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Edexcel C34 2018 January Q8
4 marks Standard +0.3
  1. Given that
$$y = 8 \tan ( 2 x ) , \quad - \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } x } { \mathrm {~d} y } = \frac { A } { B + y ^ { 2 } }$$ where \(A\) and \(B\) are integers to be found.
Edexcel C34 2018 January Q9
9 marks Standard +0.3
  1. (a) Show that
$$\frac { \cot ^ { 2 } x } { 1 + \cot ^ { 2 } x } \equiv \cos ^ { 2 } x$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\frac { \cot ^ { 2 } x } { 1 + \cot ^ { 2 } x } = 8 \cos 2 x + 2 \cos x$$ Give each solution in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2018 January Q10
12 marks Standard +0.2
  1. It is given that
$$\begin{gathered} \mathrm { f } ( x ) = \mathrm { e } ^ { - 2 x } \quad x \in \mathbb { R } \\ \mathrm {~g} ( x ) = \frac { x } { x - 3 } \quad x > 3 \end{gathered}$$
  1. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any points where the graph crosses the axes.
  2. Find the range of g
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 }\)
  4. Using algebra, find the exact value of \(x\) for which \(\operatorname { fg } ( x ) = 3\)
Edexcel C34 2018 January Q11
12 marks Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-32_858_743_118_603} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The curve \(C\) shown in Figure 3 has parametric equations $$x = 3 \cos t , \quad y = 9 \sin 2 t , \quad 0 \leqslant t \leqslant 2 \pi$$ The curve \(C\) meets the \(x\)-axis at the origin and at the points \(A\) and \(B\), as shown in Figure 3 .
  1. Write down the coordinates of \(A\) and \(B\).
  2. Find the values of \(t\) at which the curve passes through the origin.
  3. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), and hence find the gradient of the curve when \(t = \frac { \pi } { 6 }\)
  4. Show that the cartesian equation for the curve \(C\) can be written in the form $$y ^ { 2 } = a x ^ { 2 } \left( b - x ^ { 2 } \right)$$ where \(a\) and \(b\) are integers to be determined.
Edexcel C34 2018 January Q12
12 marks Standard +0.3
  1. (a) Express \(2 \sin x - 4 \cos x\) 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 3 significant figures. In a town in Norway, a student records the number of hours of daylight every day for a year. He models the number of hours of daylight, \(H\), by the continuous function given by the formula $$H = 12 + 4 \sin \left( \frac { 2 \pi t } { 365 } \right) - 8 \cos \left( \frac { 2 \pi t } { 365 } \right) , \quad 0 \leqslant t \leqslant 365$$ where \(t\) is the number of days since he began recording.
(b) Using your answer to part (a), or otherwise, find the maximum and minimum number of hours of daylight given by this formula. Give your answers to 3 significant figures.
(c) Use the formula to find the values of \(t\) when \(H = 17\), giving your answers to the nearest integer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
VIIIV SIHI NI JIIHM 10 N OCVIIV 5141 NI 3114 M I ON OCVI4V SIHIL NI JIIYM ION OC
Edexcel C34 2018 January Q13
13 marks Moderate -0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-40_495_634_207_657} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 2 x } \ln 2 x , \quad x > \frac { 1 } { 2 }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = \mathrm { e }\) and \(x = 5 \mathrm { e }\). The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { 2 x } \ln 2 x\). The values for \(y\) are given to 4 significant figures.
\(x\)e2 e3 e4 e5 e
\(y\)0.31140.21950.17120.14160.1215
  1. Use the trapezium rule with all the \(y\) values in the table to find an approximate value for the area of \(R\), giving your answer to 3 significant figures.
  2. Using the substitution \(u = \ln 2 x\), or otherwise, find \(\int \frac { 1 } { 2 x } \ln 2 x \mathrm {~d} x\)
  3. Use your answer to part (b) to find the true area of \(R\), giving your answer to 3 significant figures.
  4. Using calculus, find an equation for the tangent to the curve at the point where \(x = \frac { \mathrm { e } ^ { 2 } } { 2 }\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are exact multiples of powers of e.
Edexcel C34 2018 January Q14
12 marks Standard +0.3
14. The volume of a spherical balloon of radius \(r \mathrm {~cm}\) is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) The volume of the balloon increases with time \(t\) seconds according to the formula $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 9000 \pi } { ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$
  2. Using the chain rule, or otherwise, show that $$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { k } { r ^ { n } ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$ where \(k\) and \(n\) are constants to be found. Initially, the radius of the balloon is 3 cm .
  3. Using the values of \(k\) and \(n\) found in part (b), solve the differential equation $$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { k } { r ^ { n } ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$ to obtain a formula for \(r\) in terms of \(t\).
  4. Hence find the radius of the balloon when \(t = 175\), giving your answer to 3 significant figures.
    (1)
  5. Find the rate of increase of the radius of the balloon when \(t = 175\). Give your answer to 3 significant figures.
    END
Edexcel C34 2019 January Q1
12 marks Standard +0.3
  1. (a) Express \(7 \sin 2 \theta - 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
    (b) Hence solve, for \(0 \leqslant \theta < 90 ^ { \circ }\), the equation
$$7 \sin 2 \theta - 2 \cos 2 \theta = 4$$ giving your answers in degrees to one decimal place.
(c) Express \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\) in the form \(a \sin 2 \theta + b \cos 2 \theta + c\), where \(a\), \(b\) and \(c\) are constants to be found.
(d) Use your answers to part (a) and part (c) to deduce the exact maximum value of \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\)
Edexcel C34 2019 January Q2
10 marks Standard +0.3
2. Given that $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \equiv A + \frac { B } { x + 1 } + \frac { C } { x - 3 }$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence, or otherwise, find the series expansion of $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \quad | x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\)
    Give each coefficient as a simplified fraction.
Edexcel C34 2019 January Q3
8 marks Standard +0.8
3. The function f is defined by $$f : x \mapsto 2 x ^ { 2 } + 3 k x + k ^ { 2 } \quad x \in \mathbb { R } , - 4 k \leqslant x \leqslant 0$$ where \(k\) is a positive constant.
  1. Find, in terms of \(k\), the range of f . The function g is defined by $$\mathrm { g } : x \mapsto 2 k - 3 x \quad x \in \mathbb { R }$$ Given that \(\operatorname { gf } ( - 2 ) = - 12\)
  2. find the possible values of \(k\).
Edexcel C34 2019 January Q4
11 marks Standard +0.3
  1. The curve \(C\) has equation
$$81 y ^ { 3 } + 64 x ^ { 2 } y + 256 x = 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Hence find the coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
Edexcel C34 2019 January Q5
8 marks Standard +0.3
5. The angle \(x\) and the angle \(y\) are such that $$\tan x = m \text { and } 4 \tan y = 8 m + 5$$ where \(m\) is a constant.
Given that \(16 \sec ^ { 2 } x + 16 \sec ^ { 2 } y = 537\)
  1. find the two possible values of \(m\). Given that the angle \(x\) and the angle \(y\) are acute, find the exact value of
  2. \(\sin x\)
  3. \(\cot y\)
Edexcel C34 2019 January Q6
11 marks Standard +0.3
6. Relative to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates ( \(2,1,9 ) , ( 5,2,7 )\) and \(( 4 , - 3,3 )\) respectively. The line \(l\) passes through the points \(A\) and \(B\).
  1. Find a vector equation for the line \(l\).
  2. Find, in degrees, the acute angle between the line \(I\) and the line \(A C\). The point \(D\) lies on the line \(l\) such that angle \(A C D\) is \(90 ^ { \circ }\)
  3. Find the coordinates of \(D\).
  4. Find the exact area of triangle \(A D C\), giving your answer as a fully simplified surd.
Edexcel C34 2019 January Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae871952-f525-44e6-8bac-09308aa1964f-26_615_867_292_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { x + 7 } { \sqrt { 2 x - 3 } } \quad x > \frac { 3 } { 2 }$$ The region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 6\)
  1. Use the trapezium rule with 4 strips of equal width to find an estimate for the area of \(R\), giving your answer to 2 decimal places.
  2. Using the substitution \(u = 2 x - 3\), or otherwise, use calculus to find the exact area of \(R\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are constants to be found.
Edexcel C34 2019 January Q8
13 marks Challenging +1.2
8. A curve has parametric equations $$x = t ^ { 2 } - t , \quad y = \frac { 4 t } { 1 - t } \quad t \neq 1$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer as a simplified fraction.
  2. Find an equation for the tangent to the curve at the point \(P\) where \(t = - 1\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. The tangent to the curve at \(P\) cuts the curve at the point \(Q\).
  3. Use algebra to find the coordinates of \(Q\).
Edexcel C34 2019 January Q9
10 marks Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae871952-f525-44e6-8bac-09308aa1964f-34_1331_1589_264_182} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} (c) Find the exact value for the volume of this solid, giving your answer as a single, simplified fraction. \section*{Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x + \sin 2 x\).
The region \(R\), shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\)
The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x + \sin 2 x\). The region \(R\), shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.} \(\_\_\_\_\) simplified fraction.
Edexcel C34 2019 January Q10
10 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae871952-f525-44e6-8bac-09308aa1964f-38_570_671_310_680} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Diagram not drawn to scale Figure 3 shows a container in the shape of an inverted right circular cone which contains some water. The cone has an internal radius of 3 m and a vertical height of 5 m as shown in Figure 3. At time \(t\) seconds,the height of the water is \(h\) metres,the volume of the water is \(V \mathrm {~m} ^ { 3 }\) and water is leaking from a hole in the bottom of the container at a constant rate of \(0.02 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
[The volume of a cone of radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) .]
  1. Show that,while the water is leaking, $$h ^ { 2 } \frac { \mathrm {~d} h } { \mathrm {~d} t } = - \frac { 1 } { \mathrm { k } \pi }$$ where \(k\) is a constant to be found. Given that the container is initially full of water,
  2. express \(h\) in terms of \(t\) .
  3. Find the time taken for the container to empty,giving your answer to the nearest minute.
Edexcel C34 2019 January Q11
6 marks Standard +0.3
11. (a) Given that \(0 \leqslant \mathrm { f } ( x ) \leqslant \pi\), sketch the graph of \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \arccos ( x - 1 ) , \quad 0 \leqslant x \leqslant 2$$ The equation \(\arccos ( x - 1 ) - \tan x = 0\) has a single root \(\alpha\).
(b) Show that \(0.9 < \alpha < 1.1\) The iteration formula $$x _ { n + 1 } = \arctan \left( \arccos \left( x _ { n } - 1 \right) \right)$$ can be used to find an approximation for \(\alpha\).
(c) Taking \(x _ { 0 } = 1.1\) find, to 3 decimal places, the values of \(x _ { 1 }\) and \(x _ { 2 }\)
Edexcel C34 2019 January Q12
5 marks Moderate -0.3
12. Given that \(k\) is a positive constant,
  1. sketch the graph with equation $$y = 2 | x | - k$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis.
  2. Find, in terms of \(k\), the values of \(x\) for which $$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$
Edexcel C34 2019 January Q13
10 marks Standard +0.3
13. A scientist is studying a population of insects. The number of insects, \(N\), in the population, \(t\) days after the start of the study is modelled by the equation $$N = \frac { 240 } { 1 + k \mathrm { e } ^ { - \frac { t } { 16 } } }$$ where \(k\) is a constant.
Given that there were 50 insects at the start of the study,
  1. find the value of \(k\)
  2. use the model to find the value of \(t\) when \(N = 100\)
  3. Show that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { 1 } { p } N - \frac { 1 } { q } N ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
    END
Edexcel C34 2014 June Q1
7 marks Moderate -0.3
1. $$f ( x ) = 2 x ^ { 3 } + x - 10$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 1.5,2 ]\) The only real root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\) The iterative formula $$x _ { n + 1 } = \left( 5 - \frac { 1 } { 2 } x _ { n } \right) ^ { \frac { 1 } { 3 } } , \quad x _ { 0 } = 1.5$$ can be used to find an approximate value for \(\alpha\)
  2. Calculate \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
  3. By choosing a suitable interval, show that \(\alpha = 1.6126\) correct to 4 decimal places.
Edexcel C34 2014 June Q2
6 marks Standard +0.3
2. A curve \(C\) has the equation $$x ^ { 3 } - 3 x y - x + y ^ { 3 } - 11 = 0$$ Find an equation of the tangent to \(C\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C34 2014 June Q3
4 marks Standard +0.3
3. Given that $$y = \frac { \cos 2 \theta } { 1 + \sin 2 \theta } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \frac { a } { 1 + \sin 2 \theta } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where \(a\) is a constant to be determined.
Edexcel C34 2014 June Q4
4 marks Moderate -0.8
4. Find
  1. \(\int ( 2 x + 3 ) ^ { 12 } \mathrm {~d} x\)
  2. \(\int \frac { 5 x } { 4 x ^ { 2 } + 1 } \mathrm {~d} x\)
Edexcel C34 2014 June Q5
5 marks Standard +0.3
5. $$f ( x ) = \left( 8 + 27 x ^ { 3 } \right) ^ { \frac { 1 } { 3 } } , \quad | x | < \frac { 2 } { 3 }$$ Find the first three non-zero terms of the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\). Give each coefficient as a simplified fraction.