Questions Further Paper 2 (305 questions)

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AQA Further Paper 2 2019 June Q5
4 marks Standard +0.8
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]
AQA Further Paper 2 2019 June Q6
6 marks Challenging +1.8
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]
AQA Further Paper 2 2019 June Q7
6 marks Standard +0.3
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]
AQA Further Paper 2 2019 June Q8
9 marks Challenging +1.8
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]
AQA Further Paper 2 2019 June Q9
13 marks Challenging +1.8
  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]
AQA Further Paper 2 2019 June Q10
7 marks Standard +0.3
Prove by induction that \(f(n) = n^3 + 3n^2 + 8n\) is divisible by 6 for all integers \(n \geq 1\) [7 marks]
AQA Further Paper 2 2019 June Q11
8 marks Challenging +1.2
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]
AQA Further Paper 2 2019 June Q12
5 marks Challenging +1.2
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]
AQA Further Paper 2 2019 June Q13
10 marks Standard +0.8
  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]
AQA Further Paper 2 2019 June Q14
12 marks Challenging +1.8
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]
AQA Further Paper 2 2019 June Q15
14 marks Challenging +1.8
\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]
AQA Further Paper 2 2020 June Q1
1 marks Moderate -0.8
Three of the four expressions below are equivalent to each other. Which of the four expressions is not equivalent to any of the others? Circle your answer. [1 mark] \(\mathbf{a} \times (\mathbf{a} + \mathbf{b})\) \quad \((\mathbf{a} + \mathbf{b}) \times \mathbf{b}\) \quad \((\mathbf{a} - \mathbf{b}) \times \mathbf{b}\) \quad \(\mathbf{a} \times (\mathbf{a} - \mathbf{b})\)
AQA Further Paper 2 2020 June Q2
1 marks Standard +0.3
Given that \(\arg(a + bi) = \varphi\), where \(a\) and \(b\) are positive real numbers and \(0 < \varphi < \frac{\pi}{2}\), three of the following four statements are correct. Which statement is not correct? Tick \((\checkmark)\) one box. [1 mark] \(\arg(-a - bi) = \pi - \varphi\) \(\arg(a - bi) = -\varphi\) \(\arg(b + ai) = \frac{\pi}{2} - \varphi\) \(\arg(b - ai) = \varphi - \frac{\pi}{2}\)
AQA Further Paper 2 2020 June Q3
1 marks Moderate -0.5
Find the gradient of the tangent to the curve $$y = \sin^{-1} x$$ at the point where \(x = \frac{1}{5}\) Circle your answer. [1 mark] \(\frac{5\sqrt{6}}{12}\) \quad \(\frac{2\sqrt{6}}{5}\) \quad \(\frac{4\sqrt{3}}{25}\) \quad \(\frac{25}{24}\)
AQA Further Paper 2 2020 June Q4
3 marks Standard +0.8
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are defined as follows: $$\mathbf{A} = \begin{bmatrix} x + 1 & 2 \\ x + 2 & -3 \end{bmatrix}$$ $$\mathbf{B} = \begin{bmatrix} x - 4 & x - 2 \\ 0 & -2 \end{bmatrix}$$ Show that there is a value of \(x\) for which \(\mathbf{AB} = k\mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix and \(k\) is an integer to be found. [3 marks]
AQA Further Paper 2 2020 June Q5
5 marks Standard +0.3
Solve the inequality $$\frac{2x + 3}{x - 1} \leq x + 5$$ [5 marks]
AQA Further Paper 2 2020 June Q6
5 marks Challenging +1.2
Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]
AQA Further Paper 2 2020 June Q7
5 marks Standard +0.8
The diagram shows part of the graph of \(y = \cos^{-1} x\) \includegraphics{figure_7} The finite region enclosed by the graph of \(y = \cos^{-1} x\), the \(y\)-axis, the \(x\)-axis and the line \(x = 0.8\) is rotated by \(2\pi\) radians about the \(x\)-axis. Use Simpson's rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places. [5 marks]
AQA Further Paper 2 2020 June Q8
9 marks Hard +2.3
  1. Factorise \(\begin{vmatrix} 2u + h + x & x + h & x^2 + h^2 \\ 0 & a & -a^2 \\ a + b & b & b^2 \end{vmatrix}\) as fully as possible. [6 marks]
  2. The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{bmatrix} 13 + x & x + 3 & x^2 + 9 \\ 0 & 5 & 25 \\ 8 & 3 & 9 \end{bmatrix}$$ Under the transformation represented by \(\mathbf{M}\), a solid of volume \(0.625 \text{m}^3\) becomes a solid of volume \(300 \text{m}^3\) Use your answer to part (a) to find the possible values of \(x\). [3 marks]
AQA Further Paper 2 2020 June Q9
7 marks Challenging +1.8
The matrix \(\mathbf{C} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}\), where \(a\) and \(b\) are positive real numbers, and \(\mathbf{C}^2 = \begin{bmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}\) Use \(\mathbf{C}\) to show that \(\cos \frac{\pi}{12}\) can be written in the form \(\frac{\sqrt{m + n}}{2}\), where \(m\) and \(n\) are integers. [7 marks]
AQA Further Paper 2 2020 June Q10
6 marks Challenging +1.2
The sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 0 \quad u_{n+1} = \frac{5}{6 - u_n}$$ Prove by induction that, for all integers \(n \geq 1\), $$u_n = \frac{5^n - 5}{5^n - 1}$$ [6 marks]
AQA Further Paper 2 2020 June Q11
8 marks Challenging +1.2
  1. Starting from the series given in the formulae booklet, show that the general term of the Maclaurin series for $$\frac{\sin x}{x} - \cos x$$ is $$(-1)^{r+1} \frac{2r}{(2r + 1)!} x^{2r}$$ [4 marks]
  2. Show that $$\lim_{x \to 0} \left[ \frac{\sin x}{x} - \cos x \right] \frac{1}{1 - \cos x} = \frac{2}{3}$$ [4 marks]
AQA Further Paper 2 2020 June Q12
12 marks Challenging +1.3
  1. Given that \(I = \int_a^b e^{2t} \sin t \, dt\), show that $$I = \left[ qe^{2t} \sin t + re^{2t} \cos t \right]_a^b$$ where \(q\) and \(r\) are rational numbers to be found. [6 marks]
  2. A small object is initially at rest. The subsequent motion of the object is modelled by the differential equation $$\frac{dv}{dt} + v = 5e^t \sin t$$ where \(v\) is the velocity at time \(t\). Find the speed of the object when \(t = 2\pi\), giving your answer in exact form. [6 marks]
AQA Further Paper 2 2020 June Q13
10 marks Challenging +1.2
Charlotte is trying to solve this mathematical problem: Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + \frac{dy}{dx} - 2y = 10e^{-2x}$$ Charlotte's solution starts as follows: Particular integral: \(y = \lambda e^{-2x}\) so $$\frac{dy}{dx} = -2\lambda e^{-2x}$$ and $$\frac{d^2y}{dx^2} = 4\lambda e^{-2x}$$
  1. Show that Charlotte's method will fail to find a particular integral for the differential equation. [2 marks]
  2. Explain how Charlotte should have started her solution differently and find the general solution of the differential equation. [8 marks]
AQA Further Paper 2 2020 June Q14
11 marks Hard +2.3
The diagram shows the polar curve \(C_1\) with equation \(r = 2 \sin \theta\) The diagram also shows part of the polar curve \(C_2\) with equation \(r = 1 + \cos 2\theta\) \includegraphics{figure_14}
  1. On the diagram above, complete the sketch of \(C_2\) [2 marks]
  2. Show that the area of the region shaded in the diagram is equal to $$k\pi + m\alpha - \sin 2\alpha + q \sin 4\alpha$$ where \(\alpha = \sin^{-1} \left( \frac{\sqrt{5} - 1}{2} \right)\), and \(k\), \(m\) and \(q\) are rational numbers. [9 marks]