Questions FP2 (1279 questions)

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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]
AQA FP2 2013 January Q1
7 marks Moderate -0.3
  1. Show that $$12 \cosh x - 4 \sinh x = 4\text{e}^x + 8\text{e}^{-x}$$ [2 marks]
  2. Solve the equation $$12 \cosh x - 4 \sinh x = 33$$ giving your answers in the form \(k \ln 2\). [5 marks]
AQA FP2 2013 January Q2
10 marks Standard +0.3
Two loci, \(L_1\) and \(L_2\), in an Argand diagram are given by $$L_1 : |z + 6 - 5\text{i}| = 4\sqrt{2}$$ $$L_2 : \arg(z + \text{i}) = \frac{3\pi}{4}$$ The point \(P\) represents the complex number \(-2 + \text{i}\).
  1. Verify that the point \(P\) is a point of intersection of \(L_1\) and \(L_2\). [2 marks]
  2. Sketch \(L_1\) and \(L_2\) on one Argand diagram. [6 marks]
  3. The point \(Q\) is also a point of intersection of \(L_1\) and \(L_2\). Find the complex number that is represented by \(Q\). [2 marks]
AQA FP2 2013 January Q3
7 marks Standard +0.3
  1. Show that \(\frac{1}{5r-2} - \frac{1}{5r+3} = \frac{A}{(5r-2)(5r+3)}\), stating the value of the constant \(A\). [2 marks]
  2. Hence use the method of differences to show that $$\sum_{r=1}^{n} \frac{1}{(5r-2)(5r+3)} = \frac{n}{3(5n+3)}$$ [4 marks]
  3. Find the value of $$\sum_{r=1}^{\infty} \frac{1}{(5r-2)(5r+3)}$$ [1 mark]
AQA FP2 2013 January Q4
9 marks Standard +0.3
The roots of the equation $$z^3 - 5z^2 + kz - 4 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\).
    1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha\beta\gamma\). [2 marks]
    2. Hence find the value of \(\alpha^2\beta\gamma + \alpha\beta^2\gamma + \alpha\beta\gamma^2\). [2 marks]
  2. The value of \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\) is \(-4\).
    1. Explain why \(\alpha\), \(\beta\) and \(\gamma\) cannot all be real. [1 mark]
    2. By considering \((\alpha\beta + \beta\gamma + \gamma\alpha)^2\), find the possible values of \(k\). [4 marks]
AQA FP2 2013 January Q5
11 marks Standard +0.8
  1. Using the definition \(\tanh y = \frac{\text{e}^y - \text{e}^{-y}}{\text{e}^y + \text{e}^{-y}}\), show that, for \(|x| < 1\), $$\tanh^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$$ [3 marks]
  2. Hence, or otherwise, show that \(\frac{\text{d}}{\text{d}x}(\tanh^{-1} x) = \frac{1}{1-x^2}\). [3 marks]
  3. Use integration by parts to show that $$\int_{0}^{\frac{1}{4}} \tanh^{-1} x \, \text{d}x = \ln \left(\frac{3^m}{2^n}\right)$$ where \(m\) and \(n\) are positive integers. [5 marks]
AQA FP2 2013 January Q6
8 marks Standard +0.8
A curve is defined parametrically by $$x = t^3 + 5, \quad y = 6t^2 - 1$$ The arc length between the points where \(t = 0\) and \(t = 3\) on the curve is \(s\).
  1. Show that \(s = \int_{0}^{3} 3t\sqrt{t^2 + A} \, \text{d}t\), stating the value of the constant \(A\). [4 marks]
  2. Hence show that \(s = 61\). [4 marks]
AQA FP2 2013 January Q7
9 marks Standard +0.8
The polynomial \(\text{p}(n)\) is given by \(\text{p}(n) = (n-1)^3 + n^3 + (n+1)^3\).
    1. Show that \(\text{p}(k+1) - \text{p}(k)\), where \(k\) is a positive integer, is a multiple of 9. [3 marks]
    2. Prove by induction that \(\text{p}(n)\) is a multiple of 9 for all integers \(n \geqslant 1\). [4 marks]
  2. Using the result from part (a)(ii), show that \(n(n^2 + 2)\) is a multiple of 3 for any positive integer \(n\). [2 marks]
AQA FP2 2013 January Q8
14 marks Challenging +1.2
  1. Express \(-4 + 4\sqrt{3}\text{i}\) in the form \(r\text{e}^{\text{i}\theta}\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\). [3 marks]
    1. Solve the equation \(z^3 = -4 + 4\sqrt{3}\text{i}\), giving your answers in the form \(r\text{e}^{\text{i}\theta}\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\). [4 marks]
    2. The roots of the equation \(z^3 = -4 + 4\sqrt{3}\text{i}\) are represented by the points \(P\), \(Q\) and \(R\) on an Argand diagram. Find the area of the triangle \(PQR\), giving your answer in the form \(k\sqrt{3}\), where \(k\) is an integer. [3 marks]
  3. By considering the roots of the equation \(z^3 = -4 + 4\sqrt{3}\text{i}\), show that $$\cos\frac{2\pi}{9} + \cos\frac{4\pi}{9} + \cos\frac{8\pi}{9} = 0$$ [4 marks]
AQA FP2 2011 June Q1
8 marks Moderate -0.3
  1. Draw on the same Argand diagram:
    1. the locus of points for which $$|z - 2 - 5i| = 5$$ [3 marks]
    2. the locus of points for which $$\arg(z + 2i) = \frac{\pi}{4}$$ [3 marks]
  2. Indicate on your diagram the set of points satisfying both $$|z - 2 - 5i| \leqslant 5$$ and $$\arg(z + 2i) = \frac{\pi}{4}$$ [2 marks]
AQA FP2 2011 June Q2
10 marks Standard +0.3
  1. Use the definitions of \(\cosh \theta\) and \(\sinh \theta\) in terms of \(e^\theta\) to show that $$\cosh x \cosh y - \sinh x \sinh y = \cosh(x - y)$$ [4 marks]
  2. It is given that \(x\) satisfies the equation $$\cosh(x - \ln 2) = \sinh x$$
    1. Show that \(\tanh x = \frac{5}{4}\). [4 marks]
    2. Express \(x\) in the form \(\frac{1}{2} \ln a\). [2 marks]
AQA FP2 2011 June Q3
6 marks Standard +0.8
  1. Show that $$(r + 1)! - (r - 1)! = (r^2 + r - 1)(r - 1)!$$ [2 marks]
  2. Hence show that $$\sum_{r=1}^{n} (r^2 + r - 1)(r - 1)! = (n + 2)n! - 2$$ [4 marks]
AQA FP2 2011 June Q4
14 marks Standard +0.8
The cubic equation $$z^3 - 2z^2 + k = 0 \quad (k \neq 0)$$ has roots \(\alpha\), \(\beta\) and \(\gamma\).
    1. Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha\beta + \beta\gamma + \gamma\alpha\). [2 marks]
    2. Show that \(\alpha^2 + \beta^2 + \gamma^2 = 4\). [2 marks]
    3. Explain why \(\alpha^3 - 2\alpha^2 + k = 0\). [1 mark]
    4. Show that \(\alpha^3 + \beta^3 + \gamma^3 = 8 - 3k\). [2 marks]
  2. Given that \(\alpha^4 + \beta^4 + \gamma^4 = 0\):
    1. show that \(k = 2\); [4 marks]
    2. find the value of \(\alpha^5 + \beta^5 + \gamma^5\). [3 marks]
AQA FP2 2011 June Q5
13 marks Challenging +1.3
  1. The arc of the curve \(y^2 = x^2 + 8\) between the points where \(x = 0\) and \(x = 6\) is rotated through \(2\pi\) radians about the \(x\)-axis. Show that the area \(S\) of the curved surface formed is given by $$S = 2\sqrt{2}\pi \int_0^6 \sqrt{x^2 + 4} \, dx$$ [5 marks]
  2. By means of the substitution \(x = 2 \sinh \theta\), show that $$S = \pi(24\sqrt{5} + 4\sqrt{2} \sinh^{-1} 3)$$ [8 marks]
AQA FP2 2011 June Q6
8 marks Standard +0.3
  1. Show that $$(k + 1)(4(k + 1)^2 - 1) = 4k^3 + 12k^2 + 11k + 3$$ [2 marks]
  2. Prove by induction that, for all integers \(n \geqslant 1\), $$1^2 + 3^2 + 5^2 + \ldots + (2n - 1)^2 = \frac{1}{3}n(4n^2 - 1)$$ [6 marks]