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OCR MEI C1 2006 June Q1
3 marks Easy -1.8
1 The volume of a cone is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\). Make \(r\) the subject of this formula.
OCR MEI C1 2006 June Q2
2 marks Easy -1.2
2 One root of the equation \(x ^ { 3 } + a x ^ { 2 } + 7 = 0\) is \(x = - 2\). Find the value of \(a\).
OCR MEI C1 2006 June Q3
3 marks Easy -1.2
3 A line has equation \(3 x + 2 y = 6\). Find the equation of the line parallel to this which passes through the point \(( 2,10 )\).
OCR MEI C1 2006 June Q4
2 marks Moderate -0.8
4 In each of the following cases choose one of the statements $$\mathrm { P } \Rightarrow \mathrm { Q } \quad \mathrm { P } \Leftrightarrow \mathrm { Q } \quad \mathrm { P } \Leftarrow \mathrm { Q }$$ to describe the complete relationship between P and Q .
  1. P: \(x ^ { 2 } + x - 2 = 0\) Q: \(x = 1\)
  2. \(\mathrm { P } : y ^ { 3 } > 1\) Q: \(y > 1\)
OCR MEI C1 2006 June Q5
3 marks Easy -1.8
5 Find the coordinates of the point of intersection of the lines \(y = 3 x + 1\) and \(x + 3 y = 6\).
OCR MEI C1 2006 June Q6
4 marks Easy -1.2
6 Solve the inequality \(x ^ { 2 } + 2 x < 3\).
OCR MEI C1 2006 June Q7
5 marks Easy -1.2
7
  1. Simplify \(6 \sqrt { 2 } \times 5 \sqrt { 3 } - \sqrt { 24 }\).
  2. Express \(( 2 - 3 \sqrt { 5 } ) ^ { 2 }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers.
OCR MEI C1 2006 June Q8
4 marks Easy -1.2
8 Calculate \({ } ^ { 6 } \mathrm { C } _ { 3 }\).
Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - 2 x ) ^ { 6 }\).
OCR MEI C1 2006 June Q9
5 marks Easy -1.2
9 Simplify the following.
  1. \(\frac { 16 ^ { \frac { 1 } { 2 } } } { 81 ^ { \frac { 3 } { 4 } } }\)
  2. \(\frac { 12 \left( a ^ { 3 } b ^ { 2 } c \right) ^ { 4 } } { 4 a ^ { 2 } c ^ { 6 } }\)
OCR MEI C1 2006 June Q10
5 marks Moderate -0.8
10 Find the coordinates of the points of intersection of the circle \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(y = 3 x\). Give your answers in surd form.
OCR MEI C1 2006 June Q12
12 marks Moderate -0.3
12 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 20 x + 12\).
  1. Show that \(x = - 2\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Divide \(\mathrm { f } ( x )\) by \(x + 6\).
  3. Express \(\mathrm { f } ( x )\) in fully factorised form.
  4. Sketch the graph of \(y = \mathrm { f } ( x )\).
  5. Solve the equation \(\mathrm { f } ( x ) = 12\).
OCR MEI C1 2006 June Q13
12 marks Moderate -0.8
13 Answer the whole of this question on the insert provided.
The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
  2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
  3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
  4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).
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 \(d\).