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 2008 January Q6
6
  1. Using the binomial expansion, or otherwise:
    1. express \(( 1 + x ) ^ { 3 }\) in ascending powers of \(x\);
    2. express \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
  2. Hence, or otherwise:
    1. express \(( 1 + 4 x ) ^ { 3 }\) in ascending powers of \(x\);
    2. express \(( 1 + 3 x ) ^ { 4 }\) in ascending powers of \(x\).
  3. Show that the expansion of $$( 1 + 3 x ) ^ { 4 } - ( 1 + 4 x ) ^ { 3 }$$ can be written in the form $$p x ^ { 2 } + q x ^ { 3 } + r x ^ { 4 }$$ where \(p , q\) and \(r\) are integers.
AQA C2 2008 January Q7
7
  1. Given that $$\log _ { a } x = \log _ { a } 16 - \log _ { a } 2$$ write down the value of \(x\).
  2. Given that $$\log _ { a } y = 2 \log _ { a } 3 + \log _ { a } 4 + 1$$ express \(y\) in terms of \(a\), giving your answer in a form not involving logarithms.
AQA C2 2008 January Q8
8
  1. Sketch the graph of \(y = 3 ^ { x }\), stating the coordinates of the point where the graph crosses the \(y\)-axis.
  2. Describe a single geometrical transformation that maps the graph of \(y = 3 ^ { x }\) :
    1. onto the graph of \(y = 3 ^ { 2 x }\);
    2. onto the graph of \(y = 3 ^ { x + 1 }\).
    1. Using the substitution \(Y = 3 ^ { x }\), show that the equation $$9 ^ { x } - 3 ^ { x + 1 } + 2 = 0$$ can be written as $$( Y - 1 ) ( Y - 2 ) = 0$$
    2. Hence show that the equation \(9 ^ { x } - 3 ^ { x + 1 } + 2 = 0\) has a solution \(x = 0\) and, by using logarithms, find the other solution, giving your answer to four decimal places.
      (4 marks)
AQA C2 2008 January Q9
9
  1. Given that $$\frac { 3 + \sin ^ { 2 } \theta } { \cos \theta - 2 } = 3 \cos \theta$$ show that $$\cos \theta = - \frac { 1 } { 2 }$$
  2. Hence solve the equation $$\frac { 3 + \sin ^ { 2 } 3 x } { \cos 3 x - 2 } = 3 \cos 3 x$$ giving all solutions in degrees in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
AQA C2 2009 January Q1
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 10 cm .
\includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-2_371_378_555_824} The angle \(A O B\) is 0.8 radians.
  1. Find the area of the sector.
    1. Find the perimeter of the sector \(O A B\).
    2. The perimeter of the sector \(O A B\) is equal to the perimeter of a square. Find the area of the square.
AQA C2 2009 January Q2
2
  1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for $$\int _ { 1.5 } ^ { 6 } x ^ { 2 } \sqrt { x ^ { 2 } - 1 } \mathrm {~d} x$$ 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.
    (1 mark)
AQA C2 2009 January Q3
3 The diagram shows a triangle \(A B C\). The size of angle \(A\) is \(63 ^ { \circ }\), and the lengths of \(A B\) and \(A C\) are 7.4 m and 5.26 m respectively.
  1. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { m } ^ { 2 }\) to three significant figures.
  2. Show that the length of \(B C\) is 6.86 m , correct to three significant figures.
  3. Find the value of \(\sin \boldsymbol { B }\) to two significant figures.
AQA C2 2009 January Q4
4 The diagram shows a sketch of the curves with equations \(y = 2 x ^ { \frac { 3 } { 2 } }\) and \(y = 8 x ^ { \frac { 1 } { 2 } }\).
\includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-3_433_720_1452_644} The curves intersect at the origin and at the point \(A\), where \(x = 4\).
    1. For the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 4\).
      (2 marks)
    2. Find an equation of the normal to the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\) at the point \(A\).
    1. Find \(\int 8 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
    2. Find the area of the shaded region bounded by the two curves.
  3. Describe a single geometrical transformation that maps the graph of \(y = 2 x ^ { \frac { 3 } { 2 } }\) onto the graph of \(y = 2 ( x + 3 ) ^ { \frac { 3 } { 2 } }\).
    (2 marks)
AQA C2 2009 January Q5
5
  1. By using the binomial expansion, or otherwise, express \(( 1 + 2 x ) ^ { 4 }\) in the form $$1 + a x + b x ^ { 2 } + c x ^ { 3 } + 16 x ^ { 4 }$$ where \(a\), \(b\) and \(c\) are integers.
  2. Hence show that \(( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 } = 2 + 48 x ^ { 2 } + 32 x ^ { 4 }\).
  3. Hence show that the curve with equation $$y = ( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 }$$ has just one stationary point and state its coordinates.
AQA C2 2009 January Q6
6
  1. Write each of the following in the form \(\log _ { a } k\), where \(k\) is an integer:
    1. \(\log _ { a } 4 + \log _ { a } 10\);
    2. \(\log _ { a } 16 - \log _ { a } 2\);
    3. \(3 \log _ { a } 5\).
  2. Use logarithms to solve the equation \(( 1.5 ) ^ { 3 x } = 7.5\), giving your value of \(x\) to three decimal places.
  3. Given that \(\log _ { 2 } p = m\) and \(\log _ { 8 } q = n\), express \(p q\) in the form \(2 ^ { y }\), where \(y\) is an expression in \(m\) and \(n\).
AQA C2 2009 January Q7
7
  1. Solve the equation \(\sin x = 0.8\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your answers in radians to three significant figures.
  2. The diagram shows the graph of the curve \(y = \sin x , 0 \leqslant x \leqslant 2 \pi\) and the lines \(y = k\) and \(y = - k\).
    \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-5_497_780_552_689} The line \(y = k\) intersects the curve at the points \(P\) and \(Q\), and the line \(y = - k\) intersects the curve at the points \(R\) and \(S\). The point \(M\) is the minimum point of the curve.
    1. Write down the coordinates of the point \(M\).
    2. The \(x\)-coordinate of \(P\) is \(\alpha\). Write down the \(x\)-coordinate of the point \(Q\) in terms of \(\pi\) and \(\alpha\).
    3. Find the length of \(R S\) in terms of \(\pi\) and \(\alpha\), giving your answer in its simplest form.
  3. Sketch the graph of \(y = \sin 2 x\) for \(0 \leqslant x \leqslant 2 \pi\), indicating the coordinates of points where the graph intersects the \(x\)-axis and the coordinates of any maximum points.
AQA C2 2009 January Q8
8 The 25th term of an arithmetic series is 38 .
The sum of the first 40 terms of the series is 1250 .
  1. Show that the common difference of this series is 1.5 .
  2. Find the number of terms in the series which are less than 100 .
AQA C2 2010 January Q1
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_444_373_541_804} The radius of the circle is 15 cm and angle \(A O B = 1.2\) radians.
    1. Show that the area of the sector is \(135 \mathrm {~cm} ^ { 2 }\).
    2. Calculate the length of the arc \(A B\).
  2. The point \(P\) lies on the radius \(O B\) such that \(O P = 10 \mathrm {~cm}\), as shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_449_378_1436_799} Calculate the perimeter of the shaded region bounded by \(A P , P B\) and the arc \(A B\), giving your answer to three significant figures.
    (5 marks)
AQA C2 2010 January Q2
2 At the point \(( x , y )\) on a curve, where \(x > 0\), the gradient is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 \sqrt { x ^ { 5 } } - 4$$
  1. Write \(\sqrt { x ^ { 5 } }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
  2. Find \(\int \left( 7 \sqrt { x ^ { 5 } } - 4 \right) \mathrm { d } x\).
  3. Hence find the equation of the curve, given that the curve passes through the point \(( 1,3 )\).
AQA C2 2010 January Q3
3
  1. Find the value of \(x\) in each of the following:
    1. \(\quad \log _ { 9 } x = 0\);
    2. \(\quad \log _ { 9 } x = \frac { 1 } { 2 }\).
  2. Given that $$2 \log _ { a } n = \log _ { a } 18 + \log _ { a } ( n - 4 )$$ find the possible values of \(n\).
AQA C2 2010 January Q4
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
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
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
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
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
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
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
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
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
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.