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Edexcel C2 Q9
13 marks Moderate -0.3
9. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\).
  1. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Fully factorise \(\mathrm { f } ( x )\).
  3. Using your answer to part (b), write down the coordinates of one of the turning points of the curve \(y = \mathrm { f } ( x )\) and give a reason for your answer.
  4. Using differentiation, find the \(x\)-coordinate of the other turning point of the curve \(y = \mathrm { f } ( x )\).
Edexcel C2 Q1
4 marks Moderate -0.8
  1. Evaluate
$$\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) d x .$$
Edexcel C2 Q2
4 marks Easy -1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05006f1f-ebf0-4d70-9dbb-68221c09043e-2_510_842_534_513} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \sqrt { 4 x - 1 }\). Use the trapezium rule with five equally-spaced ordinates to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Edexcel C2 Q3
7 marks Moderate -0.3
3. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for
  1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
  2. \(\log _ { 2 } ( \sqrt { x } )\).
    (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$
Edexcel C2 Q4
9 marks Moderate -0.8
4. $$f ( x ) = 2 - x - x ^ { 3 }$$
  1. Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
  2. Verify that the point \(( 1,0 )\) lies on the curve \(y = \mathrm { f } ( x )\).
  3. Find the area of the region bounded by the curve \(y = \mathrm { f } ( x )\) and the coordinate axes.
Edexcel C2 Q5
9 marks Standard +0.3
5. Figure 2 Figure 2 shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,
  1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
  2. show that \(Q R = 2 \sqrt { 6 }\),
  3. find \(\angle P Q R\) in degrees to 1 decimal place.
Edexcel C2 Q6
10 marks Moderate -0.3
6. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,
  1. find an expression for \(b\) in terms of \(a\). Given also that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\),
  2. find the values of \(a\) and \(b\),
  3. fully factorise \(\mathrm { p } ( x )\).
Edexcel C2 Q7
10 marks Standard +0.3
7.
  1. Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < 2 \pi\) for which $$\tan \left( x + \frac { \pi } { 4 } \right) = 3 .$$
  2. Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y .$$
Edexcel C2 Q8
11 marks Moderate -0.3
  1. The point \(A\) has coordinates ( 4,6 ).
Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
Edexcel C2 Q9
11 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05006f1f-ebf0-4d70-9dbb-68221c09043e-4_325_662_1345_520} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of a design being produced by a computer program.
The program draws a series of circles with each one touching the previous one and such that their centres lie on a horizontal straight line. The radii of the circles form a geometric sequence with first term 1 mm and second term 1.5 mm . The width of the design is \(w\) as shown.
  1. Find the radius of the fourth circle to be drawn.
  2. Show that when eight circles have been drawn, \(w = 98.5 \mathrm {~mm}\) to 3 significant figures.
  3. Find the total area of the design in square centimetres when ten circles have been drawn.
AQA C3 2007 January Q1
4 marks Moderate -0.8
1 Use the mid-ordinate rule with four strips of equal width to find an estimate for \(\int _ { 1 } ^ { 5 } \frac { 1 } { 1 + \ln x } \mathrm {~d} x\), giving your answer to three significant figures.
(4 marks)
AQA C3 2007 January Q2
4 marks Standard +0.3
2 Describe a sequence of two geometrical transformations that maps the graph of \(y = \sec x\) onto the graph of \(y = 1 + \sec 3 x\).
AQA C3 2007 January Q3
9 marks Moderate -0.3
3 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 3 - x ^ { 2 } , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 2 } { x + 1 } , & \text { for real values of } x , x \neq - 1 \end{array}$$
  1. Find the range of f.
  2. The inverse of g is \(\mathrm { g } ^ { - 1 }\).
    1. Find \(\mathrm { g } ^ { - 1 } ( x )\).
    2. State the range of \(\mathrm { g } ^ { - 1 }\).
  3. The composite function gf is denoted by h .
    1. Find \(\mathrm { h } ( x )\), simplifying your answer.
    2. State the greatest possible domain of h .
AQA C3 2007 January Q4
12 marks Moderate -0.3
4
  1. Use integration by parts to find \(\int x \sin x \mathrm {~d} x\).
  2. Using the substitution \(u = x ^ { 2 } + 5\), or otherwise, find \(\int x \sqrt { x ^ { 2 } + 5 } \mathrm {~d} x\).
  3. The diagram shows the curve \(y = x ^ { 2 } - 9\) for \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-3_844_663_685_694} The shaded region \(R\) is bounded by the curve, the lines \(y = 1\) and \(y = 2\), and the \(y\)-axis. Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
AQA C3 2007 January Q5
8 marks Moderate -0.3
5
    1. Show that the equation $$2 \cot ^ { 2 } x + 5 \operatorname { cosec } x = 10$$ can be written in the form \(2 \operatorname { cosec } ^ { 2 } x + 5 \operatorname { cosec } x - 12 = 0\).
    2. Hence show that \(\sin x = - \frac { 1 } { 4 }\) or \(\sin x = \frac { 2 } { 3 }\).
  2. Hence, or otherwise, solve the equation $$2 \cot ^ { 2 } ( \theta - 0.1 ) + 5 \operatorname { cosec } ( \theta - 0.1 ) = 10$$ giving all values of \(\theta\) in radians to two decimal places in the interval \(- \pi < \theta < \pi\).
    (3 marks)
AQA C3 2007 January Q6
8 marks Moderate -0.3
6
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
    1. \(y = \left( 4 x ^ { 2 } + 3 x + 2 \right) ^ { 10 }\);
    2. \(y = x ^ { 2 } \tan x\).
    1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(x = 2 y ^ { 3 } + \ln y\).
    2. Hence find an equation of the tangent to the curve \(x = 2 y ^ { 3 } + \ln y\) at the point \(( 2,1 )\).
AQA C3 2007 January Q7
9 marks Moderate -0.3
7
  1. Sketch the graph of \(y = | 2 x |\).
  2. On a separate diagram, sketch the graph of \(y = 4 - | 2 x |\), indicating the coordinates of the points where the graph crosses the coordinate axes.
  3. Solve \(4 - | 2 x | = x\).
  4. Hence, or otherwise, solve the inequality \(4 - | 2 x | > x\).
AQA C3 2007 January Q8
7 marks Standard +0.3
8 The diagram shows the curve \(y = \cos ^ { - 1 } x\) for \(- 1 \leqslant x \leqslant 1\). \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-4_492_698_1640_671}
  1. Write down the exact coordinates of the points \(A\) and \(B\).
  2. The equation \(\cos ^ { - 1 } x = 3 x + 1\) has only one root. Given that the root of this equation is \(\alpha\), show that \(0.1 \leqslant \alpha \leqslant 0.2\).
  3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 3 } \left( \cos ^ { - 1 } x _ { n } - 1 \right)\) with \(x _ { 1 } = 0.1\) to find the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to three decimal places.
AQA C3 2007 January Q9
14 marks Standard +0.3
9 The sketch shows the graph of \(y = 4 - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\). \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-5_711_921_466_557}
    1. Find \(\int \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
      (2 marks)
    2. Hence show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x = 4 \ln 2 - \frac { 3 } { 2 }\).
    1. Write down the \(y\)-coordinate of \(A\).
    2. Show that \(x = \ln 2\) at \(B\).
  3. Find the equation of the normal to the curve \(y = 4 - \mathrm { e } ^ { 2 x }\) at the point \(B\).
  4. Find the area of the region enclosed by the curve \(y = 4 - \mathrm { e } ^ { 2 x }\), the normal to the curve at \(B\) and the \(y\)-axis.
AQA C3 2008 January Q1
7 marks Moderate -0.8
1
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
    1. \(y = \left( 2 x ^ { 2 } - 5 x + 1 \right) ^ { 20 }\);
    2. \(y = x \cos x\).
  2. Given that $$y = \frac { x ^ { 3 } } { x - 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k x ^ { 2 } ( x - 3 ) } { ( x - 2 ) ^ { 2 } }$$ where \(k\) is a positive integer.
AQA C3 2008 January Q2
8 marks Moderate -0.3
2
  1. Solve the equation \(\cot x = 2\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
  2. Show that the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\) can be written as $$2 \cot ^ { 2 } x - 3 \cot x - 2 = 0$$
  3. Solve the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
AQA C3 2008 January Q3
7 marks Standard +0.3
3 The equation $$x + ( 1 + 3 x ) ^ { \frac { 1 } { 4 } } = 0$$ has a single root, \(\alpha\).
  1. Show that \(\alpha\) lies between - 0.33 and - 0.32 .
  2. Show that the equation \(x + ( 1 + 3 x ) ^ { \frac { 1 } { 4 } } = 0\) can be rearranged into the form $$x = \frac { 1 } { 3 } \left( x ^ { 4 } - 1 \right)$$
  3. Use the iteration \(x _ { n + 1 } = \frac { \left( x _ { n } ^ { 4 } - 1 \right) } { 3 }\) with \(x _ { 1 } = - 0.3\) to find \(x _ { 4 }\), giving your answer to three significant figures.
AQA C3 2008 January Q4
9 marks Moderate -0.8
4 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 3 } , & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { x - 3 } , & \text { for real values of } x , x \neq 3 \end{array}$$
  1. State the range of f.
    1. Find fg(x).
    2. Solve the equation \(\operatorname { fg } ( x ) = 64\).
    1. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).
    2. State the range of \(\mathrm { g } ^ { - 1 }\).
AQA C3 2008 January Q5
9 marks Standard +0.3
5
    1. Given that \(y = 2 x ^ { 2 } - 8 x + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence, or otherwise, find $$\int _ { 4 } ^ { 6 } \frac { x - 2 } { 2 x ^ { 2 } - 8 x + 3 } d x$$ giving your answer in the form \(k \ln 3\), where \(k\) is a rational number.
  2. Use the substitution \(u = 3 x - 1\) to find \(\int x \sqrt { 3 x - 1 } \mathrm {~d} x\), giving your answer in terms of \(x\).
AQA C3 2008 January Q6
6 marks Moderate -0.8
6
  1. Sketch the curve with equation \(y = \operatorname { cosec } x\) for \(0 < x < \pi\).
  2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0.1 } ^ { 0.5 } \operatorname { cosec } x \mathrm {~d} x\), giving your answer to three significant figures.