Edexcel C1 (Core Mathematics 1) 2016 June

1. Find

$$\int \left( 2 x ^ { 4 } - \frac { 4 } { \sqrt { } x } + 3 \right) d x$$ giving each term in its simplest form.

Express \(9 ^ { 3 x + 1 }\) in the form \(3 ^ { y }\), giving \(y\) in the form \(a x + b\), where \(a\) and \(b\) are constants.
(2)

1. Simplify $$\sqrt { 50 } - \sqrt { 18 }$$ giving your answer in the form \(a \sqrt { 2 }\), where \(a\) is an integer.
2. Hence, or otherwise, simplify $$\frac { 12 \sqrt { 3 } } { \sqrt { 50 } - \sqrt { 18 } }$$ giving your answer in the form \(b \sqrt { c }\), where \(b\) and \(c\) are integers and \(b \neq 1\)

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum point \(A\) at \(( - 2,4 )\) and a minimum point \(B\) at \(( 3 , - 8 )\) and passes through the origin \(O\). On separate diagrams, sketch the curve with equation

1. \(y = 3 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( x ) - 4\)
   (3) On each diagram, show clearly the coordinates of the maximum and the minimum points and the coordinates of the point where the curve crosses the \(y\)-axis.

5. Solve the simultaneous equations $$\begin{gathered} y + 4 x + 1 = 0
y ^ { 2 } + 5 x ^ { 2 } + 2 x = 0 \end{gathered}$$

6. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 4
a _ { n + 1 } & = 5 - k a _ { n } , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.

1. Write down expressions for \(a _ { 2 }\) and \(a _ { 3 }\) in terms of \(k\). Find
2. \(\sum _ { r = 1 } ^ { 3 } \left( 1 + a _ { r } \right)\) in terms of \(k\), giving your answer in its simplest form,
3. \(\sum _ { r = 1 } ^ { 100 } \left( a _ { r + 1 } + k a _ { r } \right)\)

1. Given that

$$y = 3 x ^ { 2 } + 6 x ^ { \frac { 1 } { 3 } } + \frac { 2 x ^ { 3 } - 7 } { 3 \sqrt { } x } , \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Give each term in your answer in its simplified form.

8. The straight line with equation \(y = 3 x - 7\) does not cross or touch the curve with equation \(y = 2 p x ^ { 2 } - 6 p x + 4 p\), where \(p\) is a constant.

1. Show that \(4 p ^ { 2 } - 20 p + 9 < 0\)
2. Hence find the set of possible values of \(p\).
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9. On John's 10th birthday he received the first of an annual birthday gift of money from his uncle. This first gift was \(\pounds 60\) and on each subsequent birthday the gift was \(\pounds 15\) more than the year before. The amounts of these gifts form an arithmetic sequence.

1. Show that, immediately after his 12th birthday, the total of these gifts was \(\pounds 225\)
2. Find the amount that John received from his uncle as a birthday gift on his 18th birthday.
3. Find the total of these birthday gifts that John had received from his uncle up to and including his 21st birthday. When John had received \(n\) of these birthday gifts, the total money that he had received from these gifts was \(\pounds 3375\)
4. Show that \(n ^ { 2 } + 7 n = 25 \times 18\)
5. Find the value of \(n\), when he had received \(\pounds 3375\) in total, and so determine John's age at this time.

10. \begin{figure}[h] \end{figure} The points \(P ( 0,2 )\) and \(Q ( 3,7 )\) lie on the line \(l _ { 1 }\), as shown in Figure 2.
The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\), passes through \(Q\) and crosses the \(x\)-axis at the point \(R\), as shown in Figure 2. Find

1. an equation for \(l _ { 2 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers,
2. the exact coordinates of \(R\),
3. the exact area of the quadrilateral \(O R Q P\), where \(O\) is the origin.

11. The curve \(C\) has equation \(y = 2 x ^ { 3 } + k x ^ { 2 } + 5 x + 6\), where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\), where \(x = - 2\), lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the line with equation \(2 y - 17 x - 1 = 0\)
   Find
2. the value of \(k\),
3. the value of the \(y\) coordinate of \(P\),
4. the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.