Questions — Edexcel FP2 (360 questions)

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Edexcel FP2 Specimen Q2
6 marks Standard +0.8
  1. A transformation from the \(z\)-plane to the \(w\)-plane is given by
$$w = z ^ { 2 }$$
  1. Show that the line with equation \(\operatorname { Im } ( z ) = 1\) in the \(z\)-plane is mapped to a parabola in the \(w\)-plane, giving an equation for this parabola.
  2. Sketch the parabola on an Argand diagram.
Edexcel FP2 Specimen Q3
10 marks Standard +0.3
  1. The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0 \\ 1 & 2 & 0 \\ - 1 & 0 & 4 \end{array} \right)$$
  1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
  2. For each of the eigenvalues find a corresponding eigenvector.
  3. Find a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P }\) is a diagonal matrix.
Edexcel FP2 Specimen Q4
13 marks Challenging +1.8
    1. A group \(G\) contains distinct elements \(a , b\) and \(e\) where \(e\) is the identity element and the group operation is multiplication.
Given \(a ^ { 2 } b = b a\), prove \(a b \neq b a\) (ii) The set \(H = \{ 1,2,4,7,8,11,13,14 \}\) forms a group under the operation of multiplication modulo 15
  1. Find the order of each element of \(H\).
  2. Find three subgroups of \(H\) each of order 4, and describe each of these subgroups. The elements of another group \(J\) are the matrices \(\left( \begin{array} { c c } \cos \left( \frac { k \pi } { 4 } \right) & \sin \left( \frac { k \pi } { 4 } \right) \\ - \sin \left( \frac { k \pi } { 4 } \right) & \cos \left( \frac { k \pi } { 4 } \right) \end{array} \right)\) where \(k = 1,2,3,4,5,6,7,8\) and the group operation is matrix multiplication.
  3. Determine whether \(H\) and \(J\) are isomorphic, giving a reason for your answer.
Edexcel FP2 Specimen Q5
12 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-14_480_588_210_740} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} An engineering student makes a miniature arch as part of the design for a piece of coursework. The cross-section of this arch is modelled by the curve with equation $$y = A - \frac { 1 } { 2 } \cosh 2 x , \quad - \ln a \leqslant x \leqslant \ln a$$ where \(a > 1\) and \(A\) is a positive constant. The curve begins and ends on the \(x\)-axis, as shown in Figure 1.
  1. Show that the length of this curve is \(k \left( a ^ { 2 } - \frac { 1 } { a ^ { 2 } } \right)\), stating the value of the constant \(k\). The length of the curved cross-section of the miniature arch is required to be 2 m long.
  2. Find the height of the arch, according to this model, giving your answer to 2 significant figures.
  3. Find also the width of the base of the arch giving your answer to 2 significant figures.
  4. Give the equation of another curve that could be used as a suitable model for the cross-section of an arch, with approximately the same height and width as you found using the first model.
    (You do not need to consider the arc length of your curve)
Edexcel FP2 Specimen Q6
9 marks Challenging +1.2
  1. A curve has equation
$$| z + 6 | = 2 | z - 6 | \quad z \in \mathbb { C }$$
  1. Show that the curve is a circle with equation \(x ^ { 2 } + y ^ { 2 } - 20 x + 36 = 0\)
  2. Sketch the curve on an Argand diagram. The line \(l\) has equation \(a z ^ { * } + a ^ { * } z = 0\), where \(a \in \mathbb { C }\) and \(z \in \mathbb { C }\) Given that the line \(l\) is a tangent to the curve and that \(\arg a = \theta\)
  3. find the possible values of \(\tan \theta\)
Edexcel FP2 Specimen Q7
9 marks Challenging +1.2
7. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Prove that, for \(n \geqslant 2\), $$n I _ { n } = ( n - 1 ) I _ { n - 2 }$$
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-22_588_1018_630_520} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A designer is asked to produce a poster to completely cover the curved surface area of a solid cylinder which has diameter 1 m and height 0.7 m . He uses a large sheet of paper with height 0.7 m and width of \(\pi \mathrm { m }\).
    Figure 2 shows the first stage of the design, where the poster is divided into two sections by a curve. The curve is given by the equation $$y = \sin ^ { 2 } ( 4 x ) - \sin ^ { 10 } ( 4 x )$$ relative to axes taken along the bottom and left hand edge of the paper.
    The region of the poster below the curve is shaded and the region above the curve remains unshaded, as shown in Figure 2. Find the exact area of the poster which is shaded.
Edexcel FP2 Specimen Q8
9 marks Standard +0.8
  1. A staircase has \(n\) steps. A tourist moves from the bottom (step zero) to the top (step \(n\) ). At each move up the staircase she can go up either one step or two steps, and her overall climb up the staircase is a combination of such moves.
If \(u _ { n }\) is the number of ways that the tourist can climb up a staircase with \(n\) steps,
  1. explain why \(u _ { n }\) satisfies the recurrence relation $$u _ { n } = u _ { n - 1 } + u _ { n - 2 } , \text { with } u _ { 1 } = 1 \text { and } u _ { 2 } = 2$$
  2. Find the number of ways in which she can climb up a staircase when there are eight steps. A staircase at a certain tourist attraction has 400 steps.
  3. Show that the number of ways in which she could climb up to the top of this staircase is given by $$\frac { 1 } { \sqrt { 5 } } \left[ \left( \frac { 1 + \sqrt { 5 } } { 2 } \right) ^ { 401 } - \left( \frac { 1 - \sqrt { 5 } } { 2 } \right) ^ { 401 } \right]$$
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]
Edexcel FP2 Q2
5 marks Challenging +1.2
The displacement \(x\) metres of a particle at time \(t\) seconds is given by the differential equation $$\frac{d^2 x}{dt^2} + x + \cos x = 0.$$ When \(t = 0\), \(x = 0\) and \(\frac{dx}{dt} = \frac{1}{2}\). Find a Taylor series solution for \(x\) in ascending powers of \(t\), up to and including the term in \(t^4\). [5]
Edexcel FP2 Q3
7 marks Standard +0.8
  1. Find the set of values of \(x\) for which $$x + 4 > \frac{2}{x+3}$$ [6]
  2. Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \left|\frac{2}{x+3}\right|$$ [1]
Edexcel FP2 Q4
10 marks Standard +0.3
\(z = -8 + (8\sqrt{3})i\)
  1. Find the modulus of \(z\) and the argument of \(z\). [3]
Using de Moivre's theorem,
  1. find \(z^3\). [2]
  2. find the values of \(w\) such that \(w^4 = z\), giving your answers in the form \(a + ib\), where \(a, b \in \mathbb{R}\). [5]
Edexcel FP2 Q5
10 marks Standard +0.8
\includegraphics{figure_1} Figure 1 Figure 1 shows the curves given by the polar equations $$r = 2, \quad 0 \leq \theta \leq \frac{\pi}{2},$$ and $$r = 1.5 + \sin 3\theta, \quad 0 \leq \theta \leq \frac{\pi}{2}.$$
  1. Find the coordinates of the points where the curves intersect. [3]
The region \(S\), between the curves, for which \(r > 2\) and for which \(r < (1.5 + \sin 3\theta)\), is shown shaded in Figure 1.
  1. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a\pi + b\sqrt{3}\), where \(a\) and \(b\) are simplified fractions. [7]
Edexcel FP2 Q6
10 marks Standard +0.8
A complex number \(z\) is represented by the point \(P\) in the Argand diagram.
  1. Given that \(|z - 6| = |z|\), sketch the locus of \(P\). [2]
  2. Find the complex numbers \(z\) which satisfy both \(|z - 6| = |z|\) and \(|z - 3 - 4i| = 5\). [3]
The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac{30}{z}\).
  1. Show that \(T\) maps \(|z - 6| = |z|\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle. [5]
Edexcel FP2 Q7
7 marks Challenging +1.2
  1. Show that the transformation \(z = y^{\frac{1}{2}}\) transforms the differential equation $$\frac{dy}{dx} - 4y \tan x = 2y^{\frac{1}{2}}$$ [I] into the differential equation $$\frac{dz}{dx} - 2z \tan x = 1$$ [II]
  2. Solve the differential equation (II) to find \(z\) as a function of \(x\). [6]
  3. Hence obtain the general solution of the differential equation (I). [1]
Edexcel FP2 Q8
14 marks Challenging +1.2
  1. Find the value of \(z\) for which \(y = zx \sin 5x\) is a particular integral of the differential equation $$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [4]
  2. Using your answer to part (a), find the general solution of the differential equation $$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [3]
Given that at \(x = 0\), \(y = 0\) and \(\frac{dy}{dx} = 5\),
  1. find the particular solution of this differential equation, giving your solution in the form \(y = f(x)\). [5]
  2. Sketch the curve with equation \(y = f(x)\) for \(0 \leq x \leq \pi\). [2]
Edexcel FP2 Q1
7 marks Standard +0.3
Find the set of values of \(x\) for which $$\frac{3}{x+3} > \frac{x-4}{x}.$$ [7]
Edexcel FP2 Q2
7 marks Standard +0.8
$$\frac{d^2 y}{dx^2} = e^x \left(2x \frac{dy}{dx} + y^2 + 1\right).$$
  1. Show that $$\frac{d^4 y}{dx^4} = e^x \left[2x \frac{d^3 y}{dx^3} + 4 \frac{d^2 y}{dx^2} + 6y \frac{dy}{dx} + y^2 + 1\right],$$ where \(k\) is a constant to be found. [3]
Given that, at \(x = 0\), \(y = 1\) and \(\frac{dy}{dx} = 2\),
  1. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x^4\). [4]