Edexcel AS Paper 1 (AS Paper 1)

1. Find

$$\int \left( \frac { 1 } { 2 } x ^ { 2 } - 9 \sqrt { x } + 4 \right) d x$$ giving your answer in its simplest form.

2. Use a counter example to show that the following statement is false. $$\text { " } n ^ { 2 } - n + 5 \text { is a prime number, for } 2 \leq n \leq 6 \text { " }$$

1. Given that the point \(A\) has position vector \(x \mathbf { i } - \mathbf { j }\), the point B has position vector \(- 2 \mathbf { i } + y \mathbf { j }\) and \(\overrightarrow { A B } = - 3 \mathbf { i } + 4 \mathbf { j }\), find
   a. the values of \(x\) and \(y\)
   b. a unit vector in the direction of \(\overrightarrow { A B }\).
2. The line \(l _ { 1 }\) has equation \(2 x - 3 y = 9\)

The line \(l _ { 2 }\) passes through the points \(( 3 , - 1 )\) and \(( - 1,5 )\) Determine, giving full reasons for your answer, whether lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, perpendicular or neither.

5. A student is asked to solve the equation $$\log _ { 3 } x - \log _ { 3 } \sqrt { x - 2 } = 1$$ The student's attempt is shown $$\begin{aligned} \log _ { 3 } x - \log _ { 3 } \sqrt { x - 2 } & = 1
x - \sqrt { x - 2 } & = 3 ^ { 1 }
x - 3 & = \sqrt { x - 2 }
( x - 3 ) ^ { 2 } & = x - 2
x ^ { 2 } - 7 x + 11 & = 0
x = \frac { 7 + \sqrt { 5 } } { 2 } \quad \text { or } \quad x & = \frac { 7 - \sqrt { 5 } } { 2 } \end{aligned}$$ a. Identify the error made by this student, giving a brief explanation.
b. Write out the correct solution.

6. \begin{figure}[h] \end{figure} A stone is thrown over level ground from the top of a tower, \(X\).
The height, \(h\), in meters, of the stone above the ground level after \(t\) seconds is modelled by the function. $$\mathrm { h } ( t ) = 7 + 21 t - 4.9 t ^ { 2 } , t \geq 0$$ A sketch of \(h\) against \(t\) is shown in Figure 1.
Using the model,
a. give a physical interpretation of the meaning of the constant term 7 in the model.
b. find the time taken after the stone is thrown for it to reach ground level.
c. Rearrange \(\mathrm { h } ( t )\) into the form \(A - B ( t - C ) ^ { 2 }\), where \(A , B\) and \(C\) are constants to be found.
d. Using your answer to part cor otherwise, find the maximum height of the stone above the ground, and the time after which this maximum height is reached.

7. In a triangle \(P Q R , P Q = 20 \mathrm {~cm} , P R = 10 \mathrm {~cm}\) and angle \(Q P R = \theta\), where \(\theta\) is measured in degrees. The area of triangle \(P Q R\) is \(80 \mathrm {~cm} ^ { 2 }\).
a. Show that the two possible values of \(\cos \theta = \pm \frac { 3 } { 5 }\) Given that \(Q R\) is the longest side of the triangle,
b. find the exact perimeter of the triangle \(P Q R\), giving your answer as a simplified surd.

8. \begin{figure}[h] \end{figure} Figure 2 shows a solid cuboid \(A B C D E F G H\).
\(A B = x \mathrm {~cm} , B C = 2 x \mathrm {~cm} , A E = h \mathrm {~cm}\)
The total surface area of the cuboid is \(180 \mathrm {~cm} ^ { 2 }\).
The volume of the cuboid is \(V \mathrm {~cm} ^ { 3 }\).
a. Show that \(V = 60 x - \frac { 4 x ^ { 3 } } { 3 }\) Given that \(x\) can vary,
b. use calculus to find, to 3 significant figures, the value of \(x\) for which \(V\) is a maximum. Justify that this value of \(x\) gives a maximum value of \(V\).
c. Find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).

9. $$f ( x ) = - 2 x ^ { 3 } - x ^ { 2 } + 4 x + 3$$ a. Use the factor theorem to show that ( \(3 - 2 x\) ) is a factor of \(\mathrm { f } ( x )\).
b. Hence show that \(\mathrm { f } ( x )\) can be written in the form \(\mathrm { f } ( x ) = ( 3 - 2 x ) ( x + a ) ^ { 2 }\) where \(a\) is an integer to be found. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
c. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
i. \(\mathrm { f } ( x ) \leq 0\)
ii. \(\mathrm { f } \left( \frac { x } { 2 } \right) = 0\)

10. Prove, from the first principles, that the derivative of \(5 x ^ { 2 }\) is \(10 x\).

11. The first 3 terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + k x ) ^ { 10 }\) are given by $$1 + 15 x + p x ^ { 2 }$$ where \(k\) and \(p\) are constants.
a. Find the value of \(k\)
b. Find the value of \(p\)
c. Given that, in the expansion of \(( 1 + k x ) ^ { 10 }\), the coefficient of \(x ^ { 4 }\) is \(q\), find the value of \(q\).

12. a. Explain mathematically why there are no values of \(\theta\) that satisfy the equation $$( 3 \cos \theta - 4 ) ( 2 \cos \theta + 5 ) = 0$$ b. Giving your solutions to one decimal place, where appropriate, solve the equation $$3 \sin y + 2 \tan y = 0 \quad \text { for } 0 \leq y \leq \pi$$ (Solutions based entirely on graphical or numerical methods are not acceptable.)

13. \begin{figure}[h] \end{figure} The value of a sculpture, \(\pounds V\), is modelled by the equation \(V = A p ^ { t }\), where \(A\) and \(p\) are constants and \(t\) is the number of years since the value of the painting was first recorded on \(1 ^ { \text {st } }\) January 1960. The line \(l\) shown in Figure 4 illustrates the linear relationship between \(t\) and \(\log _ { 10 } V\) for \(t \geq 0\). The line \(l\) passes through the point \(\left( 0 , \log _ { 10 } 20 \right)\) and \(\left( 50 , \log _ { 10 } 2000 \right)\).
a. Write down the equation of the line \(l\).
b. Using your answer to part a or otherwise, find the values of \(A\) and \(p\).
c. With reference to the model, interpret the values of the constant \(A\) and \(p\).
d. Use your model, to predict the value of the sculpture, on \(1 { } ^ { \text {st } }\) January 2020, giving your answer to the nearest pounds.

14. A curve with centre \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 2 x - 6 y - 40 = 0$$ a. i. State the coordinates of \(C\).
ii. Find the radius of the circle, giving your answer as \(r = n \sqrt { 2 }\).
b. The line \(l\) is a tangent to the circle and has gradient - 7 . Find two possible equations for \(l\), giving your answers in the form \(y = m x + c\).

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(y = 2 x + \frac { 8 } { x ^ { 2 } } - 5 , x > 0\).
The point \(A \left( 4 , \frac { 7 } { 2 } \right)\) lies on C . The line \(l\) is the tangent to \(C\) at the point A .
The region \(\boldsymbol { R }\), shown shaded in figure 5 is bounded by the line \(l\), the curve \(C\), the line with equation \(x = 1\) and the \(x\)-axis. Find the exact area of \(\boldsymbol { R }\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(Total for Question 15 is 9 marks)