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Edexcel C12 2017 October Q15

14 marks Standard +0.3

15. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-42_695_1450_251_246} \captionsetup{labelformat=empty} \caption{Figure 5}

\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. 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 $y = \mathrm { f } ( x ) , x > 0$
Use algebra to show that the $x$ coordinate of $Q$ satisfies the equation $$x ^ { 2 } - 2 x - 15 = 0$$
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 $P Q$.
Use calculus to find the exact area of $S$.

Edexcel C12 2017 October Q16

5 marks Moderate -0.3

$\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 2 x - 5$, 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)
VILIV SIMI NI III IM I ON OC
VILV SIHI NI JAHMMION OC
VALV SIHI NI JIIIM ION OC

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Edexcel C12 2018 October Q1

5 marks Easy -1.2

(i) Given that $125 \sqrt { 5 } = 5 ^ { a }$, find the value of $a$.
(ii) Show that $\frac { 16 } { 4 - \sqrt { 8 } } = 8 + 4 \sqrt { 2 }$

You must show all stages of your working.

Edexcel C12 2018 October Q2

7 marks Moderate -0.8

2. Use algebra to solve the simultaneous equations $$\begin{aligned} x + y & = 5 \\ x ^ { 2 } + x + y ^ { 2 } & = 51 \end{aligned}$$ You must show all stages of your working.

VIIIV SIHI NI III IM ION OC

VIIV SIHI NI JIIIM ION OC

VI4V SIHI NI JIIIM ION OO

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Edexcel C12 2018 October Q3

7 marks Easy -1.2

3. Given that $y = 2 x ^ { 3 } - \frac { 5 } { 3 x ^ { 2 } } + 7 , x \neq 0$, find in its simplest form

$\frac { \mathrm { d } y } { \mathrm {~d} x }$,
$\int y \mathrm {~d} x$.
VIIN SIHI NI IIIIM ION OC VIIN SIHI NI JYHM IONOO VI4V SIHI NI JIIIM ION OC

Edexcel C12 2018 October Q4

6 marks Moderate -0.8

4. A sequence of numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ satisfies $$u _ { n } = k n - 3 ^ { n }$$ where $k$ is a constant. Given that $u _ { 2 } = u _ { 4 }$

find the value of $k$
evaluate $\sum _ { r = 1 } ^ { 4 } u _ { r }$

Edexcel C12 2018 October Q5

6 marks Moderate -0.3

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

$$\left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of $x ^ { 3 }$ in the expansion of $$\left( 3 + 5 x - 2 x ^ { 2 } \right) \left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$

Edexcel C12 2018 October Q6

5 marks Easy -1.3

6. (a) Sketch the graph of $y = \left( \frac { 1 } { 2 } \right) ^ { x } , x \in \mathbb { R }$, showing the coordinates of the point at which the graph crosses the $y$-axis. The table below gives corresponding values of $x$ and $y$, for $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ The values of $y$ are rounded to 3 decimal places.

$x$	- 0.9	- 0.8	- 0.7	- 0.6	- 0.5
$y$	1.866	1.741	1.625	1.516	1.414

(b) Use the trapezium rule with all the values of $y$ from the table to find an approximate value for $$\int _ { - 0.9 } ^ { - 0.5 } \left( \frac { 1 } { 2 } \right) ^ { x } d x$$ II

Edexcel C12 2018 October Q7

8 marks Moderate -0.8

7. The point $A$ has coordinates $( - 1,5 )$ and the point $B$ has coordinates $( 4,1 )$. The line $l$ passes through the points $A$ and $B$.

Find the gradient of $l$.
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 point $M$ is the midpoint of $A B$. The point $C$ has coordinates $( 5 , k )$ where $k$ is a constant.
Given that the distance from $M$ to $C$ is $\sqrt { 13 }$
find the exact possible values of the constant $k$.

Edexcel C12 2018 October Q8

9 marks Moderate -0.3

8. $$f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } + p x + q$$ where $p$ and $q$ are constants.
When $\mathrm { f } ( x )$ is divided by $( x - 1 )$, the remainder is - 6

Use the remainder theorem to show that $p + q = - 5$ Given also that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$,
find the value of $p$ and the value of $q$.
Factorise $\mathrm { f } ( \mathrm { x } )$ completely.

Edexcel C12 2018 October Q9

7 marks Easy -1.2

9. A car manufacturer currently makes 1000 cars each week. The manufacturer plans to increase the number of cars it makes each week. The number of cars made will be increased by 20 each week from 1000 in week 1, to 1020 in week 2, to 1040 in week 3 and so on, until 1500 cars are made in week $N$.

Find the value of $N$. The car manufacturer then plans to continue to make 1500 cars each week.
Find the total number of cars that will be made in the first 50 weeks starting from and including week 1.

Edexcel C12 2018 October Q10

11 marks Standard +0.3

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-28_826_1632_264_153} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The finite region $R$, which is shown shaded in Figure 1, is bounded by the coordinate axes, the straight line $l$ with equation $y = \frac { 1 } { 3 } x + 5$ and the curve $C$ with equation $y = 4 x ^ { \frac { 1 } { 2 } } - x + 5 , x \geqslant 0$ The line $l$ meets the curve $C$ at the point $D$ on the $y$-axis and at the point $E$, as shown in Figure 1.

Use algebra to find the coordinates of the points $D$ and $E$. The curve $C$ crosses the $x$-axis at the point $F$.
Verify that the $x$ coordinate of $F$ is 25
Use algebraic integration to find the exact area of the shaded region $R$.

Edexcel C12 2018 October Q11

8 marks Moderate -0.3

11. The equation $7 x ^ { 2 } + 2 k x + k ^ { 2 } = k + 7$, where $k$ is a constant, has two distinct real roots.

Show that $k$ satisfies the inequality $$6 k ^ { 2 } - 7 k - 49 < 0$$
Find the range of possible values for $k$.

Find the coordinates of the centre of $C$. Given that the radius of $C$ is 10
find the value of $k$. The point $A ( a , - 16 )$, where $a > 0$, lies on the circle $C$. The tangent to $C$ at the point $A$ crosses the $x$-axis at the point $D$ and crosses the $y$-axis at the point $E$.
Find the exact area of triangle $O D E$.

Edexcel C12 2018 October Q15

11 marks Standard +0.3

15. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-46_396_591_251_664} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a plan for a garden.
The garden consists of two identical rectangles of width $y \mathrm {~m}$ and length $x \mathrm {~m}$, joined to a sector of a circle with radius $x \mathrm {~m}$ and angle 0.8 radians, as shown in Figure 2. The area of the garden is $60 \mathrm {~m} ^ { 2 }$.

Show that the perimeter, $P \mathrm {~m}$, of the garden is given by $$P = 2 x + \frac { 120 } { x }$$
Use calculus to find the exact minimum value for $P$, giving your answer in the form $a \sqrt { b }$, where $a$ and $b$ are integers.
Justify that the value of $P$ found in part (b) is the minimum. \includegraphics[max width=\textwidth, alt={}, center]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-49_83_59_2636_1886}

Edexcel C12 2018 October Q16

9 marks Moderate -0.3

16. The first three terms of a geometric series are $( k + 5 ) , k$ and $( 2 k - 24 )$ respectively, where $k$ is a constant.

Show that $k ^ { 2 } - 14 k - 120 = 0$
Hence find the possible values of $k$.
Given that the series is convergent, find
1. the common ratio,
2. the sum to infinity.
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Edexcel C12 Specimen Q1

3 marks Easy -1.5

Simplify fully

$\left( 25 x ^ { 4 } \right) ^ { \frac { 1 } { 2 } }$,
$\left( 25 x ^ { 4 } \right) ^ { - \frac { 3 } { 2 } }$.

Edexcel C12 Specimen Q3

6 marks Easy -1.3

3. Answer this question without the use of a calculator and show all your working.

Show that $$( 5 - \sqrt { 8 } ) ( 1 + \sqrt { 2 } ) \equiv a + b \sqrt { 2 }$$ giving the values of the integers $a$ and $b$.
Show that $$\sqrt { 80 } + \frac { 30 } { \sqrt { 5 } } \equiv c \sqrt { 5 } , \text { where } c \text { is an integer. }$$