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AQA C1 2009 June Q1
8 marks Moderate -0.8
1 The line \(A B\) has equation \(3 x + 5 y = 11\).
    1. Find the gradient of \(A B\).
    2. The point \(A\) has coordinates (2,1). Find an equation of the line which passes through the point \(A\) and which is perpendicular to \(A B\).
  2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 8\) at the point \(C\). Find the coordinates of \(C\).
AQA C1 2009 June Q2
7 marks Easy -1.2
2
  1. Express \(\frac { 5 + \sqrt { 7 } } { 3 - \sqrt { 7 } }\) in the form \(m + n \sqrt { 7 }\), where \(m\) and \(n\) are integers.
  2. The diagram shows a right-angled triangle. The hypotenuse has length \(2 \sqrt { 5 } \mathrm {~cm}\). The other two sides have lengths \(3 \sqrt { 2 } \mathrm {~cm}\) and \(x \mathrm {~cm}\). Find the value of \(x\).
AQA C1 2009 June Q3
13 marks Moderate -0.8
3 The curve with equation \(y = x ^ { 5 } + 20 x ^ { 2 } - 8\) passes through the point \(P\), where \(x = - 2\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Verify that the point \(P\) is a stationary point of the curve.
    1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
    2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
  4. Find an equation of the tangent to the curve at the point where \(x = 1\).
AQA C1 2009 June Q4
17 marks Moderate -0.8
4
  1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\).
    1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
    2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
    3. Express \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
    4. The equation \(\mathrm { p } ( x ) = 0\) has one root equal to - 2 . Show that the equation has no other real roots.
  2. The curve with equation \(y = x ^ { 3 } - x + 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{5f1ff5fa-b6e8-4c4f-aef7-63eb947b299f-3_529_702_945_667} The curve cuts the \(x\)-axis at the point \(A ( - 2,0 )\) and the \(y\)-axis at the point \(B\).
    1. State the \(y\)-coordinate of the point \(B\).
    2. Find \(\int _ { - 2 } ^ { 0 } \left( x ^ { 3 } - x + 6 \right) \mathrm { d } x\).
    3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - x + 6\) and the line \(A B\).
AQA C1 2009 June Q5
11 marks Moderate -0.8
5 A circle with centre \(C\) has equation $$( x - 5 ) ^ { 2 } + ( y + 12 ) ^ { 2 } = 169$$
  1. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle.
    1. Verify that the circle passes through the origin \(O\).
    2. Given that the circle also passes through the points \(( 10,0 )\) and \(( 0 , p )\), sketch the circle and find the value of \(p\).
  3. The point \(A ( - 7 , - 7 )\) lies on the circle.
    1. Find the gradient of \(A C\).
    2. Hence find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
AQA C1 2009 June Q6
10 marks Moderate -0.3
6
    1. Express \(x ^ { 2 } - 8 x + 17\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 17\).
    3. State the value of \(x\) for which the minimum value of \(x ^ { 2 } - 8 x + 17\) occurs.
      (1 mark)
  2. The point \(A\) has coordinates (5,4) and the point \(B\) has coordinates ( \(x , 7 - x\) ).
    1. Expand \(( x - 5 ) ^ { 2 }\).
    2. Show that \(A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)\).
    3. Use your results from part (a) to find the minimum value of the distance \(A B\) as \(x\) varies.
AQA C1 2009 June Q7
9 marks Standard +0.3
7 The curve \(C\) has equation \(y = k \left( x ^ { 2 } + 3 \right)\), where \(k\) is a constant.
The line \(L\) has equation \(y = 2 x + 2\).
  1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$k x ^ { 2 } - 2 x + 3 k - 2 = 0$$
  2. The curve \(C\) and the line \(L\) intersect in two distinct points.
    1. Show that $$3 k ^ { 2 } - 2 k - 1 < 0$$
    2. Hence find the possible values of \(k\).
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 \(a\).
AQA C3 Q2
Moderate -0.3
2 Use Simpson's rule with 5 ordinates ( 4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.