Questions C2 (1410 questions)

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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)
AQA C2 2012 June Q1
5 marks Moderate -0.8
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
6 marks Moderate -0.8
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
7 marks Moderate -0.8
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
8 marks Moderate -0.8
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
9 marks Standard +0.3
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.
AQA C2 2012 June Q6
10 marks Moderate -0.8
6 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 4 } { x ^ { 2 } } - 11$$ The point \(P ( 2,1 )\) lies on the curve.
    1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 2\).
      (l mark)
    2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 2\).
    3. Hence state whether \(P\) is a maximum point or a minimum point, giving a reason for your answer.
  2. Find the equation of the curve.
AQA C2 2012 June Q7
7 marks Moderate -0.3
7 It is given that \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\).
  1. Find the possible values of \(\tan \theta\).
  2. Hence solve the equation \(( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0\), giving all solutions for \(\theta\), in degrees, in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
AQA C2 2012 June Q8
8 marks Standard +0.3
8
  1. Sketch the curve with equation \(y = 7 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
  2. The curve \(C _ { 1 }\) has equation \(y = 7 ^ { x }\). The curve \(C _ { 2 }\) has equation \(y = 7 ^ { 2 x } - 12\).
    1. By forming and solving a quadratic equation, prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at exactly one point. State the \(y\)-coordinate of this point.
    2. Use logarithms to find the \(x\)-coordinate of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\), giving your answer to three significant figures.
      (2 marks)
AQA C2 2012 June Q9
15 marks Standard +0.3
9 The diagram shows part of a curve whose equation is \(y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)\).
\includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-5_355_451_367_799}
  1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 1 } \log _ { 10 } \left( x ^ { 2 } + 1 \right) d x$$ giving your answer to three significant figures.
  2. The graph of \(y = 2 \log _ { 10 } x\) can be transformed into the graph of \(y = 1 + 2 \log _ { 10 } x\) by means of a translation. Write down the vector of the translation.
    1. Show that \(\log _ { 10 } \left( 10 x ^ { 2 } \right) = 1 + 2 \log _ { 10 } x\).
    2. Show that the graph of \(y = 2 \log _ { 10 } x\) can also be transformed into the graph of \(y = 1 + 2 \log _ { 10 } x\) by means of a stretch, and describe the stretch.
    3. The curve with equation \(y = 1 + 2 \log _ { 10 } x\) intersects the curve \(y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)\) at the point \(P\). Given that the \(x\)-coordinate of \(P\) is positive, find the gradient of the line \(O P\), where \(O\) is the origin. Give your answer in the form \(\log _ { 10 } \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.
AQA C2 2013 June Q1
5 marks Easy -1.2
1 A geometric series has first term 80 and common ratio \(\frac { 1 } { 2 }\).
  1. Find the third term of the series.
  2. Find the sum to infinity of the series.
  3. Find the sum of the first 12 terms of the series, giving your answer to two decimal places.
AQA C2 2013 June Q2
8 marks Standard +0.3
2 The diagram shows a sector \(O A B\) of a circle with centre \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_341_371_968_815} The radius of the circle is 20 cm and the angle \(A O B = 0.8\) radians.
  1. Find the length of the arc \(A B\).
  2. Find the area of the sector \(O A B\).
  3. A line from \(B\) meets the radius \(O A\) at the point \(D\), as shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_344_371_1747_815} The length of \(B D\) is 15 cm . Find the size of the obtuse angle \(O D B\), in radians, giving your answer to three significant figures.
AQA C2 2013 June Q3
9 marks Moderate -0.3
3
    1. Using the binomial expansion, or otherwise, express \(( 2 + y ) ^ { 3 }\) in the form \(a + b y + c y ^ { 2 } + y ^ { 3 }\), where \(a , b\) and \(c\) are integers.
    2. Hence show that \(\left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 }\) can be expressed in the form \(p + q x ^ { - 4 }\), where \(p\) and \(q\) are integers.
    1. Hence find \(\int \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x\).
    2. Hence find the value of \(\int _ { 1 } ^ { 2 } \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x\).
AQA C2 2013 June Q4
5 marks Moderate -0.8
4
  1. Sketch the graph of \(y = 9 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
    (2 marks)
  2. Use logarithms to solve the equation \(9 ^ { x } = 15\), giving your value of \(x\) to three significant figures.
  3. The curve \(y = 9 ^ { x }\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
    (l mark)
AQA C2 2013 June Q5
9 marks Moderate -0.3
5
  1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } \sqrt { 8 x ^ { 3 } + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
  2. Describe the single transformation that maps the graph of \(y = \sqrt { 8 x ^ { 3 } + 1 }\) onto the graph of \(y = \sqrt { x ^ { 3 } + 1 }\).
  3. The curve with equation \(y = \sqrt { x ^ { 3 } + 1 }\) is translated by \(\left[ \begin{array} { c } 2 \\ - 0.7 \end{array} \right]\) to give the curve with equation \(y = \mathrm { g } ( x )\). Find the value of \(\mathrm { g } ( 4 )\).
    (3 marks)
AQA C2 2013 June Q6
12 marks Moderate -0.3
6 A curve has the equation $$y = \frac { 12 + x ^ { 2 } \sqrt { x } } { x } , \quad x > 0$$
  1. Express \(\frac { 12 + x ^ { 2 } \sqrt { x } } { x }\) in the form \(12 x ^ { p } + x ^ { q }\).
    1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find an equation of the normal to the curve at the point on the curve where \(x = 4\).
    3. The curve has a stationary point \(P\). Show that the \(x\)-coordinate of \(P\) can be written in the form \(2 ^ { k }\), where \(k\) is a rational number.
AQA C2 2013 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 \(u _ { 1 } = 96\) and \(u _ { 2 } = 72\).
The limit of \(u _ { n }\) as \(n\) tends to infinity is 24 .
  1. Show that \(p = \frac { 2 } { 3 }\).
  2. Find the value of \(u _ { 3 }\).
AQA C2 2013 June Q8
7 marks Standard +0.3
8
  1. Given that \(\log _ { a } b = c\), express \(b\) in terms of \(a\) and \(c\).
  2. By forming a quadratic equation, show that there is only one value of \(x\) which satisfies the equation \(2 \log _ { 2 } ( x + 7 ) - \log _ { 2 } ( x + 5 ) = 3\).
AQA C2 2013 June Q9
14 marks Standard +0.3
9
    1. On the axes given below, sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    2. Solve the equation \(\tan x = - 1\), giving all values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    1. Given that \(6 \tan \theta \sin \theta = 5\), show that \(6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0\).
    2. Hence solve the equation \(6 \tan 3 x \sin 3 x = 5\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
      \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-5_720_1367_806_390}
AQA C2 2014 June Q1
5 marks Easy -1.2
1 The diagram shows a triangle \(A B C\). The size of angle \(B A C\) is \(47 ^ { \circ }\) and the lengths of \(A B\) and \(A C\) are 5 cm and 12 cm respectively.
  1. Calculate the area of the triangle \(A B C\), giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\).
  2. Calculate the length of \(B C\), giving your answer, in cm , to one decimal place.
    [0pt] [3 marks]
AQA C2 2014 June Q2
8 marks Moderate -0.8
2
  1. Find \(\int \left( 1 + 3 x ^ { \frac { 1 } { 2 } } + x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
    1. The expression \(( 1 + y ) ^ { 3 }\) can be written in the form \(1 + 3 y + n y ^ { 2 } + y ^ { 3 }\). Write down the value of the constant \(n\).
    2. Hence, or otherwise, expand \(( 1 + \sqrt { x } ) ^ { 3 }\).
  3. Hence find the exact value of \(\int _ { 0 } ^ { 1 } ( 1 + \sqrt { x } ) ^ { 3 } \mathrm {~d} x\).