Questions — Edexcel FP1 (310 questions)

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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]