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AQA C2 2008 June Q1
9 marks Moderate -0.8
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
6 marks Standard +0.3
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
7 marks Moderate -0.8
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
8 marks Moderate -0.8
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 marks Easy -1.2
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
9 marks Moderate -0.3
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
9 marks Moderate -0.8
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
14 marks Moderate -0.3
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
8 marks Moderate -0.3
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 2010 June Q1
7 marks Moderate -0.8
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.
AQA C2 2010 June Q2
5 marks Moderate -0.8
2 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = 6 + \frac { 2 } { 5 } u _ { n }$$ The first term of the sequence is given by \(u _ { 1 } = 2\).
  1. Find the value of \(u _ { 2 }\) and the value of \(u _ { 3 }\).
  2. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).
AQA C2 2010 June Q3
6 marks Easy -1.2
3 The triangle \(A B C\), shown in the diagram, is such that \(A B = 6 \mathrm {~cm} , B C = 15 \mathrm {~cm}\), angle \(B A C = 150 ^ { \circ }\) and angle \(A C B = \theta\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-4_376_867_406_584}
  1. Show that \(\theta = 11.5 ^ { \circ }\), 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 2010 June Q4
8 marks Moderate -0.8
4
  1. The expression \(\left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 }\) can be written in the form $$1 + \frac { p } { x ^ { 2 } } + \frac { q } { x ^ { 4 } } - \frac { 1 } { x ^ { 6 } }$$ Find the values of the integers \(p\) and \(q\).
    1. Hence find \(\int \left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
    2. Hence find the value of \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
      REFERENREFERENCE
      \(\_\_\_\_\)
      \includegraphics[max width=\textwidth, alt={}]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-5_40_1567_2637_272}
AQA C2 2010 June Q5
11 marks Standard +0.3
5
  1. An infinite geometric series has common ratio \(r\).
    The first term of the series is 10 and its sum to infinity is 50 .
    1. Show that \(r = \frac { 4 } { 5 }\).
    2. Find the second term of the series.
  2. The first and second terms of the geometric series in part (a) have the same values as the 4th and 8th terms respectively of an arithmetic series.
    1. Find the common difference of the arithmetic series.
    2. The \(n\)th term of the arithmetic series is \(u _ { n }\). Find the value of \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).
AQA C2 2010 June Q6
13 marks Moderate -0.8
6 A curve \(C\) has the equation $$y = \frac { x ^ { 3 } + \sqrt { x } } { x } , \quad x > 0$$
  1. Express \(\frac { x ^ { 3 } + \sqrt { x } } { x }\) in the form \(x ^ { p } + x ^ { q }\).
    1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find an equation of the normal to the curve \(C\) at the point on the curve where \(x = 1\).
    1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    2. Hence deduce that the curve \(C\) has no maximum points. \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-7_1463_1707_1244_153}
AQA C2 2010 June Q7
8 marks Moderate -0.3
7
  1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leqslant x \leqslant 2 \pi\). State the values of the intercepts with the coordinate axes.
    1. Given that $$\sin ^ { 2 } \theta = \cos \theta ( 2 - \cos \theta )$$ prove that \(\cos \theta = \frac { 1 } { 2 }\).
    2. Hence solve the equation $$\sin ^ { 2 } 2 x = \cos 2 x ( 2 - \cos 2 x )$$ in the interval \(0 \leqslant x \leqslant \pi\), giving your answers in radians to three significant figures.
AQA C2 2010 June Q8
17 marks Moderate -0.3
8 The diagram shows a sketch of the curve \(y = 2 ^ { 4 x }\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-9_435_814_374_623} The curve intersects the \(y\)-axis at the point \(A\).
  1. Find the value of the \(y\)-coordinate of \(A\).
  2. Use the trapezium rule with six ordinates (five strips) to find an approximate value for \(\int _ { 0 } ^ { 1 } 2 ^ { 4 x } \mathrm {~d} x\), giving your answer to two decimal places.
  3. Describe the geometrical transformation that maps the graph of \(y = 2 ^ { 4 x }\) onto the graph of \(y = 2 ^ { 4 x - 3 }\).
  4. The curve \(y = 2 ^ { 4 x }\) is translated by the vector \(\left[ \begin{array} { c } 1 \\ - \frac { 1 } { 2 } \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). The curve \(y = \mathrm { g } ( x )\) crosses the \(x\)-axis at the point \(Q\). Find the \(x\)-coordinate of \(Q\).
    1. Given that $$\log _ { a } k = 3 \log _ { a } 2 + \log _ { a } 5 - \log _ { a } 4$$ show that \(k = 10\).
    2. The line \(y = \frac { 5 } { 4 }\) crosses the curve \(y = 2 ^ { 4 x - 3 }\) at the point \(P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 1 } { 4 \log _ { 10 } 2 }\).
AQA C2 2011 June Q1
6 marks Moderate -0.8
1 The triangle \(A B C\), shown in the diagram, is such that \(A C = 9 \mathrm {~cm} , B C = 10 \mathrm {~cm}\), angle \(A B C = 54 ^ { \circ }\) and the acute angle \(B A C = \theta\).
  1. Show that \(\theta = 64 ^ { \circ }\), correct to the nearest degree.
  2. Calculate the area of triangle \(A B C\), giving your answer to the nearest square centimetre.
AQA C2 2011 June Q2
6 marks Easy -1.2
2 The diagram shows a sector \(O A B\) of a circle with centre \(O\). \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-2_440_392_1500_826} The radius of the circle is 6 cm and the angle \(A O B = 0.5\) radians.
  1. Find the area of the sector \(O A B\).
    1. Find the length of the arc \(A B\).
    2. Hence show that the perimeter of the sector \(O A B = k \times\) the length of the \(\operatorname { arc } A B\) where \(k\) is an integer.
AQA C2 2011 June Q3
10 marks Standard +0.3
3
  1. The expression \(\left( 2 + x ^ { 2 } \right) ^ { 3 }\) can be written in the form $$8 + p x ^ { 2 } + q x ^ { 4 } + x ^ { 6 }$$ Show that \(p = 12\) and find the value of the integer \(q\).
    1. Hence find \(\int \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x\).
      (5 marks)
    2. Hence find the exact value of \(\int _ { 1 } ^ { 2 } \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x\).
      (2 marks)
AQA C2 2011 June Q4
10 marks Moderate -0.3
4
  1. Sketch the curve with equation \(y = 4 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
    (2 marks)
  2. Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { x } - 5\).
    1. Use the substitution \(Y = 2 ^ { x }\) to show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) can be written as \(Y ^ { 2 } - 4 Y - 5 = 0\).
    2. Hence show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) has only one real solution. Use logarithms to find this solution, giving your answer to three decimal places.
      (4 marks)
AQA C2 2011 June Q5
13 marks Moderate -0.3
5 The diagram shows part of a curve with a maximum point \(M\). \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-4_480_645_354_694} The curve is defined for \(x \geqslant 0\) by the equation $$y = 6 x - 2 x ^ { \frac { 3 } { 2 } }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    (3 marks)
    1. Hence find the coordinates of the maximum point \(M\).
    2. Write down the equation of the normal to the curve at \(M\).
  3. The point \(P \left( \frac { 9 } { 4 } , \frac { 27 } { 4 } \right)\) lies on the curve.
    1. Find an equation of the normal to the curve at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are positive integers.
    2. The normals to the curve at the points \(M\) and \(P\) intersect at the point \(R\). Find the coordinates of \(R\). \(6 \quad\) A curve \(C\), defined for \(0 \leqslant x \leqslant 2 \pi\) by the equation \(y = \sin x\), where \(x\) is in radians, is sketched below. The region bounded by the curve \(C\), the \(x\)-axis from 0 to 2 and the line \(x = 2\) is shaded. \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-5_441_789_466_612}
AQA C2 2011 June Q7
6 marks Standard +0.3
7 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 two terms of the sequence are given by \(u _ { 1 } = 60\) and \(u _ { 2 } = 48\).
The limit of \(u _ { n }\) as \(n\) tends to infinity is 12 .
  1. Show that \(p = \frac { 3 } { 4 }\) and find the value of \(q\).
  2. Find the value of \(u _ { 3 }\).
AQA C2 2011 June Q8
4 marks Moderate -0.8
8 Prove that, for all values of \(x\), the value of the expression $$( 3 \sin x + \cos x ) ^ { 2 } + ( \sin x - 3 \cos x ) ^ { 2 }$$ is an integer and state its value.
AQA C2 2011 June Q9
10 marks Moderate -0.8
9 The first term of a geometric series is 12 and the common ratio of the series is \(\frac { 3 } { 8 }\).
  1. Find the sum to infinity of the series.
  2. Show that the sixth term of the series can be written in the form \(\frac { 3 ^ { 6 } } { 2 ^ { 13 } }\).
  3. The \(n\)th term of the series is \(u _ { n }\).
    1. Write down an expression for \(u _ { n }\) in terms of \(n\).
    2. Hence show that $$\log _ { a } u _ { n } = n \log _ { a } 3 - ( 3 n - 5 ) \log _ { a } 2$$ (4 marks)