Questions — Edexcel FP1 (310 questions)

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Edexcel FP1 Q13
9 marks Standard +0.3
$$z = \frac{a + 3i}{2 + ai}, \quad a \in \mathbb{R}.$$
  1. Given that \(a = 4\), find \(|z|\). [3]
  2. Show that there is only one value of \(a\) for which \(\arg z = \frac{\pi}{4}\), and find this value. [6]
Edexcel FP1 Q14
6 marks Standard +0.3
$$f(n) = (2n + 1)7^n - 1.$$ Prove by induction that, for all positive integers \(n\), \(f(n)\) is divisible by 4. [6]
Edexcel FP1 Q15
10 marks Standard +0.3
Given that \(z = 2 - 2i\) and \(w = -\sqrt{3} + i\),
  1. find the modulus and argument of \(wz^2\). [6]
  2. Show on an Argand diagram the points \(A\), \(B\) and \(C\) which represent \(z\), \(w\) and \(wz^2\) respectively, and determine the size of angle \(BOC\). [4]
Edexcel FP1 Q16
6 marks Standard +0.3
  1. Show that \(\sum_{r=1}^{n} (r + 1)(r + 5) = \frac{1}{6} n(n + 7)(2n + 7)\). [4]
  2. Hence calculate the value of \(\sum_{r=10}^{40} (r + 1)(r + 5)\). [2]
Edexcel FP1 Q17
2 marks Moderate -0.8
$$f(x) = 2^x + x - 4.$$ The equation \(f(x) = 0\) has a root \(\alpha\) in the interval \([1, 2]\). Use linear interpolation on the values at the end points of this interval to find an approximation to \(\alpha\). [2]
Edexcel FP1 Q18
6 marks Standard +0.3
The complex number \(z = a + ib\), where \(a\) and \(b\) are real numbers, satisfies the equation $$z^2 + 16 - 30i = 0.$$
  1. Show that \(ab = 15\). [2]
  2. Write down a second equation in \(a\) and \(b\) and hence find the roots of \(z^2 + 16 - 30i = 0\). [4]
Edexcel FP1 Q19
11 marks Moderate -0.3
Given that \(z = 1 + \sqrt{3}i\) and that \(\frac{w}{z} = 2 + 2i\), find
  1. \(w\) in the form \(a + ib\), where \(a, b \in \mathbb{R}\), [3]
  2. the argument of \(w\), [2]
  3. the exact value for the modulus of \(w\). [2]
On an Argand diagram, the point \(A\) represents \(z\) and the point \(B\) represents \(w\).
  1. Draw the Argand diagram, showing the points \(A\) and \(B\). [2]
  2. Find the distance \(AB\), giving your answer as a simplified surd. [2]
Edexcel FP1 Q20
5 marks Standard +0.3
Show that the normal to the rectangular hyperbola \(xy = c^2\), at the point \(P \left( ct, \frac{c}{t} \right)\), \(t \neq 0\) has equation $$y = t^2 x + \frac{c}{t} - ct^3.$$ [5]
Edexcel FP1 Q21
13 marks Standard +0.3
Given that \(z = -2\sqrt{2} + 2\sqrt{2}i\) and \(w = 1 - i\sqrt{3}\), find
  1. \(\left|\frac{z}{w}\right|\), [3]
  2. \(\arg \left( \frac{z}{w} \right)\). [3]
  1. On an Argand diagram, plot points \(A\), \(B\), \(C\) and \(D\) representing the complex numbers \(z\), \(w\), \(\left( \frac{z}{w} \right)\) and 4, respectively. [3]
  2. Show that \(\angle AOC = \angle DOB\). [2]
  3. Find the area of triangle \(AOC\). [2]
Edexcel FP1 Q22
6 marks Standard +0.3
Given that \(-2\) is a root of the equation \(z^3 + 6z + 20 = 0\),
  1. Find the other two roots of the equation, [3]
  2. show, on a single Argand diagram, the three points representing the roots of the equation, [1]
  3. prove that these three points are the vertices of a right-angled triangle. [2]
Edexcel FP1 Q23
3 marks Moderate -0.8
$$f(x) = 1 - e^x + 3 \sin 2x$$ The equation \(f(x) = 0\) has a root \(\alpha\) in the interval \(1.0 < x < 1.4\). Starting with the interval \((1.0, 1.4)\), use interval bisection three times to find the value of \(\alpha\) to one decimal place. [3]
Edexcel FP1 Q24
9 marks Moderate -0.3
$$z = -4 + 6i.$$
  1. Calculate \(\arg z\), giving your answer in radians to 3 decimal places. [2]
The complex number \(w\) is given by \(w = \frac{A}{2 - i}\), where \(A\) is a positive constant. Given that \(|w| = \sqrt{20}\),
  1. find \(w\) in the form \(a + ib\), where \(a\) and \(b\) are constants, [4]
  2. calculate \(\arg \frac{w}{z}\). [3]
Edexcel FP1 Q25
5 marks Standard +0.3
The point \(P(ap^2, 2ap)\) lies on the parabola \(M\) with equation \(y^2 = 4ax\), where \(a\) is a positive constant.
  1. Show that an equation of the tangent to \(M\) at \(P\) is \(py = x + ap^2\). [3]
The point \(Q(16ap^2, 8ap)\) also lies on \(M\).
  1. Write down an equation of the tangent to \(M\) at \(Q\). [2]
Edexcel FP1 Q26
5 marks Standard +0.3
  1. Express \(\frac{6x + 10}{x + 3}\) in the form \(p + \frac{q}{x + 3}\), where \(p\) and \(q\) are integers to be found. [1]
The sequence of real numbers \(u_1, u_2, u_3, ...\) is such that \(u_1 = 5.2\) and \(u_{n+1} = \frac{6u_n + 10}{u_n + 3}\).
  1. Prove by induction that \(u_n > 5\), for \(n \in \mathbb{Z}^+\). [4]
Edexcel FP1 Q27
6 marks Standard +0.3
Prove that \(\sum_{r=1}^{n} (r - 1)(r + 2) = \frac{1}{3} (n - 1)n(n + 4)\). [6]
Edexcel FP1 Q28
10 marks Standard +0.3
Given that \(\frac{z + 2i}{z - \lambda i} = i\), where \(\lambda\) is a positive, real constant,
  1. show that \(z = \left( \frac{\lambda}{2} + 1 \right) + i \left( \frac{\lambda}{2} - 1 \right)\). [5]
Given also that \(\arg z = \arctan \frac{1}{3}\), calculate
  1. the value of \(\lambda\), [3]
  2. the value of \(|z|^2\). [2]
Edexcel FP1 Q29
5 marks Standard +0.3
The temperature \(\theta\) °C of a room \(t\) hours after a heating system has been turned on is given by $$\theta = t + 26 - 20e^{-0.5t}, \quad t \geq 0.$$ The heating system switches off when \(\theta = 20\). The time \(t = \alpha\), when the heating system switches off, is the solution of the equation \(\theta - 20 = 0\), where \(\alpha\) lies in the interval \([1.8, 2]\).
  1. Using the end points of the interval \([1.8, 2]\), find, by linear interpolation, an approximation to \(\alpha\). Give your answer to 2 decimal places. [4]
  2. Use your answer to part (a) to estimate, giving your answer to the nearest minute, the time for which the heating system was on. [1]
Edexcel FP1 Q30
9 marks Standard +0.3
The parabola \(C\) has equation \(y^2 = 4ax\), where \(a\) is a constant.
  1. Show that an equation for the normal to \(C\) at the point \(P(ap^2, 2ap)\) is \(y + px = 2ap + ap^3\). [4]
The normals to \(C\) at the points \(P(ap^2, 2ap)\) and \(Q(aq^2, 2aq)\), \(p \neq q\), meet at the point \(R\).
  1. Find, in terms of \(a\), \(p\) and \(q\), the coordinates of \(R\). [5]
Edexcel FP1 Q31
2 marks Standard +0.3
A transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix $$\mathbf{A} = \begin{pmatrix} -4 & 2 \\ 2 & -1 \end{pmatrix}, \text{ where } k \text{ is a constant.}$$ Find the image under \(T\) of the line with equation \(y = 2x + 1\). [2]
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]