Questions — Edexcel C2 (579 questions)

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Edexcel C2 Q9
13 marks Moderate -0.3
9. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\).
  1. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Fully factorise \(\mathrm { f } ( x )\).
  3. Using your answer to part (b), write down the coordinates of one of the turning points of the curve \(y = \mathrm { f } ( x )\) and give a reason for your answer.
  4. Using differentiation, find the \(x\)-coordinate of the other turning point of the curve \(y = \mathrm { f } ( x )\).
Edexcel C2 Q1
4 marks Moderate -0.8
  1. Evaluate
$$\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) d x .$$
Edexcel C2 Q2
4 marks Easy -1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05006f1f-ebf0-4d70-9dbb-68221c09043e-2_510_842_534_513} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \sqrt { 4 x - 1 }\). Use the trapezium rule with five equally-spaced ordinates to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Edexcel C2 Q3
7 marks Moderate -0.3
3. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for
  1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
  2. \(\log _ { 2 } ( \sqrt { x } )\).
    (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$
Edexcel C2 Q4
9 marks Moderate -0.8
4. $$f ( x ) = 2 - x - x ^ { 3 }$$
  1. Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
  2. Verify that the point \(( 1,0 )\) lies on the curve \(y = \mathrm { f } ( x )\).
  3. Find the area of the region bounded by the curve \(y = \mathrm { f } ( x )\) and the coordinate axes.
Edexcel C2 Q5
9 marks Standard +0.3
5. Figure 2 Figure 2 shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,
  1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
  2. show that \(Q R = 2 \sqrt { 6 }\),
  3. find \(\angle P Q R\) in degrees to 1 decimal place.
Edexcel C2 Q6
10 marks Moderate -0.3
6. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,
  1. find an expression for \(b\) in terms of \(a\). Given also that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\),
  2. find the values of \(a\) and \(b\),
  3. fully factorise \(\mathrm { p } ( x )\).
Edexcel C2 Q7
10 marks Standard +0.3
7. (a) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < 2 \pi\) for which $$\tan \left( x + \frac { \pi } { 4 } \right) = 3 .$$ (b) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y .$$
Edexcel C2 Q8
11 marks Moderate -0.3
  1. The point \(A\) has coordinates ( 4,6 ).
Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
Edexcel C2 Q9
11 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05006f1f-ebf0-4d70-9dbb-68221c09043e-4_325_662_1345_520} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of a design being produced by a computer program.
The program draws a series of circles with each one touching the previous one and such that their centres lie on a horizontal straight line. The radii of the circles form a geometric sequence with first term 1 mm and second term 1.5 mm . The width of the design is \(w\) as shown.
  1. Find the radius of the fourth circle to be drawn.
  2. Show that when eight circles have been drawn, \(w = 98.5 \mathrm {~mm}\) to 3 significant figures.
  3. Find the total area of the design in square centimetres when ten circles have been drawn.
Edexcel C2 2013 June Q7
9 marks Moderate -0.3
  1. Find by calculation the \(x\)-coordinate of \(A\) and the \(x\)-coordinate of \(B\). The shaded region \(R\) is bounded by the line with equation \(y = 10\) and the curve as shown in Figure 1.
  2. Use calculus to find the exact area of \(R\).
Edexcel C2 Q7
11 marks Moderate -0.3
  1. Find the coordinates of the points where the curve and line intersect.
  2. Find the area of the shaded region bounded by the curve and line.
Edexcel C2 Q1
4 marks Easy -1.2
Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((3 + 2x)^5\), giving each term in its simplest form. [4]
Edexcel C2 Q2
6 marks Moderate -0.8
The points \(A\) and \(B\) have coordinates \((5, -1)\) and \((13, 11)\) respectively.
  1. Find the coordinates of the mid-point of \(AB\). [2]
Given that \(AB\) is a diameter of the circle \(C\),
  1. find an equation for \(C\). [4]
Edexcel C2 Q3
7 marks Moderate -0.3
Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which
  1. \(3^x = 5\), [3]
  2. \(\log_2(2x + 1) - \log_2 x = 2\). [4]
Edexcel C2 Q4
7 marks Moderate -0.3
  1. Show that the equation $$5 \cos^2 x = 3(1 + \sin x)$$ can be written as $$5 \sin^2 x + 3 \sin x - 2 = 0.$$ [2]
  2. Hence solve, for \(0 \leq x < 360°\), the equation $$5 \cos^2 x = 3(1 + \sin x),$$ giving your answers to 1 decimal place where appropriate. [5]
Edexcel C2 Q5
8 marks Moderate -0.8
\(f(x) = x^3 - 2x^2 + ax + b\), where \(a\) and \(b\) are constants. When \(f(x)\) is divided by \((x - 2)\), the remainder is 1. When \(f(x)\) is divided by \((x + 1)\), the remainder is 28.
  1. Find the value of \(a\) and the value of \(b\). [6]
  2. Show that \((x - 3)\) is a factor of \(f(x)\). [2]
Edexcel C2 Q6
8 marks Moderate -0.3
The second and fourth terms of a geometric series are 7.2 and 5.832 respectively. The common ratio of the series is positive. For this series, find
  1. the common ratio, [2]
  2. the first term, [2]
  3. the sum of the first 50 terms, giving your answer to 3 decimal places, [2]
  4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places. [2]
Edexcel C2 Q7
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the triangle \(ABC\), with \(AB = 8\) cm, \(AC = 11\) cm and \(\angle BAC = 0.7\) radians. The arc \(BD\), where \(D\) lies on \(AC\), is an arc of a circle with centre \(A\) and radius 8 cm. The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(BC\) and \(CD\) and the arc \(BD\). Find
  1. the length of the arc \(BD\), [2]
  2. the perimeter of \(R\), giving your answer to 3 significant figures, [4]
  3. the area of \(R\), giving your answer to 3 significant figures. [5]
Edexcel C2 Q8
12 marks Moderate -0.3
\includegraphics{figure_2} The line with equation \(y = 3x + 20\) cuts the curve with equation \(y = x^2 + 6x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.
  1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). [5]
The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.
  1. Use calculus to find the exact area of \(S\). [7]
Edexcel C2 Q9
12 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the stage is \(2x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2x\) metres. The perimeter of the stage is 80 m.
  1. Show that the area, \(A\) m², of the stage is given by $$A = 80x - \left(2 + \frac{\pi}{2}\right)x^2.$$ [4]
  2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value. [4]
  3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\). [2]
  4. Calculate, to the nearest m², the maximum area of the stage. [2]
Edexcel C2 Q10
12 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows part of a curve \(C\) with equation \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 4 respectively. The region \(R\), shaded in Figure 1, is bounded by \(C\) and the straight line joining \(P\) and \(Q\).
  1. Find the exact area of \(R\). [8]
  2. Use calculus to show that \(y\) is increasing for \(x > 2\). [4]
Edexcel C2 Q1
4 marks Easy -1.2
Find the coordinates of the stationary point on the curve with equation \(y = 2x^2 - 12x\). [4]
Edexcel C2 Q2
6 marks Moderate -0.8
Solve
  1. \(5^x = 8\), giving your answer to 3 significant figures, [3]
  2. \(\log_2(x + 1) - \log_2 x = \log_2 7\). [3]
Edexcel C2 Q3
6 marks Moderate -0.8
  1. Use the factor theorem to show that \((x + 4)\) is a factor of \(2x^3 + x^2 - 25x + 12\). [2]
  2. Factorise \(2x^3 + x^2 - 25x + 12\) completely. [4]