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Edexcel C3 Q5
12 marks Standard +0.2
5. The function f is defined by $$\mathrm { f } : x \rightarrow 3 \mathrm { e } ^ { x - 1 } , \quad x \in \mathbb { R } .$$
  1. State the range of f.
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$g : x \rightarrow 5 x - 2 , \quad x \in \mathbb { R } .$$ Find, in terms of e,
  3. the value of \(\mathrm { gf } ( \ln 2 )\),
  4. the solution of the equation $$\mathrm { f } ^ { - 1 } \mathrm {~g} ( x ) = 4 .$$
Edexcel C3 Q6
13 marks Standard +0.3
6. $$f ( x ) = 2 x ^ { 2 } + 3 \ln ( 2 - x ) , \quad x \in \mathbb { R } , \quad x < 2 .$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = 2 - \mathrm { e } ^ { k x ^ { 2 } } ,$$ where \(k\) is a constant to be found. The root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) is 1.9 correct to 1 decimal place.
  2. Use the iteration formula $$x _ { n + 1 } = 2 - \mathrm { e } ^ { k x _ { n } ^ { 2 } } ,$$ with \(x _ { 0 } = 1.9\) and your value of \(k\), to find \(\alpha\) to 3 decimal places and justify the accuracy of your answer.
  3. Solve the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\).
Edexcel C3 Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7c3dd501-0545-4166-aaf9-5e1ac1f369c5-4_552_771_248_470} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 45,7\) ) and a minimum point at \(( 135 , - 1 )\).
  1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
    1. \(y = \mathrm { f } ( | x | )\),
    2. \(y = 1 + 2 \mathrm { f } ( x )\). Given that $$f ( x ) = A + 2 \sqrt { 2 } \cos x ^ { \circ } - 2 \sqrt { 2 } \sin x ^ { \circ } , \quad x \in \mathbb { R } , \quad - 180 \leq x \leq 180 ,$$ where \(A\) is a constant,
  2. show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = A + R \cos ( x + \alpha ) ^ { \circ } ,$$ where \(R > 0\) and \(0 < \alpha < 90\),
  3. state the value of \(A\),
  4. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.
Edexcel C3 Q1
6 marks Moderate -0.3
  1. \(f ( x ) \equiv \frac { 2 x - 3 } { x - 2 } , \quad x \in \mathbb { R } , \quad x > 2\).
    1. Find the range of f .
    2. Show that \(\operatorname { ff } ( x ) = x\) for all \(x > 2\).
    3. Hence, write down an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    4. Solve each equation, giving your answers in exact form.
    5. \(\mathrm { e } ^ { 4 x - 3 } = 2\)
    6. \(\quad \ln ( 2 y - 1 ) = 1 + \ln ( 3 - y )\)
    7. The curve \(C\) has the equation \(y = 2 \mathrm { e } ^ { x } - 6 \ln x\) and passes through the point \(P\) with \(x\)-coordinate 1.
    8. Find an equation for the tangent to \(C\) at \(P\).
    The tangent to \(C\) at \(P\) meets the coordinate axes at the points \(Q\) and \(R\).
  2. Show that the area of triangle \(O Q R\), where \(O\) is the origin, is \(\frac { 9 } { 3 - \mathrm { e } }\).
Edexcel C3 Q4
9 marks Standard +0.3
4. (a) Express $$\frac { x - 10 } { ( x - 3 ) ( x + 4 ) } - \frac { x - 8 } { ( x - 3 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form.
(b) Hence, show that the equation $$\frac { x - 10 } { ( x - 3 ) ( x + 4 ) } - \frac { x - 8 } { ( x - 3 ) ( 2 x - 1 ) } = 1$$ has no real roots.
Edexcel C3 Q5
9 marks Standard +0.8
5. Find the values of \(x\) in the interval \(- 180 < x < 180\) for which $$\tan ( x + 45 ) ^ { \circ } - \tan x ^ { \circ } = 4$$ giving your answers to 1 decimal place.
Edexcel C3 Q6
10 marks Standard +0.8
6. (a) Sketch on the same diagram the graphs of \(y = | x | - a\) and \(y = | 3 x + 5 a |\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes.
(b) Solve the equation $$| x | - a = | 3 x + 5 a | .$$
Edexcel C3 Q7
12 marks Standard +0.3
  1. (a) Use the identity
$$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to prove that $$\cos x \equiv 1 - 2 \sin ^ { 2 } \frac { x } { 2 }$$ (b) Prove that, for \(\sin x \neq 0\), $$\frac { 1 - \cos x } { \sin x } \equiv \tan \frac { x } { 2 }$$ (c) Find the values of \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) for which $$\frac { 1 - \cos x } { \sin x } = 2 \sec ^ { 2 } \frac { x } { 2 } - 5$$ giving your answers to 1 decimal place where appropriate.
Edexcel C3 Q8
14 marks Standard +0.3
8. A curve has the equation \(y = ( 2 x + 3 ) \mathrm { e } ^ { - x }\).
  1. Find the exact coordinates of the stationary point of the curve. The curve crosses the \(y\)-axis at the point \(P\).
  2. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) meets the curve again at \(Q\).
  3. Show that the \(x\)-coordinate of \(Q\) lies in the interval \([ - 2 , - 1 ]\).
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 3 - 3 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { x _ { n } } - 2 }$$ with \(x _ { 0 } = - 1\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 2 decimal places.
  5. Show that your value for \(x _ { 4 }\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. END
Edexcel C4 Q1
6 marks Standard +0.3
  1. The function \(f\) is given by
$$f ( x ) = \frac { 3 ( x + 1 ) } { ( x + 2 ) ( x - 1 ) } , x \in \mathbb { R } , x \neq - 2 , x \neq 1$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, or otherwise, prove that \(\mathrm { f } ^ { \prime } ( x ) < 0\) for all values of \(x\) in the domain.
Edexcel C4 Q2
4 marks Moderate -0.3
2. The curve \(C\) is described by the parametric equations $$x = 3 \cos t , \quad y = \cos 2 t , \quad 0 \leq t \leq \pi .$$
  1. Find a cartesian equation of the curve \(C\).
  2. Draw a sketch of the curve \(C\).
Edexcel C4 Q3
6 marks Standard +0.3
3. Use the substitution \(x = \sin \theta\) to show that, for \(| x | \leq 1\), $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c \text {, where } c \text { is an arbitrary constant. }$$
Edexcel C4 Q4
6 marks Moderate -0.3
  1. A measure of the effective voltage, \(M\) volts, in an electrical circuit is given by
$$M ^ { 2 } = \int _ { 0 } ^ { 1 } V ^ { 2 } \mathrm {~d} t$$ where \(V\) volts is the voltage at time \(t\) seconds. Pairs of values of \(V\) and \(t\) are given in the following table.
\(t\)00.250.50.751
\(V\)- 4820737- 161- 29
\(V ^ { 2 }\)
Use the trapezium rule with five values of \(V ^ { 2 }\) to estimate the value of \(M\).
(6)
Edexcel C4 Q5
8 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{964070ca-a2c0-4935-8a5b-f1f656495f2e-3_771_1049_251_477}
\end{figure} Figure 1 shows part of the curve with equation \(y = 1 + \frac { 1 } { 2 \sqrt { x } }\). The shaded region \(R\), bounded by the curve, that \(x\)-axis and the lines \(x = 1\) and \(x = 4\), is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Using integration, show that the volume of the solid generated is \(\pi \left( 5 + \frac { 1 } { 2 } \ln 2 \right)\).
(8)
Edexcel C4 Q6
11 marks Moderate -0.3
6. Liquid is poured into a container at a constant rate of \(30 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds liquid is leaking from the container at a rate of \(\frac { 2 } { 15 } V \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\), where \(V \mathrm {~cm} ^ { 3 }\) is the volume of liquid in the container at that time.
  1. Show that $$- 15 \frac { \mathrm {~d} V } { \mathrm {~d} t } = 2 V - 450$$ Given that \(V = 1000\) when \(t = 0\),
  2. find the solution of the differential equation, in the form \(V = \mathrm { f } ( t )\).
  3. Find the limiting value of \(V\) as \(t \rightarrow \infty\).
Edexcel C4 Q7
11 marks Standard +0.3
7. The curve \(C\) has equation \(y = \frac { x } { 4 + x ^ { 2 } }\).
  1. Use calculus to find the coordinates of the turning points of \(C\). Using the result \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }\), or otherwise,
  2. determine the nature of each of the turning points.
  3. Sketch the curve \(C\).
Edexcel C4 Q8
13 marks Standard +0.3
8. (i) Given that \(\cos ( x + 30 ) ^ { \circ } = 3 \cos ( x - 30 ) ^ { \circ }\), prove that tan \(x ^ { \circ } = - \frac { \sqrt { 3 } } { 2 }\).
(ii) (a) Prove that \(\frac { 1 - \cos 2 \theta } { \sin 2 \theta } \equiv \tan \theta\).
(b) Verify that \(\theta = 180 ^ { \circ }\) is a solution of the equation \(\sin 2 \theta = 2 - 2 \cos 2 \theta\).
(c) Using the result in part (a), or otherwise, find the other two solutions, \(0 < \theta < 360 ^ { \circ }\), of the equation using \(\sin 2 \theta = 2 - 2 \cos 2 \theta\).
Edexcel C4 Q9
14 marks Standard +0.8
9. The equations of the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) , \\ l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where \(\lambda\) and \(\mu\) are parameters.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of \(Q\), their point of intersection.
  2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\). The point \(P\) with \(x\)-coordinate 3 lies on the line \(l _ { 1 }\) and the point \(R\) with \(x\)-coordinate 4 lies on the line \(l _ { 2 }\).
  3. Find, in its simplest form, the exact area of the triangle \(P Q R\). END
Edexcel C4 Q1
6 marks Standard +0.3
  1. Use integration by parts to find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x \mathrm {~d} x\).
    (6)
  2. Fluid flows out of a cylindrical tank with constant cross section. At time \(t\) minutes, \(t \geq 0\), the volume of fluid remaining in the tank is \(V \mathrm {~m} ^ { 3 }\). The rate at which the fluid flows, in \(\mathrm { m } ^ { 3 } \mathrm {~min} ^ { - 1 }\), is proportional to the square root of \(V\).
    1. Show that the depth \(h\) metres of fluid in the tank satisfies the differential equation
    $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - k \sqrt { } h , \quad \text { where } k \text { is a positive constant. }$$
  3. Show that the general solution of the differential equation may be written as $$h = ( A - B t ) ^ { 2 } , \quad \text { where } A \text { and } B \text { are constants. }$$ Given that at time \(t = 0\) the depth of fluid in the tank is 1 m , and that 5 minutes later the depth of fluid has reduced to 0.5 m ,
  4. find the time, \(T\) minutes, which it takes for the tank to empty.
  5. Find the depth of water in the tank at time \(0.5 T\) minutes.
Edexcel C4 Q3
14 marks Challenging +1.2
3. (a) Use the identity for \(\cos ( A + B )\) to prove that \(\cos 2 A = 2 \cos ^ { 2 } A - 1\).
(b) Use the substitution \(x = 2 \sqrt { } 2 \sin \theta\) to prove that $$\int _ { 2 } ^ { \sqrt { 6 } } \sqrt { \left( 8 - x ^ { 2 } \right) } \mathrm { d } x = \frac { 1 } { 3 } ( \pi + 3 \sqrt { } 3 - 6 ) .$$ A curve is given by the parametric equations $$x = \sec \theta , \quad y = \ln ( 1 + \cos 2 \theta ) , \quad 0 \leq \theta < \frac { \pi } { 2 } .$$ (c) Find an equation of the tangent to the curve at the point where \(\theta = \frac { \pi } { 3 }\).
Edexcel C4 Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{07bc7f2d-c2b9-4502-91cd-a76afb1ca6c0-3_717_863_248_737}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac { 4 } { x - 3 } , x \neq 3\). The points \(A\) and \(B\) on the curve have \(x\)-coordinates 3.25 and 5 respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\).
  2. Show that an equation of \(C\) is \(\frac { 3 y + 4 } { y } , y \neq 0\). The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\) axis. The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis to form a solid shape \(S\).
  3. Find the volume of \(S\), giving your answer in the form \(\pi ( a + b \ln c )\), where \(a , b\) and \(c\) are integers. The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
  4. show that the volume of the tower is approximately \(15500 \mathrm {~m} ^ { 3 }\).
Edexcel C4 Q5
11 marks Standard +0.3
5. Relative to a fixed origin \(O\), the point \(A\) has position vector \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\), the point \(B\) has position vector \(5 \mathbf { i } + \mathbf { j } + \mathbf { k }\), and the point \(C\) has position vector \(7 \mathbf { i } - \mathbf { j }\).
  1. Find the cosine of angle \(A B C\).
  2. Find the exact value of the area of triangle \(A B C\). The point \(D\) has position vector \(7 \mathbf { i } + 3 \mathbf { k }\).
  3. Show that \(A C\) is perpendicular to \(C D\).
  4. Find the ratio \(A D : D B\).
Edexcel C4 Q6
9 marks Moderate -0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{07bc7f2d-c2b9-4502-91cd-a76afb1ca6c0-5_809_1226_201_303}
\end{figure} Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation \(y = 8 \sqrt { \left( \sin \frac { \pi x } { 10 } \right) }\), in the interval \(0 \leq x \leq\) 10. The concrete surround is represented by the shaded area bounded by the curve, the \(x\)-axis and the lines \(x = - 2 , x = 12\) and \(y = 10\). The units on both axes are metres.
  1. Using this model, copy and complete the table below, giving the values of \(y\) to 2 decimal places.
    \(x\)0246810
    \(y\)06.130
    The area of the cross-section of the tunnel is given by \(\int _ { 0 } ^ { 10 } y \mathrm {~d} x\).
  2. Estimate this area, using the trapezium rule with all the values from your table.
  3. Deduce an estimate of the cross-sectional area of the concrete surround.
  4. State, with a reason, whether your answer in part (c) over-estimates or under-estimates the true value.
    (2)
Edexcel C4 Q7
16 marks Standard +0.8
7. $$f ( x ) = \frac { 25 } { ( 3 + 2 x ) ^ { 2 } ( 1 - x ) } , \quad | x | < 1$$
  1. Express \(\mathrm { f } ( x )\) as a sum of partial fractions.
  2. Hence find \(\int f ( x ) d x\).
  3. Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\). Give each coefficient as a simplified fraction. END
Edexcel C4 Q1
6 marks Moderate -0.8
  1. The following is a table of values for \(y = \sqrt { } ( 1 + \sin x )\), where \(x\) is in radians.
\(x\)00.511.52
\(y\)11.216\(p\)1.413\(q\)
  1. Find the value of \(p\) and the value of \(q\).
    (2)
  2. Use the trapezium rule and all the values of \(y\) in the completed table to obtain an estimate of \(I\), where $$I = \int _ { 0 } ^ { 2 } \sqrt { } ( 1 + \sin x ) \mathrm { d } x$$ (4)