Questions C2 (1410 questions)

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AQA C2 2013 January Q4
3 marks Moderate -0.8
4 Given that $$\log _ { a } N - \log _ { a } x = \frac { 3 } { 2 }$$ express \(x\) in terms of \(a\) and \(N\), giving your answer in a form not involving logarithms.
(3 marks)
AQA C2 2013 January Q5
12 marks Moderate -0.3
5 The point \(P ( 2,8 )\) lies on a curve, and the point \(M\) is the only stationary point of the curve. The curve has equation \(y = 6 + 2 x - \frac { 8 } { x ^ { 2 } }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the normal to the curve at the point \(P ( 2,8 )\) has equation \(x + 4 y = 34\).
    1. Show that the stationary point \(M\) lies on the \(x\)-axis.
    2. Hence write down the equation of the tangent to the curve at \(M\).
  4. The tangent to the curve at \(M\) and the normal to the curve at \(P\) intersect at the point \(T\). Find the coordinates of \(T\).
AQA C2 2013 January Q6
10 marks Moderate -0.8
6
  1. A geometric series begins \(420 + 294 + 205.8 + \ldots\).
    1. Show that the common ratio of the series is 0.7 .
    2. Find the sum to infinity of the series.
    3. Write the \(n\)th term of the series in the form \(p \times q ^ { n }\), where \(p\) and \(q\) are constants.
  2. The first term of an arithmetic series is 240 and the common difference of the series is - 8 . The \(n\)th term of the series is \(u _ { n }\).
    1. Write down an expression for \(u _ { n }\).
    2. Given that \(u _ { k } = 0\), find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
AQA C2 2013 January Q7
9 marks Moderate -0.3
7
  1. Describe a geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 3 \times 4 ^ { x }\).
  2. Sketch the curve with equation \(y = 3 \times 4 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
  3. The curve with equation \(y = 4 ^ { - x }\) intersects the curve \(y = 3 \times 4 ^ { x }\) at the point \(P\). Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.
AQA C2 2013 January Q8
9 marks Moderate -0.3
8
  1. Expand \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 }\).
  2. The first four terms of the binomial expansion of \(\left( 1 + \frac { x } { 4 } \right) ^ { 8 }\) in ascending powers of \(x\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\).
  3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 } \left( 1 + \frac { x } { 4 } \right) ^ { 8 }\).
AQA C2 2013 January Q9
12 marks Standard +0.3
9
  1. Write down the two solutions of the equation \(\tan \left( x + 30 ^ { \circ } \right) = \tan 79 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    (2 marks)
  2. Describe a single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x + 30 ^ { \circ } \right)\).
    1. Given that \(5 + \sin ^ { 2 } \theta = ( 5 + 3 \cos \theta ) \cos \theta\), show that \(\cos \theta = \frac { 3 } { 4 }\).
    2. Hence solve the equation \(5 + \sin ^ { 2 } 2 x = ( 5 + 3 \cos 2 x ) \cos 2 x\) in the interval \(0 < x < 2 \pi\), giving your values of \(x\) in radians to three significant figures.
AQA C2 2005 June Q1
5 marks Moderate -0.8
1 The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_423_707_612_657} The lengths of \(A C\) and \(B C\) are 5 cm and 4.8 cm respectively.
The size of the angle \(B C A\) is \(30 ^ { \circ }\).
  1. Calculate the area of the triangle \(A B C\).
  2. Calculate the length of \(A B\), giving your answer to three significant figures.
AQA C2 2005 June Q2
5 marks Moderate -0.8
2 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]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_486_381_1686_739} The angle \(A O B\) is 1.5 radians. The perimeter of the sector is 56 cm .
  1. Show that \(r = 16\).
  2. Find the area of the sector.
AQA C2 2005 June Q3
6 marks Easy -1.2
3 The \(n\)th term of an arithmetic sequence is \(u _ { n }\), where $$u _ { n } = 90 - 3 n$$
  1. Find the value of \(u _ { 1 }\) and the value of \(u _ { 2 }\).
  2. Write down the common difference of the arithmetic sequence.
  3. Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).
AQA C2 2005 June Q4
19 marks Moderate -0.3
4 The diagram shows a curve \(C\) with equation \(y = \sqrt { x }\). The point \(O\) is the origin \(( 0,0 )\).
\includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-3_488_1136_1009_443} The region bounded by the curve \(C\), the \(x\)-axis and the vertical lines \(x = 1\) and \(x = 4\) is shown shaded in the diagram.
    1. Write \(\sqrt { x }\) in the form \(x ^ { p }\), where \(p\) is a constant.
    2. Find \(\int \sqrt { x } \mathrm {~d} x\).
    3. Hence find the area of the shaded region.
  2. The point on \(C\) for which \(x = 4\) is \(P\). The tangent to \(C\) at the point \(P\) intersects the \(x\)-axis and the \(y\)-axis at the points \(A\) and \(B\) respectively.
    1. Find an equation for the tangent to the curve \(C\) at the point \(P\).
    2. Find the area of the triangle \(A O B\).
  3. Describe the single geometrical transformation by which the curve with equation \(y = \sqrt { x - 1 }\) can be obtained from the curve \(C\).
  4. Use the trapezium rule with four ordinates (three strips) to find an approximation for \(\int _ { 1 } ^ { 4 } \sqrt { x - 1 } \mathrm {~d} x\), giving your answer to three significant figures.
AQA C2 2005 June Q5
12 marks Standard +0.3
5 The sum to infinity of a geometric series is four times the first term of the series.
  1. Show that the common ratio, \(r\), of the geometric series is \(\frac { 3 } { 4 }\).
  2. The first term of the geometric series is 48 . Find the sum of the first 10 terms of the series, giving your answer to four decimal places.
  3. The \(n\)th term of the geometric series is \(u _ { n }\) and the ( \(2 n\) )th term of the series is \(u _ { 2 n }\).
    1. Write \(u _ { n }\) and \(u _ { 2 n }\) in terms of \(n\).
    2. Hence show that \(\log _ { 10 } \left( u _ { n } \right) - \log _ { 10 } \left( u _ { 2 n } \right) = n \log _ { 10 } \left( \frac { 4 } { 3 } \right)\).
    3. Hence show that the value of $$\log _ { 10 } \left( \frac { u _ { 100 } } { u _ { 200 } } \right)$$ is 12.5 correct to three significant figures.
AQA C2 2005 June Q6
9 marks Moderate -0.8
6
  1. Using the binomial expansion, or otherwise, express \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
    1. Hence show that \(( 1 + \sqrt { 5 } ) ^ { 4 } = 56 + 24 \sqrt { 5 }\).
    2. Hence show that \(\log _ { 2 } ( 1 + \sqrt { 5 } ) ^ { 4 } = k + \log _ { 2 } ( 7 + 3 \sqrt { 5 } )\), where \(k\) is an integer.
AQA C2 2005 June Q7
9 marks Moderate -0.8
7 A curve is defined, for \(x > 0\), by the equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { x ^ { 8 } - 1 } { x ^ { 3 } }$$
  1. Express \(\frac { x ^ { 8 } - 1 } { x ^ { 3 } }\) in the form \(x ^ { p } - x ^ { q }\), where \(p\) and \(q\) are integers.
    1. Hence differentiate \(\mathrm { f } ( x )\) to find \(\mathrm { f } ^ { \prime } ( x )\).
    2. Hence show that f is an increasing function.
  3. Find the gradient of the normal to the curve at the point \(( 1,0 )\).
AQA C2 2005 June Q8
10 marks Moderate -0.3
8
    1. Show that the equation $$4 \tan \theta \sin \theta = 15$$ can be written as $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ (1 mark)
    2. Use an appropriate identity to show that the equation $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ can be written as $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$
    1. Solve the equation \(4 c ^ { 2 } + 15 c - 4 = 0\).
    2. Hence explain why the only value of \(\cos \theta\) which satisfies the equation $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$ is \(\cos \theta = \frac { 1 } { 4 }\).
    3. Hence solve the equation \(4 \tan \theta \sin \theta = 15\) giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  3. Write down all the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\) for which $$4 \tan 4 x \sin 4 x = 15$$ giving your answers to the nearest degree.
AQA C2 2006 June Q1
5 marks Moderate -0.8
1 The diagram shows a sector of a circle of radius 5 cm and angle \(\theta\) radians.
\includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-2_327_358_571_842} The area of the sector is \(8.1 \mathrm {~cm} ^ { 2 }\).
  1. Show that \(\theta = 0.648\).
  2. Find the perimeter of the sector.
AQA C2 2006 June Q2
6 marks Easy -1.2
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
7 marks Moderate -0.8
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
8 marks Moderate -0.3
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
6 marks Moderate -0.8
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
13 marks Moderate -0.3
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
21 marks Moderate -0.8
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
11 marks Moderate -0.3
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
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 \(25 \times 0.8 ^ { n }\).