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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 \(B\).
  1. Find the gradient of \(l\).
  2. 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 \(\sqrt { 13 }\)
  3. 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
  1. Use the remainder theorem to show that \(p + q = - 5\) Given also that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(p\) and the value of \(q\).
  3. 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 \(N\).
  1. Find the value of \(N\). The car manufacturer then plans to continue to make 1500 cars each week.
  2. 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.
  1. Use algebra to find the coordinates of the points \(D\) and \(E\). The curve \(C\) crosses the \(x\)-axis at the point \(F\).
  2. Verify that the \(x\) coordinate of \(F\) is 25
  3. 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.
  1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } - 7 k - 49 < 0$$
  2. Find the range of possible values for \(k\).
Edexcel C12 2018 October Q12
8 marks Standard +0.3
12.
  1. Show that the equation $$6 \cos x - 5 \tan x = 0$$ may be expressed in the form $$6 \sin ^ { 2 } x + 5 \sin x - 6 = 0$$
  2. Hence solve for \(0 \leqslant \theta < 360 ^ { \circ }\) $$6 \cos \left( 2 \theta - 10 ^ { \circ } \right) - 5 \tan \left( 2 \theta - 10 ^ { \circ } \right) = 0$$ giving your answers to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2018 October Q13
7 marks Moderate -0.3
13.
  1. Find the value of \(x\) for which $$4 ^ { 3 x + 2 } = 3 ^ { 600 }$$ giving your answer to 4 significant figures.
  2. Given that $$\log _ { a } ( 3 b - 2 ) - 2 \log _ { a } 5 = 4 , \quad a > 0 , a \neq 1 , b > \frac { 2 } { 3 }$$ find an expression for \(b\) in terms of \(a\).
Edexcel C12 2018 October Q14
11 marks Standard +0.8
14. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 16 y + k = 0$$ where \(k\) is a constant.
  1. Find the coordinates of the centre of \(C\). Given that the radius of \(C\) is 10
  2. 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 \(E\).
  3. 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 \(60 \mathrm {~m} ^ { 2 }\).
  1. Show that the perimeter, \(P \mathrm {~m}\), of the garden is given by $$P = 2 x + \frac { 120 } { x }$$
  2. 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.
  3. 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.
  1. Show that \(k ^ { 2 } - 14 k - 120 = 0\)
  2. Hence find the possible values of \(k\).
  3. Given that the series is convergent, find
    1. the common ratio,
    2. the sum to infinity.
      Leave blankQ16
      END
Edexcel C12 Specimen Q1
3 marks Easy -1.5
Simplify fully
  1. \(\left( 25 x ^ { 4 } \right) ^ { \frac { 1 } { 2 } }\),
  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.
  1. Show that $$( 5 - \sqrt { 8 } ) ( 1 + \sqrt { 2 } ) \equiv a + b \sqrt { 2 }$$ giving the values of the integers \(a\) and \(b\).
  2. Show that $$\sqrt { 80 } + \frac { 30 } { \sqrt { 5 } } \equiv c \sqrt { 5 } , \text { where } c \text { is an integer. }$$
Edexcel C12 Specimen Q4
7 marks Easy -1.2
4. Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form,
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\int y \mathrm {~d} x\).
Edexcel C12 Specimen Q5
4 marks Easy -1.2
5. $$y = \frac { 5 } { 3 x ^ { 2 } - 2 }$$ The table below gives values of \(y\) rounded to 3 decimal places where necessary.
\(x\)22.252.52.753
\(y\)0.50.3790.2990.2420.2
Use the trapezium rule, with all the values of \(y\) from the table above, to find an approximate value for $$\int _ { 2 } ^ { 3 } \frac { 5 } { 3 x ^ { 2 } - 2 } d x$$ © Pearson Education Limited 2013
Sample Assessment Materials
Edexcel C12 Specimen Q6
7 marks Moderate -0.5
6. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is 7
  1. Show that \(a + b = 3\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ), the remainder is - 8
  2. Find the value of \(a\) and the value of \(b\).
Edexcel C12 Specimen Q7
5 marks Moderate -0.3
7. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{gathered} a _ { 1 } = 2 \\ a _ { n + 1 } = 3 a _ { n } - c \end{gathered}$$ where \(c\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
  2. find the value of \(c\).
Edexcel C12 Specimen Q8
7 marks Moderate -0.5
8. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
  2. Hence find the set of possible values of \(k\).
Edexcel C12 Specimen Q9
6 marks Standard +0.3
9. Given that \(y = 3 x ^ { 2 }\),
  1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
  2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$
Edexcel C12 Specimen Q10
8 marks Moderate -0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-18_508_812_306_644} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 2 } ( 9 - 2 x ) .$$ There is a minimum at the origin, a maximum at the point \(( 3,27 )\) and \(C\) cuts the \(x\)-axis at the point \(A\).
  1. Write down the coordinates of the point \(A\).
  2. On separate diagrams sketch the curve with equation
    1. \(y = \mathrm { f } ( x + 3 )\),
    2. \(y = \mathrm { f } ( 3 x )\). On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant, has a maximum point at \(( 3,10 )\).
  3. Write down the value of \(k\).
Edexcel C12 Specimen Q11
11 marks Moderate -0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-22_337_892_278_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line with equation \(y = x + 4\) cuts the curve with equation \(y = - x ^ { 2 } + 2 x + 24\) at the points \(A\) and \(B\), as shown in Figure 2.
  1. Use algebra to find the coordinates of the points \(A\) and \(B\). The finite region \(R\) is bounded by the straight line and the curve and is shown shaded in Figure 2.
  2. Use calculus to find the exact area of \(R\).
Edexcel C12 Specimen Q12
11 marks Standard +0.3
12. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\)
  1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
  2. Find an equation for \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
    Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
  3. find the length of \(P Q\), giving your answer in its simplest surd form.
Edexcel C12 Specimen Q13
11 marks Standard +0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-28_374_410_278_776} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius \(x\) metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to \(x\) metres and width equal to \(y\) metres. Given that the area of the flowerbed is \(4 \mathrm {~m} ^ { 2 }\),
  1. show that $$y = \frac { 16 - \pi x ^ { 2 } } { 8 x }$$
  2. Hence show that the perimeter \(P\) metres of the flowerbed is given by the equation $$P = \frac { 8 } { x } + 2 x$$
  3. Use calculus to find the minimum value of \(P\).
Edexcel C12 Specimen Q14
10 marks Moderate -0.3
In this question you must show all stages of your working. (Solutions based entirely on graphical or numerical methods are not acceptable.)
  1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers in degrees to 1 decimal place, $$3 \sin \left( x + 45 ^ { \circ } \right) = 2$$
  2. Find, for \(0 \leqslant x < 2 \pi\), all the solutions of $$2 \sin ^ { 2 } x + 2 = 7 \cos x$$ giving your answers in radians. \includegraphics[max width=\textwidth, alt={}, center]{1528bec3-7a7a-42c5-bac2-756ff3493818-35_108_95_2572_1804}
Edexcel C12 Specimen Q15
12 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-36_394_608_287_676} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The triangle \(X Y Z\) in Figure 4 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 4 is a major arc of the circle with centre \(X\) and radius 4 cm .
  1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
  2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region enclosed by the major arc \(Z W\) of the circle and the lines \(W Y\) and \(Y Z\) is shown shaded in Figure 4. Calculate
  3. the area of this shaded region,
  4. the perimeter \(Z W Y Z\) of this shaded region. \includegraphics[max width=\textwidth, alt={}, center]{1528bec3-7a7a-42c5-bac2-756ff3493818-39_90_54_2576_1868}
Edexcel C12 Specimen Q16
13 marks Moderate -0.8
16. Maria trains for a triathlon, which involves swimming, cycling and running. On the first day of training she swims 1.5 km and then she swims 1.5 km on each of the following days.
  1. Find the total distance that Maria swims in the first 17 days of training. Maria also runs 1.5 km on the first day of training and on each of the following days she runs 0.25 km further than on the previous day. So she runs 1.75 km on the second day and 2 km on the third day and so on.
  2. Find how far Maria runs on the 17th day of training. Maria also cycles 1.5 km on the first day of training and on each of the following days she cycles \(5 \%\) further than on the previous day.
  3. Find the total distance that Maria cycles in the first 17 days of training.
  4. Find the total distance Maria travels by swimming, running and cycling in the first 17 days of training. Maria needs to cycle 40 km in the triathlon.
  5. On which day of training does Maria first cycle more than 40 km ?