Questions C12 (247 questions)

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Edexcel C12 2015 January Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-16_608_952_267_495} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \sin \left( x - 60 ^ { \circ } \right) , - 360 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\)
  1. Write down the exact coordinates of the points at which \(C\) meets the two coordinate axes.
  2. Solve, for \(- 360 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), $$4 \sin \left( x - 60 ^ { \circ } \right) = \sqrt { 6 } - \sqrt { 2 }$$ showing each stage of your working.
Edexcel C12 2015 January Q12
12. A business is expected to have a yearly profit of \(\pounds 275000\) for the year 2016. The profit is expected to increase by \(10 \%\) per year, so that the expected yearly profits form a geometric sequence with common ratio 1.1
  1. Show that the difference between the expected profit for the year 2020 and the expected profit for the year 2021 is \(\pounds 40300\) to the nearest hundred pounds.
  2. Find the first year for which the expected yearly profit is more than one million pounds.
  3. Find the total expected profits for the years 2016 to 2026 inclusive, giving your answer to the nearest hundred pounds.
Edexcel C12 2015 January Q13
13. The curve \(C\) has equation $$y = 3 x ^ { 2 } - 4 x + 2$$ The line \(l _ { 1 }\) is the normal to the curve \(C\) at the point \(P ( 1,1 )\)
  1. Show that \(l _ { 1 }\) has equation $$x + 2 y - 3 = 0$$ The line \(l _ { 1 }\) meets curve \(C\) again at the point \(Q\).
  2. By solving simultaneous equations, determine the coordinates of the point \(Q\). Another line \(l _ { 2 }\) has equation \(k x + 2 y - 3 = 0\), where \(k\) is a constant.
  3. Show that the line \(l _ { 2 }\) meets the curve \(C\) once only when $$k ^ { 2 } - 16 k + 40 = 0$$
  4. Find the two exact values of \(k\) for which \(l _ { 2 }\) is a tangent to \(C\).
Edexcel C12 2015 January Q14
14. In this question, solutions based entirely on graphical or numerical methods are not acceptable.
  1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$3 \sin x + 7 \cos x = 0$$ Give each solution, in degrees, to one decimal place.
  2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$10 \cos ^ { 2 } \theta + \cos \theta = 11 \sin ^ { 2 } \theta - 9$$ Give each solution, in radians, to 3 significant figures.
Edexcel C12 2015 January Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-24_591_570_255_678} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } + 10 x ^ { \frac { 3 } { 2 } } + k x , \quad x \geqslant 0$$ where \(k\) is a constant.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\) on the curve \(C\) is a minimum turning point.
    Given that the \(x\) coordinate of \(P\) is 4
  2. show that \(k = - 78\) The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\).
    The finite region \(R\), shown shaded in Figure 5, is bounded by \(C\), the \(y\)-axis and \(P N\).
  3. Use integration to find the area of \(R\).
Edexcel C12 2016 January Q1
  1. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) satisfies
$$u _ { n + 1 } = 2 u _ { n } - 6 , \quad n \geqslant 1$$ Given that \(u _ { 1 } = 2\)
  1. find the value of \(u _ { 3 }\)
  2. evaluate \(\sum _ { i = 1 } ^ { 4 } u _ { i }\)
Edexcel C12 2016 January Q2
2. (i) Given that \(\frac { 49 } { \sqrt { 7 } } = 7 ^ { a }\), find the value of \(a\).
(ii) Show that \(\frac { 10 } { \sqrt { 18 } - 4 } = 15 \sqrt { 2 } + 20\) You must show all stages of your working.
Edexcel C12 2016 January Q3
3. Find, using calculus and showing each step of your working, $$\int _ { 1 } ^ { 4 } \left( 6 x - 3 - \frac { 2 } { \sqrt { x } } \right) \mathrm { d } x$$
Edexcel C12 2016 January Q4
4. The \(4 ^ { \text {th } }\) term of an arithmetic sequence is 3 and the sum of the first 6 terms is 27 Find the first term and the common difference of this sequence.
Edexcel C12 2016 January Q5
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
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. (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
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
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. 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. \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
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
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
  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
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
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
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
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
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
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 $$2 \log _ { 3 } \left( \frac { x } { 9 } \right) - \log _ { 3 } \sqrt { x } = 2$$