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Edexcel C12 2016 January Q5
5 marks Moderate -0.8
5. (a) Sketch the graph of \(y = \sin 2 x , \quad 0 \leqslant x \leqslant \frac { 3 \pi } { 2 }\) Show the coordinates of the points where your graph crosses the \(x\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = \sin 2 x\).
The values of \(y\) are rounded to 3 decimal places where necessary.
\(x\)0\(\frac { \pi } { 12 }\)\(\frac { \pi } { 6 }\)\(\frac { \pi } { 4 }\)
\(y\)00.50.8661
(b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for
Edexcel C12 2016 January Q6
7 marks Moderate -0.8
6. $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$
  1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Factorise \(\mathrm { f } ( x )\).
  3. Hence find exact values for all the solutions of the equation \(\mathrm { f } ( x ) = 0\)
Edexcel C12 2016 January Q7
7 marks Moderate -0.8
7. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { 8 }\), where \(k\) is a non-zero constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 3 }\) in this expansion is 1512
(b) find the value of \(k\).
Edexcel C12 2016 January Q8
6 marks Moderate -0.3
8. (a) Given that \(7 \sin x = 3 \cos x\), find the exact value of \(\tan x\).
(b) Hence solve for \(0 \leqslant \theta < 360 ^ { \circ }\) $$7 \sin \left( 2 \theta + 30 ^ { \circ } \right) = 3 \cos \left( 2 \theta + 30 ^ { \circ } \right)$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2016 January Q9
7 marks Standard +0.3
9. The resident population of a city is 130000 at the end of Year 1 A model predicts that the resident population of the city will increase by \(2 \%\) each year, with the populations at the end of each year forming a geometric sequence.
  1. Show that the predicted resident population at the end of Year 2 is 132600
  2. Write down the value of the common ratio of the geometric sequence. The model predicts that Year \(N\) will be the first year which will end with the resident population of the city exceeding 260000
  3. Show that $$N > \frac { \log _ { 10 } 2 } { \log _ { 10 } 1.02 } + 1$$
  4. Find the value of \(N\).
Edexcel C12 2016 January Q10
10 marks Moderate -0.3
10. The curve \(C\) has equation $$y = 12 x ^ { \frac { 5 } { 4 } } - \frac { 5 } { 18 } x ^ { 2 } - 1000 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence find the coordinates of the stationary point on \(C\).
  3. Use \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) to determine the nature of this stationary point.
Edexcel C12 2016 January Q11
11 marks Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{88ed9a17-97a5-4548-80bb-70b4b901a39d-13_625_1155_285_456} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a triangle \(X Y Z\) with \(X Y = 10 \mathrm {~cm} , Y Z = 16 \mathrm {~cm}\) and \(Z X = 12 \mathrm {~cm}\).
  1. Find the size of the angle \(Y X Z\), giving your answer in radians to 3 significant figures. The point \(A\) lies on the line \(X Y\) and the point \(B\) lies on the line \(X Z\) and \(A X = B X = 5 \mathrm {~cm} . A B\) is the arc of a circle with centre \(X\). The shaded region \(S\), shown in Figure 1, is bounded by the lines \(B Z , Z Y , Y A\) and the arc \(A B\). Find
  2. the perimeter of the shaded region to 3 significant figures,
  3. the area of the shaded region to 3 significant figures.
Edexcel C12 2016 January Q12
10 marks Moderate -0.8
12. $$f ( x ) = \frac { ( 4 + 3 \sqrt { } x ) ^ { 2 } } { x } , \quad x > 0$$
  1. Show that \(\mathrm { f } ( x ) = A x ^ { - 1 } + B x ^ { k } + C\), where \(A , B , C\) and \(k\) are constants to be determined.
  2. Hence find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Find an equation of the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = 4\) 2. LIIIII
Edexcel C12 2016 January Q13
8 marks Standard +0.3
13. The equation \(k \left( 3 x ^ { 2 } + 8 x + 9 \right) = 2 - 6 x\), where \(k\) is a real constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$11 k ^ { 2 } - 30 k - 9 > 0$$
  2. Find the range of possible values for \(k\).
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Edexcel C12 2016 January Q14
8 marks Moderate -0.3
  1. (i) Given that
$$\log _ { a } x + \log _ { a } 3 = \log _ { a } 27 - 1 , \text { where } a \text { is a positive constant }$$ find, in its simplest form, an expression for \(x\) in terms of \(a\).
(ii) Solve the equation $$\left( \log _ { 5 } y \right) ^ { 2 } - 7 \left( \log _ { 5 } y \right) + 12 = 0$$ showing each step of your working.
Edexcel C12 2016 January Q15
10 marks Moderate -0.3
15. The points \(A\) and \(B\) have coordinates \(( - 8 , - 8 )\) and \(( 12,2 )\) respectively. \(A B\) is the diameter of a circle \(C\).
  1. Find an equation for the circle \(C\). The point \(( 4,8 )\) also lies on \(C\).
  2. Find an equation of the tangent to \(C\) at the point ( 4,8 ), giving your answer in the form \(a x + b y + c = 0\)
Edexcel C12 2016 January Q16
15 marks Moderate -0.3
16. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{88ed9a17-97a5-4548-80bb-70b4b901a39d-19_835_922_303_513} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line \(l\) with equation \(y = \frac { 1 } { 2 } x + 1\) cuts the curve \(C\), with equation \(y = x ^ { 2 } - 4 x + 3\), at the points \(P\) and \(Q\), as shown in Figure 2
  1. Use algebra to find the coordinates of the points \(P\) and \(Q\). The curve \(C\) crosses the \(x\)-axis at the points \(T\) and \(S\).
  2. Write down the coordinates of the points \(T\) and \(S\). The finite region \(R\) is shown shaded in Figure 2. This region \(R\) is bounded by the line segment \(P Q\), the line segment \(T S\), and the \(\operatorname { arcs } P T\) and \(S Q\) of the curve.
  3. Use integration to find the exact area of the shaded region \(R\).
Edexcel C12 2017 January Q1
7 marks Moderate -0.8
Given \(y = \frac { x ^ { 3 } } { 3 } - 2 x ^ { 2 } + 3 x + 5\)
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying each term.
  2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 0\)
Edexcel C12 2017 January Q3
8 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-04_629_1061_260_555} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shape \(P O Q A B C P\), as shown in Figure 1, consists of a triangle \(P O C\), a sector \(O Q A\) of a circle with radius 7 cm and centre \(O\), joined to a rectangle \(O A B C\). The points \(P , O\) and \(Q\) lie on a straight line. \(P O = 4 \mathrm {~cm} , C O = 5 \mathrm {~cm}\) and angle \(A O Q = 0.8\) radians.
  1. Find the length of arc \(A Q\).
  2. Find the size of angle \(P O C\) in radians, giving your answer to 3 decimal places.
    (2)
  3. Find the perimeter of the shape \(P O Q A B C P\), in cm , giving your answer to 2 decimal places.
    (4)
Edexcel C12 2017 January Q4
6 marks Moderate -0.5
4. An arithmetic series has first term \(a\) and common difference \(d\). Given that the sum of the first 9 terms is 54
  1. show that $$a + 4 d = 6$$ Given also that the 8th term is half the 7th term,
  2. find the values of \(a\) and \(d\).
Edexcel C12 2017 January Q5
7 marks Moderate -0.8
5. (a) Given that $$y = \log _ { 3 } x$$ find expressions in terms of \(y\) for
  1. \(\log _ { 3 } \left( \frac { x } { 9 } \right)\)
  2. \(\log _ { 3 } \sqrt { x }\) Write each answer in its simplest form.
    (b) Hence or otherwise solve $$2 \log _ { 3 } \left( \frac { x } { 9 } \right) - \log _ { 3 } \sqrt { x } = 2$$
Edexcel C12 2017 January Q6
11 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-08_906_1100_127_388} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line \(l _ { 1 }\) has equation \(2 y = 3 x + 5\) The line \(l _ { 1 }\) cuts the \(x\)-axis at the point \(A\), as shown in Figure 2.
    1. State the gradient of \(l _ { 1 }\)
    2. Write down the \(x\) coordinate of point \(A\). Another straight line \(l _ { 2 }\) intersects \(l _ { 1 }\) at the point \(B\) with \(x\) coordinate 1 and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\)
  2. find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers,
  3. find the exact area of triangle \(A B C\).
Edexcel C12 2017 January Q7
9 marks Moderate -0.8
7. (i) Find $$\int \frac { 2 + 4 x ^ { 3 } } { x ^ { 2 } } \mathrm {~d} x$$ giving each term in its simplest form.
(ii) Given that \(k\) is a constant and $$\int _ { 2 } ^ { 4 } \left( \frac { 4 } { \sqrt { x } } + k \right) \mathrm { d } x = 30$$ find the exact value of \(k\).
Edexcel C12 2017 January Q8
10 marks Standard +0.3
8. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } - 23 x - 10$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 3\) ).
  2. Show that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Hence fully factorise \(\mathrm { f } ( x )\).
  4. Hence solve $$2 \left( 3 ^ { 3 t } \right) - 5 \left( 3 ^ { 2 t } \right) - 23 \left( 3 ^ { t } \right) = 10$$ giving your answer to 3 decimal places.
Edexcel C12 2017 January Q9
8 marks Moderate -0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-14_609_744_223_593} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 8 } { x } + \frac { 1 } { 2 } x - 5 , \quad 0 < x \leqslant 12$$ The curve crosses the \(x\)-axis at \(( 2,0 )\) and \(( 8,0 )\) and has a minimum point at \(A\).
  1. Use calculus to find the coordinates of point \(A\).
  2. State
    1. the roots of the equation \(2 \mathrm { f } ( x ) = 0\)
    2. the coordinates of the turning point on the curve \(y = \mathrm { f } ( x ) + 2\)
    3. the roots of the equation \(\mathrm { f } ( 4 x ) = 0\)
Edexcel C12 2017 January Q10
6 marks Moderate -0.8
10. The first 3 terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + a x ) ^ { 20 }\) are given by $$1 + 4 x + p x ^ { 2 }$$ where \(a\) and \(p\) are constants.
  1. Find the value of \(a\).
  2. Find the value of \(p\). One of the terms in the binomial expansion of \(( 1 + a x ) ^ { 20 }\) is \(q x ^ { 4 }\), where \(q\) is a constant.
  3. Find the value of \(q\).
Edexcel C12 2017 January Q11
10 marks Moderate -0.3
11. In this question solutions based entirely on graphical or numerical methods are not acceptable.
  1. Solve, for \(0 \leqslant x < 2 \pi\), $$3 \cos ^ { 2 } x + 1 = 4 \sin ^ { 2 } x$$ giving your answers in radians to 2 decimal places.
  2. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$5 \sin \left( \theta + 10 ^ { \circ } \right) = \cos \left( \theta + 10 ^ { \circ } \right)$$ giving your answers in degrees to one decimal place.
Edexcel C12 2017 January Q12
11 marks Standard +0.3
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-20_775_1015_260_459} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 3 } { 4 } x ^ { 2 } - 4 \sqrt { x } + 7 , \quad x > 0$$ The point \(P\) lies on \(C\) and has coordinates \(( 4,11 )\).
Line \(l\) is the tangent to \(C\) at the point \(P\).
  1. Use calculus to show that \(l\) has equation \(y = 5 x - 9\) The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line \(x = 1\), the \(x\)-axis and the line \(l\).
  2. Find, by using calculus, the area of \(R\), giving your answer to 2 decimal places.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2017 January Q13
11 marks Standard +0.3
13. (a) On separate axes sketch the graphs of
  1. \(y = c ^ { 2 } - x ^ { 2 }\)
  2. \(y = x ^ { 2 } ( x - 3 c )\) where \(c\) is a positive constant.
    Show clearly the coordinates of the points where each graph crosses or meets the \(x\)-axis and the \(y\)-axis.
    (b) Prove that the \(x\) coordinate of any point of intersection of $$y = c ^ { 2 } - x ^ { 2 } \text { and } y = x ^ { 2 } ( x - 3 c )$$ where \(c\) is a positive constant, is given by a solution of the equation $$x ^ { 3 } + ( 1 - 3 c ) x ^ { 2 } - c ^ { 2 } = 0$$ Given that the graphs meet when \(x = 2\) (c) find the exact value of \(c\), writing your answer as a fully simplified surd.
Edexcel C12 2017 January Q14
9 marks Moderate -0.3
14. A geometric series has a first term \(a\) and a common ratio \(r\).
  1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ A liquid is to be stored in a barrel. Due to evaporation, the volume of the liquid in a barrel at the end of a year is \(7 \%\) less than the volume at the start of the year. At the start of the first year, a barrel is filled with 180 litres of the liquid.
  2. Show that the amount of the liquid in this barrel at the end of 5 years is approximately 125.2 litres. At the start of each year a new identical barrel is filled with 180 litres of the liquid so that, at the end of 20 years, there are 20 barrels containing varying amounts of the liquid.
  3. Calculate the total amount of the liquid, to the nearest litre, in the 20 barrels at the end of 20 years.