Edexcel C12 (Core Mathematics 1 & 2) 2018 June

1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { \sqrt { ( x + 1 ) } }\), with the values
   for \(y\) rounded to 3 decimal places where necessary.

1. Complete the table by giving the value of \(y\) corresponding to \(x = 15\)
2. Use the trapezium rule with all the values of \(y\) from the completed table to find an approximate value for $$\int _ { 0 } ^ { 15 } \frac { 1 } { \sqrt { ( x + 1 ) } } \mathrm { d } x$$ giving your answer to 2 decimal places.

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

1. Show that $$a + 4 b = 28$$ When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 17
2. Find the value of \(a\) and the value of \(b\).

3. The line \(l _ { 1 }\) passes through the points \(A ( - 1,4 )\) and \(B ( 5 , - 8 )\)

1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is perpendicular to the line \(l _ { 1 }\) and passes through the point \(B ( 5 , - 8 )\)
2. Find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
   II
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4. Given that $$y = \frac { 64 x ^ { 6 } } { 25 } , x > 0$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are constants.

1. \(y ^ { - \frac { 1 } { 2 } }\)
2. \(( 25 y ) ^ { \frac { 2 } { 3 } }\)

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

$$\left( 1 + \frac { x } { 3 } \right) ^ { 18 }$$ giving each term in its simplest form.
(b) Use the answer to part (a) to find an estimated value for \(\left( \frac { 31 } { 30 } \right) ^ { 18 }\), stating the value of \(x\) that you have used and showing your working. Give your estimate to 4 decimal places. II

6. Find the exact values of \(x\) for which $$2 \log _ { 5 } ( x + 5 ) - \log _ { 5 } ( 2 x + 2 ) = 2$$ Give your answers as simplified surds.

7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = u _ { n } - 5 , \quad n \geqslant 1 \end{aligned}$$ Find the values of

1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. \(u _ { 100 }\)
3. \(\sum _ { i = 1 } ^ { 100 } u _ { i }\)

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

1. Show that \(k\) satisfies the inequality $$k ^ { 2 } - 6 k + 4 > 0$$
2. Find the exact range of possible values for \(k\).

9. A cyclist aims to travel a total of 1200 km over a number of days. He cycles 12 km on day 1
He increases the distance that he cycles each day by \(6 \%\) of the distance cycled on the previous day, until he reaches the total of 1200 km .

1. Show that on day 8 he cycles approximately 18 km . He reaches his total of 1200 km on day \(N\), where \(N\) is a positive integer.
2. Find the value of \(N\). The cyclist stops when he reaches 1200 km .
3. Find the distance that he cycles on day \(N\). Give your answer to the nearest km .

10. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 1 shows a semicircle with centre \(O\) and radius \(3 \mathrm {~cm} . X Y\) is the diameter of this semicircle. The point Z is on the circumference such that angle \(X O Z = 1.3\) radians. The shaded region enclosed by the chord \(X Z\), the arc \(Z Y\) and the diameter \(X Y\) is a template for a badge. Find, giving each answer to 3 significant figures,

1. the length of the chord \(X Z\),
2. the perimeter of the template \(X Z Y X\),
3. the area of the template.

11. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where $$f ^ { \prime } ( x ) = \frac { 5 x ^ { 2 } + 4 } { 2 \sqrt { x } } - 5$$ It is given that the point \(P ( 4,14 )\) lies on \(C\).

1. Find \(\mathrm { f } ( x )\), writing each term in a simplified form.
2. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

12. [In this question solutions based entirely on graphical or numerical methods are not acceptable.]

1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin \left( x + 65 ^ { \circ } \right) + 2 = 0$$ giving your answers in degrees to one decimal place.
2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$12 \sin ^ { 2 } \theta + \cos \theta = 6$$ giving your answers in radians to 3 significant figures.

13. The point \(A ( 9 , - 13 )\) lies on a circle \(C\) with centre the origin and radius \(r\).

1. Find the exact value of \(r\).
2. Find an equation of the circle \(C\). A straight line through point \(A\) has equation \(2 y + 3 x = k\), where \(k\) is a constant.
3. Find the value of \(k\). This straight line cuts the circle again at the point \(B\).
4. Find the exact coordinates of point \(B\).

14. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ^ { 2 } ( 2 x + 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(\left( - \frac { 1 } { 2 } , 0 \right)\), touches it at \(( 2,0 )\) and crosses the \(y\)-axis at ( 0,4 ). There is a maximum turning point at the point marked \(P\).

1. Use \(\mathrm { f } ^ { \prime } ( x )\) to find the exact coordinates of the turning point \(P\). A second curve \(C _ { 2 }\) has equation \(y = \mathrm { f } ( x + 1 )\).
2. Write down an equation of the curve \(C _ { 2 }\) You may leave your equation in a factorised form.
3. Use your answer to part (b) to find the coordinates of the point where the curve \(C _ { 2 }\) meets the \(y\)-axis.
4. Write down the coordinates of the two turning points on the curve \(C _ { 2 }\)
5. Sketch the curve \(C _ { 2 }\), with equation \(y = \mathrm { f } ( x + 1 )\), giving the coordinates of the points where the curve crosses or touches the \(x\)-axis.

15. \begin{figure}[h] \end{figure} A design for a logo consists of two finite regions \(R _ { 1 }\) and \(R _ { 2 }\), shown shaded in Figure 3 .
The region \(R _ { 1 }\) is bounded by the straight line \(l\) and the curve \(C\).
The region \(R _ { 2 }\) is bounded by the straight line \(l\), the curve \(C\) and the line with equation \(x = 5\)
The line \(l\) has equation \(y = 8 x + 38\)
The curve \(C\) has equation \(y = 4 x ^ { 2 } + 6\)
Given that the line \(l\) meets the curve \(C\) at the points \(( - 2,22 )\) and \(( 4,70 )\), use integration to find

1. the area of the larger lower region, labelled \(R _ { 1 }\)
2. the exact value of the total area of the two shaded regions. Given that $$\frac { \text { Area of } R _ { 1 } } { \text { Area of } R _ { 2 } } = k$$
3. find the value of \(k\).

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