Questions — AQA C2 (184 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 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 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.
AQA C2 2010 June Q2
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
AQA C2 2012 June Q1
1 The arithmetic series $$23 + 32 + 41 + 50 + \ldots + 2534$$ has 280 terms.
  1. Write down the common difference of the series.
  2. Find the 100th term of the series.
  3. Find the sum of the 280 terms of the series.
AQA C2 2012 June Q2
2 The triangle \(A B C\), shown in the diagram, is such that \(A B = 26 \mathrm {~cm}\) and \(B C = 31.5 \mathrm {~cm}\). The acute angle \(A B C\) is \(\theta\), where \(\sin \theta = \frac { 5 } { 13 }\).
  1. Calculate the area of triangle \(A B C\).
  2. Find the exact value of \(\cos \theta\).
  3. Calculate the length of \(A C\).
AQA C2 2012 June Q3
3
  1. \(\quad\) Expand \(\left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 }\).
  2. Hence find \(\int \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).
  3. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { \frac { 3 } { 2 } } - 1 \right) ^ { 2 } \mathrm {~d} x\).
AQA C2 2012 June Q4
4 The \(n\)th term of a geometric series is \(u _ { n }\), where \(u _ { n } = 48 \left( \frac { 1 } { 4 } \right) ^ { n }\).
  1. Find the value of \(u _ { 1 }\) and the value of \(u _ { 2 }\).
  2. Find the value of the common ratio of the series.
  3. Find the sum to infinity of the series.
  4. Find the value of \(\sum _ { n = 4 } ^ { \infty } u _ { n }\).
AQA C2 2012 June Q5
5 The diagram shows a sector \(O P Q\) of a circle with centre \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-3_305_531_1105_758} The radius of the circle is 18 m and the angle \(P O Q\) is \(\frac { 2 \pi } { 3 }\) radians.
  1. Find the length of the arc \(P Q\), giving your answer as a multiple of \(\pi\).
  2. The tangents to the circle at the points \(P\) and \(Q\) meet at the point \(T\), and the angles \(T P O\) and \(T Q O\) are both right angles, as shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-3_597_529_1848_758}
    1. Angle \(P T Q = \alpha\) radians. Find \(\alpha\) in terms of \(\pi\).
    2. Find the area of the shaded region bounded by the \(\operatorname { arc } P Q\) and the tangents \(T P\) and \(T Q\), giving your answer to three significant figures.