Edexcel AS Paper 1 (AS Paper 1) 2024 June

1. Find

$$\int \frac { 2 \sqrt { x } - 3 } { x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.

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

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + b x + 8 a$$ where \(a\) and \(b\) are constants.
Given that ( \(x - 4\) ) is a factor of \(\mathrm { f } ( x )\),

1. use the factor theorem to show that $$10 a = 32 + b$$ Given also that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),
2. express \(\mathrm { f } ( x )\) in the form $$f ( x ) = ( 2 x + k ) ( x - 4 ) ( x - 2 )$$ where \(k\) is a constant to be found.
3. Hence,
   1. state the number of real roots of the equation \(\mathrm { f } ( x ) = 0\)
   2. write down the largest root of the equation \(\mathrm { f } \left( \frac { 1 } { 3 } x \right) = 0\)

1. Relative to a fixed origin \(O\),

• point \(P\) has position vector \(9 \mathbf { i } - 8 \mathbf { j }\)
• point \(Q\) has position vector \(3 \mathbf { i } - 5 \mathbf { j }\)
  1. Find \(\overrightarrow { P Q }\)

Given that \(R\) is the point such that \(\overrightarrow { Q R } = 9 \mathbf { i } + 18 \mathbf { j }\)

show that angle \(P Q R = 90 ^ { \circ }\) Given also that \(S\) is the point such that \(\overrightarrow { P S } = 3 \overrightarrow { Q R }\)

find the exact area of \(P Q R S\)

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of triangle \(A B D\) and triangle \(B C D\)
Given that

• \(A D C\) is a straight line
• \(B D = ( x + 3 ) \mathrm { cm }\)
• \(B C = x \mathrm {~cm}\)
• angle \(B D C = 30 ^ { \circ }\)
• angle \(B C D = 140 ^ { \circ }\)
  1. show that \(x = 10.5\) correct to 3 significant figures.

Given also that \(A D = ( x - 2 ) \mathrm { cm }\)

find the length of \(A B\), giving your answer to 3 significant figures.

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

$$y = \frac { 6 } { x } + 3$$

1. 1. Sketch \(C _ { 1 }\) stating the coordinates of any points where the curve cuts the coordinate axes.
   2. State the equations of any asymptotes to the curve \(C _ { 1 }\) The curve \(C _ { 2 }\) has equation $$y = 3 x ^ { 2 } - 4 x - 10$$
2. Show that \(C _ { 1 }\) and \(C _ { 2 }\) intersect when $$3 x ^ { 3 } - 4 x ^ { 2 } - 13 x - 6 = 0$$ Given that the \(x\) coordinate of one of the points of intersection is \(- \frac { 2 } { 3 }\)
3. use algebra to find the \(x\) coordinates of the other points of intersection between \(C _ { 1 }\) and \(C _ { 2 }\)
   (Solutions relying on calculator technology are not acceptable.)

1. The binomial expansion of

$$( 1 + a x ) ^ { 12 }$$ up to and including the term in \(x ^ { 2 }\) is $$1 - \frac { 15 } { 2 } x + k x ^ { 2 }$$ where \(a\) and \(k\) are constants.

1. Show that \(a = - \frac { 5 } { 8 }\)
2. Hence find the value of \(k\) Using the expansion and making your method clear,
3. find an estimate for the value of \(\left( \frac { 17 } { 16 } \right) ^ { 12 }\), giving your answer to 4 decimal places.

7. \begin{figure}[h] \end{figure} A chimney emits smoke particles.
On a particular day, the concentration of smoke particles in the air emitted by this chimney, \(P\) parts per million, is measured at various distances, \(x \mathrm {~km}\), from the chimney. Figure 2 shows a sketch of the linear relationship between \(\log _ { 10 } P\) and \(x\) that is used to model this situation. The line passes through the point ( \(0,3.3\) ) and the point ( \(6,2.1\) )

1. Find a complete equation for the model in the form $$P = a b ^ { x }$$ where \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) each to 4 significant figures.
2. With reference to the model, interpret the value of \(a b\)

8. \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 3 shows a sketch of the curve \(C\) with equation $$y = x ^ { 3 } - 14 x + 23$$ The line \(l\) is the tangent to \(C\) at the point \(A\), also shown in Figure 3.
Given that \(l\) has equation \(y = - 2 x + 7\)

1. show, using calculus, that the \(x\) coordinate of \(A\) is 2 The line \(l\) cuts \(C\) again at the point \(B\).
2. Verify that the \(x\) coordinate of \(B\) is - 4 The finite region, \(R\), shown shaded in Figure 3, is bounded by \(C\) and \(l\).
   Using algebraic integration,
3. show that the area of \(R\) is 108

9. $$\begin{aligned} p & = \log _ { a } 16
q & = \log _ { a } 25 \end{aligned}$$ where \(a\) is a constant.
Find in terms of \(p\) and/or \(q\),

1. \(\log _ { a } 256\)
2. \(\log _ { a } 100\)
3. \(\log _ { a } 80 \times \log _ { a } 3.2\)

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the circle \(C\)

• the point \(P ( - 1 , k + 8 )\) is the centre of \(C\)
• the point \(Q \left( 3 , k ^ { 2 } - 2 k \right)\) lies on \(C\)
• \(k\) is a positive constant
• the line \(l\) is the tangent to \(C\) at \(Q\)

Given that the gradient of \(l\) is - 2

1. show that $$k ^ { 2 } - 3 k - 10 = 0$$
2. Hence find an equation for \(C\)

1. The prices of two precious metals are being monitored.

The price per gram of metal \(A , \pounds V _ { A }\), is modelled by the equation $$V _ { A } = 100 + 20 \mathrm { e } ^ { 0.04 t }$$ where \(t\) is the number of months after monitoring began.
The price per gram of metal \(B , \pounds V _ { B }\), is modelled by the equation $$V _ { B } = p \mathrm { e } ^ { - 0.02 t }$$ where \(p\) is a positive constant and \(t\) is the number of months after monitoring began.
Given that \(V _ { B } = 2 V _ { A }\) when \(t = 0\)

1. find the value of \(p\) When \(t = T\), the rate of increase in the price per gram of metal \(A\) was equal to the rate of decrease in the price per gram of metal \(B\)
2. Find the value of \(T\), giving your answer to one decimal place.
   (Solutions based entirely on calculator technology are not acceptable.)

12. \begin{figure}[h] \end{figure} Figure 5 shows the plan view of the design for a swimming pool.
The pool is modelled as a quarter of a circle joined to two equal sized rectangles as shown. Given that

• the quarter circle has radius \(x\) metres
• the rectangles each have length \(x\) metres and width \(y\) metres
• the total surface area of the swimming pool is \(100 \mathrm {~m} ^ { 2 }\)
  1. show that, according to the model, the perimeter \(P\) metres of the swimming pool is given by

$$P = 2 x + \frac { 200 } { x }$$

Use calculus to find the value of \(x\) for which \(P\) has a stationary value.

Prove, by further calculus, that this value of \(x\) gives a minimum value for \(P\) Access to the pool is by side \(A B\) shown in Figure 5.
Given that \(A B\) must be at least one metre,

determine, according to the model, whether the swimming pool with the minimum perimeter would be suitable.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
   1. Show that the equation
   $$\sin \theta ( 7 \sin \theta - 4 \cos \theta ) = 4$$ can be written as $$3 \tan ^ { 2 } \theta - 4 \tan \theta - 4 = 0$$
2. Hence solve, for \(0 < x < 360 ^ { \circ }\) $$\sin x ( 7 \sin x - 4 \cos x ) = 4$$ giving your answers to one decimal place.
3. Hence find the smallest solution of the equation $$\sin 4 \alpha ( 7 \sin 4 \alpha - 4 \cos 4 \alpha ) = 4$$ in the range \(720 ^ { \circ } < \alpha < 1080 ^ { \circ }\), giving your answer to one decimal place.

1. Prove, using algebra, that

$$n ^ { 2 } + 5 n$$ is even for all \(n \in \mathbb { N }\)