Edexcel AS Paper 1 (AS Paper 1) 2019 June

1. The line \(l _ { 1 }\) has equation \(2 x + 4 y - 3 = 0\)

The line \(l _ { 2 }\) has equation \(y = m x + 7\), where \(m\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,

1. find the value of \(m\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\).
2. Find the \(x\) coordinate of \(P\).
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1. Find, using algebra, all real solutions to the equation
   1. \(16 a ^ { 2 } = 2 \sqrt { a }\)
   2. \(b ^ { 4 } + 7 b ^ { 2 } - 18 = 0\)

1. (a) Given that \(k\) is a constant, find

$$\int \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x$$ simplifying your answer.
(b) Hence find the value of \(k\) such that $$\int _ { 0.5 } ^ { 2 } \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x = 8$$

1. A tree was planted in the ground.

Its height, \(H\) metres, was measured \(t\) years after planting.
Exactly 3 years after planting, the height of the tree was 2.35 metres.
Exactly 6 years after planting, the height of the tree was 3.28 metres.
Using a linear model,

1. find an equation linking \(H\) with \(t\). The height of the tree was approximately 140 cm when it was planted.
2. Explain whether or not this fact supports the use of the linear model in part (a).

1. A curve has equation

$$y = 3 x ^ { 2 } + \frac { 24 } { x } + 2 \quad x > 0$$

1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the exact range of values of \(x\) for which the curve is increasing.

6. Figure 1 Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = 2 x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C\) is \(18 \sqrt { 3 } \mathrm {~cm} ^ { 2 }\)

1. show that \(x = 2 \sqrt { 3 }\)
2. Hence find the exact length of BC, giving your answer as a simplified surd.

1. The curve \(C\) has equation

$$y = \frac { k ^ { 2 } } { x } + 1 \quad x \in \mathbb { R } , x \neq 0$$ where \(k\) is a constant.

1. Sketch \(C\) stating the equation of the horizontal asymptote. The line \(l\) has equation \(y = - 2 x + 5\)
2. Show that the \(x\) coordinate of any point of intersection of \(l\) with \(C\) is given by a solution of the equation $$2 x ^ { 2 } - 4 x + k ^ { 2 } = 0$$
3. Hence find the exact values of \(k\) for which \(l\) is a tangent to \(C\).

1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 2 + \frac { 3 x } { 4 } \right) ^ { 6 }$$ giving each term in its simplest form.
(b) Explain how you could use your expansion to estimate the value of \(1.925 ^ { 6 }\) You do not need to perform the calculation.

1. A company started mining tin in Riverdale on 1st January 2019.

A model to find the total mass of tin that will be mined by the company in Riverdale is given by the equation $$T = 1200 - 3 ( n - 20 ) ^ { 2 }$$ where \(T\) tonnes is the total mass of tin mined in the \(n\) years after the start of mining.
Using this model,

1. calculate the mass of tin that will be mined up to 1st January 2020,
2. deduce the maximum total mass of tin that could be mined,
3. calculate the mass of tin that will be mined in 2023.
4. State, giving reasons, the limitation on the values of \(n\).

1. A circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 4 x + 8 y - 8 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\). The straight line with equation \(x = k\), where \(k\) is a constant, is a tangent to \(C\).
2. Find the possible values for \(k\).

11. $$f ( x ) = 2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 48$$

1. Prove that \(( x - 4 )\) is a factor of \(\mathrm { f } ( x )\).
2. Hence, using algebra, show that the equation \(\mathrm { f } ( x ) = 0\) has only two distinct roots. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{deba6a2b-1821-4110-bde8-bde18a5f9be9-24_727_1059_566_504} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
3. Deduce, giving reasons for your answer, the number of real roots of the equation $$2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 46 = 0$$ Given that \(k\) is a constant and the curve with equation \(y = \mathrm { f } ( x + k )\) passes through the origin, (d) find the two possible values of \(k\).

1. (a) Show that

$$\frac { 10 \sin ^ { 2 } \theta - 7 \cos \theta + 2 } { 3 + 2 \cos \theta } \equiv 4 - 5 \cos \theta$$ (b) Hence, or otherwise, solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$\frac { 10 \sin ^ { 2 } x - 7 \cos x + 2 } { 3 + 2 \cos x } = 4 + 3 \sin x$$

13. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 2 x ^ { 3 } - 17 x ^ { 2 } + 40 x$$ The curve has a minimum turning point at \(x = k\).
The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = k\). Show that the area of \(R\) is \(\frac { 256 } { 3 }\)
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. The value of a car, \(\pounds V\), can be modelled by the equation

$$V = 15700 \mathrm { e } ^ { - 0.25 t } + 2300 \quad t \in \mathbb { R } , t \geqslant 0$$ where the age of the car is \(t\) years.
Using the model,

1. find the initial value of the car. Given the model predicts that the value of the car is decreasing at a rate of \(\pounds 500\) per year at the instant when \(t = T\),
2. 1. show that $$3925 \mathrm { e } ^ { - 0.25 T } = 500$$
   2. Hence find the age of the car at this instant, giving your answer in years and months to the nearest month.
      (Solutions based entirely on graphical or numerical methods are not acceptable.) The model predicts that the value of the car approaches, but does not fall below, \(\pounds A\).
3. State the value of \(A\).
4. State a limitation of this model.

1. Given \(n \in \mathbb { N }\), prove that \(n ^ { 3 } + 2\) is not divisible by 8

1. (i) Two non-zero vectors, \(\mathbf { a }\) and \(\mathbf { b }\), are such that

$$| \mathbf { a } + \mathbf { b } | = | \mathbf { a } | + | \mathbf { b } |$$ Explain, geometrically, the significance of this statement.
(ii) Two different vectors, \(\mathbf { m }\) and \(\mathbf { n }\), are such that \(| \mathbf { m } | = 3\) and \(| \mathbf { m } - \mathbf { n } | = 6\) The angle between vector \(\mathbf { m }\) and vector \(\mathbf { n }\) is \(30 ^ { \circ }\)
Find the angle between vector \(\mathbf { m }\) and vector \(\mathbf { m } - \mathbf { n }\), giving your answer, in degrees, to one decimal place.