Edexcel C12 (Core Mathematics 1 & 2) 2015 June

1. The line \(l _ { 1 }\) has equation

$$10 x - 2 y + 7 = 0$$

1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 1 } { 3 } , \frac { 4 } { 3 } \right)\).
2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

2. $$f ( x ) = x ^ { 4 } - x ^ { 3 } + 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 22
Find the value of \(a\) and the value of \(b\).

3. Given that $$y = \frac { 1 } { 27 } x ^ { 3 }$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are constants.

1. \(y ^ { \frac { 1 } { 3 } }\)
2. \(3 y ^ { - 1 }\)
3. \(\sqrt { ( 27 y ) }\)

4. (a) Sketch the graph of \(y = \frac { 1 } { x } , x > 0\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { x }\), with the values for \(y\) rounded to 3 decimal places where necessary. (b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value, to 2 decimal places, for \(\int _ { 1 } ^ { 3 } \frac { 1 } { x } \mathrm {~d} x\)

1. (i) Find, giving your answer to 3 significant figures, the value of \(y\) for which

$$3 ^ { y } = 12$$ (ii) Solve, giving an exact answer, the equation $$\log _ { 2 } ( x + 3 ) - \log _ { 2 } ( 2 x + 4 ) = 4$$ (You should show each step in your working.)

6. (a) Find the first 3 terms in ascending powers of \(x\) of the binomial expansion of $$( 2 + a x ) ^ { 6 }$$ where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\)
(b) find the value of \(a\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a circle with centre \(O\) and radius 9 cm . The points \(A\) and \(B\) lie on the circumference of this circle. The minor sector \(O A B\) has perimeter 30 cm and the angle between the radii \(O A\) and \(O B\) of this sector is \(\theta\) radians. Find

1. the length of the arc \(A B\),
2. the value of \(\theta\),
3. the area of the minor sector \(O A B\),
4. the area of triangle \(O A B\), giving your answer to 3 significant figures.

8. A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find

1. the number of houses built in 1986, the first year of the building programme,
2. the total number of houses built in the 25 years of the programme.

9. The equation \(x ^ { 2 } + ( 6 k + 4 ) x + 3 = 0\), where \(k\) is a constant, has no real roots.

1. Show that \(k\) satisfies the inequality $$9 k ^ { 2 } + 12 k + 1 < 0$$
2. Find the range of possible values for \(k\), giving your boundaries as fully simplified surds.

10. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 4
u _ { n + 1 } & = \frac { 2 u _ { n } } { 3 } , \quad n \geqslant 1 \end{aligned}$$

1. Find the exact values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. Find the value of \(u _ { 20 }\), giving your answer to 3 significant figures.
3. Evaluate $$12 - \sum _ { i = 1 } ^ { 16 } u _ { i }$$ giving your answer to 3 significant figures.
4. Explain why \(\sum _ { i = 1 } ^ { N } u _ { i } < 12\) for all positive integer values of \(N\).

1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where

$$f ^ { \prime } ( x ) = 3 \sqrt { x } - \frac { 9 } { \sqrt { x } } + 2$$ Given that the point \(P ( 9,14 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\), simplifying your answer,
2. find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

12. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the \(x\)-axis at the origin and at the point \(( 6,0 )\). The curve has maximum points at \(( 1,6 )\) and \(( 5,6 )\) and has a minimum point at \(( 3,2 )\). On separate diagrams sketch the curve with equation

1. \(y = - \mathrm { f } ( x )\)
2. \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\)
3. \(y = \mathrm { f } ( x + 4 )\) On each diagram show clearly the coordinates of the maximum and minimum points, and the coordinates of the points where the curve crosses the \(x\)-axis.

1. (i) Showing each step in your reasoning, prove that

$$( \sin x + \cos x ) ( 1 - \sin x \cos x ) \equiv \sin ^ { 3 } x + \cos ^ { 3 } x$$ (ii) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$3 \sin \theta = \tan \theta$$ giving your answers in degrees to 1 decimal place, as appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

14. \begin{figure}[h] \end{figure} The finite region \(R\), which is shown shaded in Figure 3, is bounded by the straight line \(l\) with equation \(y = 4 x + 3\) and the curve \(C\) with equation \(y = 2 x ^ { \frac { 3 } { 2 } } - 2 x + 3 , x \geqslant 0\) The line \(l\) meets the curve \(C\) at the point \(A\) on the \(y\)-axis and \(l\) meets \(C\) again at the point \(B\), as shown in Figure 3.

1. Use algebra to find the coordinates of \(A\) and \(B\).
2. Use integration to find the area of the shaded region \(R\).

15. \begin{figure}[h] \end{figure} Diagram not drawn to scale The circle shown in Figure 4 has centre \(P ( 5,6 )\) and passes through the point \(A ( 12,7 )\). Find

1. the exact radius of the circle,
2. an equation of the circle,
3. an equation of the tangent to the circle at the point \(A\). The circle also passes through the points \(B ( 0,1 )\) and \(C ( 4,13 )\).
4. Use the cosine rule on triangle \(A B C\) to find the size of the angle \(B C A\), giving your answer in degrees to 3 significant figures.

1. \hspace{0pt} [In this question you may assume the formula for the area of a circle and the following formulae:
   a sphere of radius \(r\) has volume \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and surface area \(S = 4 \pi r ^ { 2 }\)
   a cylinder of radius \(r\) and height \(h\) has volume \(V = \pi r ^ { 2 } h\) and curved surface area \(S = 2 \pi r h ]\)

\begin{figure}[h] \end{figure} Figure 5 shows the model for a building. The model is made up of three parts. The roof is modelled by the curved surface of a hemisphere of radius \(R \mathrm {~cm}\). The walls are modelled by the curved surface of a circular cylinder of radius \(R \mathrm {~cm}\) and height \(H \mathrm {~cm}\). The floor is modelled by a circular disc of radius \(R \mathrm {~cm}\). The model is made of material of negligible thickness, and the walls are perpendicular to the base. It is given that the volume of the model is \(800 \pi \mathrm {~cm} ^ { 3 }\) and that \(0 < R < 10.6\)

1. Show that $$H = \frac { 800 } { R ^ { 2 } } - \frac { 2 } { 3 } R$$
2. Show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the model is given by $$A = \frac { 5 \pi R ^ { 2 } } { 3 } + \frac { 1600 \pi } { R }$$
3. Use calculus to find the value of \(R\), to 3 significant figures, for which \(A\) is a minimum.
4. Prove that this value of \(R\) gives a minimum value for \(A\).
5. Find, to 3 significant figures, the value of \(H\) which corresponds to this value for \(R\).