Questions — Pre-U (885 questions)

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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 9794/2 2012 June Q1
5 marks Easy -1.3
  1. Solve the equation \(x^2 - 8x + 4 = 0\), giving your answer in the form \(p \pm q\sqrt{3}\), where \(p\) and \(q\) are integers. [2]
  2. Expand and simplify \((6 + 2\sqrt{3})(2 - \sqrt{3})\). [3]
Pre-U Pre-U 9794/2 2012 June Q2
9 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows a triangle \(ABC\). The vertices have coordinates \(A(3, -7)\), \(B(9, 1)\) and \(C(-1, -5)\).
    1. Find the length of the side \(AB\). [2]
    2. Find the coordinates of the mid-point of \(AB\). [1]
    3. A circle has diameter \(AB\). Find the equation of the circle in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\), \(b\) and \(r\) are constants to be found. [3]
  2. Find the equation of the line \(l\) passing through \(B\) parallel to \(AC\). [3]
Pre-U Pre-U 9794/2 2012 June Q3
4 marks Easy -1.2
Find the exact value of \(\int_0^1 (e^x - x) dx\). [4]
Pre-U Pre-U 9794/2 2012 June Q4
4 marks Easy -1.2
Use logarithms to solve the equation \(2^{2x-1} = 5\). [4]
Pre-U Pre-U 9794/2 2012 June Q5
3 marks Easy -2.0
Sketch, on separate diagrams, the graphs of the following functions for \(0 \leqslant x \leqslant 2\pi\) giving the coordinates of all points of intersection with the axes.
  1. \(y = \sin x\). [1]
  2. \(y = \sin\left(x + \frac{1}{6}\pi\right)\). [2]
Pre-U Pre-U 9794/2 2012 June Q6
8 marks Moderate -0.8
  1. An arithmetic sequence has first term 5 and fifth term 37.
    1. Find an expression for \(u_n\), the \(n\)th term of the sequence, in terms of \(n\). [4]
    2. Find an expression for \(S_n\), the sum of the first \(n\) terms of this sequence, in terms of \(n\). [2]
  2. Hence, or otherwise, calculate \(\sum_{n=5}^{25} (8n - 3)\). [2]
Pre-U Pre-U 9794/2 2012 June Q7
5 marks Moderate -0.8
Let \(y = (2x - 3)e^{-2x}\).
  1. Find \(\frac{dy}{dx}\), giving your answer in the form \(e^{-2x}(ax + b)\), where \(a\) and \(b\) are integers. [3]
  2. Determine the set of values of \(x\) for which \(y\) is increasing. [2]
Pre-U Pre-U 9794/2 2012 June Q8
6 marks Moderate -0.3
Solve the differential equation \(\frac{dy}{dx} = -y^2 x^3\), where \(y = 2\) when \(x = 1\), expressing your solution in the form \(y = f(x)\). [6]
Pre-U Pre-U 9794/2 2012 June Q9
9 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows a sector of a circle, \(OMN\). The angle \(MON\) is \(2x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.
  1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and perimeter, \(P\), of the sector. [2]
  2. Given that \(P = 20\), show that \(A = \frac{100x}{(1 + x)^2}\). [2]
  3. Find \(\frac{dA}{dx}\), and hence find the value of \(x\) for which the area of the sector is a maximum. [5]
Pre-U Pre-U 9794/2 2012 June Q10
12 marks Standard +0.3
    1. Find \(\int \frac{e^x}{1 + e^x} dx\). [2]
    2. Hence evaluate \(\int_0^{\ln 3} \frac{e^x}{1 + e^x} dx\), giving your answer in the form \(\ln k\), where \(k\) is an integer. [3]
    1. Using the substitution \(u = 1 + e^x\), find \(\int \left(\frac{e^x}{1 + e^x}\right)^2 dx\). [5]
    2. Hence find the exact volume of the solid of revolution generated when the curve given by \(y = \frac{e^x}{1 + e^x}\), between \(x = -\ln 3\) and \(x = \ln 3\), is rotated through \(2\pi\) radians about the \(x\)-axis. [2]
Pre-U Pre-U 9794/2 2012 June Q11
15 marks Challenging +1.2
The function f is defined by \(f : t \mapsto 2 \sin t + \cos 2t\) for \(0 \leqslant t < 2\pi\).
  1. Show that \(\frac{df}{dt} = 2 \cos t(1 - 2 \sin t)\). [2]
  2. Determine the range of f. [5]
A curve \(C\) is given parametrically by \(x = 2 \cos t + \sin 2t\), \(y = f(t)\) for \(0 \leqslant t < 2\pi\).
  1. Show that \(x^2 + y^2 = 5 + 4 \sin 3t\). [3]
  2. Deduce that \(C\) lies between two circles centred at the origin, and touches both. [2]
  3. Find the gradient of the tangent to \(C\) at the point at which \(t = 0\). [3]
Pre-U Pre-U 9794/3 2013 November Q1
5 marks Easy -1.2
  1. Given that \(X \sim \text{Geo}\left(\frac{1}{6}\right)\), write down the values of E(\(X\)) and Var(\(X\)). [2]
  2. \(Y \sim \text{B}(n, p)\). Given that E(\(Y\)) = 4 and Var(\(Y\)) = \(\frac{8}{3}\), find the values of \(n\) and \(p\). [3]