Questions — Edexcel C2 (579 questions)

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Edexcel C2 2018 June Q8
9 marks Moderate -0.3
8 In this question solutions based entirely on graphical or numerical methods are not acceptable.
  1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \cos \left( x + 70 ^ { \circ } \right) = 3$$ giving your answers in degrees to one decimal place.
  2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$6 \cos ^ { 2 } \theta - 5 = 6 \sin ^ { 2 } \theta + \sin \theta$$ giving your answers in radians to 3 significant figures.
Edexcel C2 2018 June Q9
12 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8daf56fa-bfce-454e-bbb8-fecd8170d77e-28_751_876_214_539} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 7 x ^ { 2 } ( 5 - 2 \sqrt { x } ) , \quad x \geqslant 0$$ The curve has a turning point at the point \(A\), where \(x > 0\), as shown in Figure 3.
  1. Using calculus, find the coordinates of the point \(A\). The curve crosses the \(x\)-axis at the point \(B\), as shown in Figure 3.
  2. Use algebra to find the \(x\) coordinate of the point \(B\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line through \(A\) parallel to the \(x\)-axis and the line through \(B\) parallel to the \(y\)-axis.
  3. Use integration to find the area of the region \(R\), giving your answer to 2 decimal places.
    END
Edexcel C2 Specimen Q1
4 marks Moderate -0.8
Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 + 3 x ) ^ { 6 }\).
(4)
Edexcel C2 Specimen Q4
7 marks Standard +0.3
4. Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(3 \sin ^ { 2 } x = 1 + \cos x\), giving your answers to the nearest degree.
Edexcel C2 Specimen Q5
7 marks Moderate -0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{afaf76d8-2a1f-4239-8275-70fad4f418c1-1_351_663_1529_693}
\end{figure} The shaded area in Fig. 1 shows a badge \(A B C\), where \(A B\) and \(A C\) are straight lines, with \(A B = A C = 8 \mathrm {~mm}\). The curve \(B C\) is an arc of a circle, centre \(O\), where \(O B = O C =\) 8 mm and \(O\) is in the same plane as \(A B C\). The angle \(B A C\) is 0.9 radians.
  1. Find the perimeter of the badge.
  2. Find the area of the badge.
Edexcel C2 Specimen Q6
10 marks Moderate -0.3
6. At the beginning of the year 2000 a company bought a new machine for \(\pounds 15000\). Each year the value of the machine decreases by \(20 \%\) of its value at the start of the year.
  1. Show that at the start of the year 2002, the value of the machine was \(\pounds 9600\). When the value of the machine falls below \(\pounds 500\), the company will replace it.
  2. Find the year in which the machine will be replaced. To plan for a replacement machine, the company pays \(\pounds 1000\) at the start of each year into a savings account. The account pays interest at a fixed rate of \(5 \%\) per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced.
  3. Using your answer to part (b), find how much the savings account will be worth immediately after the payment at the start of the year in which the machine is replaced.
Edexcel C2 Specimen Q7
12 marks Standard +0.3
7. (a) Use the factor theorem to show that \(( x + 1 )\) is a factor of \(x ^ { 3 } - x ^ { 2 } - 10 x - 8\).
(b) Find all the solutions of the equation \(x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0\).
(c) Prove that the value of \(x\) that satisfies $$2 \log _ { 2 } x + \log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( 5 x + 4 )$$ is a solution of the equation $$x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0$$ (d) State, with a reason, the value of \(x\) that satisfies equation (I).
Edexcel C2 Specimen Q8
12 marks Moderate -0.3
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{afaf76d8-2a1f-4239-8275-70fad4f418c1-2_616_712_1658_713}
\end{figure} The line with equation \(y = x + 5\) cuts the curve with equation \(y = x ^ { 2 } - 3 x + 8\) at the points \(A\) and \(B\), as shown in Fig. 2.
  1. Find the coordinates of the points \(A\) and \(B\).
  2. Find the area of the shaded region between the curve and the line, as shown in Fig. 2.
Edexcel C2 Specimen Q9
13 marks Standard +0.3
9. Figure 3 $$( x + 1 ) ^ { 2 }$$ Figure 3 shows a triangle \(P Q R\). The size of angle \(Q P R\) is \(30 ^ { \circ }\), the length of \(P Q\) is \(( x + 1 )\) and the length of \(P R\) is \(( 4 - x ) ^ { 2 }\), where \(X \in \Re\).
  1. Show that the area \(A\) of the triangle is given by \(A = \frac { 1 } { 4 } \left( x ^ { 3 } - 7 x ^ { 2 } + 8 x + 16 \right)\)
  2. Use calculus to prove that the area of \(\triangle P Q R\) is a maximum when \(x = \frac { 2 } { 3 }\). Explain clearly how you know that this value of \(x\) gives the maximum area.
  3. Find the maximum area of \(\triangle P Q R\).
  4. Find the length of \(Q R\) when the area of \(\triangle P Q R\) is a maximum. END
Edexcel C2 2005 January Q2
6 marks Easy -1.2
The points \(A\) and \(B\) have coordinates \(( 5 , - 1 )\) and \(( 13,11 )\) respectively.
  1. Find the coordinates of the mid-point of \(A B\). Given that \(A B\) is a diameter of the circle \(C\),
  2. find an equation for \(C\).
Edexcel C2 2009 January Q2
5 marks Moderate -0.5
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-03_870_1027_205_406} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = ( 1 + x ) ( 4 - x )\).
The curve intersects the \(x\)-axis at \(x = - 1\) and \(x = 4\). The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis. Use calculus to find the exact area of \(R\).
Edexcel C2 2005 June Q2
6 marks Moderate -0.8
Solve
  1. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
  2. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).
Edexcel C2 2005 June Q3
6 marks Moderate -0.8
  1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\).
  2. Factorise \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\) completely.
Edexcel C2 2005 June Q4
6 marks Standard +0.3
  1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + p x ) ^ { 12 }\), where \(p\) is a non-zero constant. Given that, in the expansion of \(( 1 + p x ) ^ { 12 }\), the coefficient of \(x\) is \(( - q )\) and the coefficient of \(x ^ { 2 }\) is \(11 q\),
  2. find the value of \(p\) and the value of \(q\).
Edexcel C2 2006 June Q2
5 marks Easy -1.2
Use calculus to find the exact value of \(\int _ { 1 } ^ { 2 } \left( 3 x ^ { 2 } + 5 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
Edexcel C2 2006 June Q3
4 marks Easy -1.2
  1. Write down the value of \(\log _ { 6 } 36\).
  2. Express \(2 \log _ { a } 3 + \log _ { a } 11\) as a single logarithm to base \(a\).
Edexcel C2 2006 June Q4
8 marks Moderate -0.8
$$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2007 June Q2
6 marks Moderate -0.8
$$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 16 x + 12$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ). Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 Specimen Q2
4 marks Easy -1.2
The circle \(C\) has centre \(( 3,4 )\) and passes through the point \(( 8 , - 8 )\). Find an equation for C
Edexcel C2 Specimen Q3
6 marks Moderate -0.8
The trapezium rule, with the table below, was used to estimate the area between the curve \(y = \sqrt { x ^ { 3 } + 1 }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis.
\(x\)11.522.53
\(y\)1.4142.0923.000
  1. Calculate, to 3 decimal places, the values of \(y\) for \(x = 2.5\) and \(x = 3\).
  2. Use the values from the table and your answers to part (a) to find an estimate, to 2 decimal places, for this area.
Edexcel C2 Q1
6 marks Moderate -0.8
  1. Given that \(p = \log _ { q } 16\), express in terms of \(p\),
    1. \(\log _ { q } 2\),
    2. \(\log _ { q } ( 8 q )\).
      [0pt] [P2 January 2002 Question 2]
    3. \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 7 x + c\), where \(c\) is a constant.
    Given that \(\mathrm { f } ( 4 ) = 0\),
  2. find the value of \(c\),
  3. factorise \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
  4. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(\mathrm { f } ( x ) = 0\).
Edexcel C2 Q3
8 marks Moderate -0.3
3. Find the values of \(\theta\), to 1 decimal place, in the interval \(- 180 \leq \theta < 180\) for which $$2 \sin ^ { 2 } \theta ^ { \circ } - 2 \sin \theta ^ { \circ } = \cos ^ { 2 } \theta ^ { \circ }$$ [P1 January 2002 Question 3]
Edexcel C2 Q4
9 marks Challenging +1.2
4. A population of deer is introduced into a park. The population \(P\) at \(t\) years after the deer have been introduced is modelled by \(P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } }\), where \(a\) is a constant. Given that there are 800 deer in the park after 6 years,
  1. calculate, to 4 decimal places, the value of \(a\),
  2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
  3. With reference to this model, give a reason why the population of deer cannot exceed 2000.
Edexcel C2 Q5
11 marks Standard +0.3
5. (a) Given that \(( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = A + B x ^ { 2 } + C x ^ { 4 }\), find the values of the constants \(A , B\) and \(C\).
(b) Using the substitution \(y = x ^ { 2 }\) and your answers to part (a), solve, $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = 349$$
Edexcel C2 Q6
10 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1033051d-18bf-4734-a556-4c8e1c789992-3_842_963_280_392}
\end{figure} Fig. 1 shows a gardener's design for the shape of a flower bed with perimeter \(A B C D . A D\) is an arc of a circle with centre \(O\) and radius \(5 \mathrm {~m} . B C\) is an arc of a circle with centre \(O\) and radius \(7 \mathrm {~m} . O A B\) and \(O D C\) are straight lines and the size of \(\angle A O D\) is \(\theta\) radians.
  1. Find, in terms of \(\theta\), an expression for the area of the flower bed. Given that the area of the flower bed is \(15 \mathrm {~m} ^ { 2 }\),
  2. show that \(\theta = 1.25\),
  3. calculate, in m , the perimeter of the flower bed. The gardener now decides to replace arc \(A D\) with the straight line \(A D\).
  4. Find, to the nearest cm , the reduction in the perimeter of the flower bed.