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Edexcel C12 2014 January Q9
7 marks Moderate -0.3
9. In the first month after opening, a mobile phone shop sold 300 phones. A model for future sales assumes that the number of phones sold will increase by \(5 \%\) per month, so that \(300 \times 1.05\) will be sold in the second month, \(300 \times 1.05 ^ { 2 }\) in the third month, and so on. Using this model, calculate
  1. the number of phones sold in the 24th month,
  2. the total number of phones sold over the whole 24 months. This model predicts that, in the \(N\) th month, the number of phones sold in that month exceeds 3000 for the first time.
  3. Find the value of \(N\).
Edexcel C12 2014 January Q10
9 marks Moderate -0.8
10. The curve \(C\) has equation \(y = \cos \left( x - \frac { \pi } { 3 } \right) , 0 \leqslant x \leqslant 2 \pi\)
  1. In the space below, sketch the curve \(C\).
  2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
  3. Solve, for \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\), $$\cos \left( x - \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number.
Edexcel C12 2014 January Q11
8 marks Moderate -0.8
11. The first three terms of an arithmetic series are \(60,4 p\) and \(2 p - 6\) respectively.
  1. Show that \(p = 9\)
  2. Find the value of the 20th term of this series.
  3. Prove that the sum of the first \(n\) terms of this series is given by the expression $$12 n ( 6 - n )$$ \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-27_106_68_2615_1877}
Edexcel C12 2014 January Q12
11 marks Standard +0.3
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e878227b-d625-4ef2-ac49-a9dc05c5321a-28_549_542_212_703} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram NOT drawn to scale Figure 1 shows the plan for a pond and platform. The platform is shown shaded in the figure and is labelled \(A B C D\). The pond and platform together form a circle of radius 22 m with centre \(O\). \(O A\) and \(O D\) are radii of the circle. Point \(B\) lies on \(O A\) such that the length of \(O B\) is 10 m and point \(C\) lies on \(O D\) such that the length of \(O C\) is 10 m . The length of \(B C\) is 15 m . The platform is bounded by the arc \(A D\) of the circle, and the straight lines \(A B , B C\) and \(C D\). Find
  1. the size of the angle \(B O C\), giving your answer in radians to 3 decimal places,
  2. the perimeter of the platform to 3 significant figures,
  3. the area of the platform to 3 significant figures.
Edexcel C12 2014 January Q13
14 marks Moderate -0.3
13. The curve \(C\) has equation $$y = \frac { ( x - 3 ) ( 3 x - 25 ) } { x } , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in a fully simplified form.
  2. Hence find the coordinates of the turning point on the curve \(C\).
  3. Determine whether this turning point is a minimum or maximum, justifying your answer. The point \(P\), with \(x\) coordinate \(2 \frac { 1 } { 2 }\), lies on the curve \(C\).
  4. Find the equation of the normal at \(P\), in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-35_90_72_2631_1873}
Edexcel C12 2014 January Q14
12 marks Standard +0.3
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e878227b-d625-4ef2-ac49-a9dc05c5321a-36_652_791_223_548} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram NOT drawn to scale Figure 2 shows part of the line \(l\) with equation \(y = 2 x - 3\) and part of the curve \(C\) with equation \(y = x ^ { 2 } - 2 x - 15\) The line \(l\) and the curve \(C\) intersect at the points \(A\) and \(B\) as shown.
  1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). In Figure 2, the shaded region \(R\) is bounded by the line \(l\), the curve \(C\) and the positive \(x\)-axis.
  2. Use integration to calculate an exact value for the area of \(R\).
Edexcel C12 2014 January Q15
14 marks Moderate -0.8
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e878227b-d625-4ef2-ac49-a9dc05c5321a-40_883_824_212_568} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Diagram NOT drawn to scale The points \(X\) and \(Y\) have coordinates \(( 0,3 )\) and \(( 6,11 )\) respectively. \(X Y\) is a chord of a circle \(C\) with centre \(Z\), as shown in Figure 3.
  1. Find the gradient of \(X Y\). The point \(M\) is the midpoint of \(X Y\).
  2. Find an equation for the line which passes through \(Z\) and \(M\). Given that the \(y\) coordinate of \(Z\) is 10 ,
  3. find the \(x\) coordinate of \(Z\),
  4. find the equation of the circle \(C\), giving your answer in the form $$x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$$ where \(a\), \(b\) and \(c\) are constants.
Edexcel C12 2015 January Q1
3 marks Easy -1.8
Simplify the following expressions fully.
  1. \(\left( x ^ { 6 } \right) ^ { \frac { 1 } { 3 } }\)
  2. \(\sqrt { 2 } \left( x ^ { 3 } \right) \div \sqrt { \frac { 32 } { x ^ { 2 } } }\)
Edexcel C12 2015 January Q3
6 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-05_645_933_258_463} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points ( \(2.5,0\) ) and ( 0,9 ), has a stationary point at ( 1,11 ), and has an asymptote \(y = 3\) On separate diagrams, sketch the curve with equation
  1. \(y = 3 \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } ( - x )\) 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.
Edexcel C12 2015 January Q4
7 marks Moderate -0.8
  1. (a) Find the first 4 terms in ascending powers of \(x\) of the binomial expansion of
$$\left( 2 + \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Use your expansion to find an estimated value for \(2.025 ^ { 10 }\), stating the value of \(x\) which you have used and showing your working.
Edexcel C12 2015 January Q5
7 marks Moderate -0.8
5. (a) Prove that the sum of the first \(n\) terms of an arithmetic series is given by the formula $$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$ where \(a\) is the first term of the series and \(d\) is the common difference between the terms.
(b) Find the sum of the integers which are divisible by 7 and lie between 1 and 500
Edexcel C12 2015 January Q6
7 marks Standard +0.3
6. Given that $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$
  1. show that $$4 x ^ { 2 } - 16 x - 9 = 0$$
  2. Hence solve the equation $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$
Edexcel C12 2015 January Q7
8 marks Moderate -0.8
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } + 10 x - 6 y + 18 = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the radius of \(C\). The circle \(C\) meets the line with equation \(x = - 3\) at two points.
  3. Find the exact values for the \(y\) coordinates of these two points, giving your answers as fully simplified surds.
Edexcel C12 2015 January Q8
9 marks Moderate -0.8
  1. A sequence is defined by
$$\begin{aligned} u _ { 1 } & = k \\ u _ { n + 1 } & = 3 u _ { n } - 12 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Write down fully simplified expressions for \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\) in terms of \(k\). Given that \(u _ { 4 } = 15\)
  2. find the value of \(k\),
  3. find \(\sum _ { i = 1 } ^ { 4 } u _ { i }\), giving an exact numerical answer.
Edexcel C12 2015 January Q9
8 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-13_460_698_269_625} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} In Figure 3, the points \(A\) and \(B\) are the centres of the circles \(C _ { 1 }\) and \(C _ { 2 }\) respectively. The circle \(C _ { 1 }\) has radius 10 cm and the circle \(C _ { 2 }\) has radius 5 cm . The circles intersect at the points \(X\) and \(Y\), as shown in the figure. Given that the distance between the centres of the circles is 12 cm ,
  1. calculate the size of the acute angle \(X A B\), giving your answer in radians to 3 significant figures,
  2. find the area of the major sector of circle \(C _ { 1 }\), shown shaded in Figure 3,
  3. find the area of the kite \(A Y B X\).
Edexcel C12 2015 January Q10
9 marks Moderate -0.3
10. $$f ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x - 5$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) there is no remainder.
When \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is - 15
  1. Find the value of \(a\) and the value of \(b\).
  2. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C12 2015 January Q11
8 marks Moderate -0.8
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-16_608_952_267_495} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \sin \left( x - 60 ^ { \circ } \right) , - 360 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\)
  1. Write down the exact coordinates of the points at which \(C\) meets the two coordinate axes.
  2. Solve, for \(- 360 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), $$4 \sin \left( x - 60 ^ { \circ } \right) = \sqrt { 6 } - \sqrt { 2 }$$ showing each stage of your working.
Edexcel C12 2015 January Q12
10 marks Moderate -0.8
12. A business is expected to have a yearly profit of \(\pounds 275000\) for the year 2016. The profit is expected to increase by \(10 \%\) per year, so that the expected yearly profits form a geometric sequence with common ratio 1.1
  1. Show that the difference between the expected profit for the year 2020 and the expected profit for the year 2021 is \(\pounds 40300\) to the nearest hundred pounds.
  2. Find the first year for which the expected yearly profit is more than one million pounds.
  3. Find the total expected profits for the years 2016 to 2026 inclusive, giving your answer to the nearest hundred pounds.
Edexcel C12 2015 January Q13
15 marks Standard +0.3
13. The curve \(C\) has equation $$y = 3 x ^ { 2 } - 4 x + 2$$ The line \(l _ { 1 }\) is the normal to the curve \(C\) at the point \(P ( 1,1 )\)
  1. Show that \(l _ { 1 }\) has equation $$x + 2 y - 3 = 0$$ The line \(l _ { 1 }\) meets curve \(C\) again at the point \(Q\).
  2. By solving simultaneous equations, determine the coordinates of the point \(Q\). Another line \(l _ { 2 }\) has equation \(k x + 2 y - 3 = 0\), where \(k\) is a constant.
  3. Show that the line \(l _ { 2 }\) meets the curve \(C\) once only when $$k ^ { 2 } - 16 k + 40 = 0$$
  4. Find the two exact values of \(k\) for which \(l _ { 2 }\) is a tangent to \(C\).
Edexcel C12 2015 January Q14
10 marks Standard +0.3
14. In this question, solutions based entirely on graphical or numerical methods are not acceptable.
  1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$3 \sin x + 7 \cos x = 0$$ Give each solution, in degrees, to one decimal place.
  2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$10 \cos ^ { 2 } \theta + \cos \theta = 11 \sin ^ { 2 } \theta - 9$$ Give each solution, in radians, to 3 significant figures.
Edexcel C12 2015 January Q15
11 marks Moderate -0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-24_591_570_255_678} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } + 10 x ^ { \frac { 3 } { 2 } } + k x , \quad x \geqslant 0$$ where \(k\) is a constant.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\) on the curve \(C\) is a minimum turning point.
    Given that the \(x\) coordinate of \(P\) is 4
  2. show that \(k = - 78\) The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\).
    The finite region \(R\), shown shaded in Figure 5, is bounded by \(C\), the \(y\)-axis and \(P N\).
  3. Use integration to find the area of \(R\).
Edexcel C12 2016 January Q1
5 marks Easy -1.2
  1. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies
$$u _ { n + 1 } = 2 u _ { n } - 6 , \quad n \geqslant 1$$ Given that \(u _ { 1 } = 2\)
  1. find the value of \(u _ { 3 }\)
  2. evaluate \(\sum _ { i = 1 } ^ { 4 } u _ { i }\)
Edexcel C12 2016 January Q2
5 marks Easy -1.2
2. (i) Given that \(\frac { 49 } { \sqrt { 7 } } = 7 ^ { a }\), find the value of \(a\).
(ii) Show that \(\frac { 10 } { \sqrt { 18 } - 4 } = 15 \sqrt { 2 } + 20\) You must show all stages of your working.
Edexcel C12 2016 January Q3
5 marks Easy -1.2
3. Find, using calculus and showing each step of your working, $$\int _ { 1 } ^ { 4 } \left( 6 x - 3 - \frac { 2 } { \sqrt { x } } \right) \mathrm { d } x$$
Edexcel C12 2016 January Q4
6 marks Moderate -0.5
4. The \(4 ^ { \text {th } }\) term of an arithmetic sequence is 3 and the sum of the first 6 terms is 27 Find the first term and the common difference of this sequence.