Edexcel FP1 (Further Pure Mathematics 1)

Given that \(z = 22 + 4i\) and \(\frac{z}{w} = 6 - 8i\), find

1. \(w\) in the form \(a + bi\), where \(a\) and \(b\) are real, [3]
2. the argument of \(z\), in radians to 2 decimal places. [2]

1. Prove that \(\sum_{r=1}^{n} (r + 1)(r - 1) = \frac{1}{6} n (n - 1)(2n + 5)\). [5]
2. Deduce that \(n(n - 1)(2n + 5)\) is divisible by 6 for all \(n > 1\). [2]

$$f(x) = x^3 + x - 3.$$ The equation \(f(x) = 0\) has a root, \(\alpha\), between 1 and 2.

1. By considering \(f'(x)\), show that \(\alpha\) is the only real root of the equation \(f(x) = 0\). [3]
2. Taking 1.2 as your first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(f(x)\) to obtain a second approximation to \(\alpha\). Give your answer to 3 significant figures. [2]
3. Prove that your answer to part (b) gives the value of \(\alpha\) correct to 3 significant figures. [2]

Given that \(2 + i\) is a root of the equation $$z^2 + bz + c = 0, \text{ where } b \text{ and } c \text{ are real constants,}$$

1. write down the other root of the equation,
2. find the value of \(b\) and the value of \(c\). [5]

Prove that $$\sum_{r=1}^{n} 6(r^2 - 1) = (n - 1)n(2n + 5).$$ [4]

Given that \(z = 3 + 4i\) and \(w = -1 + 7i\).

1. find \(|w|\). [1]

The complex numbers \(z\) and \(w\) are represented by the points \(A\) and \(B\) on an Argand diagram.

1. Show points \(A\) and \(B\) on an Argand diagram. [1]
2. Prove that \(\triangle OAB\) is an isosceles right-angled triangle. [5]
3. Find the exact value of \(\arg \left( \frac{z}{w} \right)\). [3]

The point \(P \left( 2p, \frac{2}{p} \right)\) and the point \(Q \left( 2q, \frac{2}{q} \right)\), where \(p \neq -q\), lie on the rectangular hyperbola with equation \(xy = 4\). The tangents to the curve at the points \(P\) and \(Q\) meet at the point \(R\). Show that at the point \(R\), $$x = \frac{4pq}{p + q} \text{ and } y = \frac{4}{p + q}.$$ [8]

For \(n \in \mathbb{Z}^+\) prove that

1. \(2^{3n + 2} + 5^{n + 1}\) is divisible by 3, [9]
2. \(\begin{pmatrix} -2 & -1 \\ 9 & 4 \end{pmatrix}^n = \begin{pmatrix} 1-3n & -n \\ 9n & 3n+1 \end{pmatrix}\). [7]

$$f(x) = 2 \sin 2x + x - 2.$$ The root \(\alpha\) of the equation \(f(x) = 0\) lies in the interval \([2, \pi]\). Using the end points of this interval find, by linear interpolation, an approximation to \(\alpha\). [4]

Given that \(z = 3 - 3i\) express, in the form \(a + ib\), where \(a\) and \(b\) are real numbers,

1. \(z^2\), [2]
2. \(\frac{1}{z}\), [2]
3. Find the exact value of each of \(|z|\), \(|z^2|\) and \(\left|\frac{1}{z}\right|\). [2]

The complex numbers \(z\), \(z^2\) and \(\frac{1}{z}\) are represented by the points \(A\), \(B\) and \(C\) respectively on an Argand diagram. The real number 1 is represented by the point \(D\), and \(O\) is the origin.

1. Show the points \(A\), \(B\), \(C\) and \(D\) on an Argand diagram. [2]
2. Prove that \(\triangle OAB\) is similar to \(\triangle OCD\). [3]

1. Using that 3 is the real root of the cubic equation \(x^3 - 27 = 0\), show that the complex roots of the cubic satisfy the quadratic equation \(x^2 + 3x + 9 = 0\). [2]
2. Hence, or otherwise, find the three cube roots of 27, giving your answers in the form \(a + ib\), where \(a, b \in \mathbb{R}\). [3]
3. Show these roots on an Argand diagram. [2]

$$f(x) = 3^x - x - 6.$$

1. Show that \(f(x) = 0\) has a root \(\alpha\) between \(x = 1\) and \(x = 2\). [2]
2. Starting with the interval \((1, 2)\), use interval bisection three times to find an interval of width 0.125 which contains \(\alpha\). [2]

$$z = \frac{a + 3i}{2 + ai}, \quad a \in \mathbb{R}.$$

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]

$$f(n) = (2n + 1)7^n - 1.$$ Prove by induction that, for all positive integers \(n\), \(f(n)\) is divisible by 4. [6]

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]

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]

$$f(x) = 2^x + x - 4.$$ 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]

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]

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]

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]

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]

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]

$$f(x) = 1 - e^x + 3 \sin 2x$$ 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]

$$z = -4 + 6i.$$

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]

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]

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]

Prove that \(\sum_{r=1}^{n} (r - 1)(r + 2) = \frac{1}{3} (n - 1)n(n + 4)\). [6]

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]

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]

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]

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]

Prove by induction that, for \(n \in \mathbb{Z}^+\), \(\sum_{r=1}^{n} r 2^r = 2\{1 + (n - 1)2^n\}\). [5]

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]

$$f(x) = 0.25x - 2 + 4 \sin \sqrt{x}.$$

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]

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]

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]

$$f (x) = x^3 + 8x - 19.$$

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]

$$z = \sqrt{3} - i.$$ \(z^*\) 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]

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]

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]

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]

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]

$$f(x) = 3x^2 + x - \tan \left( \frac{x}{2} \right) - 2, \quad -\pi < x < \pi.$$ 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]

$$z = -2 + i.$$

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]

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]

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]

$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}, \text{ where } k \text{ is constant.}$$ 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]