Questions C2 (1410 questions)

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AQA C2 2010 January Q4
10 marks Moderate -0.3
4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 31 terms of the series is 310 .
  1. Show that \(a + 15 d = 10\).
  2. Given also that the 21st term is twice the 16th term, find the value of \(d\).
  3. The \(n\)th term of the series is \(u _ { n }\). Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).
AQA C2 2010 January Q5
10 marks Standard +0.3
5 A curve has equation \(y = \frac { 1 } { x ^ { 3 } } + 48 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the equation of each of the two tangents to the curve that are parallel to the \(x\)-axis.
  3. Find an equation of the normal to the curve at the point \(( 1,49 )\).
AQA C2 2010 January Q6
12 marks Moderate -0.3
6
  1. Sketch the curve with equation \(y = 2 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
    1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 2 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
    2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
  3. Describe a geometrical transformation that maps the graph of \(y = 2 ^ { x }\) onto the graph of \(y = 2 ^ { x + 7 } + 3\).
  4. The curve \(y = 2 ^ { x + k } + 3\) intersects the \(y\)-axis at the point \(A ( 0,8 )\). Show that \(k = \log _ { m } n\), where \(m\) and \(n\) are integers.
AQA C2 2010 January Q7
8 marks Moderate -0.3
7
  1. The first four terms of the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\) in ascending powers of \(x\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the integers \(a , b\) and \(c\).
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 1 - \frac { 1 } { 2 } x \right) ^ { 2 } ( 1 + 2 x ) ^ { 7 }\).
AQA C2 2010 January Q8
12 marks Standard +0.3
8
  1. Solve the equation \(\tan \left( x + 52 ^ { \circ } \right) = \tan 22 ^ { \circ }\), giving the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    1. Show that the equation $$3 \tan \theta = \frac { 8 } { \sin \theta }$$ can be written as $$3 \cos ^ { 2 } \theta + 8 \cos \theta - 3 = 0$$
    2. Find the value of \(\cos \theta\) that satisfies the equation $$3 \cos ^ { 2 } \theta + 8 \cos \theta - 3 = 0$$
    3. Hence solve the equation $$3 \tan 2 x = \frac { 8 } { \sin 2 x }$$ giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
AQA C2 2011 January Q1
4 marks Easy -1.2
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 5 cm .
\includegraphics[max width=\textwidth, alt={}, center]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-02_415_525_550_794} The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians.
The length of the \(\operatorname { arc } A B\) is 4 cm .
  1. Find the value of \(\theta\).
  2. Find the area of the sector \(O A B\).
AQA C2 2011 January Q2
5 marks Easy -1.3
2
  1. Write down the values of \(p , q\) and \(r\) given that:
    1. \(8 = 2 ^ { p }\);
    2. \(\frac { 1 } { 8 } = 2 ^ { q }\);
    3. \(\sqrt { 2 } = 2 ^ { r }\).
  2. Find the value of \(x\) for which \(\sqrt { 2 } \times 2 ^ { x } = \frac { 1 } { 8 }\).
AQA C2 2011 January Q3
8 marks Moderate -0.3
3 The triangle \(A B C\), shown in the diagram, is such that \(A B = 5 \mathrm {~cm} , A C = 8 \mathrm {~cm}\), \(B C = 10 \mathrm {~cm}\) and angle \(B A C = \theta\).
  1. Show that \(\theta = 97.9 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
    1. Calculate the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
    2. The line through \(A\), perpendicular to \(B C\), meets \(B C\) at the point \(D\). Calculate the length of \(A D\), giving your answer, in cm , to three significant figures.
AQA C2 2011 January Q4
6 marks Moderate -0.8
4
  1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } \sqrt { 27 x ^ { 3 } + 4 } \mathrm {~d} x\), giving your answer to three significant figures.
  2. The curve with equation \(y = \sqrt { 27 x ^ { 3 } + 4 }\) is stretched parallel to the \(x\)-axis with scale factor 3 to give the curve with equation \(y = \mathrm { g } ( x )\). Write down an expression for \(\mathrm { g } ( x )\).
    (2 marks)
    \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-05_1988_1717_719_150}
AQA C2 2011 January Q5
10 marks Standard +0.3
5
  1. Using the binomial expansion, or otherwise, express \(( 1 - x ) ^ { 3 }\) in ascending powers of \(x\).
  2. Show that the expansion of $$( 1 + y ) ^ { 4 } - ( 1 - y ) ^ { 3 }$$ is $$7 y + p y ^ { 2 } + q y ^ { 3 } + y ^ { 4 }$$ where \(p\) and \(q\) are constants to be found.
  3. Hence find \(\int \left[ ( 1 + \sqrt { x } ) ^ { 4 } - ( 1 - \sqrt { x } ) ^ { 3 } \right] \mathrm { d } x\), expressing each coefficient in its simplest form.
AQA C2 2011 January Q6
9 marks Moderate -0.8
6 A geometric series has third term 36 and sixth term 972.
    1. Show that the common ratio of the series is 3 .
    2. Find the first term of the series.
  2. The \(n\)th term of the series is \(u _ { n }\).
    1. Show that \(\sum _ { n = 1 } ^ { 20 } u _ { n } = 2 \left( 3 ^ { 20 } - 1 \right)\).
    2. Find the least value of \(n\) such that \(u _ { n } > 4 \times 10 ^ { 15 }\).
      \(7 \quad\) A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\) and is sketched below.
      \includegraphics[max width=\textwidth, alt={}, center]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-08_602_799_447_632}
AQA C2 2011 January Q8
7 marks Moderate -0.3
8
  1. Given that \(2 \log _ { k } x - \log _ { k } 5 = 1\), express \(k\) in terms of \(x\). Give your answer in a form not involving logarithms.
  2. Given that \(\log _ { a } y = \frac { 3 } { 2 }\) and that \(\log _ { 4 } a = b + 2\), show that \(y = 2 ^ { p }\), where \(p\) is an expression in terms of \(b\).
    \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-09_2102_1717_605_150}
AQA C2 2011 January Q9
10 marks Standard +0.3
9
  1. Solve the equation \(\tan x = - 3\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
    1. Given that $$7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6$$ show that $$\tan ^ { 2 } \theta + \tan \theta - 6 = 0$$
    2. Hence solve the equation \(7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
      (4 marks)
AQA C2 2012 January Q1
4 marks Easy -1.2
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 6 cm .
\includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-2_358_332_358_829} The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians.
The area of the sector \(O A B\) is \(21.6 \mathrm {~cm} ^ { 2 }\).
  1. Find the value of \(\theta\).
  2. Find the length of the \(\operatorname { arc } A B\).
AQA C2 2012 January Q2
5 marks Moderate -0.8
2
  1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 4 } \frac { 2 ^ { x } } { x + 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
  2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
AQA C2 2012 January Q3
3 marks Easy -1.2
3
  1. Write \(\sqrt [ 4 ] { x ^ { 3 } }\) in the form \(x ^ { k }\).
  2. Write \(\frac { 1 - x ^ { 2 } } { \sqrt [ 4 ] { x ^ { 3 } } }\) in the form \(x ^ { p } - x ^ { q }\).
AQA C2 2012 January Q4
8 marks Standard +0.3
4 The triangle \(A B C\), shown in the diagram, is such that \(A B\) is 10 metres and angle \(B A C\) is \(150 ^ { \circ }\).
\includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-3_323_746_406_648} The area of triangle \(A B C\) is \(40 \mathrm {~m} ^ { 2 }\).
  1. Show that the length of \(A C\) is 16 metres.
  2. Calculate the length of \(B C\), giving your answer, in metres, to two decimal places.
  3. Calculate the smallest angle of triangle \(A B C\), giving your answer to the nearest \(0.1 ^ { \circ }\).
AQA C2 2012 January Q5
8 marks Moderate -0.8
5
    1. Describe the geometrical transformation that maps the graph of \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) onto the graph of \(y = ( 1 + 2 x ) ^ { 6 }\).
    2. The curve \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) is translated by the vector \(\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression for \(\mathrm { g } ( x )\), simplifying your answer.
  2. The first four terms in the binomial expansion of \(\left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\), giving your answers in their simplest form.
AQA C2 2012 January Q6
10 marks Standard +0.3
6 An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 25 terms of the series is 3500 .
  1. Show that \(a + 12 d = 140\).
  2. The fifth term of this series is 100 . Find the value of \(d\) and the value of \(a\).
  3. The \(n\)th term of this series is \(u _ { n }\). Given that $$33 \left( \sum _ { n = 1 } ^ { 25 } u _ { n } - \sum _ { n = 1 } ^ { k } u _ { n } \right) = 67 \sum _ { n = 1 } ^ { k } u _ { n }$$ find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
    (3 marks)
AQA C2 2012 January Q7
10 marks Standard +0.3
7
  1. Sketch the graph of \(y = \frac { 1 } { 2 ^ { x } }\), indicating the value of the intercept on the \(y\)-axis.
  2. Use logarithms to solve the equation \(\frac { 1 } { 2 ^ { x } } = \frac { 5 } { 4 }\), giving your answer to three significant figures.
  3. Given that $$\log _ { a } \left( b ^ { 2 } \right) + 3 \log _ { a } y = 3 + 2 \log _ { a } \left( \frac { y } { a } \right)$$ express \(y\) in terms of \(a\) and \(b\).
    Give your answer in a form not involving logarithms.
AQA C2 2012 January Q8
10 marks Moderate -0.3
8
  1. Given that \(2 \sin \theta = 7 \cos \theta\), find the value of \(\tan \theta\).
    1. Use an appropriate identity to show that the equation $$6 \sin ^ { 2 } x = 4 + \cos x$$ can be written as $$6 \cos ^ { 2 } x + \cos x - 2 = 0$$
    2. Hence solve the equation \(6 \sin ^ { 2 } x = 4 + \cos x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\), giving your answers to the nearest degree.
AQA C2 2012 January Q9
17 marks Moderate -0.8
9 The diagram shows part of a curve crossing the \(x\)-axis at the origin \(O\) and at the point \(A ( 8,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-5_547_536_497_760} The curve has equation $$y = 12 x - 3 x ^ { \frac { 5 } { 3 } }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
    2. Show that the equation of the tangent at \(A ( 8,0 )\) is \(y + 8 x = 64\).
  3. Find \(\int \left( 12 x - 3 x ^ { \frac { 5 } { 3 } } \right) \mathrm { d } x\).
  4. Calculate the area of the shaded region bounded by the curve from \(O\) to \(A\) and the tangents \(O P\) and \(A P\).
AQA C2 2013 January Q1
5 marks Moderate -0.8
1 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]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-2_382_351_379_826} The angle \(A O B\) is 1.25 radians. The perimeter of the sector is 39 cm .
  1. Show that \(r = 12\).
  2. Calculate the area of the sector \(O A B\).
AQA C2 2013 January Q2
9 marks Moderate -0.8
2
  1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 1 } ^ { 5 } \frac { 1 } { x ^ { 2 } + 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
    1. Find \(\int \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving the coefficient of each term in its simplest form.
    2. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\).
AQA C2 2013 January Q3
6 marks Moderate -0.3
3 The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-3_273_622_356_708} The lengths of \(A C\) and \(B C\) are 5 cm and 6 cm respectively.
The area of triangle \(A B C\) is \(12.5 \mathrm {~cm} ^ { 2 }\), and angle \(A C B\) is obtuse.
  1. Find the size of angle \(A C B\), giving your answer to the nearest \(0.1 ^ { \circ }\).
  2. Find the length of \(A B\), giving your answer to two significant figures.