Questions — Edexcel (10514 questions)

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Edexcel FP2 Q28
16 marks Challenging +1.8
  1. Sketch the curve with polar equation $$r = 3 \cos 2\theta, \quad -\frac{\pi}{4} \leq \theta < \frac{\pi}{4}.$$ [2]
  2. Find the area of the smaller finite region enclosed between the curve and the half-line \(\theta = \frac{\pi}{6}\). [6]
  3. Find the exact distance between the two tangents which are parallel to the initial line. [8]
Edexcel FP2 Q29
7 marks Standard +0.8
Find the complete set of values of \(x\) for which $$|x^2 - 2| > 2x.$$ [7]
Edexcel FP2 Q30
11 marks Standard +0.3
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + 2y = x.$$ [5] Given that \(y = 1\) at \(x = 0\),
  2. find the exact values of the coordinates of the minimum point of the particular solution curve, [4]
  3. draw a sketch of this particular solution curve. [2]
Edexcel FP2 Q31
12 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{d^2 y}{dt^2} + 2\frac{dy}{dt} + 2y = 2e^{-t}.$$ [6]
  2. Find the particular solution that satisfies \(y = 1\) and \(\frac{dy}{dt} = 1\) at \(t = 0\). [6]
Edexcel FP2 Q32
16 marks Challenging +1.2
\includegraphics{figure_1} Figure 1 is a sketch of the two curves \(C_1\) and \(C_2\) with polar equations $$C_1 : r = 3a(1 - \cos \theta), \quad -\pi \leq \theta < \pi$$ and $$C_2 : r = a(1 + \cos \theta), \quad -\pi \leq \theta < \pi.$$ The curves meet at the pole \(O\), and at the points \(A\) and \(B\).
  1. Find, in terms of \(a\), the polar coordinates of the points \(A\) and \(B\). [4]
  2. Show that the length of the line \(AB\) is \(\frac{3\sqrt{3}}{2}a\). [2] The region inside \(C_2\) and outside \(C_1\) is shown shaded in Fig. 1.
  3. Find, in terms of \(a\), the area of this region. [7] A badge is designed which has the shape of the shaded region. Given that the length of the line \(AB\) is \(4.5\) cm,
  4. calculate the area of this badge, giving your answer to three significant figures. [3]
Edexcel FP2 Q33
8 marks Standard +0.3
Given that \(y = \tan x\),
  1. find \(\frac{dy}{dx}\), \(\frac{d^2 y}{dx^2}\) and \(\frac{d^3 y}{dx^3}\). [3]
  2. Find the Taylor series expansion of \(\tan x\) in ascending powers of \(\left(x - \frac{\pi}{4}\right)\) up to and including the term in \(\left(x - \frac{\pi}{4}\right)^3\). [3]
  3. Hence show that \(\tan \frac{3\pi}{10} \approx 1 + \frac{\pi}{10} + \frac{\pi^2}{200} + \frac{\pi^3}{3000}\). [2]
Edexcel FP2 Q34
11 marks Challenging +1.2
  1. Prove by induction that $$\frac{d^n}{dx^n}(e^x \cos x) = 2^{\frac{1}{2}n} e^x \cos\left(x + \frac{1}{4}n\pi\right), \quad n \geq 1.$$ [8]
  2. Find the Maclaurin series expansion of \(e^x \cos x\), in ascending powers of \(x\), up to and including the term in \(x^4\). [3]
Edexcel FP2 Q35
14 marks Challenging +1.3
The transformation \(T\) from the complex \(z\)-plane to the complex \(w\)-plane is given by $$w = \frac{z + 1}{z + i}, \quad z \neq -i.$$
  1. Show that \(T\) maps points on the half-line \(\arg(z) = \frac{\pi}{4}\) in the \(z\)-plane into points on the circle \(|w| = 1\) in the \(w\)-plane. [4]
  2. Find the image under \(T\) in the \(w\)-plane of the circle \(|z| = 1\) in the \(z\)-plane. [6]
  3. Sketch on separate diagrams the circle \(|z| = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane. [2]
  4. Mark on your sketches the point \(P\), where \(z = i\), and its image \(Q\) under \(T\) in the \(w\)-plane. [2]
Edexcel FP2 Q36
5 marks Moderate -0.3
  1. Sketch the graph of \(y = |x - 2a|\), given that \(a > 0\). [2]
  2. Solve \(|x - 2a| > 2x + a\), where \(a > 0\). [3]
Edexcel FP2 Q37
7 marks Standard +0.8
Find the general solution of the differential equation $$\frac{dy}{dx} + 2y \cot 2x = \sin x, \quad 0 < x < \frac{\pi}{2},$$ giving your answer in the form \(y = f(x)\). [7]
Edexcel FP2 Q38
10 marks Standard +0.3
  1. Express \(\frac{1}{r(r + 2)}\) in partial fractions. [2]
  2. Hence prove, by the method of differences, that $$\sum_{r=1}^{n} \frac{4}{r(r + 2)} = \frac{n(3n + 5)}{(n + 1)(n + 2)}.$$ [5]
  3. Find the value of \(\sum_{r=50}^{100} \frac{4}{r(r + 2)}\), to 4 decimal places. [3]
Edexcel FP2 Q39
12 marks Challenging +1.3
  1. Show that the transformation \(y = xv\) transforms the equation $$x^2\frac{d^2 y}{dx^2} - 2x\frac{dy}{dx} + (2 + 9x^2)y = x^5, \quad \text{I}$$ into the equation $$\frac{d^2 v}{dx^2} + 9v = x^2. \quad \text{II}$$ [5]
  2. Solve the differential equation II to find \(v\) as a function of \(x\). [6]
  3. Hence state the general solution of the differential equation I. [1]
Edexcel FP2 Q40
13 marks Standard +0.8
The curve \(C\) has polar equation \(r = 6 \cos \theta\), \(-\frac{\pi}{2} \leq \theta < \frac{\pi}{2}\), and the line \(D\) has polar equation \(r = 3 \sec\left(\frac{\pi}{3} - \theta\right)\), \(-\frac{\pi}{6} \leq \theta \leq \frac{5\pi}{6}\).
  1. Find a cartesian equation of \(C\) and a cartesian equation of \(D\). [5]
  2. Sketch on the same diagram the graphs of \(C\) and \(D\), indicating where each cuts the initial line. [3] The graphs of \(C\) and \(D\) intersect at the points \(P\) and \(Q\).
  3. Find the polar coordinates of \(P\) and \(Q\). [5]
Edexcel FP2 Q41
5 marks Standard +0.3
  1. By expressing \(\frac{2}{4r^2 - 1}\) in partial fractions, or otherwise, prove that $$\sum_{r=1}^{n} \frac{2}{4r^2 - 1} = 1 - \frac{1}{2n + 1}.$$ [3]
  2. Hence find the exact value of \(\sum_{r=11}^{20} \frac{2}{4r^2 - 1}\). [2]
Edexcel FP2 Q42
7 marks Standard +0.3
Find the general solution of the differential equation $$(x + 1)\frac{dy}{dx} + 2y = \frac{1}{x}, \quad x > 0.$$ giving your answer in the form \(y = f(x)\). [7]
Edexcel FP2 Q43
12 marks Standard +0.3
  1. On the same diagram, sketch the graphs of \(y = |x^2 - 4|\) and \(y = |2x - 1|\), showing the coordinates of the points where the graphs meet the axes. [4]
  2. Solve \(|x^2 - 4| = |2x - 1|\), giving your answers in surd form where appropriate. [5]
  3. Hence, or otherwise, find the set of values of \(x\) for which of \(|x^2 - 4| > |2x - 1|\). [3]
Edexcel FP2 Q44
14 marks Standard +0.3
  1. Find the general solution of the differential equation $$2\frac{d^2 x}{dt^2} + 5\frac{dx}{dt} + 2x = 2t + 9.$$ [6]
  2. Find the particular solution of this differential equation for which \(x = 3\) and \(\frac{dx}{dt} = -1\) when \(t = 0\). [4] The particular solution in part (b) is used to model the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds (\(t \geq 0\)), \(P\) is \(x\) metres from the origin \(O\).
  3. Show that the minimum distance between \(O\) and \(P\) is \(\frac{1}{2}(5 + \ln 2)\) m and justify that the distance is a minimum. [4]
Edexcel FP2 Q45
13 marks Challenging +1.3
\includegraphics{figure_1} The curve \(C\) which passes through \(O\) has polar equation $$r = 4a(1 + \cos \theta), \quad -\pi < \theta \leq \pi$$ The line \(l\) has polar equation $$r = 3a \sec \theta, \quad -\frac{\pi}{2} < \theta < \frac{\pi}{2}.$$ The line \(l\) cuts \(C\) at the points \(P\) and \(Q\), as shown in Figure 1.
  1. Prove that \(PQ = 6\sqrt{3}a\). [6] The region \(R\), shown shaded in Figure 1, is bounded by \(l\) and \(C\).
  2. Use calculus to find the exact area of \(R\). [7]
Edexcel FP2 Q46
11 marks Standard +0.3
A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that $$|z - 3i| = 3,$$
  1. sketch the locus of \(P\). [2]
  2. Find the complex number \(z\) which satisfies both \(|z - 3i| = 3\) and \(\arg (z - 3i) = \frac{3}{4}\pi\). [4] The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{2i}{w}.$$
  3. Show that \(T\) maps \(|z - 3i| = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line. [5]
Edexcel F3 2021 June Q1
6 marks Standard +0.3
  1. Using the definitions of hyperbolic functions in terms of exponentials, show that $$1 - \tanh^2 x = \operatorname{sech}^2 x$$ [3]
  2. Solve the equation $$2\operatorname{sech}^2 x + 3\tanh x = 3$$ giving your answer as an exact logarithm. [3]
Edexcel F3 2021 June Q2
7 marks Challenging +1.2
A curve has equation $$y = \sqrt{9 - x^2} \quad 0 \leq x \leq 3$$
  1. Using calculus, show that the length of the curve is \(\frac{3\pi}{2}\) [4]
The curve is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Using calculus, find the exact area of the surface generated. [3]
Edexcel F3 2021 June Q3
9 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} 3 & 1 & p \\ 1 & 1 & 2 \\ -1 & p & 2 \end{pmatrix}$$ where \(p\) is a real constant
  1. Find the exact values of \(p\) for which \(\mathbf{M}\) has no inverse. [4]
Given that \(\mathbf{M}\) does have an inverse,
  1. find \(\mathbf{M}^{-1}\) in terms of \(p\). [5]
Edexcel F3 2021 June Q4
8 marks Standard +0.8
  1. \(f(x) = x \arccos x \quad -1 \leq x \leq 1\) Find the exact value of \(f'(0.5)\). [3]
  2. \(g(x) = \arctan(e^{2x})\) Show that $$g''(x) = k \operatorname{sech}(2x) \tanh(2x)$$ where \(k\) is a constant to be found. [5]
Edexcel F3 2021 June Q5
10 marks Challenging +1.8
$$I_n = \int \sec^n x \, dx \quad n \geq 0$$
  1. Prove that for \(n \geq 2\) $$(n-1)I_n = \tan x \sec^{n-2} x + (n-2)I_{n-2}$$ [6]
  2. Hence, showing each step of your working, find the exact value of $$\int_0^{\pi/4} \sec^6 x \, dx$$ [4]
Edexcel F3 2021 June Q6
13 marks Standard +0.8
The line \(l_1\) has equation $$\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + 3\mathbf{k})$$ and the line \(l_2\) has equation $$\mathbf{r} = 2\mathbf{i} + s\mathbf{j} + \mu(\mathbf{i} - 2\mathbf{j} + \mathbf{k})$$ where \(s\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters. Given that \(l_1\) and \(l_2\) both lie in a common plane \(\Pi_1\)
  1. show that an equation for \(\Pi_1\) is \(3x + y - z = 3\) [4]
  2. Find the value of \(s\). [1]
The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot (\mathbf{i} + \mathbf{j} - 2\mathbf{k}) = 3\)
  1. Find an equation for the line of intersection of \(\Pi_1\) and \(\Pi_2\) [4]
  2. Find the acute angle between \(\Pi_1\) and \(\Pi_2\) giving your answer in degrees to 3 significant figures. [4]