Questions — Edexcel (10514 questions)

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Edexcel FP1 Q32
5 marks Standard +0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\), \(\sum_{r=1}^{n} r 2^r = 2\{1 + (n - 1)2^n\}\). [5]
Edexcel FP1 Q33
6 marks Standard +0.3
The complex numbers \(z\) and \(w\) satisfy the simultaneous equations $$2z + iw = -1,$$ $$z - w = 3 + 3i.$$
  1. Use algebra to find \(z\), giving your answers in the form \(a + ib\), where \(a\) and \(b\) are real. [4]
  2. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. [2]
Edexcel FP1 Q34
5 marks Moderate -0.8
$$f(x) = 0.25x - 2 + 4 \sin \sqrt{x}.$$
  1. Show that the equation \(f(x) = 0\) has a root \(\alpha\) between \(x = 0.24\) and \(x = 0.28\). [2]
  2. Starting with the interval \([0.24, 0.28]\), use interval bisection three times to find an interval of width 0.005 which contains \(\alpha\). [3]
Edexcel FP1 Q35
4 marks Moderate -0.8
  1. Find the roots of the equation \(z^2 + 2z + 17 = 0\), giving your answers in the form \(a + ib\), where \(a\) and \(b\) are integers. [3]
  2. Show these roots on an Argand diagram. [1]
Edexcel FP1 Q36
5 marks Moderate -0.3
The complex numbers \(z_1\) and \(z_2\) are given by $$z_1 = 5 + 3i,$$ $$z_1 = 1 + pi,$$ where \(p\) is an integer.
  1. Find \(\frac{z_2}{z_1}\), in the form \(a + ib\), where \(a\) and \(b\) are expressed in terms of \(p\). [3]
Given that \(\arg \left( \frac{z_2}{z_1} \right) = \frac{\pi}{4}\),
  1. find the value of \(p\). [2]
Edexcel FP1 Q37
11 marks Standard +0.3
$$f (x) = x^3 + 8x - 19.$$
  1. Show that the equation \(f(x) = 0\) has only one real root. [3]
  2. Show that the real root of \(f(x) = 0\) lies between 1 and 2. [2]
  3. Obtain an approximation to the real root of \(f(x) = 0\) by performing two applications of the Newton-Raphson procedure to \(f(x)\) , using \(x = 2\) as the first approximation. Give your answer to 3 decimal places. [4]
  4. By considering the change of sign of \(f(x)\) over an appropriate interval, show that your answer to part (c) is accurate to 3 decimal places. [2]
Edexcel FP1 Q38
13 marks Moderate -0.3
$$z = \sqrt{3} - i.$$ \(z^*\) is the complex conjugate of \(z\).
  1. Show that \(\frac{z}{z^*} = \frac{1}{2} - \frac{\sqrt{3}}{2} i\). [3]
  2. Find the value of \(\left| \frac{z}{z^*} \right|\). [2]
  3. Verify, for \(z = \sqrt{3} - i\), that \(\arg \frac{z}{z^*} = \arg z - \arg z^*\). [4]
  4. Display \(z\), \(z^*\) and \(\frac{z}{z^*}\) on a single Argand diagram. [2]
  5. Find a quadratic equation with roots \(z\) and \(z^*\) in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are real constants to be found. [2]
Edexcel FP1 Q39
10 marks Challenging +1.2
The points \(P(ap^2, 2ap)\) and \(Q(aq^2, 2aq)\), \(p \neq q\), lie on the parabola \(C\) with equation \(y^2 = 4ax\), where \(a\) is a constant.
  1. Show that an equation for the chord \(PQ\) is \((p + q) y = 2(x + apq)\) . [3]
The normals to \(C\) at \(P\) and \(Q\) meet at the point \(R\).
  1. Show that the coordinates of \(R\) are \((a(p^2 + q^2 + pq + 2), -apq(p + q) )\). [7]
Edexcel FP1 Q40
5 marks Moderate -0.3
Prove by induction that, for \(n \in \mathbb{Z}^+\), \(\sum_{r=1}^{n} (2r - 1)^2 = \frac{1}{3} n(2n - 1)(2n + 1)\). [5]
Edexcel FP1 Q41
7 marks Standard +0.8
Given that \(f(n) = 3^{4n} + 2^{4n + 2}\),
  1. show that, for \(k \in \mathbb{Z}^+\), \(f(k + 1) - f(k)\) is divisible by 15, [4]
  2. prove that, for \(n \in \mathbb{Z}^+\), \(f (n)\) is divisible by 5. [3]
Edexcel FP1 Q42
6 marks Standard +0.3
Given that \(x = -\frac{1}{2}\) is the real solution of the equation $$2x^3 - 11x^2 + 14x + 10 = 0,$$ find the two complex solutions of this equation. [6]
Edexcel FP1 Q43
4 marks Standard +0.3
$$f(x) = 3x^2 + x - \tan \left( \frac{x}{2} \right) - 2, \quad -\pi < x < \pi.$$ The equation \(f(x) = 0\) has a root \(\alpha\) in the interval \([0.7, 0.8]\). Use linear interpolation, on the values at the end points of this interval, to obtain an approximation to \(\alpha\). Give your answer to 3 decimal places. [4]
Edexcel FP1 Q44
10 marks Moderate -0.8
$$z = -2 + i.$$
  1. Express in the form \(a + ib\)
    1. \(\frac{1}{z}\)
    2. \(z^2\). [4]
  2. Show that \(|z^2 - z| = 5\sqrt{2}\). [2]
  3. Find \(\arg (z^2 - z)\). [2]
  4. Display \(z\) and \(z^2 - z\) on a single Argand diagram. [2]
Edexcel FP1 Q45
7 marks Moderate -0.8
  1. Write down the value of the real root of the equation \(x^3 - 64 = 0\). [1]
  2. Find the complex roots of \(x^3 - 64 = 0\) , giving your answers in the form \(a + ib\), where \(a\) and \(b\) are real. [4]
  3. Show the three roots of \(x^3 - 64 = 0\) on an Argand diagram. [2]
Edexcel FP1 Q46
7 marks Moderate -0.3
The complex number \(z\) is defined by $$z = \frac{a + 2i}{a - 1}, \quad a \in \mathbb{R}, a > 0 .$$ Given that the real part of \(z\) is \(\frac{1}{2}\) , find
  1. the value of \(a\), [4]
  2. the argument of \(z\), giving your answer in radians to 2 decimal places. [3]
Edexcel FP1 Q47
11 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}, \text{ where } k \text{ is constant.}$$ A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\).
  1. Find the value of \(k\) for which the line \(y = 2x\) is mapped onto itself under \(T\). [3]
  2. Show that \(\mathbf{A}\) is non-singular for all values of \(k\). [3]
  3. Find \(\mathbf{A}^{-1}\) in terms of \(k\). [2]
A point \(P\) is mapped onto a point \(Q\) under \(T\). The point \(Q\) has position vector \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\) relative to an origin \(O\). Given that \(k = 3\),
  1. find the position vector of \(P\). [3]
Edexcel FP2 Q1
6 marks Standard +0.8
  1. Express \(\frac{1}{t(t+2)}\) in partial fractions. [1]
  2. Hence show that \(\sum_{n=1}^{\infty} \frac{4}{n(n+2)} = \frac{n(3n+5)}{(n+1)(n+2)}\) [5]
Edexcel FP2 Q2
6 marks Standard +0.3
Solve the equation $$z^2 = 4\sqrt{2} - 4\sqrt{2}i,$$ giving your answers in the form \(r(\cos \theta + i \sin \theta)\), where \(-\pi < \theta \leq \pi\). [6]
Edexcel FP2 Q3
8 marks Standard +0.8
Find the general solution of the differential equation $$\sin x \frac{dy}{dx} - y \cos x = \sin 2x \sin x$$ giving your answer in the form \(y = f(x)\). [8]
Edexcel FP2 Q4
8 marks Challenging +1.2
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3\cos \theta, \quad a > 0, \quad 0 \leq \theta < 2\pi.$$ The area enclosed by the curve is \(\frac{10\pi}{2}\). Find the value of \(a\). [8]
Edexcel FP2 Q5
10 marks Challenging +1.2
\(y = \sec^2 x\)
  1. Show that \(\frac{d^2 y}{dx^2} = 6 \sec^4 x - 4 \sec^2 x\). [4]
  2. Find a Taylor series expansion of \(\sec^2 x\) in ascending powers of \(\left(x - \frac{\pi}{4}\right)\), up to and including the term in \(\left(x - \frac{\pi}{4}\right)^3\). [6]
Edexcel FP2 Q6
10 marks Challenging +1.2
A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z}{z-i}, \quad z \neq i.$$ The circle with equation \(|z| = 3\) is mapped by \(T\) onto the curve \(C\).
  1. Show that \(C\) is a circle and find its centre and radius. [8]
The region \(|z| < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
  1. Shade the region \(R\) on an Argand diagram. [2]
Edexcel FP2 Q7
12 marks Standard +0.8
  1. Sketch the graph of \(y = |x^2 - a^2|\), where \(a > 1\), showing the coordinates of the points where the graph meets the axes. [2]
  2. Solve \(|x^2 - a^2| = a^2 - x\), \(a > 1\). [6]
  3. Find the set of values of \(x\) for which \(|x^2 - a^2| > a^2 - x\), \(a > 1\). [4]
Edexcel FP2 Q8
15 marks Standard +0.8
$$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 6x = 2e^{-t}.$$ Given that \(x = 0\) and \(\frac{dx}{dt} = 2\) at \(t = 0\),
  1. find \(x\) in terms of \(t\). [8]
The solution to part (a) is used to represent the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds, where \(t > 0\), \(P\) is \(x\) metres from the origin \(O\).
  1. Show that the maximum distance between \(O\) and \(P\) is \(\frac{2\sqrt{3}}{9}\) m and justify that this distance is a maximum. [7]
Edexcel FP2 Q1
7 marks Standard +0.3
  1. Express \(\frac{3}{(3r-1)(3r+2)}\) in partial fractions. [2]
  2. Using your answer to part (a) and the method of differences, show that $$\sum_{r=1}^n \frac{3}{(3r-1)(3r+2)} = \frac{3n}{2(3n+2)}$$ [3]
  3. Evaluate \(\sum_{r=1}^{30} \frac{3}{(3r-1)(3r+2)}\), giving your answer to 3 significant figures. [2]