Edexcel C4 (Core Mathematics 4)

1. The curve \(C\) has equation \(5 x ^ { 2 } + 2 x y - 3 y ^ { 2 } + 3 = 0\). The point \(P\) on the curve \(C\) has coordinates \(( 1,2 )\).
   1. Find the gradient of the curve at \(P\).
   2. Find the equation of the normal to the curve \(C\) at \(P\), in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6e307391-198f-4ea9-99ed-6ef184fca0f7-2_674_895_900_392}
   \end{figure} In Fig. 1, the curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x + \frac { 2 } { x ^ { 2 } } , \quad x > 0$$ The shaded region is bounded by \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 2\). The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis. Using calculus, calculate the volume of the solid generated. Give your answer in the form \(\pi ( a + \ln b )\), where \(a\) and \(b\) are constants.
   (8)

3.
\includegraphics[max width=\textwidth, alt={}, center]{6e307391-198f-4ea9-99ed-6ef184fca0f7-3_826_873_246_539} Figure 2 shows part of the curve with equation $$y = \mathrm { e } ^ { x } \cos x , 0 \leq x \leq \frac { \pi } { 2 }$$ The finite region \(R\) is bounded by the curve and the coordinate axes.

1. Calculate, to 2 decimal places, the \(y\)-coordinates of the points on the curve where \(x = 0 , \frac { \pi } { 6 } , \frac { \pi } { 3 }\) and \(\frac { \pi } { 2 }\).
   (3)
2. Using the trapezium rule and all the values calculated in part (a), find an approximation for the area of \(R\).
   (4)
3. State, with a reason, whether your approximation underestimates or overestimates the area of \(R\).
   (2)

4. A curve is given parametrically by the equations $$x = 5 \cos t , \quad y = - 2 + 4 \sin t , \quad 0 \leq t < 2 \pi$$

1. Find the coordinates of all the points at which \(C\) intersects the coordinate axes, giving your answers in surd form where appropriate.
2. Sketch the graph at \(C\).
   \(P\) is the point on \(C\) where \(t = \frac { 1 } { 6 } \pi\).
3. Show that the normal to \(C\) at \(P\) has equation $$8 \sqrt { } 3 y = 10 x - 25 \sqrt { } 3$$

5. \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). Figure 1 shows the part of \(C\) for which \(0 \leq x \leq 2\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } - 2 x ^ { 2 } ,$$ and that \(C\) has a single maximum, at \(x = k\),

1. show that \(1.48 < k < 1.49\). Given also that the point \(( 0,5 )\) lies on \(C\),
2. find \(\mathrm { f } ( x )\). The finite region \(R\) is bounded by \(C\), the coordinate axes and the line \(x = 2\).
3. Use integration to find the exact area of \(R\).
   (4)

6. When \(( 1 + a x ) ^ { n }\) is expanded as a series in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 2 }\) are - 6 and 27 respectively.

1. Find the value of \(a\) and the value of \(n\).
2. Find the coefficient of \(x ^ { 3 }\).
3. State the set of values of \(x\) for which the expansion is valid.

7. Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l _ { 1 }\) and \(l _ { 2 }\), along which they travel are $$\text { and } \quad \begin{aligned} & \mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )
& \mathbf { r } = 9 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + \mu ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) , \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalars.

1. Show that the submarines are moving in perpendicular directions.
2. Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\), find the position vector of \(A\). The point \(b\) has position vector \(10 \mathbf { j } - 11 \mathbf { k }\).
3. Show that only one of the submarines passes through the point \(B\).
4. Given that 1 unit on each coordinate axis represents 100 m , find, in km , the distance \(A B\).

8. In a chemical reaction two substances combine to form a third substance. At time \(t , t \geq 0\), the concentration of this third substance is \(x\) and the reaction is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 1 - 2 x ) ( 1 - 4 x ) \text {, where } k \text { is a positive constant. }$$

1. Solve this differential equation and hence show that $$\ln \left| \frac { 1 - 2 x } { 1 - 4 x } \right| = 2 k t + c \text {, where } c \text { is an arbitrary constant. }$$
2. Given that \(x = 0\) when \(t = 0\), find an expression for \(x\) in terms of \(k\) and \(t\).
3. Find the limiting value of the concentration \(x\) as \(t\) becomes very large. END