Edexcel C12 (Core Mathematics 1 & 2) 2016 October

1. $$f ( x ) = 3 x ^ { 2 } + x - \frac { 4 } { \sqrt { x } } + 6 x ^ { - 3 } , \quad x > 0$$ Find \(\int \mathrm { f } ( x ) \mathrm { d } x\), simplifying each term.

2. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which

1. \(7 ^ { 2 x } = 14\)
2. \(\log _ { 5 } ( 3 x + 1 ) = - 2\)

3. Answer this question without the use of a calculator and show your method clearly.

1. Show that $$\sqrt { 45 } - \frac { 20 } { \sqrt { 5 } } + \sqrt { 6 } \sqrt { 30 } = 5 \sqrt { 5 }$$
2. Show that $$\frac { 17 \sqrt { 2 } } { \sqrt { 2 } + 6 } = 3 \sqrt { 2 } - 1$$

4. $$f ( x ) = 6 x ^ { 3 } - 7 x ^ { 2 } - 43 x + 30$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
   1. \(2 x + 1\)
   2. \(x - 3\)
2. Hence factorise \(\mathrm { f } ( x )\) completely.

5. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 3 - \frac { a x } { 2 } \right) ^ { 5 }$$ where \(a\) is a positive constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 3 }\),
(b) find the exact value of \(a\) in its simplest form.

6. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 36
u _ { n + 1 } & = \frac { 2 } { 3 } u _ { n } , \quad n \geqslant 1 \end{aligned}$$

1. Find the exact simplified values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. Write down the common ratio of the sequence.
3. Find, giving your answer to 4 significant figures, the value of \(u _ { 11 }\)
4. Find the exact value of \(\sum _ { i = 1 } ^ { 6 } u _ { i }\)
5. Find the value of \(\sum _ { i = 1 } ^ { \infty } u _ { i }\)

1. (a) Sketch the graph of \(y = 3 ^ { x - 2 } , x \in \mathbb { R }\)

Give the exact values for the coordinates of the point where your graph crosses the \(y\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = 3 ^ { x - 2 }\)
The values of \(y\) are 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 for $$\int _ { 0.5 } ^ { 3 } 3 ^ { x - 2 } \mathrm {~d} x$$ Give your answer to 2 decimal places.

8. \begin{figure}[h] \end{figure} The compound shape \(A B C D A\), shown in Figure 1, consists of a triangle \(A B D\) joined along its edge \(B D\) to a sector \(D B C\) of a circle with centre \(B\) and radius 6 cm . The points \(A , B\) and \(C\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B C = 6 \mathrm {~cm}\). Angle \(D A B = 1.1\) radians.

1. Show that angle \(A B D = 1.20\) radians to 3 significant figures.
2. Find the area of the compound shape, giving your answer to 3 significant figures.

1. In a large theatre there are 20 rows of seats.

The number of seats in the first row is \(a\), where \(a\) is a constant. In the second row the number of seats is \(( a + d )\), where \(d\) is a constant. In the third row the number of seats is \(( a + 2 d )\), and on each subsequent row there are \(d\) more seats than on the previous row. The number of seats in each row forms an arithmetic sequence. The total number of seats in the first 10 rows is 395

1. Use this information to show that \(10 a + 45 d = 395\) The total number of seats in the first 18 rows is 927
2. Use this information to write down a second simplified equation relating \(a\) and \(d\).
3. Solve these equations to find the value of \(a\) and the value of \(d\).
4. Find the number of seats in the 20th row of the theatre.

10. (a) Given that $$8 \tan x = - 3 \cos x$$ show that $$3 \sin ^ { 2 } x - 8 \sin x - 3 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$8 \tan 2 \theta = - 3 \cos 2 \theta$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-29_124_37_2615_1882}

11. The equation \(5 x ^ { 2 } + 6 = k \left( 13 x ^ { 2 } - 12 x \right)\), where \(k\) is a constant, has two distinct real roots.

1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } + 13 k - 5 > 0$$
2. Find the set of possible values for \(k\).

12. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { x ^ { 3 } - 9 x ^ { 2 } - 81 x } { 27 }$$ The curve crosses the \(x\)-axis at the point \(A\), the point \(B\) and the origin \(O\). The curve has a maximum turning point at \(C\) and a minimum turning point at \(D\).

1. Use algebra to find exact values for the \(x\) coordinates of the points \(A\) and \(B\).
2. Use calculus to find the coordinates of the points \(C\) and \(D\). The graph of \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, has its minimum turning point on the \(y\)-axis.
3. Write down the value of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-35_29_37_182_1914}

13. The circle \(C\) has centre \(A ( 1 , - 3 )\) and passes through the point \(P ( 8 , - 2 )\).

1. Find an equation for the circle \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(P\).
2. Find an equation for \(l _ { 1 }\), giving your answer in the form \(y = m x + c\) The line \(l _ { 2 }\), with equation \(y = x + 6\), is the tangent to \(C\) at the point \(Q\).
3. Find the coordinates of the point \(Q\).

14. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = - x ^ { 2 } + 6 x - 8\). The normal to \(C\) at the point \(P ( 5 , - 3 )\) is the line \(l\), which is also shown in Figure 3.

1. 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 finite region \(R\), shown shaded in Figure 3, is bounded below by the line \(l\) and the curve \(C\), and is bounded above by the \(x\)-axis.
2. Find the exact value of the area of \(R\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

15. \begin{figure}[h] \end{figure} Figure 4 shows a solid wooden block. The block is a right prism with length \(h \mathrm {~cm}\). The cross-section of the block is a semi-circle with radius \(r \mathrm {~cm}\). The total surface area of the block, including the curved surface, the two semi-circular ends and the rectangular base, is \(200 \mathrm {~cm} ^ { 2 }\)

1. Show that the volume \(V \mathrm {~cm} ^ { 3 }\) of the block is given by $$V = \frac { \pi r \left( 200 - \pi r ^ { 2 } \right) } { 4 + 2 \pi }$$
2. Use calculus to find the maximum value of \(V\). Give your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify, by further differentiation, that the value of \(V\) that you have found is a maximum.