Edexcel C12 (Core Mathematics 1 & 2) 2018 October

1. (i) Given that \(125 \sqrt { 5 } = 5 ^ { a }\), find the value of \(a\).
   (ii) Show that \(\frac { 16 } { 4 - \sqrt { 8 } } = 8 + 4 \sqrt { 2 }\)

You must show all stages of your working.

2. Use algebra to solve the simultaneous equations $$\begin{aligned} x + y & = 5
x ^ { 2 } + x + y ^ { 2 } & = 51 \end{aligned}$$ You must show all stages of your working.
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3. Given that \(y = 2 x ^ { 3 } - \frac { 5 } { 3 x ^ { 2 } } + 7 , x \neq 0\), find in its simplest form

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\int y \mathrm {~d} x\).

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4. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies $$u _ { n } = k n - 3 ^ { n }$$ where \(k\) is a constant. Given that \(u _ { 2 } = u _ { 4 }\)

1. find the value of \(k\)
2. evaluate \(\sum _ { r = 1 } ^ { 4 } u _ { r }\)

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

$$\left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$\left( 3 + 5 x - 2 x ^ { 2 } \right) \left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$

6. (a) Sketch the graph of \(y = \left( \frac { 1 } { 2 } \right) ^ { x } , x \in \mathbb { R }\), showing the coordinates of the point at which the graph crosses the \(y\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) The values of \(y\) are rounded to 3 decimal places. (b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for $$\int _ { - 0.9 } ^ { - 0.5 } \left( \frac { 1 } { 2 } \right) ^ { x } d x$$ II

7. The point \(A\) has coordinates \(( - 1,5 )\) and the point \(B\) has coordinates \(( 4,1 )\). The line \(l\) passes through the points \(A\) and \(B\).

1. Find the gradient of \(l\).
2. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. The point \(M\) is the midpoint of \(A B\). The point \(C\) has coordinates \(( 5 , k )\) where \(k\) is a constant.
   Given that the distance from \(M\) to \(C\) is \(\sqrt { 13 }\)
3. find the exact possible values of the constant \(k\).
   

8. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + p x + q$$ where \(p\) and \(q\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is - 6

1. Use the remainder theorem to show that \(p + q = - 5\) Given also that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(p\) and the value of \(q\).
3. Factorise \(\mathrm { f } ( \mathrm { x } )\) completely.
   

9. A car manufacturer currently makes 1000 cars each week. The manufacturer plans to increase the number of cars it makes each week. The number of cars made will be increased by 20 each week from 1000 in week 1, to 1020 in week 2, to 1040 in week 3 and so on, until 1500 cars are made in week \(N\).

1. Find the value of \(N\). The car manufacturer then plans to continue to make 1500 cars each week.
2. Find the total number of cars that will be made in the first 50 weeks starting from and including week 1.

10. \begin{figure}[h] \end{figure} The finite region \(R\), which is shown shaded in Figure 1, is bounded by the coordinate axes, the straight line \(l\) with equation \(y = \frac { 1 } { 3 } x + 5\) and the curve \(C\) with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - x + 5 , x \geqslant 0\) The line \(l\) meets the curve \(C\) at the point \(D\) on the \(y\)-axis and at the point \(E\), as shown in Figure 1.

1. Use algebra to find the coordinates of the points \(D\) and \(E\). The curve \(C\) crosses the \(x\)-axis at the point \(F\).
2. Verify that the \(x\) coordinate of \(F\) is 25
3. Use algebraic integration to find the exact area of the shaded region \(R\).

11. The equation \(7 x ^ { 2 } + 2 k x + k ^ { 2 } = k + 7\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } - 7 k - 49 < 0$$
2. Find the range of possible values for \(k\).

12. (a) Show that the equation $$6 \cos x - 5 \tan x = 0$$ may be expressed in the form $$6 \sin ^ { 2 } x + 5 \sin x - 6 = 0$$ (b) Hence solve for \(0 \leqslant \theta < 360 ^ { \circ }\) $$6 \cos \left( 2 \theta - 10 ^ { \circ } \right) - 5 \tan \left( 2 \theta - 10 ^ { \circ } \right) = 0$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

13. (i) Find the value of \(x\) for which $$4 ^ { 3 x + 2 } = 3 ^ { 600 }$$ giving your answer to 4 significant figures.
(ii) Given that $$\log _ { a } ( 3 b - 2 ) - 2 \log _ { a } 5 = 4 , \quad a > 0 , a \neq 1 , b > \frac { 2 } { 3 }$$ find an expression for \(b\) in terms of \(a\).

14. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 16 y + k = 0$$ where \(k\) is a constant.

1. Find the coordinates of the centre of \(C\). Given that the radius of \(C\) is 10
2. find the value of \(k\). The point \(A ( a , - 16 )\), where \(a > 0\), lies on the circle \(C\). The tangent to \(C\) at the point \(A\) crosses the \(x\)-axis at the point \(D\) and crosses the \(y\)-axis at the point \(E\).
3. Find the exact area of triangle \(O D E\).

15. \begin{figure}[h] \end{figure} Figure 2 shows a plan for a garden.
The garden consists of two identical rectangles of width \(y \mathrm {~m}\) and length \(x \mathrm {~m}\), joined to a sector of a circle with radius \(x \mathrm {~m}\) and angle 0.8 radians, as shown in Figure 2. The area of the garden is \(60 \mathrm {~m} ^ { 2 }\).

1. Show that the perimeter, \(P \mathrm {~m}\), of the garden is given by $$P = 2 x + \frac { 120 } { x }$$
2. Use calculus to find the exact minimum value for \(P\), giving your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
3. Justify that the value of \(P\) found in part (b) is the minimum. \includegraphics[max width=\textwidth, alt={}, center]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-49_83_59_2636_1886}

16. The first three terms of a geometric series are \(( k + 5 ) , k\) and \(( 2 k - 24 )\) respectively, where \(k\) is a constant.

1. Show that \(k ^ { 2 } - 14 k - 120 = 0\)
2. Hence find the possible values of \(k\).
3. Given that the series is convergent, find
   1. the common ratio,
   2. the sum to infinity.

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