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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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
AQA C2 2006 June Q2
2 The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-2_757_558_1409_726} The lengths of \(A C\) and \(B C\) are 4.8 cm and 12 cm respectively.
The size of the angle \(B A C\) is \(100 ^ { \circ }\).
  1. Show that angle \(A B C = 23.2 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
  2. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to three significant figures.
AQA C2 2006 June Q3
3 The first term of an arithmetic series is 1 . The common difference of the series is 6 .
  1. Find the tenth term of the series.
  2. The sum of the first \(n\) terms of the series is 7400 .
    1. Show that \(3 n ^ { 2 } - 2 n - 7400 = 0\).
    2. Find the value of \(n\).
AQA C2 2006 June Q4
4
  1. The expression \(( 1 - 2 x ) ^ { 4 }\) can be written in the form $$1 + p x + q x ^ { 2 } - 32 x ^ { 3 } + 16 x ^ { 4 }$$ By using the binomial expansion, or otherwise, find the values of the integers \(p\) and \(q\).
  2. Find the coefficient of \(x\) in the expansion of \(( 2 + x ) ^ { 9 }\).
  3. Find the coefficient of \(x\) in the expansion of \(( 1 - 2 x ) ^ { 4 } ( 2 + x ) ^ { 9 }\).
AQA C2 2006 June Q5
5
  1. Given that $$\log _ { a } x = 2 \log _ { a } 6 - \log _ { a } 3$$ show that \(x = 12\).
  2. Given that $$\log _ { a } y + \log _ { a } 5 = 7$$ express \(y\) in terms of \(a\), giving your answer in a form not involving logarithms.
    (3 marks)
AQA C2 2006 June Q6
6 The diagram shows a sketch of the curve with equation \(y = 27 - 3 ^ { x }\).
\includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-4_933_1074_376_484} The curve \(y = 27 - 3 ^ { x }\) intersects the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).
    1. Find the \(y\)-coordinate of point \(A\).
    2. Verify that the \(x\)-coordinate of point \(B\) is 3 .
  2. The region, \(R\), bounded by the curve \(y = 27 - 3 ^ { x }\) and the coordinate axes is shaded. Use the trapezium rule with four ordinates (three strips) to find an approximate value for the area of \(R\).
    1. Use logarithms to solve the equation \(3 ^ { x } = 13\), giving your answer to four decimal places.
    2. The line \(y = k\) intersects the curve \(y = 27 - 3 ^ { x }\) at the point where \(3 ^ { x } = 13\). Find the value of \(k\).
    1. Describe the single geometrical transformation by which the curve with equation \(y = - 3 ^ { x }\) can be obtained from the curve \(y = 27 - 3 ^ { x }\).
    2. Sketch the curve \(y = - 3 ^ { x }\).
AQA C2 2006 June Q7
7 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7$$
    1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 4\).
      (1 mark)
    2. Write \(\frac { 16 } { x ^ { 2 } }\) in the form \(16 x ^ { k }\), where \(k\) is an integer.
    3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    4. Hence determine whether the point where \(x = 4\) is a maximum or a minimum, giving a reason for your answer.
  2. The point \(P ( 1,8 )\) lies on the curve.
    1. Show that the gradient of the curve at the point \(P\) is 12 .
    2. Find an equation of the normal to the curve at \(P\).
    1. Find \(\int \left( 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7 \right) \mathrm { d } x\).
    2. Hence find the equation of the curve which passes through the point \(P ( 1,8 )\).
AQA C2 2006 June Q8
8
  1. Describe the single geometrical transformation by which the curve with equation \(y = \tan \frac { 1 } { 2 } x\) can be obtained from the curve \(y = \tan x\).
  2. Solve the equation \(\tan \frac { 1 } { 2 } x = 3\) in the interval \(\mathbf { 0 } < \boldsymbol { x } < \mathbf { 4 } \boldsymbol { \pi }\), giving your answers in radians to three significant figures.
  3. Solve the equation $$\cos \theta ( \sin \theta - 3 \cos \theta ) = 0$$ in the interval \(0 < \theta < 2 \pi\), giving your answers in radians to three significant figures.
    (5 marks)
AQA C2 2008 June Q1
1
  1. Write \(\sqrt { x ^ { 3 } }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
    (1 mark)
  2. A curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { 2 } - \sqrt { x ^ { 3 } }$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find the equation of the tangent to the curve at the point where \(x = 4\), giving your answer in the form \(y = m x + c\).
AQA C2 2008 June Q2
2 The diagram shows a shaded segment of a circle with centre \(O\) and radius 14 cm , where \(P Q\) is a chord of the circle.
\includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-2_423_551_1270_740} In triangle \(O P Q\), angle \(P O Q = \frac { 3 \pi } { 7 }\) radians and angle \(O P Q = \alpha\) radians.
  1. Find the length of the arc \(P Q\), giving your answer as a multiple of \(\pi\).
  2. Find \(\alpha\) in terms of \(\pi\).
  3. Find the perimeter of the shaded segment, giving your answer to three significant figures.
AQA C2 2008 June Q3
3 A geometric series begins $$20 + 16 + 12.8 + 10.24 + \ldots$$
  1. Find the common ratio of the series.
  2. Find the sum to infinity of the series.
  3. Find the sum of the first 20 terms of the series, giving your answer to three decimal places.
  4. Prove that the \(n\)th term of the series is \(25 \times 0.8 ^ { n }\).
AQA C2 2008 June Q4
4 The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-3_394_522_1062_751} The size of angle \(B A C\) is \(65 ^ { \circ }\), and the lengths of \(A B\) and \(A C\) are 7.6 m and 8.3 m respectively.
  1. Show that the length of \(B C\) is 8.56 m , correct to three significant figures.
  2. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { m } ^ { 2 }\) to three significant figures.
  3. The perpendicular from \(A\) to \(B C\) meets \(B C\) at the point \(D\). Calculate the length of \(A D\), giving your answer to the nearest 0.1 m .
AQA C2 2008 June Q5
5
  1. Write down the value of:
    1. \(\log _ { a } 1\);
    2. \(\log _ { a } a\).
  2. Given that $$\log _ { a } x = \log _ { a } 5 + \log _ { a } 6 - \log _ { a } 1.5$$ find the value of \(x\).
AQA C2 2008 June Q6
6 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants.
The first three terms of the sequence are given by $$u _ { 1 } = - 8 \quad u _ { 2 } = 8 \quad u _ { 3 } = 4$$
  1. Show that \(q = 6\) and find the value of \(p\).
  2. Find the value of \(u _ { 4 }\).
  3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\).
    1. Write down an equation for \(L\).
    2. Hence find the value of \(L\).
AQA C2 2008 June Q7
7
  1. The expression \(\left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 }\) can be written in the form $$1 + \frac { p } { x ^ { 2 } } + \frac { q } { x ^ { 4 } } + \frac { 64 } { x ^ { 6 } }$$ By using the binomial expansion, or otherwise, find the values of the integers \(p\) and \(q\).
    1. Hence find \(\int \left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
    2. Hence find the value of \(\int _ { 1 } ^ { 2 } \left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
AQA C2 2008 June Q8
8 The diagram shows a sketch of the curve with equation \(y = 6 ^ { x }\).
\includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-5_403_506_370_769}
    1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
    2. Explain, with the aid of a diagram, whether your approximate value will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\).
    1. Describe a single geometrical transformation that maps the graph of \(y = 6 ^ { x }\) onto the graph of \(y = 6 ^ { 3 x }\).
    2. The line \(y = 84\) intersects the curve \(y = 6 ^ { 3 x }\) at the point \(A\). By using logarithms, find the \(x\)-coordinate of \(A\), giving your answer to three decimal places.
      (4 marks)
  3. The graph of \(y = 6 ^ { x }\) is translated by \(\left[ \begin{array} { c } 1
    - 2 \end{array} \right]\) to give the graph of the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
AQA C2 2008 June Q9
9
  1. Solve the equation \(\sin 2 x = \sin 48 ^ { \circ }\), giving the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
  2. Solve the equation \(2 \sin \theta - 3 \cos \theta = 0\) in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\), giving your answers to the nearest \(0.1 ^ { \circ }\).
AQA C2 2009 June Q1
1 The triangle \(A B C\), shown in the diagram, is such that \(A B = 7 \mathrm {~cm} , A C = 5 \mathrm {~cm}\), \(B C = 8 \mathrm {~cm}\) and angle \(A B C = \theta\).
  1. Show that \(\theta = 38.2 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
  2. Calculate the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
AQA C2 2009 June Q2
2
  1. Write down the value of \(n\) given that \(\frac { 1 } { x ^ { 4 } } = x ^ { n }\).
  2. Expand \(\left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 }\).
  3. Hence find \(\int \left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 } \mathrm {~d} x\).
  4. Hence find the exact value of \(\int _ { 1 } ^ { 3 } \left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 } \mathrm {~d} x\).
AQA C2 2009 June Q3
3 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = k u _ { n } + 12$$ 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. Find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\).
  3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\).
    1. Write down an equation for \(L\).
    2. Hence find the value of \(L\).
AQA C2 2009 June Q5
5 The diagram shows part of a curve with a maximum point \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{22f2da99-0878-48a6-a2b7-1ba339d3c7e4-06_472_791_358_630} The equation of the curve is $$y = 15 x ^ { \frac { 3 } { 2 } } - x ^ { \frac { 5 } { 2 } }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the coordinates of the maximum point \(M\).
  3. The point \(P ( 1,14 )\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20 x - 6\).
  4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(R M\).
AQA C2 2009 June Q6
6 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{22f2da99-0878-48a6-a2b7-1ba339d3c7e4-07_447_424_360_790} The angle \(A O B\) is 1.2 radians. The area of the sector is \(33.75 \mathrm {~cm} ^ { 2 }\).
Find the perimeter of the sector.
AQA C2 2009 June Q7
7 A geometric series has second term 375 and fifth term 81.
    1. Show that the common ratio of the series is 0.6 .
    2. Find the first term of the series.
  2. Find the sum to infinity of the series.
  3. The \(n\)th term of the series is \(u _ { n }\). Find the value of \(\sum _ { n = 6 } ^ { \infty } u _ { n }\).
    □ .......... \(\_\_\_\_\)
AQA C2 2009 June Q8
8
  1. Given that \(\frac { \sin \theta - \cos \theta } { \cos \theta } = 4\), prove that \(\tan \theta = 5\).
    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. Hence solve the equation $$2 \cos ^ { 2 } x - \sin x = 1$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
      PARTREFERENCE
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      \includegraphics[max width=\textwidth, alt={}]{22f2da99-0878-48a6-a2b7-1ba339d3c7e4-09_33_1698_2682_155}
AQA C2 2009 June Q9
9
    1. Find the value of \(p\) for which \(\sqrt { 125 } = 5 ^ { p }\).
    2. Hence solve the equation \(5 ^ { 2 x } = \sqrt { 125 }\).
  2. Use logarithms to solve the equation \(3 ^ { 2 x - 1 } = 0.05\), giving your value of \(x\) to four decimal places.
  3. It is given that $$\log _ { a } x = 2 \left( \log _ { a } 3 + \log _ { a } 2 \right) - 1$$ Express \(x\) in terms of \(a\), giving your answer in a form not involving logarithms.
    (4 marks)
AQA C2 2010 June Q1
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-2_383_472_566_778} The radius of the circle is 8 m and the angle \(A O B\) is 1.4 radians.
  1. Find the area of the sector \(O A B\).
    1. Find the perimeter of the sector \(O A B\).
    2. The perimeter of the sector \(O A B\) is equal to the circumference of a circle of radius \(x \mathrm {~m}\). Calculate the value of \(x\) to three significant figures.