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Edexcel C1 2014 June Q11
10 marks Moderate -0.5
11. Given that $$f ( x ) = 2 x ^ { 2 } + 8 x + 3$$
  1. find the value of the discriminant of \(\mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(p ( x + q ) ^ { 2 } + r\) where \(p , q\) and \(r\) are integers to be found. The line \(y = 4 x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = \mathrm { f } ( x )\).
  3. Calculate the value of \(c\).
Edexcel C34 2016 June Q1
9 marks Standard +0.3
  1. (a) Express \(3 \cos \theta + 5 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), 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 \(0 \leqslant \theta < 360 ^ { \circ }\), the equation
$$3 \cos \theta + 5 \sin \theta = 2$$ Give your answers to one decimal place.
(c) Use your solutions to parts (a) and (b) to deduce the smallest positive value of \(\theta\) for which $$3 \cos \theta - 5 \sin \theta = 2$$
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Edexcel C34 2016 June Q2
6 marks Standard +0.3
2. The point \(P\) with coordinates \(\left( \frac { \pi } { 2 } , 1 \right)\) lies on the curve with equation $$4 x \sin x = \pi y ^ { 2 } + 2 x , \quad \frac { \pi } { 6 } \leqslant x \leqslant \frac { 5 \pi } { 6 }$$ Find an equation of the normal to the curve at \(P\).
Edexcel C34 2016 June Q3
9 marks Standard +0.3
3. (a) Find the binomial expansion of $$( 1 + a x ) ^ { - 3 } , \quad | a x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient as simply as possible in terms of the constant \(a\). $$f ( x ) = \frac { 2 + 3 x } { ( 1 + a x ) ^ { 3 } } , \quad | a x | < 1$$ In the series expansion of \(\mathrm { f } ( x )\), the coefficient of \(x ^ { 2 }\) is 3
Given that \(a < 0\) (b) find the value of the constant \(a\),
(c) find the coefficient of \(x ^ { 3 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.
Edexcel C34 2016 June Q4
9 marks Standard +0.3
4. $$\mathrm { g } ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 7 x ^ { 2 } + 8 x - 48 } { x ^ { 2 } + x - 12 } , \quad x > 3 , \quad x \in \mathbb { R }$$
  1. Given that $$\frac { x ^ { 4 } + x ^ { 3 } - 7 x ^ { 2 } + 8 x - 48 } { x ^ { 2 } + x - 12 } \equiv x ^ { 2 } + A + \frac { B } { x - 3 }$$ find the values of the constants \(A\) and \(B\).
  2. Hence, or otherwise, find the equation of the tangent to the curve with equation \(y = \mathrm { g } ( x )\) at the point where \(x = 4\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined.
    (5)
Edexcel C34 2016 June Q5
6 marks Standard +0.3
5.Use integration by parts to find the exact value of $$\int _ { 0 } ^ { 2 } x 2 ^ { x } \mathrm {~d} x$$ Write your answer as a single simplified fraction.
Edexcel C34 2016 June Q6
9 marks Standard +0.3
6. Given that \(a\) and \(b\) are constants and that \(a > b > 0\)
  1. on separate diagrams, sketch the graph with equation
    1. \(y = | x - a |\)
    2. \(y = | x - a | - b\) Show on each sketch the coordinates of each point at which the graph crosses or meets the \(x\)-axis and the \(y\)-axis.
  2. Hence or otherwise find the complete set of values of \(x\) for which $$| x - a | - b < \frac { 1 } { 2 } x$$ giving your answer in terms of \(a\) and \(b\).
Edexcel C34 2016 June Q7
9 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-13_695_986_121_497} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram not drawn to scale Figure 1 shows a sketch of part of the curve with equation \(y = \frac { 1 } { \sqrt { 2 x + 5 } } , x > - 2.5\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 5\)
  1. Use the trapezium rule with three strips of equal width to find an estimate for the area of \(R\), giving your answer to 3 decimal places.
  2. Use calculus to find the exact area of \(R\).
  3. Hence calculate the magnitude of the error of the estimate found in part (a), giving your answer to one significant figure.
Edexcel C34 2016 June Q8
11 marks Standard +0.8
8. (a) Prove that $$\sin 2 x - \tan x \equiv \tan x \cos 2 x , \quad x \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta < \frac { \pi } { 2 }\)
  1. \(\sin 2 \theta - \tan \theta = \sqrt { 3 } \cos 2 \theta\)
  2. \(\tan ( \theta + 1 ) \cos ( 2 \theta + 2 ) - \sin ( 2 \theta + 2 ) = 2\) Give your answers in radians to 3 significant figures, as appropriate.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2016 June Q9
11 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-17_574_1333_260_303} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The population of a species of animal is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be $$P = \frac { 9000 \mathrm { e } ^ { k t } } { 3 \mathrm { e } ^ { k t } + 7 } , \quad t \geqslant 0$$ where \(k\) is a positive constant.
A sketch of the graph of \(P\) against \(t\) is shown in Figure 2 .
Use the given equation to
  1. find the population at the start of the study,
  2. find the value for the upper limit of the population. Given that \(P = 2500\) when \(t = 4\)
  3. calculate the value of \(k\), giving your answer to 3 decimal places. Using this value for \(k\),
  4. find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is increasing when \(t = 10\) Give your answer to the nearest integer.
Edexcel C34 2016 June Q10
9 marks Standard +0.3
10. (a) Given that \(- \frac { \pi } { 2 } < \mathrm { g } ( x ) < \frac { \pi } { 2 }\), sketch the graph of \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = \arctan x , \quad x \in \mathbb { R }$$ (b) Find the exact value of \(x\) for which $$3 g ( x + 1 ) - \pi = 0$$ The equation \(\arctan x - 4 + \frac { 1 } { 2 } x = 0\) has a positive root at \(x = \alpha\) radians.
(c) Show that \(5 < \alpha < 6\) The iteration formula $$x _ { n + 1 } = 8 - 2 \arctan x _ { n }$$ can be used to find an approximation for \(\alpha\) (d) Taking \(x _ { 0 } = 5\), use this formula to find \(x _ { 1 }\) and \(x _ { 2 }\), giving each answer to 3 decimal places.
Edexcel C34 2016 June Q11
12 marks Standard +0.3
11. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 7 \\ 4 \\ 9 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6 \\ - 7 \\ 3 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 4 \\ b \end{array} \right) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(X\),
  1. show that \(b = - 3\) and find the coordinates of \(X\). The point \(A\) lies on \(l _ { 1 }\) and has coordinates (6, 3, 5)
    The point \(B\) lies on \(l _ { 2 }\) and has coordinates \(( 14,9 , - 9 )\)
  2. Show that angle \(A X B = \arccos \left( - \frac { 1 } { 10 } \right)\)
  3. Using the result obtained in part (b), find the exact area of triangle \(A X B\). Write your answer in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers to be determined.
Edexcel C34 2016 June Q12
11 marks Standard +0.8
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-23_503_1333_267_301} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with parametric equations $$x = 3 \sin t , \quad y = 2 \sin 2 t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The finite region \(S\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { 3 } { 2 }\) The shaded region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  1. Show that the volume of the solid of revolution is given by $$k \int _ { 0 } ^ { a } \sin ^ { 2 } t \cos ^ { 3 } t \mathrm {~d} t$$ where \(k\) and \(a\) are constants to be given in terms of \(\pi\).
  2. Use the substitution \(u = \sin t\), or otherwise, to find the exact value of this volume, giving your answer in the form \(\frac { p \pi } { q }\) where \(p\) and \(q\) are integers. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2016 June Q13
14 marks Standard +0.8
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d67f716-c8af-4460-8a6b-62073ba9b825-25_362_697_246_612} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a hemispherical bowl containing some water.
At \(t\) seconds, the height of the water is \(h \mathrm {~cm}\) and the volume of the water is \(V \mathrm {~cm} ^ { 3 }\), where $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 30 - h ) , \quad 0 < h \leqslant 10$$ The water is leaking from a hole in the bottom of the bowl. Given that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = - \frac { 1 } { 10 } V\)
  1. show that \(\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { h ( 30 - h ) } { 30 ( 20 - h ) }\)
  2. Write \(\frac { 30 ( 20 - h ) } { h ( 30 - h ) }\) in partial fraction form. Given that \(h = 10\) when \(t = 0\),
  3. use your answers to parts (a) and (b) to find the time taken for the height of the water to fall to 5 cm . Give your answer in seconds to 2 decimal places.
Edexcel F1 2021 June Q1
10 marks Moderate -0.3
1.(i) $$f ( x ) = x ^ { 3 } + 4 x - 6$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval[1,1.5]
  2. Taking 1.5 as a first approximation,apply the Newton Raphson process twice to \(\mathrm { f } ( x )\) to obtain an approximate value of \(\alpha\) .Give your answer to 3 decimal places. Show your working clearly.
    (ii) $$g ( x ) = 4 x ^ { 2 } + x - \tan x$$ where \(x\) is measured in radians. The equation \(\mathrm { g } ( x ) = 0\) has a single root \(\beta\) in the interval[1.4,1.5]
    Use linear interpolation on the values at the end points of this interval to obtain an approximation to \(\beta\) .Give your answer to 3 decimal places.
Edexcel F1 2021 June Q2
7 marks Standard +0.3
2. The complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are given by $$\mathrm { z } _ { 1 } = 2 - \mathrm { i } \quad \mathrm { z } _ { 2 } = p - \mathrm { i } \quad \mathrm { z } _ { 3 } = p + \mathrm { i }$$ where \(p\) is a real number.
  1. Find \(\frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } } \right| = 2 \sqrt { 5 }\)
  2. find the possible values of \(p\).
Edexcel F1 2021 June Q3
10 marks Moderate -0.8
  1. The triangle \(T\) has vertices \(A ( 2,1 ) , B ( 2,3 )\) and \(C ( 0,1 )\).
The triangle \(T ^ { \prime }\) is the image of \(T\) under the transformation represented by the matrix $$\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$
  1. Find the coordinates of the vertices of \(T ^ { \prime }\)
  2. Describe fully the transformation represented by \(\mathbf { P }\) The \(2 \times 2\) matrix \(\mathbf { Q }\) represents a reflection in the \(x\)-axis and the \(2 \times 2\) matrix \(\mathbf { R }\) represents a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
  3. Write down the matrix \(\mathbf { Q }\) and the matrix \(\mathbf { R }\)
  4. Find the matrix \(\mathbf { R Q }\)
  5. Give a full geometrical description of the single transformation represented by the answer to part (d).
Edexcel F1 2021 June Q4
8 marks Standard +0.8
  1. A rectangular hyperbola \(H\) has equation \(x y = 25\)
The point \(P \left( 5 t , \frac { 5 } { t } \right) , t \neq 0\), is a general point on \(H\).
  1. Show that the equation of the tangent to \(H\) at \(P\) is \(t ^ { 2 } y + x = 10 t\) The distinct points \(Q\) and \(R\) lie on \(H\). The tangent to \(H\) at the point \(Q\) and the tangent to \(H\) at the point \(R\) meet at the point \(( 15 , - 5 )\).
  2. Find the coordinates of the points \(Q\) and \(R\).
Edexcel F1 2021 June Q5
7 marks Moderate -0.3
5. $$f ( x ) = \left( 9 x ^ { 2 } + d \right) \left( x ^ { 2 } - 8 x + ( 10 d + 1 ) \right)$$ where \(d\) is a positive constant.
  1. Find the four roots of \(\mathrm { f } ( x )\) giving your answers in terms of \(d\). Given \(d = 4\)
  2. Express these four roots in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
  3. Show these four roots on a single Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{d7689f4a-a41e-45be-911b-4a74e81997eb-21_2647_1840_118_111}
Edexcel F1 2021 June Q6
16 marks Standard +0.8
6. The parabola \(C\) has Cartesian equation \(y ^ { 2 } = 8 x\) The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) and the point \(Q \left( 2 q ^ { 2 } , 4 q \right)\), where \(p , q \neq 0 , p \neq q\), are points on \(C\).
  1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = 2 p ^ { 3 } + 4 p$$
  2. Write down an equation of the normal to \(C\) at \(Q\) The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(N\)
  3. Show that \(N\) has coordinates $$\left( 2 \left( p ^ { 2 } + p q + q ^ { 2 } + 2 \right) , - 2 p q ( p + q ) \right)$$ The line \(O N\), where \(O\) is the origin, is perpendicular to the line \(P Q\)
  4. Find the value of \(( p + q ) ^ { 2 } - 3 p q\)
Edexcel F1 2021 June Q7
11 marks Moderate -0.3
7. (a) Prove by induction that for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { n } { 6 } ( n + 1 ) ( 2 n + 1 )$$ (b) Hence show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 2 \right) = \frac { n } { 6 } \left( a n ^ { 2 } + b n + c \right)$$ where \(a , b\) and \(c\) are integers to be found.
(c) Using your answers to part (b), find the value of $$\sum _ { r = 10 } ^ { 25 } \left( r ^ { 2 } + 2 \right)$$
Edexcel F1 2021 June Q8
6 marks Standard +0.8
8. Prove by induction that \(4 ^ { n + 2 } + 5 ^ { 2 n + 1 }\) is divisible by 21 for all positive integers \(n\).
\includegraphics[max width=\textwidth, alt={}]{d7689f4a-a41e-45be-911b-4a74e81997eb-32_2644_1837_118_114}
Edexcel F2 2018 June Q1
5 marks Moderate -0.5
  1. Use algebra to find the set of values of \(x\) for which
$$\frac { 1 } { x - 2 } > \frac { 2 } { x }$$
Edexcel F2 2018 June Q2
8 marks Standard +0.3
  1. (a) Find the general solution of the differential equation
$$\left( x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + x y - x = 0$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
(b) Find the particular solution for which \(y = 2\) when \(x = 3\)
Edexcel F2 2018 June Q3
10 marks Challenging +1.2
3. $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - x y = 1$$
  1. Show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = \frac { 1 } { 2 } \left( a \frac { \mathrm {~d} y } { \mathrm {~d} x } + b x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + c \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } \right)$$ where \(a , b\) and \(c\) are constants to be found. Given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 2\)
  2. find a series solution for \(y\) in ascending powers of ( \(x - 2\) ), up to and including the term in \(( x - 2 ) ^ { 4 }\). Write each term in its simplest form.
  3. Use the solution to part (b) to find an approximate value for \(y\) when \(x = 2.1\), giving your answer to 3 decimal places.