Edexcel AS Paper 1 (AS Paper 1) Specimen

1. A curve has equation

$$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation.

   VIIIV SIHI NI JIIYM IONOO

   VIUV SIHI NI JIIAM ION OO

   VI4V SIHI NI JIIIM I ON OO

1. The quadrilateral \(O A B C\) has \(\overrightarrow { O A } = 4 \mathbf { i } + 2 \mathbf { j } , \overrightarrow { O B } = 6 \mathbf { i } - 3 \mathbf { j }\) and \(\overrightarrow { O C } = 8 \mathbf { i } - 20 \mathbf { j }\).
   1. Find \(\overrightarrow { A B }\).
   2. Show that quadrilateral \(O A B C\) is a trapezium.

1. A tank, which contained water, started to leak from a hole in its base.

The volume of water in the tank 24 minutes after the leak started was \(4 \mathrm {~m} ^ { 3 }\) The volume of water in the tank 60 minutes after the leak started was \(2.8 \mathrm {~m} ^ { 3 }\) The volume of water, \(V \mathrm {~m} ^ { 3 }\), in the tank \(t\) minutes after the leak started, can be described by a linear model between \(V\) and \(t\).

1. Find an equation linking \(V\) with \(t\). Use this model to find
2. 1. the initial volume of water in the tank,
   2. the time taken for the tank to empty.
3. Suggest a reason why this linear model may not be suitable.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = g ( x )\).
The curve has a single turning point, a minimum, at the point \(M ( 4 , - 1.5 )\).
The curve crosses the \(x\)-axis at two points, \(P ( 2,0 )\) and \(Q ( 7,0 )\).
The curve crosses the \(y\)-axis at a single point \(R ( 0,5 )\).

1. State the coordinates of the turning point on the curve with equation \(y = 2 \mathrm {~g} ( x )\).
2. State the largest root of the equation $$g ( x + 1 ) = 0$$
3. State the range of values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) \leqslant 0\) Given that the equation \(\mathrm { g } ( x ) + k = 0\), where \(k\) is a constant, has no real roots,
4. state the range of possible values for \(k\).

5. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12$$

1. Using the factor theorem, explain why \(\mathrm { f } ( x )\) is divisible by \(( x + 3 )\).
2. Hence fully factorise \(\mathrm { f } ( x )\).
3. Show that \(\frac { x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12 } { x ^ { 3 } + 5 x ^ { 2 } + 6 x }\) can be written in the form \(A + \frac { B } { x }\) where \(A\) and \(B\) are integers to be found.

1. (i) Use a counter example to show that the following statement is false.

$$" n ^ { 2 } - n - 1 \text { is a prime number, for } 3 \leqslant n \leqslant 10 \text {." }$$ (ii) Prove that the following statement is always true.
"The difference between the cube and the square of an odd number is even."
For example \(5 ^ { 3 } - 5 ^ { 2 } = 100\) is even.
\includegraphics[max width=\textwidth, alt={}, center]{fa7abe9f-f5c0-4578-afd1-73176c717536-12_2255_51_314_1978}

1. (a) Expand \(\left( 1 + \frac { 3 } { x } \right) ^ { 2 }\) simplifying each term.
   (b) Use the binomial expansion to find, in ascending powers of \(x\), the first four terms in the expansion of

$$\left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$ simplifying each term.
(c) Hence find the coefficient of \(x\) in the expansion of $$\left( 1 + \frac { 3 } { x } \right) ^ { 2 } \left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \sqrt { x } , x \geqslant 0\)
The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = a\), where \(a\) is a constant. Given that the area of \(R\) is 10

1. find, in simplest form, the value of
   1. \(\int _ { 1 } ^ { a } \sqrt { 8 x } \mathrm {~d} x\)
   2. \(\int _ { 0 } ^ { a } \sqrt { x } \mathrm {~d} x\)
2. show that \(a = 2 ^ { k }\), where \(k\) is a rational constant to be found.

1. Find any real values of \(x\) such that

$$2 \log _ { 4 } ( 2 - x ) - \log _ { 4 } ( x + 5 ) = 1$$

1. A circle \(C\) has centre \(( 2,5 )\). Given that the point \(P ( - 2,3 )\) lies on \(C\).
   1. find an equation for \(C\).
   The line \(l\) is the tangent to \(C\) at the point \(P\). The point \(Q ( 2 , k )\) lies on \(l\).
2. Find the value of \(k\).

1. (i) Solve, for \(- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }\), the equation,

$$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$ giving your answers to one decimal place.
(ii) (a) A student's attempt at the question
"Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(7 \tan x = 8 \sin x\) " is set out below. $$\begin{gathered} 7 \tan x = 8 \sin x
7 \times \frac { \sin x } { \cos x } = 8 \sin x
7 \sin x = 8 \sin x \cos x
7 = 8 \cos x
\cos x = \frac { 7 } { 8 }
x = 29.0 ^ { \circ } \text { (to } 3 \text { sf) } \end{gathered}$$ Identify two mistakes made by this student, giving a brief explanation of each mistake.
(b) Find the smallest positive solution to the equation $$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$

12. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3 x - 2 \sqrt { x } , x \geqslant 0\) and the line \(l\) with equation \(y = 8 x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.

1. Using algebra, find the \(x\) coordinate of point \(A\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-24_636_780_1585_644} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\).

1. The growth of pond weed on the surface of a pond is being investigated.

The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), can be modelled by the equation $$A = 0.2 \mathrm { e } ^ { 0.3 t }$$ where \(t\) is the number of days after the start of the investigation.

1. State the surface area of the pond covered by the weed at the start of the investigation.
2. Find the rate of increase of the surface area of the pond covered by the weed, in \(\mathrm { m } ^ { 2 } /\) day, exactly 5 days after the start of the investigation. Given that the pond has a surface area of \(100 \mathrm {~m} ^ { 2 }\),
3. find, to the nearest hour, the time taken, according to the model, for the surface of the pond to be fully covered by the weed. The pond is observed for one month and by the end of the month \(90 \%\) of the surface area of the pond was covered by the weed.
4. Evaluate the model in light of this information, giving a reason for your answer.

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = ( x - 2 ) ^ { 2 } ( x + 3 )\) The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)