Edexcel C12 (Core Mathematics 1 & 2) 2017 October

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

$$8 x + 2 y - 15 = 0$$

1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 3 } { 4 } , 16 \right)\).
2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

2. The point \(P ( 2,3 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\). State the coordinates of the image of \(P\) under the transformation represented by the curve with equation

1. \(y = \mathrm { f } ( x + 2 )\)
2. \(y = - \mathrm { f } ( x )\)
3. \(2 y = f ( x )\)
4. \(y = \mathrm { f } ( x ) - 4\)
   State the coordinates of the image of \(P\) under the transformation represented by the curve
   with equation (a) \(y = \mathrm { f } ( x + 2 )\)

3. (a) Express \(\frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } }\) in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants.
(b) Hence find $$\int \frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } } d x$$ simplifying your answer.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C = 24 \sqrt { 3 }\)

1. show that \(x = 4 \sqrt { 2 }\)
2. Hence find the exact length of \(B C\), giving your answer as a simplified surd.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 27 \sqrt { x } - 2 x ^ { 2 } , \quad x \in \mathbb { R } , x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The curve has a maximum turning point \(P\), as shown in Figure 2.
2. Use the answer to part (a) to find the exact coordinates of \(P\).

1. Each year Lin pays into a savings scheme. In year 1 she pays in \(\pounds 600\). Her payments then increase by \(\pounds 80\) a year, so that she pays \(\pounds 680\) into the savings scheme in year \(2 , \pounds 760\) in year 3 and so on. In year \(N\), Lin pays \(\pounds 1000\) into the savings scheme.
   1. Find the value of \(N\).
   2. Find the total amount that Lin pays into the savings scheme from year 1 to year 15 inclusive.
   Saima starts paying into a different savings scheme at the same time as Lin starts paying into her savings scheme. In year 1 she pays in \(\pounds A\). Her payments increase by \(\pounds A\) each year so that she pays \(\pounds 2 A\) in year \(2 , \pounds 3 A\) in year 3 and so on. Given that Saima and Lin have each paid, in total, the same amount of money into their savings schemes after 15 years,
2. find the value of \(A\).

7. $$g ( x ) = 2 x ^ { 3 } + a x ^ { 2 } - 18 x - 8$$ Given that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\),

1. show that \(a = - 3\)
2. Hence, using algebra, fully factorise \(\mathrm { g } ( x )\). Using your answer to part (b),
3. solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$2 \sin ^ { 3 } \theta - 3 \sin ^ { 2 } \theta - 18 \sin \theta = 8$$ giving each answer, in radians, as a multiple of \(\pi\).
   

8. \begin{figure}[h] \end{figure} Figure 3 shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circumference of this circle. The minor arc \(A B\) subtends an angle \(\theta\) radians at \(O\), as shown in Figure 3.
Given the length of minor \(\operatorname { arc } A B\) is 6 cm and the area of minor sector \(O A B\) is \(20 \mathrm {~cm} ^ { 2 }\),

1. write down two different equations in \(r\) and \(\theta\).
2. Hence find the value of \(r\) and the value of \(\theta\).
   

1. (a) Given that \(a\) is a constant, \(a > 1\), sketch the graph of

$$y = a ^ { x } , \quad x \in \mathbb { R }$$ On your diagram show the coordinates of the point where the graph crosses the \(y\)-axis.
(2) The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 ^ { x }\) (b) Use the trapezium rule, with all of the values of \(y\) from the table, to find an approximate value, to 2 decimal places, for $$\int _ { - 4 } ^ { 4 } 2 ^ { x } \mathrm {~d} x$$ (c) Use the answer to part (b) to find an approximate value for

1. \(\int _ { - 4 } ^ { 4 } 2 ^ { x + 2 } \mathrm {~d} x\)
2. \(\int _ { - 4 } ^ { 4 } \left( 3 + 2 ^ { x } \right) \mathrm { d } x\)
   \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-23_86_47_2617_1886}

10. \begin{figure}[h] \end{figure} Diagram NOT drawn to scale The points \(A ( 7 , - 3 ) , B ( 7,20 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 4. The point \(D ( 10,5 )\) is the midpoint of \(A C\).

1. Find the value of \(p\) and the value of \(q\). The line \(l\) passes through \(D\) and is perpendicular to \(A C\).
2. Find an equation for \(l\), in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. Given that the line \(l\) intersects \(A B\) at \(E\),
3. find the exact coordinates of \(E\).

11. \(\mathrm { f } ( x ) = ( a - x ) ( 3 + a x ) ^ { 5 }\), where \(a\) is a positive constant

1. Find the first 3 terms, in ascending powers of \(x\), in the binomial expansion of $$( 3 + a x ) ^ { 5 }$$ Give each term in its simplest form. Given that in the expansion of \(\mathrm { f } ( x )\) the coefficient of \(x\) is zero,
2. find the exact value of \(a\).

12. (i) Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 2 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers, in degrees, to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(ii) (a) Given that $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ show that $$\tan ^ { 2 } x = k , \quad \text { where } k \text { is a constant. }$$ (b) Hence solve, for \(0 < x \leqslant 2 \pi\), $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ giving your answers, in radians, to 3 decimal places.

1. The circle \(C\) has equation

$$( x - 3 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 30$$ Write down

1. 1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). Given that the point \(P\) with coordinates \(( 6 , k )\), where \(k\) is a constant, lies inside circle \(C\), (b) show that $$k ^ { 2 } + 8 k - 5 < 0$$
2. Hence find the exact set of values of \(k\) for which \(P\) lies inside \(C\).
   \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-34_2256_52_315_1978}

1. A new mineral has been discovered and is going to be mined over a number of years.

A model predicts that the mass of the mineral mined each year will decrease by \(15 \%\) per year, so that the mass of the mineral mined each year forms a geometric sequence. Given that the mass of the mineral mined during year 1 is 8000 tonnes,

1. show that, according to the model, the mass of the mineral mined during year 6 will be approximately 3550 tonnes. According to the model, there is a limit to the total mass of the mineral that can be mined.
2. With reference to the geometric series, state why this limit exists.
3. Calculate the value of this limit. It is decided that a total mass of 40000 tonnes of the mineral is required. This is going to be mined from year 1 to year \(N\) inclusive.
4. Assuming the model, find the value of \(N\).
   

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { ( x - 3 ) ^ { 2 } ( x + 4 ) } { 2 } , \quad x \in \mathbb { R }$$ The graph cuts the \(y\)-axis at the point \(P\) and meets the positive \(x\)-axis at the point \(R\), as shown in Figure 5.

1. 1. State the \(y\) coordinate of \(P\).
   2. State the \(x\) coordinate of \(R\). The line segment \(P Q\) is parallel to the \(x\)-axis. Point \(Q\) lies on \(y = \mathrm { f } ( x ) , x > 0\)
2. Use algebra to show that the \(x\) coordinate of \(Q\) satisfies the equation $$x ^ { 2 } - 2 x - 15 = 0$$
3. Use part (b) to find the coordinates of \(Q\). The region \(S\), shown shaded in Figure 5, is bounded by the curve \(y = \mathrm { f } ( x )\) and the line segment \(P Q\).
4. Use calculus to find the exact area of \(S\).

1. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 2 x - 5\), where \(a\) and \(b\) are constants The point \(P ( 1,4 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\).

The tangent to \(y = \mathrm { f } ( x )\) at the point \(P\) has equation \(y = 12 x - 8\) Calculate the value of \(a\) and the value of \(b\).
(5)
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