Edexcel Paper 2 (Paper 2) 2024 June

1. $$y = 4 x ^ { 3 } - 7 x ^ { 2 } + 5 x - 10$$

1. Find in simplest form
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Hence find the exact value of \(x\) when \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\)

1. Jamie takes out an interest-free loan of \(\pounds 8100\)

Jamie makes a payment every month to pay back the loan.
Jamie repays \(\pounds 400\) in month \(1 , \pounds 390\) in month \(2 , \pounds 380\) in month 3 , and so on, so that the amounts repaid each month form an arithmetic sequence.

1. Show that Jamie repays \(\pounds 290\) in month 12 After Jamie's \(N\) th payment, the loan is completely paid back.
2. Show that \(N ^ { 2 } - 81 N + 1620 = 0\)
3. Hence find the value of \(N\).

1. The point \(P ( 3 , - 2 )\) lies on the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\)

Find the coordinates of the point to which \(P\) is mapped when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation

1. \(y = \mathrm { f } ( x - 2 )\)
2. \(y = \mathrm { f } ( 2 x )\)
3. \(y = 3 \mathrm { f } ( - x ) + 5\)

1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$\begin{aligned} u _ { n + 1 } & = k u _ { n } - 5
u _ { 1 } & = 6 \end{aligned}$$ where \(k\) is a positive constant.
Given that \(u _ { 3 } = - 1\)

1. show that $$6 k ^ { 2 } - 5 k - 4 = 0$$
2. Hence
   1. find the value of \(k\),
   2. find the value of \(\sum _ { r = 1 } ^ { 3 } u _ { r }\)

1. Given that \(\theta\) is small and in radians, use the small angle approximations to find an approximate numerical value of

$$\frac { \theta \tan 2 \theta } { 1 - \cos 3 \theta }$$

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves with equations \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) where $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 4 x ^ { 2 } - 1 } & x > 0
\mathrm {~g} ( x ) = 8 \ln x & x > 0 \end{array}$$

1. Find
   1. \(\mathrm { f } ^ { \prime } ( x )\)
   2. \(\mathrm { g } ^ { \prime } ( x )\) Given that \(\mathrm { f } ^ { \prime } ( x ) = \mathrm { g } ^ { \prime } ( x )\) at \(x = \alpha\)
2. show that \(\alpha\) satisfies the equation $$4 x ^ { 2 } + 2 \ln x - 1 = 0$$ The iterative formula $$x _ { n + 1 } = \sqrt { \frac { 1 - 2 \ln x _ { n } } { 4 } }$$ is used with \(x _ { 1 } = 0.6\) to find an approximate value for \(\alpha\)
3. Calculate, giving each answer to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the value of \(\alpha\)

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the straight line \(l\).
Line \(l\) passes through the points \(A\) and \(B\).
Relative to a fixed origin \(O\)

• the point \(A\) has position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 8 \mathbf { k }\)
  1. Find \(\overrightarrow { A B }\)

Given that a point \(P\) lies on \(l\) such that $$| \overrightarrow { A P } | = 2 | \overrightarrow { B P } |$$

find the possible position vectors of \(P\).

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Prove that $$\frac { 1 } { \operatorname { cosec } \theta - 1 } + \frac { 1 } { \operatorname { cosec } \theta + 1 } \equiv 2 \tan \theta \sec \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$
2. Hence solve, for \(0 < x < 90 ^ { \circ }\), the equation $$\frac { 1 } { \operatorname { cosec } 2 x - 1 } + \frac { 1 } { \operatorname { cosec } 2 x + 1 } = \cot 2 x \sec 2 x$$ Give each answer, in degrees, to one decimal place.

9. \begin{figure}[h] \end{figure} The graph in Figure 3 shows the path of a small ball.
The ball travels in a vertical plane above horizontal ground.
The ball is thrown from the point represented by \(A\) and caught at the point represented by \(B\). The height, \(H\) metres, of the ball above the ground has been plotted against the horizontal distance, \(x\) metres, measured from the point where the ball was thrown. With respect to a fixed origin \(O\), the point \(A\) has coordinates \(( 0,2 )\) and the point \(B\) has coordinates (20, 0.8), as shown in Figure 3. The ball reaches its maximum height when \(x = 9\)
A quadratic function, linking \(H\) with \(x\), is used to model the path of the ball.

1. Find \(H\) in terms of \(x\).
2. Give one limitation of the model. Chandra is standing directly under the path of the ball at a point 16 m horizontally from \(O\). Chandra can catch the ball if the ball is less than 2.5 m above the ground.
3. Use the model to determine if Chandra can catch the ball.

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = ( t + 3 ) ^ { 2 } \quad y = 1 - t ^ { 3 } \quad - 2 \leqslant t \leqslant 1$$ The point \(P\) with coordinates \(( 4,2 )\) lies on \(C\).

1. Using parametric differentiation, show that the tangent to \(C\) at \(P\) has equation $$3 x + 4 y = 20$$ The curve \(C\) is used to model the profile of a slide at a water park.
   Units are in metres, with \(y\) being the height of the slide above water level.
2. Find, according to the model, the greatest height of the slide above water level.

11. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = 8 x ^ { 2 } \mathrm { e } ^ { - 3 x } \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 5, is bounded by

• the curve \(C\)
• the line with equation \(x = 1\)
• the \(x\)-axis

Find the exact area of \(R\), giving your answer in the form $$A + B \mathrm { e } ^ { - 3 }$$ where \(A\) and \(B\) are rational numbers to be found.

1. (a) Express \(\frac { 1 } { V ( 25 - V ) }\) in partial fractions.

The volume, \(V\) microlitres, of a plant cell \(t\) hours after the plant is watered is modelled by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 1 } { 10 } V ( 25 - V )$$ The plant cell has an initial volume of 20 microlitres.
(b) Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres.
(c) Show that $$V = \frac { A } { \mathrm { e } ^ { - k t } + B }$$ where \(A , B\) and \(k\) are constants to be found. The model predicts that there is an upper limit, \(L\) microlitres, on the volume of the plant cell.
(d) Find the value of \(L\), giving a reason for your answer.

1. The world human population, \(P\) billions, is modelled by the equation

$$P = a b ^ { t }$$ where \(a\) and \(b\) are constants and \(t\) is the number of years after 2004
Using the estimated population figures for the years from 2004 to 2007, a graph is plotted of \(\log _ { 10 } P\) against \(t\). The points lie approximately on a straight line with

• gradient 0.0054
• intercept 0.81 on the \(\log _ { 10 } P\) axis
  1. Estimate, to 3 decimal places, the value of \(a\) and the value of \(b\).

In the context of the model,

1. interpret the value of the constant \(a\),
2. interpret the value of the constant \(b\).

Use the model to estimate the world human population in 2030

Comment on the reliability of the answer to part (c).

1. The circle \(C _ { 1 }\) has equation

$$x ^ { 2 } + y ^ { 2 } - 6 x + 14 y + 33 = 0$$

1. Find
   1. the coordinates of the centre of \(C _ { 1 }\)
   2. the radius of \(C _ { 1 }\) A different circle \(C _ { 2 }\)
      • has centre with coordinates (-6, -8)
2. has radius \(k\), where \(k\) is a constant
Given that \(C _ { 1 }\) and \(C _ { 2 }\) intersect at 2 distinct points,
3. find the range of values of \(k\), writing your answer in set notation.

1. The curve \(C\) has equation

$$( x + y ) ^ { 3 } = 3 x ^ { 2 } - 3 y - 2$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P ( 1,0 )\) lies on \(C\).
2. Show that the normal to \(C\) at \(P\) has equation $$y = - 2 x + 2$$
3. Prove that the normal to \(C\) at \(P\) does not meet \(C\) again. You should use algebra for your proof and make your reasoning clear.