Questions FP1 (1491 questions)

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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]
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]