Questions — Pre-U Pre-U 9795/1 (179 questions)

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Pre-U Pre-U 9795/1 Specimen Q11
14 marks Challenging +1.2
11 The complex number \(z\) is defined as \(z = \cos \theta + \mathrm { i } \sin \theta\).
  1. Show that \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
  2. By expanding \(\left( z + z ^ { - 1 } \right) ^ { 5 }\), show that \(16 \cos ^ { 5 } \theta = \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta\).
  3. Hence find \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 5 } \theta \mathrm {~d} \theta\).
  4. Sketch the graphs of \(\mathrm { f } ( \theta ) = \sin ^ { 5 } \theta\) and \(\mathrm { f } ( \theta ) = \cos ^ { 5 } \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), and hence give the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 5 } \theta \mathrm {~d} \theta$$
Pre-U Pre-U 9795/1 Specimen Q12
14 marks Challenging +1.2
12 The curve \(C\) is defined parametrically by $$x = t + \ln ( \cosh t ) , \quad y = \sinh t$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - t } \cosh ^ { 2 } t\).
  2. Hence show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \cosh ^ { 2 } t ( 2 \sinh t - \cosh t )\).
  3. Find the exact value of \(t\) at the point on \(C\) where \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).
Pre-U Pre-U 9795/1 2011 June Q1
4 marks Standard +0.8
Given that the matrix \(\mathbf{A} = \begin{pmatrix} 2 & k \\ 1 & -3 \end{pmatrix}\), where \(k\) is real, is such that \(\mathbf{A}^3 = \mathbf{I}\), find the value of \(k\) and the numerical value of \(\det \mathbf{A}\). [4]
Pre-U Pre-U 9795/1 2011 June Q2
5 marks Standard +0.3
The cubic equation \(x^3 + x^2 + 7x - 1 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
  1. Show that \(\alpha^2 + \beta^2 + \gamma^2 = -13\). [3]
  2. State what can be deduced about the nature of these roots. [2]
Pre-U Pre-U 9795/1 2011 June Q3
5 marks Challenging +1.2
  1. Express \(\text{f}(r - 1) - \text{f}(r)\) as a single algebraic fraction, where \(\text{f}(r) = \frac{1}{(2r + 1)^2}\). [1]
  2. Hence, using the method of differences, show that $$\sum_{r=1}^{n} \frac{r}{(4r^2 - 1)^2} = \frac{n(n + 1)}{2(2n + 1)^2}$$ for all positive integers \(n\). [4]
Pre-U Pre-U 9795/1 2011 June Q4
8 marks Standard +0.8
  1. On a single diagram, sketch the graphs of \(y = \tanh x\) and \(y = \cosh x - 1\), and use your diagram to explain why the equation \(\text{f}(x) = 0\) has exactly two roots, where $$\text{f}(x) = 1 + \tanh x - \cosh x.$$ [3]
  2. The non-zero root of \(\text{f}(x) = 0\) is \(\alpha\).
    1. Verify that \(1 < \alpha < 1.5\). [1]
    2. Taking \(x_1 = 1.25\) as an initial approximation to \(\alpha\), use the Newton-Raphson iterative method to find \(x_3\), giving your answer to 5 decimal places. [4]
Pre-U Pre-U 9795/1 2011 June Q5
7 marks Standard +0.8
Find the general solution of the differential equation \(\frac{d^2 y}{dx^2} + y = 8x^2\). [7]
Pre-U Pre-U 9795/1 2011 June Q6
9 marks Challenging +1.2
Consider the set \(S\) of all matrices of the form \(\begin{pmatrix} p & p \\ p & p \end{pmatrix}\), where \(p\) is a non-zero rational number.
  1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). [5]
  2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3. [4]
Pre-U Pre-U 9795/1 2011 June Q7
11 marks Challenging +1.2
Sketch the curve with equation \(y = \frac{x^2 + 4x}{2x - 1}\), justifying all significant features. [11]
Pre-U Pre-U 9795/1 2011 June Q8
7 marks Challenging +1.2
  1. Determine the two values of \(k\) for which the system of equations \begin{align} x + 2y + 3z &= 4
    2x + 3y + kz &= 9
    x + ky + 6z &= 1 \end{align} has no unique solution. [3]
  2. Show that the system is consistent for one of these values of \(k\) and inconsistent for the other. [4]
Pre-U Pre-U 9795/1 2011 June Q9
11 marks Standard +0.3
  1. The points \(A\), \(B\) and \(C\) have position vectors $$\mathbf{a} = \begin{pmatrix} 19 \\ 3 \\ 10 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 12 \\ 7 \\ -1 \end{pmatrix} \quad \text{and} \quad \mathbf{c} = \begin{pmatrix} 5 \\ 15 \\ 3 \end{pmatrix}$$ respectively, and \(O\) is the origin. Calculate the volume of the tetrahedron \(OABC\). [3]
    1. The plane \(\Pi_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 6 \\ 2 \\ 5 \end{pmatrix}\). Determine an equation for \(\Pi_1\) in the form \(\mathbf{r} \cdot \mathbf{n} = d\). [4]
    2. A second plane, \(\Pi_2\), has equation \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 4 \\ 7 \end{pmatrix} = 13\). Find a vector equation for the line of intersection of \(\Pi_1\) and \(\Pi_2\). [4]
Pre-U Pre-U 9795/1 2011 June Q10
10 marks Challenging +1.2
  1. Use de Moivre's theorem to show that \(\cos 3\theta = 4\cos^3 \theta - 3\cos \theta\). [2]
  2. The sequence \(\{u_n\}\) is such that \(u_0 = 1\), \(u_1 = \cos \theta\) and, for \(n \geqslant 1\), $$u_{n+1} = (2\cos \theta)u_n - u_{n-1}.$$
    1. Determine \(u_2\) and \(u_3\) in terms of powers of \(\cos \theta\) only. [2]
    2. Suggest a simple expression for \(u_n\), the \(n\)th term of the sequence, and prove it for all positive integers \(n\) using induction. [6]
Pre-U Pre-U 9795/1 2011 June Q11
15 marks Hard +2.3
  1. Let \(I_n = \int_0^{\frac{\pi}{2}} \sec^n t \, dt\) for positive integers \(n\). Prove that, for \(n \geqslant 2\), $$(n - 1)I_n = \frac{2^{n-2}}{(\sqrt{3})^{n-1}} + (n - 2)I_{n-2}.$$ [5]
  2. The curve with parametric equations \(x = \tan t\), \(y = \frac{1}{4}\sec^2 t\), for \(0 \leqslant t \leqslant \frac{1}{4}\pi\), is rotated through \(2\pi\) radians about the \(x\)-axis to form a surface of revolution of area \(S\). Show that \(S = \pi I_5\) and evaluate \(S\) exactly. [10]
Pre-U Pre-U 9795/1 2011 June Q12
10 marks Challenging +1.2
The complex number \(z_1\) is such that \(z_1 = a + ib\), where \(a\) and \(b\) are positive real numbers.
  1. Given that \(z_1^2 = 2 + 2i\), show that \(a = \sqrt{\sqrt{2} + 1}\) and find the exact value of \(b\) in a similar form. [5]
The complex number \(z_2\) is such that \(z_2 = -a + ib\).
    1. Determine \(\arg z_2\) as a rational multiple of \(\pi\). [You may use the result \(\tan(\frac{1}{8}\pi) = \sqrt{2} - 1\).] [2]
    2. The point \(P_n\) in an Argand diagram represents the complex number \(z_2^n\), for positive integers \(n\). Find the least value of \(n\) for which \(P_n\) lies on the half-line with equation $$\arg(z) = \frac{1}{4}\pi.$$ [3]
Pre-U Pre-U 9795/1 2011 June Q13
18 marks Challenging +1.8
    1. Given that \(t = \tan x\), prove that \(\frac{2}{2 - \sin 2x} = \frac{1 + t^2}{1 - t + t^2}\). [2]
    2. Hence determine the value of the constant \(k\) for which $$\frac{d}{dx}\left\{\tan^{-1}\left(\frac{1 - 2\tan x}{\sqrt{3}}\right)\right\} = \frac{k}{2 - \sin 2x}.$$ [4]
  2. The curve \(C\) has cartesian equation \(x^2 - xy + y^2 = 72\).
    1. Determine a polar equation for \(C\) in the form \(r^2 = f(\theta)\), and deduce the polar coordinates \((r, \theta)\), where \(0 \leqslant \theta < 2\pi\), of the points on \(C\) which are furthest from the pole \(O\). [7]
    2. Find the exact area of the region of the plane in the first quadrant bounded by \(C\), the \(x\)-axis and the line \(y = x\). Deduce the total area of the region of the plane which lies inside \(C\) and within the first quadrant. [5]
Pre-U Pre-U 9795/1 2013 November Q1
4 marks Moderate -0.8
For real values of \(t\), the non-singular matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A}^{-1} = \begin{pmatrix} t & 5 \\ 2 & 8 \end{pmatrix} \quad \text{and} \quad \mathbf{B}^{-1} = \begin{pmatrix} 2 & -t \\ 3 & -1 \end{pmatrix}.$$
  1. Determine the values which \(t\) cannot take. [2]
  2. Without finding either \(\mathbf{A}\) or \(\mathbf{B}\), determine \((\mathbf{AB})^{-1}\) in terms of \(t\). [2]
Pre-U Pre-U 9795/1 2013 November Q2
5 marks Standard +0.3
Use de Moivre's theorem to express \(\cos 3\theta\) in terms of powers of \(\cos \theta\) only, and deduce the identity \(\cos 6x \equiv \cos 2x(2\cos 4x - 1)\). [5]
Pre-U Pre-U 9795/1 2013 November Q3
7 marks Standard +0.3
The curve \(C\) has equation \(y = \frac{2x}{x^2 + 1}\).
  1. Write down the equation of the asymptote of \(C\) and the coordinates of any points where \(C\) meets the coordinate axes. [2]
  2. Show that the curve meets the line \(y = k\) if and only if \(-1 \leqslant k \leqslant 1\). Deduce the coordinates of the turning points of the curve. [5]
[Note: You are NOT required to sketch \(C\).]
Pre-U Pre-U 9795/1 2013 November Q4
4 marks Standard +0.8
Let \(f(n) = 2(5^{n-1} + 1)\) for integers \(n = 1, 2, 3, \ldots\).
  1. Prove that, if \(f(n)\) is divisible by 8, then \(f(n + 1)\) is also divisible by 8. [3]
  2. Explain why this result does not imply that the statement '\(f(n)\) is divisible by 8 for all positive integers \(n\)' follows by mathematical induction. [1]
Pre-U Pre-U 9795/1 2013 November Q5
8 marks Challenging +1.2
The curve \(S\) has polar equation \(r = 1 + \sin \theta + \sin^2 \theta\) for \(0 \leqslant \theta < 2\pi\).
  1. Determine the polar coordinates of the points on \(S\) where \(\frac{dr}{d\theta} = 0\). [5]
  2. Sketch \(S\). [3]
Pre-U Pre-U 9795/1 2013 November Q6
8 marks Challenging +1.2
\(G\) is the set \(\{2, 4, 6, 8\}\), \(H\) is the set \(\{1, 5, 7, 11\}\) and \(\times_n\) denotes the operation of multiplication modulo \(n\).
  1. Construct the multiplication tables for \((G, \times_{10})\) and \((H, \times_{12})\). [2]
  2. By verifying the four group axioms, show that \(G\) and \(H\) are groups under their respective binary operations, and determine whether \(G\) and \(H\) are isomorphic. [6]
[You may assume that \(\times_n\) is associative.]
Pre-U Pre-U 9795/1 2013 November Q7
8 marks Standard +0.3
Relative to an origin \(O\), the points \(P\), \(Q\) and \(R\) have position vectors $$\mathbf{p} = \mathbf{i} + 2\mathbf{j} - 7\mathbf{k}, \quad \mathbf{q} = -3\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{r} = 6\mathbf{i} + 4\mathbf{j} + \alpha\mathbf{k}$$ respectively.
  1. Determine \(\mathbf{p} \times \mathbf{q}\). [2]
  2. Deduce the value of \(\alpha\) for which
    1. \(OR\) is normal to the plane \(OPQ\), [1]
    2. the volume of tetrahedron \(OPQR\) is 50, [3]
    3. \(R\) lies in the plane \(OPQ\). [2]
Pre-U Pre-U 9795/1 2013 November Q8
10 marks Standard +0.8
  1. Determine \(x\) and \(y\) given that the complex number \(z = x + \text{i}y\) simultaneously satisfies $$|z - 1| = 1 \quad \text{and} \quad \arg(z + 1) = \frac{1}{6}\pi.$$ [4]
  2. On an Argand diagram, shade the region whose points satisfy $$1 \leqslant |z - 1| \leqslant 2 \quad \text{and} \quad \frac{1}{6}\pi \leqslant \arg(z + 1) \leqslant \frac{1}{4}\pi.$$ [6]
Pre-U Pre-U 9795/1 2013 November Q9
10 marks Challenging +1.2
  1. Show that there is exactly one value of \(k\) for which the system of equations \begin{align} kx + 2y + kz &= 4
    3x + 10y + 2z &= m
    (k - 1)x - 4y + z &= k \end{align} does not have a unique solution. [4]
  2. Given that the system of equations is consistent for this value of \(k\), find the value of \(m\). [4]
  3. Explain the geometrical significance of a non-unique solution to a \(3 \times 3\) system of linear equations. [2]
Pre-U Pre-U 9795/1 2013 November Q10
8 marks Standard +0.8
The roots of the equation \(x^4 - 2x^3 + 2x^2 + x - 3 = 0\) are \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\). Determine the values of
  1. \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\), [2]
  2. \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}\), [2]
  3. \(\alpha^3 + \beta^3 + \gamma^3 + \delta^3\). [4]