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Edexcel C12 2014 June Q3
7 marks Moderate -0.3
3. Solve, giving each answer to 3 significant figures, the equations
  1. \(4 ^ { a } = 20\)
  2. \(3 + 2 \log _ { 2 } b = \log _ { 2 } ( 30 b )\) (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2014 June Q4
8 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-05_716_725_219_603} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } + \frac { 16 } { x } , \quad x > 0$$ The curve has a minimum turning point at \(A\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Hence find the coordinates of \(A\).
  3. Use your answer to part (b) to write down the turning point of the curve with equation
    1. \(y = \mathrm { f } ( x + 1 )\),
    2. \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).
Edexcel C12 2014 June Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-07_953_929_219_422} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Diagram not drawn to scale Figure 3 shows the points \(P , Q\) and \(R\). Points \(P\) and \(Q\) have coordinates ( \(- 1,4\) ) and ( 4,7 ) respectively.
  1. Find an equation for the straight line passing through points \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The point \(R\) has coordinates ( \(p , - 3\) ), where \(p\) is a positive constant. Given that angle \(Q P R = 90 ^ { \circ }\),
  2. find the value of \(p\).
Edexcel C12 2014 June Q6
7 marks Moderate -0.3
6.
  1. Show that $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv 1 - \tan ^ { 2 } x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$
  2. Hence solve, for \(0 \leqslant x < 2 \pi\), $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } + 2 = 0$$ Give your answers in terms of \(\pi\).
Edexcel C12 2014 June Q7
10 marks Moderate -0.8
7.
  1. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,3 )\). Given that $$f ^ { \prime } ( x ) = \frac { 4 } { x ^ { 3 } } + 2 x - 1$$ find the value of \(\mathrm { f } ( 1 )\).
  2. Given that $$\int _ { 1 } ^ { 4 } ( 3 \sqrt { x } + A ) \mathrm { d } x = 21$$ find the exact value of the constant \(A\).
Edexcel C12 2014 June Q8
7 marks Moderate -0.3
8. Given that $$1 + 12 x + 70 x ^ { 2 } + \ldots$$ is the binomial expansion, in ascending powers of \(x\) of \(( 1 + b x ) ^ { n }\), where \(n \in \mathbb { N }\) and \(b\) is a constant,
  1. show that \(n b = 12\)
  2. find the values of the constants \(b\) and \(n\).
Edexcel C12 2014 June Q9
7 marks Moderate -0.8
9.
  1. Find the value of \(\sum _ { r = 1 } ^ { 20 } ( 3 + 5 r )\)
  2. Given that \(\sum _ { r = 0 } ^ { \infty } \frac { a } { 4 ^ { r } } = 16\), find the value of the constant \(a\).
Edexcel C12 2014 June Q10
7 marks Moderate -0.3
10. The equation $$k x ^ { 2 } + 4 x + k = 2 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 4 < 0$$
  2. Hence find the set of all possible values of \(k\).
Edexcel C12 2014 June Q11
15 marks Challenging +1.2
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-17_1000_956_264_500} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the circle \(C\) with centre \(Q\) and equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 2 y + 5 = 0$$
  1. Find
    1. the coordinates of \(Q\),
    2. the exact value of the radius of \(C\). The tangents to \(C\) from the point \(T ( 8,4 )\) meet \(C\) at the points \(M\) and \(N\), as shown in Figure 4.
  2. Show that the obtuse angle \(M Q N\) is 2.498 radians to 3 decimal places. The region \(R\), shown shaded in Figure 4, is bounded by the tangent \(T N\), the minor arc \(N M\), and the tangent \(M T\).
  3. Find the area of region \(R\).
Edexcel C12 2014 June Q12
15 marks Standard +0.3
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-19_1011_1349_237_310} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 } C\) touches the \(x\)-axis at the origin and cuts the \(x\)-axis at the point \(A\).
  1. Show that the coordinates of \(A\) are \(( 3,0 )\).
  2. Show that the equation of the tangent to \(C\) at the point \(A\) is \(y = - 3 x + 9\) The tangent to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 5.
  3. Use algebra to find the \(x\) coordinate of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the curve \(C\) and the tangent to \(C\) at \(A\).
  4. Find, by using calculus, the area of region \(R\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2014 June Q13
10 marks Standard +0.3
13. The height of sea water, \(h\) metres, on a harbour wall at time \(t\) hours after midnight is given by $$h = 3.7 + 2.5 \cos ( 30 t - 40 ) ^ { \circ } , \quad 0 \leqslant t < 24$$
  1. Calculate the maximum value of \(h\) and the exact time of day when this maximum first occurs. Fishing boats cannot enter the harbour if \(h\) is less than 3
  2. Find the times during the morning between which fishing boats cannot enter the harbour.
    Give these times to the nearest minute.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2014 June Q14
15 marks Standard +0.3
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-23_650_1182_212_383} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} Figure 6 shows a solid triangular prism \(A B C D E F\) in which \(A B = 2 x \mathrm {~cm}\) and \(C D = l \mathrm {~cm}\). The cross section \(A B C\) is an equilateral triangle. The rectangle \(B C D F\) is horizontal and the triangles \(A B C\) and \(D E F\) are vertical.
The total surface area of the prism is \(S \mathrm {~cm} ^ { 2 }\) and the volume of the prism is \(V \mathrm {~cm} ^ { 3 }\).
  1. Show that \(S = 2 x ^ { 2 } \sqrt { 3 } + 6 x l\) Given that \(S = 960\),
  2. show that \(V = 160 x \sqrt { 3 } - x ^ { 3 }\)
  3. Use calculus to find the maximum value of \(V\), giving your answer to the nearest integer.
  4. Justify that the value of \(V\) found in part (c) is a maximum. \includegraphics[max width=\textwidth, alt={}, center]{b85872d4-00b2-499b-9765-f7559d3de66a-24_63_52_2690_1886}
Edexcel C12 2015 June Q1
4 marks Easy -1.2
  1. The line \(l _ { 1 }\) has equation
$$10 x - 2 y + 7 = 0$$
  1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 1 } { 3 } , \frac { 4 } { 3 } \right)\).
  2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C12 2015 June Q2
5 marks Moderate -0.5
2. $$f ( x ) = x ^ { 4 } - x ^ { 3 } + 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 4
When \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) the remainder is 22
Find the value of \(a\) and the value of \(b\).
Edexcel C12 2015 June Q3
3 marks Easy -1.2
3. Given that $$y = \frac { 1 } { 27 } x ^ { 3 }$$ express each of the following in the form \(k x ^ { n }\) where \(k\) and \(n\) are constants.
  1. \(y ^ { \frac { 1 } { 3 } }\)
  2. \(3 y ^ { - 1 }\)
  3. \(\sqrt { ( 27 y ) }\)
Edexcel C12 2015 June Q4
6 marks Easy -1.2
4.
  1. Sketch the graph of \(y = \frac { 1 } { x } , x > 0\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { x }\), with the values for \(y\) rounded to 3 decimal places where necessary.
    \(x\)11.522.53
    \(y\)10.6670.50.40.333
  2. Use the trapezium rule with all the values of \(y\) from the table to find an approximate value, to 2 decimal places, for \(\int _ { 1 } ^ { 3 } \frac { 1 } { x } \mathrm {~d} x\)
Edexcel C12 2015 June Q5
6 marks Moderate -0.8
  1. Find, giving your answer to 3 significant figures, the value of \(y\) for which $$3 ^ { y } = 12$$
  2. Solve, giving an exact answer, the equation $$\log _ { 2 } ( x + 3 ) - \log _ { 2 } ( 2 x + 4 ) = 4$$ (You should show each step in your working.)
Edexcel C12 2015 June Q6
6 marks Moderate -0.8
6.
  1. Find the first 3 terms in ascending powers of \(x\) of the binomial expansion of $$( 2 + a x ) ^ { 6 }$$ where \(a\) is a non-zero constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\)
  2. find the value of \(a\).
Edexcel C12 2015 June Q7
7 marks Moderate -0.8
7.
[diagram]
Figure 1 shows a circle with centre \(O\) and radius 9 cm . The points \(A\) and \(B\) lie on the circumference of this circle. The minor sector \(O A B\) has perimeter 30 cm and the angle between the radii \(O A\) and \(O B\) of this sector is \(\theta\) radians. Find
  1. the length of the arc \(A B\),
  2. the value of \(\theta\),
  3. the area of the minor sector \(O A B\),
  4. the area of triangle \(O A B\), giving your answer to 3 significant figures.
Edexcel C12 2015 June Q8
7 marks Moderate -0.8
8. A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find
  1. the number of houses built in 1986, the first year of the building programme,
  2. the total number of houses built in the 25 years of the programme.
Edexcel C12 2015 June Q9
7 marks Moderate -0.3
9. The equation \(x ^ { 2 } + ( 6 k + 4 ) x + 3 = 0\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$9 k ^ { 2 } + 12 k + 1 < 0$$
  2. Find the range of possible values for \(k\), giving your boundaries as fully simplified surds.
Edexcel C12 2015 June Q10
8 marks Moderate -0.3
10. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 4 \\ u _ { n + 1 } & = \frac { 2 u _ { n } } { 3 } , \quad n \geqslant 1 \end{aligned}$$
  1. Find the exact values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. Find the value of \(u _ { 20 }\), giving your answer to 3 significant figures.
  3. Evaluate $$12 - \sum _ { i = 1 } ^ { 16 } u _ { i }$$ giving your answer to 3 significant figures.
  4. Explain why \(\sum _ { i = 1 } ^ { N } u _ { i } < 12\) for all positive integer values of \(N\).
Edexcel C12 2015 June Q11
11 marks Moderate -0.3
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\), where
$$f ^ { \prime } ( x ) = 3 \sqrt { x } - \frac { 9 } { \sqrt { x } } + 2$$ Given that the point \(P ( 9,14 )\) lies on \(C\),
  1. find \(\mathrm { f } ( x )\), simplifying your answer,
  2. find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
Edexcel C12 2015 June Q12
9 marks Moderate -0.3
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ea81408b-e292-4529-b1e2-e3246503a3ac-17_679_1241_274_500} \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 \(x\)-axis at the origin and at the point \(( 6,0 )\). The curve has maximum points at \(( 1,6 )\) and \(( 5,6 )\) and has a minimum point at \(( 3,2 )\). On separate diagrams sketch the curve with equation
  1. \(y = - \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\)
  3. \(y = \mathrm { f } ( x + 4 )\) On each diagram show clearly the coordinates of the maximum and minimum points, and the coordinates of the points where the curve crosses the \(x\)-axis.
Edexcel C12 2015 June Q13
9 marks Standard +0.3
  1. Showing each step in your reasoning, prove that $$( \sin x + \cos x ) ( 1 - \sin x \cos x ) \equiv \sin ^ { 3 } x + \cos ^ { 3 } x$$
  2. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$3 \sin \theta = \tan \theta$$ giving your answers in degrees to 1 decimal place, as appropriate.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)