Questions — Edexcel C2 (476 questions)

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Edexcel C2 Q7
7. The points \(P\) and \(Q\) have coordinates \(( - 2,6 )\) and \(( 4 , - 1 )\) respectively. Given that \(P Q\) is a diameter of circle \(C\),
  1. find the coordinates of the centre of \(C\),
  2. show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 2 x - 5 y - 14 = 0 .$$ The point \(R\) has coordinates (2, 7).
  3. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle P R Q\) in degrees.
Edexcel C2 Q8
8. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.
  1. Find the common ratio of the series.
  2. Show that the first term of the series is \(\log _ { 3 } 2\).
  3. Find, to 3 significant figures, the sum of the first six terms of the series.
Edexcel C2 Q9
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
  1. Evaluate
$$\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) d x .$$
Edexcel C2 Q2
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
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
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
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
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
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
  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
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 Q1
  1. A geometric series has first term 75 and second term - 15 .
    1. Find the common ratio of the series.
    2. Find the sum to infinity of the series.
    3. A circle has the equation
    $$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y + k = 0 ,$$ where \(k\) is a constant.
  2. Find the coordinates of the centre of the circle. Given that the \(x\)-axis is a tangent to the circle,
  3. find the value of \(k\).
Edexcel C2 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a7285542-c32a-45b1-921b-d528676ad6b5-2_586_513_1219_593} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a circle of radius \(r\) and centre \(O\) in which \(A D\) is a diameter.
The points \(B\) and \(C\) lie on the circle such that \(O B\) and \(O C\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(O B C\) is \(\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )\).
Edexcel C2 Q4
4. (a) Sketch on the same diagram the graphs of \(y = \sin 2 x\) and \(y = \tan \frac { x } { 2 }\) for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\).
(b) Hence state how many solutions exist to the equation $$\sin 2 x = \tan \frac { x } { 2 }$$ for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) and give a reason for your answer.
Edexcel C2 Q5
5. (a) Find the value of \(a\) such that $$\log _ { a } 27 = 3 + \log _ { a } 8$$ (b) Solve the equation $$2 ^ { x + 3 } = 6 ^ { x - 1 }$$ giving your answer to 3 significant figures.
Edexcel C2 Q6
6. (a) Expand \(( 2 + x ) ^ { 4 }\) in ascending powers of \(x\), simplifying each coefficient.
(b) Find the integers \(A , B\) and \(C\) such that $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } \equiv A + B x ^ { 2 } + C x ^ { 4 }$$ (c) Find the real values of \(x\) for which $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } = 136$$
Edexcel C2 Q7
7. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + x + 2$$
  1. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Fully factorise \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\).
  4. Find the values of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) for which $$2 \sin ^ { 3 } \theta - 5 \sin ^ { 2 } \theta + \sin \theta + 2 = 0$$ giving your answers in terms of \(\pi\).
Edexcel C2 Q8
8. The curve \(C\) has the equation $$y = 3 - x ^ { \frac { 1 } { 2 } } - 2 x ^ { - \frac { 1 } { 2 } } , \quad x > 0 .$$
  1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis.
  2. Find the exact coordinates of the stationary point of \(C\).
  3. Determine the nature of the stationary point.
  4. Sketch the curve \(C\).
Edexcel C2 Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a7285542-c32a-45b1-921b-d528676ad6b5-4_620_872_895_504} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve \(C\) with equation \(y = 3 x - 4 \sqrt { x } + 2\) and the tangent to \(C\) at the point \(A\). Given that \(A\) has \(x\)-coordinate 4,
  1. show that the tangent to \(C\) at \(A\) has the equation \(y = 2 x - 2\). The shaded region is bounded by \(C\), the tangent to \(C\) at \(A\) and the positive coordinate axes.
  2. Find the area of the shaded region.
Edexcel C2 2013 June Q7
  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 Q2
  1. Find, in surd form, the length \(A B\).
  2. Find, in terms of \(\pi\), the area of the badge.
  3. Prove that the perimeter of the badge is \(\frac { 2 \sqrt { 3 } } { 3 } ( \pi + 6 ) \mathrm { cm }\).
Edexcel C2 Q2
  1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). \end{enumerate} A second circle has centre at the point \(( 15,12 )\) and radius 10.
  2. Sketch both circles on a single diagram and find the coordinates of the point where they touch.
    (4)
    [0pt] [P3 June 2003 Question 3]
Edexcel C2 Q2
  1. find the first 4 terms, simplifying each term.
  2. Find, in its simplest form, the term independent of \(x\) in this expansion.
    [0pt] [P2 June 2004 Question 3] \item The curve \(C\) has equation \(y = \cos \left( x + \frac { \pi } { 4 } \right) , 0 \leq x \leq 2 \pi\).
  3. Sketch \(C\).
  4. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
  5. Solve, for \(x\) in the interval \(0 \leq x \leq 2 \pi , \cos \left( x + \frac { \pi } { 4 } \right) = 0.5\), giving your answers in terms of \(\pi\). \item Given that \(\log _ { 2 } x = a\), find, in terms of \(a\), the simplest form of
  6. \(\log _ { 2 } ( 16 x )\),
  7. \(\log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right)\).
  8. Hence, or otherwise, solve \(\log _ { 2 } ( 16 x ) - \log _ { 2 } \left( \frac { x ^ { 4 } } { 2 } \right) = \frac { 1 } { 2 }\), giving your answer in its simplest surd form. \item (a) Given that \(3 \sin x = 8 \cos x\), find the value of \(\tan x\).
  9. Find, to 1 decimal place, all the solutions of \(3 \sin x - 8 \cos x = 0\) in the interval \(0 \leq x < 360 ^ { \circ }\).
  10. Find, to 1 decimal place, all the solutions of \(3 \sin ^ { 2 } y - 8 \cos y = 0\) in the interval \(0 \leq y < 360 ^ { \circ }\). \item \end{enumerate} $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
  11. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
  12. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).
  13. Verify that the graph of \(y = \mathrm { f } ( x )\) has stationary points at \(x = \pm \sqrt { } 3\).
  14. Determine whether the stationary value at \(x = \sqrt { } 3\) is a maximum or a minimum.
Edexcel C2 Q7
  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.