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OCR MEI C2 2009 January Q5
4 marks Moderate -0.8
5 Answer this question on the insert provided. Fig. 5 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-3_979_1077_422_536} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} On the insert, draw the graph of
  1. \(y = \mathrm { f } ( x - 2 )\),
  2. \(y = 3 \mathrm { f } ( x )\).
OCR MEI C2 2009 January Q12
12 marks Moderate -0.3
12 Answer part (ii) of this question on the insert provided. The proposal for a major building project was accepted, but actual construction was delayed. Each year a new estimate of the cost was made. The table shows the estimated cost, \(\pounds y\) million, of the project \(t\) years after the project was first accepted.
Years after proposal accepted \(( t )\)12345
Cost \(( \pounds y\) million \()\)250300360440530
The relationship between \(y\) and \(t\) is modelled by \(y = a b ^ { t }\), where \(a\) and \(b\) are constants.
  1. Show that \(y = a b ^ { t }\) may be written as $$\log _ { 10 } y = \log _ { 10 } a + t \log _ { 10 } b$$
  2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(t\), drawing by eye a line of best fit.
  3. Use your graph and the results of part (i) to find the values of \(\log _ { 10 } a\) and \(\log _ { 10 } b\) and hence \(a\) and \(b\).
  4. According to this model, what was the estimated cost of the project when it was first accepted?
  5. Find the value of \(t\) given by this model when the estimated cost is \(\pounds 1000\) million. Give your answer rounded to 1 decimal place.
AQA C2 2007 January Q1
5 marks Easy -1.2
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\). The radius of the circle is 6 cm and the angle \(A O B\) is 1.2 radians.
  1. Find the area of the sector \(O A B\).
  2. Find the perimeter of the sector \(O A B\).
AQA C2 2007 January Q2
4 marks Moderate -0.8
2 Use the trapezium rule with four ordinates (three strips) to find an approximate value for $$\int _ { 0 } ^ { 3 } \sqrt { 2 ^ { x } } \mathrm {~d} x$$ giving your answer to three decimal places.
AQA C2 2007 January Q3
5 marks Easy -1.3
3
  1. Write down the values of \(p , q\) and \(r\) given that:
    1. \(64 = 8 ^ { p }\);
    2. \(\frac { 1 } { 64 } = 8 ^ { q }\);
    3. \(\sqrt { 8 } = 8 ^ { r }\).
  2. Find the value of \(x\) for which $$\frac { 8 ^ { x } } { \sqrt { 8 } } = \frac { 1 } { 64 }$$
AQA C2 2007 January Q4
8 marks Moderate -0.8
4 The triangle \(A B C\), shown in the diagram, is such that \(B C = 6 \mathrm {~cm} , A C = 5 \mathrm {~cm}\) and \(A B = 4 \mathrm {~cm}\). The angle \(B A C\) is \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-3_442_652_452_678}
  1. Use the cosine rule to show that \(\cos \theta = \frac { 1 } { 8 }\).
  2. Hence use a trigonometrical identity to show that \(\sin \theta = \frac { 3 \sqrt { 7 } } { 8 }\).
  3. Hence find the area of the triangle \(A B C\).
AQA C2 2007 January Q5
7 marks Moderate -0.8
5 The second term of a geometric series is 48 and the fourth term is 3 .
  1. Show that one possible value for the common ratio, \(r\), of the series is \(- \frac { 1 } { 4 }\) and state the other value.
  2. In the case when \(r = - \frac { 1 } { 4 }\), find:
    1. the first term;
    2. the sum to infinity of the series.
AQA C2 2007 January Q6
16 marks Moderate -0.3
6 A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\) and is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-4_545_784_420_628}
    1. Given that \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
    3. Find an equation of the normal to \(C\) at the point ( 1,6 ).
    1. Find \(\int \left( x + 1 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
    2. Hence find the area of the region bounded by the curve \(C\), the lines \(x = 1\) and \(x = 4\) and the \(x\)-axis.
AQA C2 2007 January Q7
7 marks Moderate -0.8
7
  1. The first four terms of the binomial expansion of \(( 1 + 2 x ) ^ { 8 }\) 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 \(\left( 1 + \frac { 1 } { 2 } x \right) ( 1 + 2 x ) ^ { 8 }\).
AQA C2 2007 January Q8
12 marks Moderate -0.8
8
  1. Solve the equation \(\cos x = 0.3\) 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 \(y = \cos x\) for \(0 \leqslant x \leqslant 2 \pi\) and the line \(y = k\). \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-5_524_805_559_648} The line \(y = k\) intersects the curve \(y = \cos x , 0 \leqslant x \leqslant 2 \pi\), at the points \(P\) and \(Q\). 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 \(Q\) in terms of \(\pi\) and \(\alpha\).
  3. Describe the geometrical transformation that maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
  4. Solve the equation \(\cos 2 x = \cos \frac { 4 \pi } { 5 }\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving the values of \(x\) in terms of \(\pi\).
    (4 marks)
AQA C2 2007 January Q9
11 marks Moderate -0.8
9
  1. Solve the equation \(3 \log _ { a } x = \log _ { a } 8\).
  2. Show that $$3 \log _ { a } 6 - \log _ { a } 8 = \log _ { a } 27$$
    1. The point \(P ( 3 , p )\) lies on the curve \(y = 3 \log _ { 10 } x - \log _ { 10 } 8\). Show that \(p = \log _ { 10 } \left( \frac { 27 } { 8 } \right)\).
    2. The point \(Q ( 6 , q )\) also lies on the curve \(y = 3 \log _ { 10 } x - \log _ { 10 } 8\). Show that the gradient of the line \(P Q\) is \(\log _ { 10 } 2\).
AQA C2 2007 June Q1
8 marks Easy -1.3
1
  1. Simplify:
    1. \(x ^ { \frac { 3 } { 2 } } \times x ^ { \frac { 1 } { 2 } }\);
    2. \(x ^ { \frac { 3 } { 2 } } \div x\);
    3. \(\left( x ^ { \frac { 3 } { 2 } } \right) ^ { 2 }\).
    1. Find \(\int 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
    2. Hence find the value of \(\int _ { 1 } ^ { 9 } 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
AQA C2 2007 June Q2
7 marks Moderate -0.8
2 The \(n\)th term of a geometric sequence is \(u _ { n }\), where $$u _ { n } = 3 \times 4 ^ { n }$$
  1. Find the value of \(u _ { 1 }\) and show that \(u _ { 2 } = 48\).
  2. Write down the common ratio of the geometric sequence.
    1. Show that the sum of the first 12 terms of the geometric sequence is \(4 ^ { k } - 4\), where \(k\) is an integer.
    2. Hence find the value of \(\sum _ { n = 2 } ^ { 12 } u _ { n }\).
AQA C2 2007 June Q3
10 marks Moderate -0.3
3 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 20 cm . The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians. \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_453_499_429_804} The length of the \(\operatorname { arc } A B\) is 28 cm .
  1. Show that \(\theta = 1.4\).
  2. Find the area of the sector \(O A B\).
  3. The point \(D\) lies on \(O A\). The region bounded by the line \(B D\), the line \(D A\) and the arc \(A B\) is shaded. \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_440_380_1372_806} The length of \(O D\) is 15 cm .
    1. Find the area of the shaded region, giving your answer to three significant figures.
      (3 marks)
    2. Use the cosine rule to calculate the length of \(B D\), giving your answer to three significant figures.
      (3 marks)
AQA C2 2007 June Q4
7 marks Moderate -0.8
4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 29 terms is 1102.
  1. Show that \(a + 14 d = 38\).
  2. The sum of the second term and the seventh term is 13 . Find the value of \(a\) and the value of \(d\).
AQA C2 2007 June Q5
12 marks Moderate -0.8
5 A curve is defined for \(x > 0\) by the equation $$y = \left( 1 + \frac { 2 } { x } \right) ^ { 2 }$$ The point \(P\) lies on the curve where \(x = 2\).
  1. Find the \(y\)-coordinate of \(P\).
  2. Expand \(\left( 1 + \frac { 2 } { x } \right) ^ { 2 }\).
  3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  4. Hence show that the gradient of the curve at \(P\) is - 2 .
  5. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(x + b y + c = 0\), where \(b\) and \(c\) are integers.
AQA C2 2007 June Q6
10 marks Moderate -0.8
6 The diagram shows a sketch of the curve with equation \(y = 3 \left( 2 ^ { x } + 1 \right)\). \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-5_465_851_390_607} The curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) intersects the \(y\)-axis at the point \(A\).
  1. Find the \(y\)-coordinate of the point \(A\).
  2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 6 } 3 \left( 2 ^ { x } + 1 \right) d x\).
  3. The line \(y = 21\) intersects the curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) at the point \(P\).
    1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$2 ^ { x } = 6$$
    2. Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.
AQA C2 2007 June Q7
13 marks Moderate -0.8
7
  1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Write down the two solutions of the equation \(\tan x = \tan 61 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    1. Given that \(\sin \theta + \cos \theta = 0\), show that \(\tan \theta = - 1\).
    2. Hence solve the equation \(\sin \left( x - 20 ^ { \circ } \right) + \cos \left( x - 20 ^ { \circ } \right) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  4. Describe the single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x - 20 ^ { \circ } \right)\).
  5. The curve \(y = \tan x\) is stretched in the \(x\)-direction with scale factor \(\frac { 1 } { 4 }\) to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
AQA C2 2007 June Q8
8 marks Moderate -0.8
8
  1. It is given that \(n\) satisfies the equation $$\log _ { a } n = \log _ { a } 3 + \log _ { a } ( 2 n - 1 )$$ Find the value of \(n\).
  2. Given that \(\log _ { a } x = 3\) and \(\log _ { a } y - 3 \log _ { a } 2 = 4\) :
    1. express \(x\) in terms of \(a\);
    2. express \(x y\) in terms of \(a\).
OCR C2 Q1
5 marks Moderate -0.8
  1. Giving your answers in terms of \(\pi\), solve the equation
$$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).
OCR C2 Q2
6 marks Moderate -0.8
2. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for
  1. \(\quad \log _ { 2 } 45\),
  2. \(\log _ { 2 } 0.3\)
OCR C2 Q3
6 marks Moderate -0.3
3. For the binomial expansion in ascending powers of \(x\) of \(\left( 1 + \frac { 1 } { 4 } x \right) ^ { n }\), where \(n\) is an integer and \(n \geq 2\),
  1. find and simplify the first three terms,
  2. find the value of \(n\) for which the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\).
OCR C2 Q4
8 marks Standard +0.3
4. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-1_572_803_1336_461} The diagram shows the curves with equations \(y = 7 - 2 x - 3 x ^ { 2 }\) and \(y = \frac { 2 } { x }\).
The two curves intersect at the points \(P , Q\) and \(R\).
  1. Show that the \(x\)-coordinates of \(P , Q\) and \(R\) satisfy the equation $$3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 = 0$$ Given that \(P\) has coordinates \(( - 2 , - 1 )\),
  2. find the coordinates of \(Q\) and \(R\).
OCR C2 Q5
8 marks Moderate -0.8
5. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { x ^ { 3 } } , \quad x \neq 0$$
  1. Find \(\mathrm { f } ( x )\).
  2. Show that the area of the finite region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is \(4 \frac { 1 } { 2 }\).
OCR C2 Q6
8 marks Standard +0.3
6. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-2_577_970_799_360} The diagram shows triangle \(A B C\) in which \(A C = 14 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(\angle A B C = 1.7\) radians.
  1. Find the size of \(\angle A C B\) in radians. The point \(D\) lies on \(A C\) such that \(B D\) is an arc of a circle, centre \(C\).
  2. Find the perimeter of the shaded region bounded by the arc \(B D\) and the straight lines \(A B\) and \(A D\).