Questions — AQA C2 (185 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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AQA C2 2007 June Q8
8 marks Moderate -0.8
8
  1. It is given that \(n\) satisfies the equation $$\log _ { a } n = \log _ { a } 3 + \log _ { a } ( 2 n - 1 )$$ Find the value of \(n\).
  2. Given that \(\log _ { a } x = 3\) and \(\log _ { a } y - 3 \log _ { a } 2 = 4\) :
    1. express \(x\) in terms of \(a\);
    2. express \(x y\) in terms of \(a\).
AQA C2 2009 June Q1
5 marks Moderate -0.8
The triangle \(ABC\), shown in the diagram, is such that \(AB = 7\) cm, \(AC = 5\) cm, \(BC = 8\) cm and angle \(ABC = \theta\). \includegraphics{figure_1}
  1. Show that \(\theta = 38.2°\), correct to the nearest \(0.1°\). [3]
  2. Calculate the area of triangle \(ABC\), giving your answer, in cm\(^2\), to three significant figures. [2]
AQA C2 2009 June Q2
8 marks Moderate -0.8
  1. Write down the value of \(n\) given that \(\frac{1}{x^3} = x^n\). [1]
  2. Expand \(\left(1 + \frac{3}{x^2}\right)^2\). [2]
  3. Hence find \(\int \left(1 + \frac{3}{x^2}\right)^2 dx\). [3]
  4. Hence find the exact value of \(\int_1^3 \left(1 + \frac{3}{x^2}\right)^2 dx\). [2]
AQA C2 2009 June Q3
7 marks Moderate -0.3
The \(n\)th term of a sequence is \(u_n\). The sequence is defined by $$u_{n+1} = ku_n + 12$$ where \(k\) is a constant. The first two terms of the sequence are given by $$u_1 = 16 \quad u_2 = 24$$
  1. Show that \(k = 0.75\). [2]
  2. Find the value of \(u_3\) and the value of \(u_4\). [2]
  3. The limit of \(u_n\) as \(n\) tends to infinity is \(L\).
    1. Write down an equation for \(L\). [1]
    2. Hence find the value of \(L\). [2]
AQA C2 2009 June Q4
6 marks Moderate -0.3
  1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int_0^6 \sqrt{x^3 + 1} dx\), giving your answer to four significant figures. [4]
  2. The curve with equation \(y = \sqrt{x^3 + 1}\) is stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\) to give the curve with equation \(y = f(x)\). Write down an expression for \(f(x)\). [2]
AQA C2 2009 June Q5
13 marks Standard +0.3
The diagram shows part of a curve with a maximum point \(M\). \includegraphics{figure_5} The equation of the curve is $$y = 15x^{\frac{3}{2}} - x^{\frac{5}{2}}$$
  1. Find \(\frac{dy}{dx}\). [3]
  2. Hence find the coordinates of the maximum point \(M\). [4]
  3. The point \(P(1, 14)\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20x - 6\). [3]
  4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(RM\). [3]
AQA C2 2009 June Q6
6 marks Moderate -0.3
The diagram shows a sector \(OAB\) of a circle with centre \(O\) and radius \(r\) cm. \includegraphics{figure_6} The angle \(AOB\) is \(1.2\) radians. The area of the sector is \(33.75\) cm\(^2\). Find the perimeter of the sector. [6]
AQA C2 2009 June Q7
11 marks Moderate -0.3
A geometric series has second term \(375\) and fifth term \(81\).
    1. Show that the common ratio of the series is \(0.6\). [3]
    2. Find the first term of the series. [2]
  2. Find the sum to infinity of the series. [2]
  3. The \(n\)th term of the series is \(u_n\). Find the value of \(\sum_{n=6}^{\infty} u_n\). [4]
AQA C2 2009 June Q8
9 marks Moderate -0.3
  1. Given that \(\frac{\sin \theta - \cos \theta}{\cos \theta} = 4\), prove that \(\tan \theta = 5\). [2]
    1. Use an appropriate identity to show that the equation $$2 \cos^2 x - \sin x = 1$$ can be written as $$2 \sin^2 x + \sin x - 1 = 0$$ [2]
    2. Hence solve the equation $$2 \cos^2 x - \sin x = 1$$ giving all solutions in the interval \(0° \leq x \leq 360°\). [5]
AQA C2 2009 June Q9
10 marks Moderate -0.8
    1. Find the value of \(p\) for which \(\sqrt{125} = 5^p\). [2]
    2. Hence solve the equation \(5^{2x} = \sqrt{125}\). [1]
  2. Use logarithms to solve the equation \(3^{2x-1} = 0.05\), giving your value of \(x\) to four decimal places. [3]
  3. It is given that $$\log_a x = 2(\log_a 3 + \log_a 2) - 1$$ Express \(x\) in terms of \(a\), giving your answer in a form not involving logarithms. [4]