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Edexcel M5 Specimen Q6
11 marks Challenging +1.2
A uniform rod \(AB\) of mass \(m\) and length \(4a\) is free to rotate in a vertical plane about a horizontal axis through the point \(O\) of the rod, where \(OA = a\). The rod is slightly disturbed from rest when \(B\) is vertically above \(A\).
  1. Find the magnitude of the angular acceleration of the rod when it is horizontal. [4]
  2. Find the angular speed of the rod when it is horizontal. [2]
  3. Calculate the magnitude of the force acting on the rod at \(O\) when the rod is horizontal. [5]
Edexcel M5 Specimen Q7
12 marks Challenging +1.2
As a hailstone falls under gravity in still air, its mass increases. At time \(t\) the mass of the hailstone is \(m\). The hailstone is modelled as a uniform sphere of radius \(r\) such that $$\frac{dr}{dt} = kr,$$ where \(k\) is a positive constant.
  1. Show that \(\frac{dm}{dt} = 3km\). [2]
Assuming that there is no air resistance,
  1. show that the speed \(v\) of the hailstone at time \(t\) satisfies $$\frac{dv}{dt} = g - 3kv.$$ [4]
Given that the speed of the hailstone at time \(t = 0\) is \(u\),
  1. find an expression for \(v\) in terms of \(t\). [5]
  2. Hence show that the speed of the hailstone approaches the limiting value \(\frac{g}{3k}\). [1]
Edexcel M5 Specimen Q8
13 marks Challenging +1.3
A particle \(P\) moves in the \(x\)-\(y\) plane and has position vector \(\mathbf{r}\) metres relative to a fixed origin \(O\) at time \(t\) s. Given that \(\mathbf{r}\) satisfies the vector differential equation $$\frac{d^2\mathbf{r}}{dt^2} + 9\mathbf{r} = 8\sin t \mathbf{i}$$ and that when \(t = 0\) s, \(P\) is at \(O\) and moving with velocity \((\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\),
  1. find \(\mathbf{r}\) at time \(t\). [11]
  2. Hence find when \(P\) next returns to \(O\). [2]
AQA FP1 2014 June Q1
5 marks Moderate -0.8
A curve passes through the point \((9, 6)\) and satisfies the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2 + \sqrt{x}}$$ Use a step-by-step method with a step length of \(0.25\) to estimate the value of \(y\) at \(x = 9.5\). Give your answer to four decimal places. [5 marks]
AQA FP1 2014 June Q2
11 marks Standard +0.3
The quadratic equation $$2x^2 + 8x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
    1. Find the value of \(\alpha^2 + \beta^2\). [2 marks]
    2. Hence, or otherwise, show that \(\alpha^4 + \beta^4 = \frac{449}{2}\). [2 marks]
  3. Find a quadratic equation, with integer coefficients, which has roots $$2\alpha^4 + \frac{1}{\beta^2} \text{ and } 2\beta^4 + \frac{1}{\alpha^2}$$ [5 marks]
AQA FP1 2014 June Q3
4 marks Standard +0.3
Use the formulae for \(\sum_{r=1}^{n} r^3\) and \(\sum_{r=1}^{n} r^2\) to find the value of $$\sum_{r=3}^{60} r^2(r - 6)$$ [4 marks]
AQA FP1 2014 June Q4
6 marks Standard +0.3
Find the complex number \(z\) such that $$5iz + 3z^* + 16 = 8i$$ Give your answer in the form \(a + bi\), where \(a\) and \(b\) are real. [6 marks]
AQA FP1 2014 June Q5
5 marks Moderate -0.8
A curve \(C\) has equation \(y = x(x + 3)\).
  1. Find the gradient of the line passing through the point \((-5, 10)\) and the point on \(C\) with \(x\)-coordinate \(-5 + h\). Give your answer in its simplest form. [3 marks]
  2. Show how the answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((-5, 10)\). State the value of this gradient. [2 marks]
AQA FP1 2014 June Q6
10 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{1}{x(x + 2)}\).
  1. Write down the equations of all the asymptotes of \(C\). [2 marks]
  2. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is \(-1\).
    1. Find the \(y\)-coordinate of the stationary point. [1 mark]
    2. Sketch the curve \(C\). [2 marks]
  3. Solve the inequality $$\frac{1}{x(x + 2)} \leqslant \frac{1}{8}$$ [5 marks]
AQA FP1 2014 June Q7
10 marks Moderate -0.3
  1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
    1. a reflection in the line \(y = -x\); [1 mark]
    2. a stretch parallel to the \(y\)-axis of scale factor \(7\). [1 mark]
  2. Hence find the matrix corresponding to the combined transformation of a reflection in the line \(y = -x\) followed by a stretch parallel to the \(y\)-axis of scale factor \(7\). [2 marks]
  3. The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} -3 & -\sqrt{3} \\ -\sqrt{3} & 3 \end{bmatrix}\).
    1. Show that \(\mathbf{A}^2 = k\mathbf{I}\), where \(k\) is a constant and \(\mathbf{I}\) is the \(2 \times 2\) identity matrix. [1 mark]
    2. Show that the matrix \(\mathbf{A}\) corresponds to a combination of an enlargement and a reflection. State the scale factor of the enlargement and state the equation of the line of reflection in the form \(y = (\tan \theta)x\). [5 marks]
AQA FP1 2014 June Q8
9 marks Standard +0.3
  1. Find the general solution of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ giving your answer for \(x\) in terms of \(\pi\). [5 marks]
  2. Use your general solution to find the sum of all the solutions of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ that lie in the interval \(0 \leqslant x \leqslant 20\pi\). Give your answer in the form \(k\pi\), stating the exact value of \(k\). [4 marks]
AQA FP1 2014 June Q9
15 marks Standard +0.8
An ellipse \(E\) has equation $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$
  1. Sketch the ellipse \(E\), showing the values of the intercepts on the coordinate axes. [2 marks]
  2. Given that the line with equation \(y = x + k\) intersects the ellipse \(E\) at two distinct points, show that \(-5 < k < 5\). [5 marks]
  3. The ellipse \(E\) is translated by the vector \(\begin{bmatrix} a \\ b \end{bmatrix}\) to form another ellipse whose equation is \(9x^2 + 16y^2 + 18x - 64y = c\). Find the values of the constants \(a\), \(b\) and \(c\). [5 marks]
  4. Hence find an equation for each of the two tangents to the ellipse \(9x^2 + 16y^2 + 18x - 64y = c\) that are parallel to the line \(y = x\). [3 marks]
AQA FP1 2016 June Q1
7 marks Moderate -0.3
The quadratic equation \(x^2 - 6x + 14 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
  2. Find a quadratic equation, with integer coefficients, which has roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). [5 marks]
AQA FP1 2016 June Q2
5 marks Moderate -0.8
A curve \(C\) has equation \(y = (2 - x)(1 + x) + 3\).
  1. A line passes through the point \((2, 3)\) and the point on \(C\) with \(x\)-coordinate \(2 + h\). Find the gradient of the line, giving your answer in its simplest form. [3 marks]
  2. Show how your answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((2, 3)\). State the value of this gradient. [2 marks]
AQA FP1 2016 June Q3
7 marks Moderate -0.3
The variables \(y\) and \(x\) are related by an equation of the form $$y = a(b^x)$$ where \(a\) and \(b\) are positive constants. Let \(Y = \log_{10} y\).
  1. Show that there is a linear relationship between \(Y\) and \(x\). [2 marks]
  2. The graph of \(Y\) against \(x\), shown below, passes through the points \((0, 2.5)\) and \((5, 0.5)\). \includegraphics{figure_3}
    1. Find the gradient of the line. [1 mark]
    2. Find the value of \(a\) and the value of \(b\), giving each answer to three significant figures. [4 marks]
AQA FP1 2016 June Q4
7 marks Moderate -0.3
  1. Given that \(\sin \frac{\pi}{3} = \cos \frac{\pi}{k}\), state the value of the integer \(k\). [1 mark]
  2. Hence, or otherwise, find the general solution of the equation $$\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}$$ giving your answer, in its simplest form, in terms of \(\pi\). [4 marks]
  3. Hence, given that \(\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}\), show that there is only one finite value for \(\tan x\) and state its exact value. [2 marks]
AQA FP1 2016 June Q5
9 marks Standard +0.8
  1. Use the formulae for \(\sum_{r=1}^n r^2\) and \(\sum_{r=1}^n r\) to show that \(\sum_{r=1}^n (6r - 3)^2 = 3n(4n^2 - 1)\). [5 marks]
  2. Hence express \(\sum_{r=1}^{2n} r^3 - \sum_{r=1}^n (6r - 3)^2\) as a product of four linear factors in terms of \(n\). [4 marks]
AQA FP1 2016 June Q6
9 marks Standard +0.8
A parabola with equation \(y^2 = 4ax\), where \(a\) is a constant, is translated by the vector \(\begin{bmatrix} 2 \\ 3 \end{bmatrix}\) to give the curve \(C\). The curve \(C\) passes through the point \((4, 7)\).
  1. Show that \(a = 2\). [3 marks]
  2. Find the values of \(k\) for which the line \(ky = x\) does not meet the curve \(C\). [6 marks]
AQA FP1 2016 June Q7
10 marks Standard +0.3
  1. Solve the equation \(x^2 + 4x + 20 = 0\), giving your answers in the form \(c + di\), where \(c\) and \(d\) are integers. [3 marks]
  2. The roots of the quadratic equation $$z^2 + (4 + i + qi)z + 20 = 0$$ are \(w\) and \(w^*\).
    1. In the case where \(q\) is real, explain why \(q\) must be \(-1\). [2 marks]
    2. In the case where \(w = p + 2i\), where \(p\) is real, find the possible values of \(q\). [5 marks]
AQA FP1 2016 June Q8
10 marks Standard +0.3
The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\).
    1. Find the matrix \(\mathbf{A}^2\). [1 mark]
    2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf{A}^2\). [1 mark]
  2. Given that the matrix \(\mathbf{B}\) represents a reflection in the line \(x + \sqrt{3}y = 0\), find the matrix \(\mathbf{B}\), giving the exact values of any trigonometric expressions. [2 marks]
  3. Hence find the coordinates of the point \(P\) which is mapped onto \((0, -4)\) under the transformation represented by \(\mathbf{A}^2\) followed by a reflection in the line \(x + \sqrt{3}y = 0\). [6 marks]
AQA FP1 2016 June Q9
11 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{x - 1}{(x - 2)(2x - 1)}\). The line \(L\) has equation \(y = \frac{1}{2}(x - 1)\).
  1. Write down the equations of the asymptotes of \(C\). [2 marks]
  2. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\). [3 marks]
  3. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes. [3 marks]
  4. Hence solve the inequality \(\frac{x - 1}{(x - 2)(2x - 1)} \geqslant \frac{1}{2}(x - 1)\). [3 marks]
OCR FP1 Q1
6 marks Moderate -0.5
Use the standard results for \(\sum_{r=1}^n r\) and \(\sum_{r=1}^n r^2\) to show that, for all positive integers \(n\), $$\sum_{r=1}^n (6r^2 + 2r + 1) = n(2n^2 + 4n + 3).$$ [6]
OCR FP1 Q2
6 marks Standard +0.3
The matrices \(\mathbf{A}\) and \(\mathbf{I}\) are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}\) and \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) respectively.
  1. Find \(\mathbf{A}^2\) and verify that \(\mathbf{A}^2 = 4\mathbf{A} - \mathbf{I}\). [4]
  2. Hence, or otherwise, show that \(\mathbf{A}^{-1} = 4\mathbf{I} - \mathbf{A}\). [2]
OCR FP1 Q3
7 marks Moderate -0.8
The complex numbers \(2 + 3i\) and \(4 - i\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + iy\), showing clearly how you obtain your answers.
  1. \(z + 5w\), [2]
  2. \(z*w\), where \(z*\) is the complex conjugate of \(z\), [3]
  3. \(\frac{1}{w}\). [2]
OCR FP1 Q4
6 marks Standard +0.3
Use an algebraic method to find the square roots of the complex number \(21 - 20i\). [6]