Edexcel AS Paper 1 (AS Paper 1) 2020 June

1. A curve has equation

$$y = 2 x ^ { 3 } - 4 x + 5$$ Find the equation of the tangent to the curve at the point \(P ( 2,13 )\).
Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found.
Solutions relying on calculator technology are not acceptable.
(5)

1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]

A coastguard station \(O\) monitors the movements of a small boat.
At 10:00 the boat is at the point \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }\) relative to \(O\).
At 12:45 the boat is at the point \(( - 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km }\) relative to \(O\).
The motion of the boat is modelled as that of a particle moving in a straight line at constant speed.

1. Calculate the bearing on which the boat is moving, giving your answer in degrees to one decimal place.
2. Calculate the speed of the boat, giving your answer in \(\mathrm { kmh } ^ { - 1 }\)

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

Solutions relying on calculator technology are not acceptable.

1. Solve the equation $$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
2. Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$

1. In 1997 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(190 \mathrm {~g} / \mathrm { km }\).

In 2005 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK had fallen to \(169 \mathrm {~g} / \mathrm { km }\).
Given \(\mathrm { Ag } / \mathrm { km }\) is the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK \(n\) years after 1997 and using a linear model,

1. form an equation linking \(A\) with \(n\). In 2016 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(120 \mathrm {~g} / \mathrm { km }\).
2. Comment on the suitability of your model in light of this information.

5. \begin{figure}[h] \end{figure} Figure 1 shows the design for a structure used to support a roof.
The structure consists of four steel beams, \(A B , B D , B C\) and \(A D\).
Given \(A B = 12 \mathrm {~m} , B C = B D = 7 \mathrm {~m}\) and angle \(B A C = 27 ^ { \circ }\)

1. find, to one decimal place, the size of angle \(A C B\). The steel beams can only be bought in whole metre lengths.
2. Find the minimum length of steel that needs to be bought to make the complete structure.

1. (a) Find the first 4 terms, in ascending powers of \(x\), in the binomial expansion of

$$( 1 + k x ) ^ { 10 }$$ where \(k\) is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of \(( 1 + k x ) ^ { 10 }\) the coefficient \(x ^ { 3 }\) is 3 times the coefficient of \(x\), (b) find the possible values of \(k\).

1. Given that \(k\) is a positive constant and \(\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4\)
   1. show that \(3 k + 5 \sqrt { k } - 12 = 0\)
   2. Hence, using algebra, find any values of \(k\) such that
   $$\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4$$

1. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of a cup of tea \(t\) minutes after it was placed on a table in a room, is modelled by the equation

$$\theta = 18 + 65 \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ Find, according to the model,

1. the temperature of the cup of tea when it was placed on the table,
2. the value of \(t\), to one decimal place, when the temperature of the cup of tea was \(35 ^ { \circ } \mathrm { C }\).
3. Explain why, according to this model, the temperature of the cup of tea could not fall to \(15 ^ { \circ } \mathrm { C }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bcbd842f-b2e2-4587-ab4c-15a57a449e5d-16_675_951_973_573} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The temperature, \(\mu ^ { \circ } \mathrm { C }\), of a second cup of tea \(t\) minutes after it was placed on a table in a different room, is modelled by the equation $$\mu = A + B \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ where \(A\) and \(B\) are constants.
   Figure 2 shows a sketch of \(\mu\) against \(t\) with two data points that lie on the curve.
   The line \(l\), also shown on Figure 2, is the asymptote to the curve.
   Using the equation of this model and the information given in Figure 2
4. find an equation for the asymptote \(l\).

9. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve with equation \(y = 3 \cos x ^ { \circ }\).
The point \(P ( c , d )\) is a minimum point on the curve with \(c\) being the smallest negative value of \(x\) at which a minimum occurs.

1. State the value of \(c\) and the value of \(d\).
2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3 \cos x ^ { \circ }\) to the curve with equation
   1. \(y = 3 \cos \left( \frac { x ^ { \circ } } { 4 } \right)\)
   2. \(y = 3 \cos ( x - 36 ) ^ { \circ }\)
3. Solve, for \(450 ^ { \circ } \leqslant \theta < 720 ^ { \circ }\), $$3 \cos \theta = 8 \tan \theta$$ giving your solution to one decimal place.
   In part (c) you must show all stages of your working.
   Solutions relying entirely on calculator technology are not acceptable.

10. $$g ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 41 x - 70$$

1. Use the factor theorem to show that \(\mathrm { g } ( x )\) is divisible by \(( x - 5 )\).
2. Hence, showing all your working, write \(\mathrm { g } ( x )\) as a product of three linear factors. The finite region \(R\) is bounded by the curve with equation \(y = \mathrm { g } ( x )\) and the \(x\)-axis, and lies below the \(x\)-axis.
3. Find, using algebraic integration, the exact value of the area of \(R\).

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

$$x ^ { 2 } + y ^ { 2 } + 18 x - 2 y + 30 = 0$$ The line \(l\) is the tangent to \(C _ { 1 }\) at the point \(P ( - 5,7 )\).
Find an equation of \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
(ii) A different circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 12 y + k = 0$$ where \(k\) is a constant.
Given that \(C _ { 2 }\) lies entirely in the 4th quadrant, find the range of possible values for \(k\).

1. An advertising agency is monitoring the number of views of an online advert.

The equation $$\log _ { 10 } V = 0.072 t + 2.379 \quad 1 \leqslant t \leqslant 30 , t \in \mathbb { N }$$ is used to model the total number of views of the advert, \(V\), in the first \(t\) days after the advert went live.

1. Show that \(V = a b ^ { t }\) where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest whole number and give the value of \(b\) to 3 significant figures.
2. Interpret, with reference to the model, the value of \(a b\). Using this model, calculate
3. the total number of views of the advert in the first 20 days after the advert went live. Give your answer to 2 significant figures.

1. (a) Prove that for all positive values of \(a\) and \(b\)

$$\frac { 4 a } { b } + \frac { b } { a } \geqslant 4$$ (b) Prove, by counter example, that this is not true for all values of \(a\) and \(b\).

1. A curve has equation \(y = \mathrm { g } ( x )\).

Given that

• \(\mathrm { g } ( x )\) is a cubic expression in which the coefficient of \(x ^ { 3 }\) is equal to the coefficient of \(x\)
• the curve with equation \(y = \mathrm { g } ( x )\) passes through the origin
• the curve with equation \(y = \mathrm { g } ( x )\) has a stationary point at \(( 2,9 )\)
  1. find \(\mathrm { g } ( x )\),
  2. prove that the stationary point at \(( 2,9 )\) is a maximum.