Edexcel C12 (Core Mathematics 1 & 2) 2019 June

1. The 4th term of a geometric series is 125 and the 7th term is 8
   1. Show that the common ratio of this series is \(\frac { 2 } { 5 }\)
   2. Hence find, to 3 decimal places, the difference between the sum to infinity and the sum of the first 10 terms of this series.
      

1. Find the value of \(a\) and the value of \(b\) for which \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } \equiv 2 ^ { a x + b }\)
2. Hence solve the equation \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } = 2 \sqrt { 2 }\)

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points \(( - 6,0 )\) and \(( 0,3 )\), has a stationary point at \(( - 3,9 )\) and has an asymptote with equation \(y = 1\) On separate diagrams, sketch the curve with equation

1. \(y = - \mathrm { f } ( x )\)
2. \(y = \mathrm { f } \left( \frac { 3 } { 2 } x \right)\) On each diagram, show clearly the coordinates of the points of intersection of the curve with the two coordinate axes, the coordinates of the stationary point, and the equation of the asymptote. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-07_2255_45_316_36}

4. Given that $$y = 5 x ^ { 2 } + \frac { 1 } { 2 x } + \frac { 2 x ^ { 4 } - 8 } { 5 \sqrt { x } } \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
(6)

5. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = k - \frac { 8 } { u _ { n } } \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.

1. Write down expressions for \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\). Given that \(u _ { 3 } = 6\)
2. find the possible values of \(k\).

6. (a) Find, in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), the binomial expansion of $$\left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x\) in the expansion of $$\left( 3 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { 1 } { 4 } x \right) ^ { 12 }$$

7. (a) Sketch the graph of \(y = \sin \left( x + \frac { \pi } { 6 } \right) , \quad 0 \leqslant x \leqslant 2 \pi\) Show the coordinates of the points where the graph crosses the \(x\)-axis. The table below gives corresponding values of \(x\) and \(y\) for \(y = \sin \left( x + \frac { \pi } { 6 } \right)\).
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 $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin \left( x + \frac { \pi } { 6 } \right) \mathrm { d } x$$ Give your answer to 2 decimal places.

8. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows the design for a company logo. The design consists of a triangle \(A B E\) joined to a sector \(B C D E\) of a circle with radius 6 cm and centre \(E\). The line \(A E\) is perpendicular to the line \(D E\) and the length of \(A E\) is 9 cm . The size of angle \(D E B\) is 3.5 radians, as shown in Figure 2.

1. Find the length of the arc BCD. Find, to one decimal place,
2. the perimeter of the logo,
3. the area of the logo.

9. \(\mathrm { f } ( x ) = ( x + k ) \left( 3 x ^ { 2 } + 4 x - 16 \right) + 32 , \quad\) where \(k\) is a constant (a) Write down the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + k )\). When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is 15
(b) Show that \(k = 2\)
(c) Hence factorise \(\mathrm { f } ( x )\) completely. \section*{9.} " .
\(\mathrm { f } ( x ) = ( x + k ) \left( 3 x ^ { 2 } + 4 x - 16 \right) + 32 , \quad\) where \(k\) is a constant

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 4 x + p y + 123 = 0$$ where \(p\) is a constant. Given that the point \(( 1,16 )\) lies on \(C\),

1. find
   1. the value of \(p\),
   2. the coordinates of the centre of \(C\),
   3. the radius of \(C\).
2. Find an equation of the tangent to \(C\) at the point ( 1,16 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-31_33_19_2668_1896}

11. The straight line \(l\) has equation \(y = m x - 2\), where \(m\) is a constant. The curve \(C\) has equation \(y = 2 x ^ { 2 } + x + 6\) The line \(l\) does not cross or touch the curve \(C\).

1. Show that \(m\) satisfies the inequality $$m ^ { 2 } - 2 m - 63 < 0$$
2. Hence find the range of possible values of \(m\).

12. (a) Show that $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ may be expressed in the form $$a \cos ^ { 2 } x + b \cos x + c = 0$$ where \(a , b\) and \(c\) are constants to be found.
(b) Hence solve, for \(0 \leqslant x < 2 \pi\), the equation $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ giving your answers in radians to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
\includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-37_81_65_2640_1886}

13. Given that \(p = \log _ { a } 9\) and \(q = \log _ { a } 10\), where \(a\) is a constant, find in terms of \(p\) and \(q\),

1. \(\log _ { a } 900\)
2. \(\log _ { a } 0.3\)

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14. The 5 th term of an arithmetic series is \(4 k\), where \(k\) is a constant. The sum of the first 8 terms of this series is \(20 k + 16\)

1. 1. Find, in terms of \(k\), an expression for the common difference of the series.
   2. Show that the first term of the series is \(16 - 8 k\) Given that the 9th term of the series is 24, find
2. the value of \(k\),
3. the sum of the first 20 terms.
   \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-40_2257_54_314_1977}

15. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 3 shows the plan view of a garden. The shape of this garden consists of a rectangle joined to a semicircle. The rectangle has length \(x\) metres and width \(y\) metres.
The area of the garden is \(100 \mathrm {~m} ^ { 2 }\).

1. Show that the perimeter, \(P\) metres, of the garden is given by $$P = \frac { 1 } { 4 } x ( 4 + \pi ) + \frac { 200 } { x } \quad x > 0$$
2. Use calculus to find the exact value of \(x\) for which the perimeter of the garden is a minimum.
3. Justify that the value of \(x\) found in part (b) gives a minimum value for \(P\).
4. Find the minimum perimeter of the garden, giving your answer in metres to one decimal place.

16. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(y = 2 x ^ { 2 } - 11 x + 12\). The curve crosses the \(y\)-axis at the point \(A\) and crosses the \(x\)-axis at the points \(B\) and \(C\).

1. Find the coordinates of the points \(A , B\) and \(C\). The point \(D\) lies on the curve such that the line \(A D\) is parallel to the \(x\)-axis. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line \(A C\) and the line \(A D\).
2. Use algebraic integration to find the exact area of \(R\).