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Edexcel C34 2016 January Q10
12 marks Standard +0.3
10. (a) Express \(3 \sin 2 x + 5 \cos 2 x\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) to 3 significant figures.
(b) Solve, for \(0 < x < \pi\), $$3 \sin 2 x + 5 \cos 2 x = 4$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) $$g ( x ) = 4 ( 3 \sin 2 x + 5 \cos 2 x ) ^ { 2 } + 3$$ (c) Using your answer to part (a) and showing your working,
  1. find the greatest value of \(\mathrm { g } ( x )\),
  2. find the least value of \(\mathrm { g } ( x )\).
Edexcel C34 2016 January Q11
14 marks Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{101ec3c2-699e-4c74-bfdc-d8c610646571-16_572_1338_278_239} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }\)
The curve meets the coordinate axes at the points \(A ( 0 , - 3 )\) and \(B \left( - \frac { 1 } { 3 } \ln 4,0 \right)\) and the curve has an asymptote with equation \(y = - 4\) In separate diagrams, sketch the graph with equation
  1. \(y = | f ( x ) |\)
  2. \(y = 2 \mathrm { f } ( x ) + 6\) On each sketch, give the exact coordinates of the points where the curve crosses or meets the coordinate axes and the equation of any asymptote. Given that $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { - 3 x } - 4 , & x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \ln \left( \frac { 1 } { x + 2 } \right) , & x > - 2 \end{array}$$
  3. state the range of f,
  4. find \(\mathrm { f } ^ { - 1 } ( x )\),
  5. express \(f g ( x )\) as a polynomial in \(x\).
Edexcel C34 2016 January Q12
14 marks Standard +0.3
  1. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 12 \\ - 4 \\ 5 \end{array} \right) + \lambda \left( \begin{array} { r } 5 \\ - 4 \\ 2 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 2 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 6 \\ 3 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet, and find the position vector of their point of intersection \(A\).
  2. Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) The point \(B\) has position vector \(\left( \begin{array} { l } 7 \\ 0 \\ 3 \end{array} \right)\).
  3. Show that \(B\) lies on \(l _ { 1 }\)
  4. Find the shortest distance from \(B\) to the line \(l _ { 2 }\), giving your answer to 3 significant figures.
Edexcel C34 2016 January Q13
14 marks Standard +0.3
13. A curve \(C\) has parametric equations $$x = 6 \cos 2 t , \quad y = 2 \sin t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 }$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t\), giving the exact value of the constant \(\lambda\).
  2. Find an equation of the normal to \(C\) at the point where \(t = \frac { \pi } { 3 }\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are simplified surds. The cartesian equation for the curve \(C\) can be written in the form $$x = f ( y ) , \quad - k < y < k$$ where \(\mathrm { f } ( y )\) is a polynomial in \(y\) and \(k\) is a constant.
  3. Find \(\mathrm { f } ( y )\).
  4. State the value of \(k\).
Edexcel C34 2017 January Q1
6 marks Standard +0.3
  1. Find an equation of the tangent to the curve
$$x ^ { 3 } + 3 x ^ { 2 } y + y ^ { 3 } = 37$$ at the point \(( 1,3 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
(6)
Edexcel C34 2017 January Q2
7 marks Moderate -0.3
2. $$f ( x ) = x ^ { 3 } - 5 x + 16$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as $$x = ( a x + b ) ^ { \frac { 1 } { 3 } }$$ giving the values of the constants \(a\) and \(b\). The equation \(\mathrm { f } ( x ) = 0\) has exactly one real root \(\alpha\), where \(\alpha = - 3\) to one significant figure.
  2. Starting with \(x _ { 1 } = - 3\), use the iteration $$x _ { n + 1 } = \left( a x _ { n } + b \right) ^ { \frac { 1 } { 3 } }$$ with the values of \(a\) and \(b\) found in part (a), to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving all your answers to 3 decimal places.
  3. Using a suitable interval, show that \(\alpha = - 3.17\) correct to 2 decimal places.
Edexcel C34 2017 January Q3
9 marks Standard +0.3
3. (a) Express \(\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) }\) in partial fractions.
(b) Hence, or otherwise, find the series expansion of $$\frac { 9 + 11 x } { ( 1 - x ) ( 3 + 2 x ) } , \quad | x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a simplified fraction.
Edexcel C34 2017 January Q4
10 marks Standard +0.3
  1. Given that
$$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 4 } { 3 x + 5 } , & x > 0 \\ \mathrm {~g} ( x ) = \frac { 1 } { x } , & x > 0 \end{array}$$
  1. state the range of f,
  2. find \(\mathrm { f } ^ { - 1 } ( x )\),
  3. find \(\mathrm { fg } ( x )\).
  4. Show that the equation \(\mathrm { fg } ( x ) = \mathrm { gf } ( x )\) has no real solutions.
Edexcel C34 2017 January Q5
9 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-08_579_1038_258_452} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x \cos x , \quad x \in \mathbb { R }$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the \(x\)-axis for \(\frac { 3 \pi } { 2 } \leqslant x \leqslant \frac { 5 \pi } { 2 }\)
  1. Complete the table below with the exact value of \(y\) corresponding to \(x = \frac { 7 \pi } { 4 }\) and with the exact value of \(y\) corresponding to \(x = \frac { 9 \pi } { 4 }\)
    \(x\)\(\frac { 3 \pi } { 2 }\)\(\frac { 7 \pi } { 4 }\)\(2 \pi\)\(\frac { 9 \pi } { 4 }\)\(\frac { 5 \pi } { 2 }\)
    \(y\)0\(2 \pi\)0
  2. Use the trapezium rule, with all five \(y\) values in the completed table, to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.
  3. Find $$\int x \cos x d x$$
  4. Using your answer from part (c), find the exact area of the region \(R\).
Edexcel C34 2017 January Q6
6 marks Standard +0.3
  1. (i) Differentiate \(y = 5 x ^ { 2 } \ln 3 x , \quad x > 0\)
    (ii) Given that
$$y = \frac { x } { \sin x + \cos x } , \quad - \frac { \pi } { 4 } < x < \frac { 3 \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 1 + x ) \sin x + ( 1 - x ) \cos x } { 1 + \sin 2 x } , \quad - \frac { \pi } { 4 } < x < \frac { 3 \pi } { 4 }$$ \includegraphics[max width=\textwidth, alt={}, center]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-11_99_104_2631_1781}
Edexcel C34 2017 January Q7
7 marks Moderate -0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-12_458_433_264_781} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The point \(P \left( \frac { 1 } { 3 } , 0 \right)\) is the vertex of the graph.
The point \(Q ( 0,5 )\) is the intercept with the \(y\)-axis. Given that \(\mathrm { f } ( x ) = | a x + b |\), where \(a\) and \(b\) are constants,
    1. find all possible values for \(a\) and \(b\),
    2. hence find an equation for the graph.
  2. Sketch the graph with equation $$y = \mathrm { f } \left( \frac { 1 } { 2 } x \right) + 3$$ showing the coordinates of its vertex and its intercept with the \(y\)-axis.
Edexcel C34 2017 January Q8
9 marks Standard +0.3
8. (a) Using the trigonometric identity for \(\tan ( A + B )\), prove that $$\tan 3 x = \frac { 3 \tan x - \tan ^ { 3 } x } { 1 - 3 \tan ^ { 2 } x } , \quad x \neq ( 2 n + 1 ) 30 ^ { \circ } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(- 30 ^ { \circ } < x < 30 ^ { \circ }\), $$\tan 3 x = 11 \tan x$$ (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2017 January Q9
9 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-16_727_1491_258_239} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
  1. By using the substitution \(u = 2 x + 3\), show that $$\int _ { 0 } ^ { 12 } \frac { x } { ( 2 x + 3 ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3 - \frac { 2 } { 9 }$$ The curve \(C\) has equation $$y = \frac { 9 \sqrt { x } } { ( 2 x + 3 ) } , \quad x > 0$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 12\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Use the result of part (a) to find the exact value of the volume of the solid generated.
Edexcel C34 2017 January Q10
10 marks Standard +0.3
10. A population of insects is being studied. The number of insects, \(N\), in the population, is modelled by the equation $$N = \frac { 300 } { 3 + 17 \mathrm { e } ^ { - 0.2 t } } \quad t \in \mathbb { R } , t \geqslant 0$$ where \(t\) is the time, in weeks, from the start of the study.
Using the model,
  1. find the number of insects at the start of the study,
  2. find the number of insects when \(t = 10\),
  3. find the time from the start of the study when there are 82 insects. (Solutions based entirely on graphical or numerical methods are not acceptable.)
  4. Find, by differentiating, the rate, measured in insects per week, at which the number of insects is increasing when \(t = 5\). Give your answer to the nearest whole number.
Edexcel C34 2017 January Q11
11 marks Standard +0.3
  1. (a) Express \(35 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures.
(b) Hence solve, for \(0 \leqslant x < 2 \pi\), $$70 \sin x - 24 \cos x = 37$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) $$y = \frac { 7000 } { 31 + ( 35 \sin x - 12 \cos x ) ^ { 2 } } , \quad x > 0$$ (c) Use your answer to part (a) to calculate
  1. the minimum value of \(y\),
  2. the smallest value of \(x , x > 0\), at which this minimum value occurs.
Edexcel C34 2017 January Q12
9 marks Standard +0.3
  1. In freezing temperatures, ice forms on the surface of the water in a barrel. At time \(t\) hours after the start of freezing, the thickness of the ice formed is \(x \mathrm {~mm}\). You may assume that the thickness of the ice is uniform across the surface of the water.
At 4 pm there is no ice on the surface, and freezing begins.
At 6pm, after two hours of freezing, the ice is 1.5 mm thick.
In a simple model, the rate of increase of \(x\), in mm per hour, is assumed to be constant for a period of 20 hours. Using this simple model,
  1. express \(t\) in terms of \(x\),
  2. find the value of \(t\) when \(x = 3\) In a second model, the rate of increase of \(x\), in mm per hour, is given by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \lambda } { ( 2 x + 1 ) } \text { where } \lambda \text { is a constant and } 0 \leqslant t \leqslant 20$$ Using this second model,
  3. solve the differential equation and express \(t\) in terms of \(x\) and \(\lambda\),
  4. find the exact value for \(\lambda\),
  5. find at what time the ice is predicted to be 3 mm thick.
Edexcel C34 2017 January Q13
12 marks Standard +0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-24_515_750_264_598} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The curve \(C\) shown in Figure 4 has parametric equations $$x = 1 + \sqrt { 3 } \tan \theta , \quad y = 5 \sec \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ The curve \(C\) crosses the \(y\)-axis at \(A\) and has a minimum turning point at \(B\), as shown in Figure 4.
  1. Find the exact coordinates of \(A\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \sin \theta\), giving the exact value of the constant \(\lambda\).
  3. Find the coordinates of \(B\).
  4. Show that the cartesian equation for the curve \(C\) can be written in the form $$y = k \sqrt { \left( x ^ { 2 } - 2 x + 4 \right) }$$ where \(k\) is a simplified surd to be found.
Edexcel C34 2017 January Q14
11 marks Standard +0.3
  1. \(A B C D\) is a parallelogram with \(A B\) parallel to \(D C\) and \(A D\) parallel to \(B C\). The position vectors of \(A , B , C\), and \(D\) relative to a fixed origin \(O\) are \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) respectively.
Given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + 6 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }$$
  1. find the position vector \(\mathbf { d }\),
  2. find the angle between the sides \(A B\) and \(B C\) of the parallelogram,
  3. find the area of the parallelogram \(A B C D\). The point \(E\) lies on the line through the points \(C\) and \(D\), so that \(D\) is the midpoint of \(C E\).
  4. Use your answer to part (c) to find the area of the trapezium \(A B C E\).
Edexcel C34 2018 January Q1
6 marks Standard +0.3
  1. A curve \(C\) has equation
$$3 ^ { x } + x y = x + y ^ { 2 } , \quad y > 1$$ The point \(P\) with coordinates \(( 4,11 )\) lies on \(C\).
Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). Give your answer in the form \(a + b \ln 3\), where \(a\) and \(b\) are rational numbers.
Edexcel C34 2018 January Q2
7 marks Standard +0.3
2. $$f ( x ) = ( 125 - 5 x ) ^ { \frac { 2 } { 3 } } \quad | x | < 25$$
  1. Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving the coefficient of \(x\) and the coefficient of \(x ^ { 2 }\) as simplified fractions.
  2. Use your expansion to find an approximate value for \(120 ^ { \frac { 2 } { 3 } }\), stating the value of \(x\) which you have used and showing your working. Give your answer to 5 decimal places.
Edexcel C34 2018 January Q3
7 marks Moderate -0.3
3. $$\mathrm { f } ( x ) = \frac { x ^ { 2 } } { 4 } + \ln ( 2 x ) , \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as $$x = \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 4 } x ^ { 2 } }$$ The equation \(\mathrm { f } ( x ) = 0\) has a root near 0.5
  2. Starting with \(x _ { 1 } = 0.5\) use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 4 } x _ { n } ^ { 2 } }$$ to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 decimal places.
  3. Using a suitable interval, show that 0.473 is a root of \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.
Edexcel C34 2018 January Q4
6 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-08_771_1189_212_379} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }\)
The graph consists of two half lines that meet at the point \(P ( 2 , - 3 )\), the vertex of the graph.
The graph cuts the \(y\)-axis at the point \(( 0 , - 1 )\) and the \(x\)-axis at the points \(( - 1,0 )\) and \(( 5,0 )\).
Sketch, on separate diagrams, the graph of
  1. \(y = \mathrm { f } ( | x | )\),
  2. \(y = 2 \mathrm { f } ( x + 5 )\). In each case, give the coordinates of the points where the graph crosses or meets the coordinate axes. Also give the coordinates of any vertices corresponding to the point \(P\).
Edexcel C34 2018 January Q5
7 marks Moderate -0.3
  1. (a) Express \(\frac { 9 ( 4 + x ) } { 16 - 9 x ^ { 2 } }\) in partial fractions.
Given that $$\mathrm { f } ( x ) = \frac { 9 ( 4 + x ) } { 16 - 9 x ^ { 2 } } , \quad x \in \mathbb { R } , \quad - \frac { 4 } { 3 } < x < \frac { 4 } { 3 }$$ (b) express \(\int \mathrm { f } ( x ) \mathrm { d } x\) in the form \(\ln ( \mathrm { g } ( x ) )\), where \(\mathrm { g } ( x )\) is a rational function.
Edexcel C34 2018 January Q6
5 marks
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-14_768_712_212_616} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The curve shown in Figure 2 has equation $$y ^ { 2 } = 3 \tan \left( \frac { x } { 2 } \right) , \quad 0 < x < \pi , \quad y > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = \frac { \pi } { 3 }\) the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\)
The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to generate a solid of revolution.
Show that the exact value of the volume of the solid generated may be written as \(A \ln \left( \frac { 3 } { 2 } \right)\), where \(A\) is a constant to be found.
Edexcel C34 2018 January Q7
13 marks Standard +0.3
7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 13 \mathbf { i } + 15 \mathbf { j } - 8 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 7 \mathbf { i } - 6 \mathbf { j } + 14 \mathbf { k } ) + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection, \(B\).
  2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The point \(A\) has position vector \(- 5 \mathbf { i } - 3 \mathbf { j } + 16 \mathbf { k }\)
  3. Show that \(A\) lies on \(l _ { 1 }\) The point \(C\) lies on the line \(l _ { 1 }\) where \(\overrightarrow { A B } = \overrightarrow { B C }\)
  4. Find the position vector of \(C\).
    \section*{"}