Questions C12 (247 questions)

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Edexcel C12 2017 October Q6
  1. Each year Lin pays into a savings scheme. In year 1 she pays in \(\pounds 600\). Her payments then increase by \(\pounds 80\) a year, so that she pays \(\pounds 680\) into the savings scheme in year \(2 , \pounds 760\) in year 3 and so on. In year \(N\), Lin pays \(\pounds 1000\) into the savings scheme.
    1. Find the value of \(N\).
    2. Find the total amount that Lin pays into the savings scheme from year 1 to year 15 inclusive.
    Saima starts paying into a different savings scheme at the same time as Lin starts paying into her savings scheme. In year 1 she pays in \(\pounds A\). Her payments increase by \(\pounds A\) each year so that she pays \(\pounds 2 A\) in year \(2 , \pounds 3 A\) in year 3 and so on. Given that Saima and Lin have each paid, in total, the same amount of money into their savings schemes after 15 years,
  2. find the value of \(A\).
Edexcel C12 2017 October Q7
7. $$g ( x ) = 2 x ^ { 3 } + a x ^ { 2 } - 18 x - 8$$ Given that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\),
  1. show that \(a = - 3\)
  2. Hence, using algebra, fully factorise \(\mathrm { g } ( x )\). Using your answer to part (b),
  3. solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$2 \sin ^ { 3 } \theta - 3 \sin ^ { 2 } \theta - 18 \sin \theta = 8$$ giving each answer, in radians, as a multiple of \(\pi\).
Edexcel C12 2017 October Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-18_387_397_255_794} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circumference of this circle. The minor arc \(A B\) subtends an angle \(\theta\) radians at \(O\), as shown in Figure 3.
Given the length of minor \(\operatorname { arc } A B\) is 6 cm and the area of minor sector \(O A B\) is \(20 \mathrm {~cm} ^ { 2 }\),
  1. write down two different equations in \(r\) and \(\theta\).
  2. Hence find the value of \(r\) and the value of \(\theta\).
Edexcel C12 2017 October Q9
  1. (a) Given that \(a\) is a constant, \(a > 1\), sketch the graph of
$$y = a ^ { x } , \quad x \in \mathbb { R }$$ On your diagram show the coordinates of the point where the graph crosses the \(y\)-axis.
(2) The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 ^ { x }\)
\(x\)- 4- 2024
\(y\)0.06250.251416
(b) Use the trapezium rule, with all of the values of \(y\) from the table, to find an approximate value, to 2 decimal places, for $$\int _ { - 4 } ^ { 4 } 2 ^ { x } \mathrm {~d} x$$ (c) Use the answer to part (b) to find an approximate value for
  1. \(\int _ { - 4 } ^ { 4 } 2 ^ { x + 2 } \mathrm {~d} x\)
  2. \(\int _ { - 4 } ^ { 4 } \left( 3 + 2 ^ { x } \right) \mathrm { d } x\)
    \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-23_86_47_2617_1886}
Edexcel C12 2017 October Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-24_863_929_255_511} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Diagram NOT drawn to scale The points \(A ( 7 , - 3 ) , B ( 7,20 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 4. The point \(D ( 10,5 )\) is the midpoint of \(A C\).
  1. Find the value of \(p\) and the value of \(q\). The line \(l\) passes through \(D\) and is perpendicular to \(A C\).
  2. Find an equation for \(l\), in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. Given that the line \(l\) intersects \(A B\) at \(E\),
  3. find the exact coordinates of \(E\).
Edexcel C12 2017 October Q11
11. \(\mathrm { f } ( x ) = ( a - x ) ( 3 + a x ) ^ { 5 }\), where \(a\) is a positive constant
  1. Find the first 3 terms, in ascending powers of \(x\), in the binomial expansion of $$( 3 + a x ) ^ { 5 }$$ Give each term in its simplest form. Given that in the expansion of \(\mathrm { f } ( x )\) the coefficient of \(x\) is zero,
  2. find the exact value of \(a\).
Edexcel C12 2017 October Q12
12. (i) Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 2 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers, in degrees, to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(ii) (a) Given that $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ show that $$\tan ^ { 2 } x = k , \quad \text { where } k \text { is a constant. }$$ (b) Hence solve, for \(0 < x \leqslant 2 \pi\), $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ giving your answers, in radians, to 3 decimal places.
Edexcel C12 2017 October Q13
  1. The circle \(C\) has equation
$$( x - 3 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 30$$ Write down
    1. the coordinates of the centre of \(C\),
    2. the exact value of the radius of \(C\). Given that the point \(P\) with coordinates \(( 6 , k )\), where \(k\) is a constant, lies inside circle \(C\), (b) show that $$k ^ { 2 } + 8 k - 5 < 0$$
  2. Hence find the exact set of values of \(k\) for which \(P\) lies inside \(C\).
    \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-34_2256_52_315_1978}
Edexcel C12 2017 October Q14
  1. A new mineral has been discovered and is going to be mined over a number of years.
A model predicts that the mass of the mineral mined each year will decrease by \(15 \%\) per year, so that the mass of the mineral mined each year forms a geometric sequence. Given that the mass of the mineral mined during year 1 is 8000 tonnes,
  1. show that, according to the model, the mass of the mineral mined during year 6 will be approximately 3550 tonnes. According to the model, there is a limit to the total mass of the mineral that can be mined.
  2. With reference to the geometric series, state why this limit exists.
  3. Calculate the value of this limit. It is decided that a total mass of 40000 tonnes of the mineral is required. This is going to be mined from year 1 to year \(N\) inclusive.
  4. Assuming the model, find the value of \(N\).
Edexcel C12 2017 October Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-42_695_1450_251_246} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { ( x - 3 ) ^ { 2 } ( x + 4 ) } { 2 } , \quad x \in \mathbb { R }$$ The graph cuts the \(y\)-axis at the point \(P\) and meets the positive \(x\)-axis at the point \(R\), as shown in Figure 5.
    1. State the \(y\) coordinate of \(P\).
    2. State the \(x\) coordinate of \(R\). The line segment \(P Q\) is parallel to the \(x\)-axis. Point \(Q\) lies on \(y = \mathrm { f } ( x ) , x > 0\)
  2. Use algebra to show that the \(x\) coordinate of \(Q\) satisfies the equation $$x ^ { 2 } - 2 x - 15 = 0$$
  3. Use part (b) to find the coordinates of \(Q\). The region \(S\), shown shaded in Figure 5, is bounded by the curve \(y = \mathrm { f } ( x )\) and the line segment \(P Q\).
  4. Use calculus to find the exact area of \(S\).
Edexcel C12 2017 October Q16
  1. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 2 x - 5\), where \(a\) and \(b\) are constants The point \(P ( 1,4 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\).
The tangent to \(y = \mathrm { f } ( x )\) at the point \(P\) has equation \(y = 12 x - 8\) Calculate the value of \(a\) and the value of \(b\).
(5)
VILIV SIMI NI III IM I ON OC
VILV SIHI NI JAHMMION OC
VALV SIHI NI JIIIM ION OC
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Q16

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Edexcel C12 2018 October Q1
  1. (i) Given that \(125 \sqrt { 5 } = 5 ^ { a }\), find the value of \(a\).
    (ii) Show that \(\frac { 16 } { 4 - \sqrt { 8 } } = 8 + 4 \sqrt { 2 }\)
You must show all stages of your working.
Edexcel C12 2018 October Q2
2. Use algebra to solve the simultaneous equations $$\begin{aligned} x + y & = 5
x ^ { 2 } + x + y ^ { 2 } & = 51 \end{aligned}$$ You must show all stages of your working.
VIIIV SIHI NI III IM ION OCVIIV SIHI NI JIIIM ION OCVI4V SIHI NI JIIIM ION OO
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Edexcel C12 2018 October Q3
3. Given that \(y = 2 x ^ { 3 } - \frac { 5 } { 3 x ^ { 2 } } + 7 , x \neq 0\), find in its simplest form
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\int y \mathrm {~d} x\).
    VIIN SIHI NI IIIIM ION OCVIIN SIHI NI JYHM IONOOVI4V SIHI NI JIIIM ION OC
Edexcel C12 2018 October Q4
4. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies $$u _ { n } = k n - 3 ^ { n }$$ where \(k\) is a constant. Given that \(u _ { 2 } = u _ { 4 }\)
  1. find the value of \(k\)
  2. evaluate \(\sum _ { r = 1 } ^ { 4 } u _ { r }\)
Edexcel C12 2018 October Q5
  1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$\left( 3 + 5 x - 2 x ^ { 2 } \right) \left( 1 - \frac { 1 } { 2 } x \right) ^ { 10 }$$
Edexcel C12 2018 October Q6
6. (a) Sketch the graph of \(y = \left( \frac { 1 } { 2 } \right) ^ { x } , x \in \mathbb { R }\), showing the coordinates of the point at which the graph crosses the \(y\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) The values of \(y\) are rounded to 3 decimal places.
\(x\)- 0.9- 0.8- 0.7- 0.6- 0.5
\(y\)1.8661.7411.6251.5161.414
(b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for $$\int _ { - 0.9 } ^ { - 0.5 } \left( \frac { 1 } { 2 } \right) ^ { x } d x$$ II
Edexcel C12 2018 October Q7
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
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
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
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
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
12. (a) Show that the equation $$6 \cos x - 5 \tan x = 0$$ may be expressed in the form $$6 \sin ^ { 2 } x + 5 \sin x - 6 = 0$$ (b) 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
13. (i) Find the value of \(x\) for which $$4 ^ { 3 x + 2 } = 3 ^ { 600 }$$ giving your answer to 4 significant figures.
(ii) 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
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 \(O D E\).