Edexcel AS Paper 1 (AS Paper 1) 2018 June

1. Find

$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - 6 \sqrt { x } + 1 \right) \mathrm { d } x$$ giving your answer in its simplest form.

1. (i) Show that \(x ^ { 2 } - 8 x + 17 > 0\) for all real values of \(x\)
   (ii) "If I add 3 to a number and square the sum, the result is greater than the square of the original number."

State, giving a reason, if the above statement is always true, sometimes true or never true.

1. Given that the point \(A\) has position vector \(4 \mathbf { i } - 5 \mathbf { j }\) and the point \(B\) has position vector \(- 5 \mathbf { i } - 2 \mathbf { j }\), (a) find the vector \(\overrightarrow { A B }\),
   (b) find \(| \overrightarrow { A B } |\).

Give your answer as a simplified surd.

1. The line \(l _ { 1 }\) has equation \(4 y - 3 x = 10\)

The line \(l _ { 2 }\) passes through the points \(( 5 , - 1 )\) and \(( - 1,8 )\).
Determine, giving full reasons for your answer, whether lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, perpendicular or neither.

1. A student's attempt to solve the equation \(2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3\) is shown below.

$$\begin{aligned} & 2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3
& 2 \log _ { 2 } \left( \frac { x } { \sqrt { x } } \right) = 3
& 2 \log _ { 2 } ( \sqrt { x } ) = 3
& \log _ { 2 } x = 3
& x = 3 ^ { 2 } = 9 \end{aligned}$$ using the subtraction law for logs simplifying using the power law for logs using the definition of a log

1. Identify two errors made by this student, giving a brief explanation of each.
2. Write out the correct solution.

6. \begin{figure}[h] \end{figure} A company makes a particular type of children's toy.
The annual profit made by the company is modelled by the equation $$P = 100 - 6.25 ( x - 9 ) ^ { 2 }$$ where \(P\) is the profit measured in thousands of pounds and \(x\) is the selling price of the toy in pounds. A sketch of \(P\) against \(x\) is shown in Figure 1.
Using the model,

1. explain why \(\pounds 15\) is not a sensible selling price for the toy. Given that the company made an annual profit of more than \(\pounds 80000\)
2. find, according to the model, the least possible selling price for the toy. The company wishes to maximise its annual profit.
   State, according to the model,
3. 1. the maximum possible annual profit,
   2. the selling price of the toy that maximises the annual profit.

1. In a triangle \(A B C\), side \(A B\) has length 10 cm , side \(A C\) has length 5 cm , and angle \(B A C = \theta\) where \(\theta\) is measured in degrees. The area of triangle \(A B C\) is \(15 \mathrm {~cm} ^ { 2 }\)
   1. Find the two possible values of \(\cos \theta\)
   Given that \(B C\) is the longest side of the triangle,
2. find the exact length of \(B C\).

1. A lorry is driven between London and Newcastle.

In a simple model, the cost of the journey \(\pounds C\) when the lorry is driven at a steady speed of \(v\) kilometres per hour is $$C = \frac { 1500 } { v } + \frac { 2 v } { 11 } + 60$$

1. Find, according to this model,
   1. the value of \(v\) that minimises the cost of the journey,
   2. the minimum cost of the journey.
      (Solutions based entirely on graphical or numerical methods are not acceptable.)
2. Prove by using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) that the cost is minimised at the speed found in (a)(i).
3. State one limitation of this model.

9. $$g ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - 15 x + 50$$

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\).
2. Hence show that \(\mathrm { g } ( x )\) can be written in the form \(\mathrm { g } ( x ) = ( x + 2 ) ( a x + b ) ^ { 2 }\), where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-22_517_807_607_621} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { g } ( x )\)
3. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
   1. \(\mathrm { g } ( x ) \leqslant 0\)
   2. \(\mathrm { g } ( 2 x ) = 0\)

1. Prove, from first principles, that the derivative of \(x ^ { 3 }\) is \(3 x ^ { 2 }\)

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

$$\left( 2 - \frac { x } { 16 } \right) ^ { 9 }$$ giving each term in its simplest form. $$f ( x ) = ( a + b x ) \left( 2 - \frac { x } { 16 } \right) ^ { 9 } , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\), in the series expansion of \(\mathrm { f } ( x )\) are 128 and \(36 x\),
(b) find the value of \(a\),
(c) find the value of \(b\).

1. (a) Show that the equation

$$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 90 ^ { \circ }\) $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.)

13. \begin{figure}[h] \end{figure} The value of a rare painting, \(\pounds V\), is modelled by the equation \(V = p q ^ { t }\), where \(p\) and \(q\) are constants and \(t\) is the number of years since the value of the painting was first recorded on 1st January 1980. The line \(l\) shown in Figure 3 illustrates the linear relationship between \(t\) and \(\log _ { 10 } V\) since 1st January 1980. The equation of line \(l\) is \(\log _ { 10 } V = 0.05 t + 4.8\)

1. Find, to 4 significant figures, the value of \(p\) and the value of \(q\).
2. With reference to the model interpret
   1. the value of the constant \(p\),
   2. the value of the constant \(q\).
3. Find the value of the painting, as predicted by the model, on 1st January 2010, giving your answer to the nearest hundred thousand pounds.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0$$

1. Find
   1. the coordinates of the centre of \(C\)
   2. the radius of \(C\) The line with equation \(y = k x\), where \(k\) is a constant, cuts \(C\) at two distinct points.
2. Find the range of values for \(k\).

15. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 32 } { x ^ { 2 } } + 3 x - 8 , \quad x > 0$$ The point \(P ( 4,6 )\) lies on \(C\).
The line \(l\) is the normal to \(C\) at the point \(P\).
The region \(R\), shown shaded in Figure 4, is bounded by the line \(l\), the curve \(C\), the line with equation \(x = 2\) and the \(x\)-axis. Show that the area of \(R\) is 46
(Solutions based entirely on graphical or numerical methods are not acceptable.)