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Edexcel C12 2017 June Q8
8 marks Moderate -0.3
8.
  1. Find \(\int \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x\), simplifying each term. Given that \(b\) is a constant and $$\int _ { b } ^ { 4 } \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x = 36$$
  2. show that \(b ^ { 3 } + 2 b ^ { 2 } - 15 b = 0\)
  3. Hence find the possible values of \(b\).
Edexcel C12 2017 June Q9
10 marks Moderate -0.3
9.
  1. Find the exact value of \(x\) for which $$2 \log _ { 10 } ( x - 2 ) - \log _ { 10 } ( x + 5 ) = 0$$
  2. Given $$\log _ { p } ( 4 y + 1 ) - \log _ { p } ( 2 y - 2 ) = 1 \quad p > 2 , y > 1$$ express \(y\) in terms of \(p\).
Edexcel C12 2017 June Q10
8 marks Standard +0.3
  1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { x } { 8 } \right) ^ { 10 }$$ giving each term in its simplest form. $$\mathrm { f } ( x ) = \left( 2 - \frac { x } { 8 } \right) ^ { 10 } ( a + b x ) , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\) in the series expansion of \(\mathrm { f } ( x )\), are 256 and \(352 x\),
  2. find the value of \(a\),
  3. find the value of \(b\).
Edexcel C12 2017 June Q11
10 marks Moderate -0.8
11. Wheat is to be grown on a farm. A model predicts that the mass of wheat harvested on the farm will increase by \(1.5 \%\) per year, so that the mass of wheat harvested each year forms a geometric sequence. Given that the mass of wheat harvested during year one is 6000 tonnes,
  1. show that, according to the model, the mass of wheat harvested on the farm during year 4 will be approximately 6274 tonnes. During year \(N\), according to the model, there is predicted to be more than 8000 tonnes of wheat harvested on the farm.
  2. Find the smallest possible value of \(N\). It costs \(\pounds 5\) per tonne to harvest the wheat.
  3. Assuming the model, find the total amount that it would cost to harvest the wheat from year one to year 10 inclusive. Give your answer to the nearest \(\pounds 1000\).
Edexcel C12 2017 June Q12
12 marks Standard +0.8
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-40_814_713_219_612} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 9 x ^ { 2 } + 26 x - 18$$ The point \(P ( 4,6 )\) lies on \(C\).
  1. Use calculus to show that the normal to \(C\) at the point \(P\) has equation $$2 y + x = 16$$ The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the normal to \(C\) at \(P\).
  2. Show that \(C\) cuts the \(x\)-axis at \(( 1,0 )\)
  3. Showing all your working, use calculus to find the exact area of \(R\).
Edexcel C12 2017 June Q13
10 marks Standard +0.3
13.
  1. Show that the equation $$5 \cos x + 1 = \sin x \tan x$$ can be written in the form $$6 \cos ^ { 2 } x + \cos x - 1 = 0$$
  2. Hence solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$5 \cos 2 \theta + 1 = \sin 2 \theta \tan 2 \theta$$ giving your answers, where appropriate, to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2017 June Q14
8 marks Standard +0.3
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-48_771_812_237_575} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the circle \(C _ { 1 }\) The points \(A ( 1,4 )\) and \(B ( 7,8 )\) lie on \(C _ { 1 }\) Given that \(A B\) is a diameter of the circle \(C _ { 1 }\)
  1. find the coordinates for the centre of \(C _ { 1 }\)
  2. find the exact radius of \(C _ { 1 }\) simplifying your answer. Two distinct circles \(C _ { 2 }\) and \(C _ { 3 }\) each have centre \(( 0,0 )\).
    Given that each of these circles touch circle \(C _ { 1 }\)
  3. find the equation of circle \(C _ { 2 }\) and the equation of circle \(C _ { 3 }\)
Edexcel C12 2017 June Q15
9 marks Moderate -0.8
15. The height of water, \(H\) metres, in a harbour on a particular day is given by the equation $$H = 4 + 1.5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight, and \(\frac { \pi t } { 6 }\) is measured in radians.
  1. Show that the height of the water at 1 a.m. is 4.75 metres.
  2. Find the height of the water at 2 p.m.
  3. Find, to the nearest minute, the first two times when the height of the water is 3 metres.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2018 June Q1
5 marks Easy -1.2
  1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { \sqrt { ( x + 1 ) } }\), with the values
    for \(y\) rounded to 3 decimal places where necessary.
\(x\)03691215
\(y\)10.50.3780.3160.277
  1. Complete the table by giving the value of \(y\) corresponding to \(x = 15\)
  2. Use the trapezium rule with all the values of \(y\) from the completed table to find an approximate value for $$\int _ { 0 } ^ { 15 } \frac { 1 } { \sqrt { ( x + 1 ) } } \mathrm { d } x$$ giving your answer to 2 decimal places.
Edexcel C12 2018 June Q2
6 marks Moderate -0.3
2. $$f ( x ) = a x ^ { 3 } + 2 x ^ { 2 } + b x - 3$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is 1
  1. Show that $$a + 4 b = 28$$ When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 17
  2. Find the value of \(a\) and the value of \(b\).
Edexcel C12 2018 June Q3
6 marks Moderate -0.8
3. The line \(l _ { 1 }\) passes through the points \(A ( - 1,4 )\) and \(B ( 5 , - 8 )\)
  1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is perpendicular to the line \(l _ { 1 }\) and passes through the point \(B ( 5 , - 8 )\)
  2. Find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
    II
    "
Edexcel C12 2018 June Q4
5 marks Easy -1.2
4. Given that $$y = \frac { 64 x ^ { 6 } } { 25 } , x > 0$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are constants.
  1. \(y ^ { - \frac { 1 } { 2 } }\)
  2. \(( 25 y ) ^ { \frac { 2 } { 3 } }\)
Edexcel C12 2018 June Q5
7 marks Moderate -0.8
  1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 1 + \frac { x } { 3 } \right) ^ { 18 }$$ giving each term in its simplest form.
  2. Use the answer to part (a) to find an estimated value for \(\left( \frac { 31 } { 30 } \right) ^ { 18 }\), stating the value of \(x\) that you have used and showing your working. Give your estimate to 4 decimal places. II
Edexcel C12 2018 June Q6
7 marks Standard +0.3
6. Find the exact values of \(x\) for which $$2 \log _ { 5 } ( x + 5 ) - \log _ { 5 } ( 2 x + 2 ) = 2$$ Give your answers as simplified surds.
Edexcel C12 2018 June Q7
8 marks Easy -1.3
7. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = u _ { n } - 5 , \quad n \geqslant 1 \end{aligned}$$ Find the values of
  1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. \(u _ { 100 }\)
  3. \(\sum _ { i = 1 } ^ { 100 } u _ { i }\)
Edexcel C12 2018 June Q8
7 marks Moderate -0.3
8. The equation \(( k - 4 ) x ^ { 2 } - 4 x + k - 2 = 0\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$k ^ { 2 } - 6 k + 4 > 0$$
  2. Find the exact range of possible values for \(k\).
Edexcel C12 2018 June Q9
9 marks Standard +0.3
9. A cyclist aims to travel a total of 1200 km over a number of days. He cycles 12 km on day 1
He increases the distance that he cycles each day by \(6 \%\) of the distance cycled on the previous day, until he reaches the total of 1200 km .
  1. Show that on day 8 he cycles approximately 18 km . He reaches his total of 1200 km on day \(N\), where \(N\) is a positive integer.
  2. Find the value of \(N\). The cyclist stops when he reaches 1200 km .
  3. Find the distance that he cycles on day \(N\). Give your answer to the nearest km .
Edexcel C12 2018 June Q10
10 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce06b37a-aa57-4256-bec8-7277c8a9fc65-24_348_593_221_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram not drawn to scale Figure 1 shows a semicircle with centre \(O\) and radius \(3 \mathrm {~cm} . X Y\) is the diameter of this semicircle. The point Z is on the circumference such that angle \(X O Z = 1.3\) radians. The shaded region enclosed by the chord \(X Z\), the arc \(Z Y\) and the diameter \(X Y\) is a template for a badge. Find, giving each answer to 3 significant figures,
  1. the length of the chord \(X Z\),
  2. the perimeter of the template \(X Z Y X\),
  3. the area of the template.
Edexcel C12 2018 June Q11
10 marks Moderate -0.3
11. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where $$f ^ { \prime } ( x ) = \frac { 5 x ^ { 2 } + 4 } { 2 \sqrt { x } } - 5$$ It is given that the point \(P ( 4,14 )\) lies on \(C\).
  1. Find \(\mathrm { f } ( x )\), writing each term in a simplified form.
  2. Find the equation of the tangent to \(C\) at the point \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C12 2018 June Q12
10 marks Standard +0.3
12. [In this question solutions based entirely on graphical or numerical methods are not acceptable.]
  1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin \left( x + 65 ^ { \circ } \right) + 2 = 0$$ giving your answers in degrees to one decimal place.
  2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$12 \sin ^ { 2 } \theta + \cos \theta = 6$$ giving your answers in radians to 3 significant figures.
Edexcel C12 2018 June Q13
10 marks Moderate -0.8
13. The point \(A ( 9 , - 13 )\) lies on a circle \(C\) with centre the origin and radius \(r\).
  1. Find the exact value of \(r\).
  2. Find an equation of the circle \(C\). A straight line through point \(A\) has equation \(2 y + 3 x = k\), where \(k\) is a constant.
  3. Find the value of \(k\). This straight line cuts the circle again at the point \(B\).
  4. Find the exact coordinates of point \(B\).
Edexcel C12 2018 June Q14
15 marks Standard +0.3
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce06b37a-aa57-4256-bec8-7277c8a9fc65-40_611_1214_219_548} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ^ { 2 } ( 2 x + 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(\left( - \frac { 1 } { 2 } , 0 \right)\), touches it at \(( 2,0 )\) and crosses the \(y\)-axis at ( 0,4 ). There is a maximum turning point at the point marked \(P\).
  1. Use \(\mathrm { f } ^ { \prime } ( x )\) to find the exact coordinates of the turning point \(P\). A second curve \(C _ { 2 }\) has equation \(y = \mathrm { f } ( x + 1 )\).
  2. Write down an equation of the curve \(C _ { 2 }\) You may leave your equation in a factorised form.
  3. Use your answer to part (b) to find the coordinates of the point where the curve \(C _ { 2 }\) meets the \(y\)-axis.
  4. Write down the coordinates of the two turning points on the curve \(C _ { 2 }\)
  5. Sketch the curve \(C _ { 2 }\), with equation \(y = \mathrm { f } ( x + 1 )\), giving the coordinates of the points where the curve crosses or touches the \(x\)-axis.
Edexcel C12 2018 June Q15
10 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce06b37a-aa57-4256-bec8-7277c8a9fc65-44_851_1506_212_260} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A design for a logo consists of two finite regions \(R _ { 1 }\) and \(R _ { 2 }\), shown shaded in Figure 3 .
The region \(R _ { 1 }\) is bounded by the straight line \(l\) and the curve \(C\).
The region \(R _ { 2 }\) is bounded by the straight line \(l\), the curve \(C\) and the line with equation \(x = 5\) The line \(l\) has equation \(y = 8 x + 38\) The curve \(C\) has equation \(y = 4 x ^ { 2 } + 6\) Given that the line \(l\) meets the curve \(C\) at the points \(( - 2,22 )\) and \(( 4,70 )\), use integration to find
  1. the area of the larger lower region, labelled \(R _ { 1 }\)
  2. the exact value of the total area of the two shaded regions. Given that $$\frac { \text { Area of } R _ { 1 } } { \text { Area of } R _ { 2 } } = k$$
  3. find the value of \(k\).
    Leave
    blank
    END
Edexcel C12 2019 June Q1
6 marks Moderate -0.3
The 4th term of a geometric series is 125 and the 7th term is 8
  1. Show that the common ratio of this series is \(\frac { 2 } { 5 }\)
  2. Hence find, to 3 decimal places, the difference between the sum to infinity and the sum of the first 10 terms of this series.
Edexcel C12 2019 June Q2
6 marks Moderate -0.8
  1. Find the value of \(a\) and the value of \(b\) for which \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } \equiv 2 ^ { a x + b }\)
  2. Hence solve the equation \(\frac { 8 ^ { x } } { 2 ^ { x - 1 } } = 2 \sqrt { 2 }\)