5 \end{bmatrix}$$ Prove that the transformation represented by \(\mathbf{C}\) has no invariant lines of the form \(y = kx\) [4 marks]

Latifa and Sam are studying polynomial equations of degree greater than 2, with real coefficients and no repeated roots. Latifa says that if such an equation has exactly one real root, it must be of degree 3 Sam says that this is not correct. State, giving reasons, whether Latifa or Sam is right. [3 marks]

The transformation \(S\) is represented by the matrix \(\mathbf{M} = \begin{bmatrix} 1 & -6 \\ 2 & 7 \end{bmatrix}\) The transformation \(T\) is a reflection in the line \(y = x\sqrt{3}\) and is represented by the matrix \(\mathbf{N}\) The point \(P(x, y)\) is transformed first by \(S\), then by \(T\) The result of these transformations is the point \(Q(3, 8)\) Find the coordinates of \(P\) Give your answers to three decimal places. [5 marks]

1. Use the method of differences to show that $$\sum_{r=2}^{n} \frac{1}{(r - 1)r(r + 1)} = \frac{1}{4} - \frac{1}{2n} + \frac{1}{2(n + 1)}$$ [5 marks]
2. Find the smallest integer \(n\) such that $$\sum_{r=2}^{n} \frac{1}{(r - 1)r(r + 1)} > 0.24999$$ [3 marks]

The matrix \(\mathbf{M}\) is defined as $$\mathbf{M} = \begin{bmatrix} 5 & 2 & 1 \\ 6 & 3 & 2k + 3 \\ 2 & 1 & 5 \end{bmatrix}$$ where \(k\) is a constant.

1. Given that \(\mathbf{M}\) is a non-singular matrix, find \(\mathbf{M}^{-1}\) in terms of \(k\) [5 marks]
2. State any restrictions on the value of \(k\) [1 mark]
3. Using your answer to part (a), show that the solution to the set of simultaneous equations below is independent of the value of \(k\) \(5x + 2y + z = 1\) \(6x + 3y + (2k + 3)z = 4k + 3\) \(2x + y + 5z = 9\) [4 marks]

The diagram shows the line \(y = 5 - x\) \includegraphics{figure_15}

1. On the diagram above, sketch the graph of \(y = |x^2 - 4x|\), including all parts of the graph where it intersects the line \(y = 5 - x\) (You do not need to show the coordinates of the points of intersection.) [3 marks]
2. Find the solution of the inequality $$|x^2 - 4x| > 5 - x$$ Give your answer in an exact form. [4 marks]

The function f is defined by $$f(x) = \frac{ax + 5}{x + b}$$ where \(a\) and \(b\) are constants. The graph of \(y = f(x)\) has asymptotes \(x = -2\) and \(y = 3\)

1. Write down the value of \(a\) and the value of \(b\) [2 marks]
2. The diagram shows the graph of \(y = f(x)\) and its asymptotes. The shaded region \(R\) is enclosed by the graph of \(y = f(x)\), the \(x\)-axis and the \(y\)-axis. \includegraphics{figure_16}
   1. The shaded region \(R\) is rotated through \(360°\) about the \(x\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [3 marks]
   2. The shaded region \(R\) is rotated through \(360°\) about the \(y\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [4 marks]

The Argand diagram below shows a circle \(C\) \includegraphics{figure_17}

1. Write down the equation of the locus of \(C\) in the form $$|z - w| = a$$ where \(w\) is a complex number whose real and imaginary parts are integers, and \(a\) is an integer. [2 marks]
2. It is given that \(z_1\) is a complex number representing a point on \(C\). Of all the complex numbers which represent points on \(C\), \(z_1\) has the least argument.
   1. Find \(|z_1|\) Give your answer in an exact form. [3 marks]
   2. Show that \(\arg z_1 = \arcsin\left(\frac{6\sqrt{3} - 2}{13}\right)\) [4 marks]

In this question you may use results from the formulae booklet without proof. Use the binomial series for \((1 + x)^n\) and the Maclaurin's series for \(\sin x\) to find the series expansion for \(\frac{1}{(1 + \sin \theta)^4}\) up to and including the term in \(\theta^3\) [4 marks]

Solve the differential equation $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} - 45y = 21e^{5x} - 0.3x + 27x^2$$ given that \(y = \frac{37}{225}\) and \(\frac{dy}{dx} = 0\) when \(x = 0\) [10 marks]

The integral \(I_n\) is defined by $$I_n = \int_0^{\frac{\pi}{4}} \cos^n x \, dx \quad\quad (n \geq 0)$$

1. Show that $$I_n = \left(\frac{n-1}{n}\right)I_{n-2} + \frac{1}{n\left(2^{\frac{n}{2}}\right)} \quad\quad (n \geq 2)$$ [6 marks]
2. Use the result from part (a) to show that $$\int_0^{\frac{\pi}{4}} \cos^6 x \, dx = \frac{a\pi + b}{192}$$ where \(a\) and \(b\) are integers to be found. [3 marks]