Questions C2 (1410 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel C2 Q3
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c1a3d21d-38fe-4619-9e99-5c4788cdb891-019_675_792_287_568}
\end{figure} In Figure \(1 , A ( 4,0 )\) and \(B ( 3,5 )\) are the end points of a diameter of the circle \(C\). Find
  1. the exact length of \(A B\),
  2. the coordinates of the midpoint \(P\) of \(A B\),
  3. an equation for the circle \(C\).
Edexcel C2 2005 January Q1
Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 3 + 2 x ) ^ { 5 }\), giving each term in its simplest form.
(4)
Edexcel C2 2005 January Q3
3. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which
  1. \(3 ^ { x } = 5\),
  2. \(\log _ { 2 } ( 2 x + 1 ) - \log _ { 2 } x = 2\).
Edexcel C2 2005 January Q4
4. (a) Show that the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ can be written as $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ giving your answers to 1 decimal place where appropriate.
Edexcel C2 2005 January Q5
  1. \(\quad \mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.
When \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is 1 .
When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is 28 .
  1. Find the value of \(a\) and the value of \(b\).
  2. Show that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).
Edexcel C2 2005 January Q6
  1. 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. the first term,
  3. the sum of the first 50 terms, giving your answer to 3 decimal places,
  4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places.
Edexcel C2 2005 January Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-12_707_1072_301_434}
\end{figure} Figure 1 shows the triangle \(A B C\), with \(A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}\) and \(\angle B A C = 0.7\) radians. The \(\operatorname { arc } B D\), where \(D\) lies on \(A C\), 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 \(B C\) and \(C D\) and the \(\operatorname { arc } B D\). Find
  1. the length of the \(\operatorname { arc } B D\),
  2. the perimeter of \(R\), giving your answer to 3 significant figures,
  3. the area of \(R\), giving your answer to 3 significant figures.
Edexcel C2 2005 January Q8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-14_1102_1317_308_340}
\end{figure} The line with equation \(y = 3 x + 20\) cuts the curve with equation \(y = x ^ { 2 } + 6 x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.
  1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.
  2. Use calculus to find the exact area of \(S\).
Edexcel C2 2005 January Q9
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-16_821_958_301_516}
\end{figure} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is \(2 x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2 x\) metres. The perimeter of the stage is 80 m .
  1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 } .$$
  2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value.
  3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\).
  4. Calculate, to the nearest \(\mathrm { m } ^ { 2 }\), the maximum area of the stage.
Edexcel C2 2006 January Q1
  1. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 5 x + c\), where \(c\) is a constant.
Given that \(\mathrm { f } ( 1 ) = 0\),
  1. find the value of \(c\),
  2. factorise \(\mathrm { f } ( x )\) completely,
  3. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 3\) ).
Edexcel C2 2006 January Q2
2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 1 + p x ) ^ { 9 }$$ where \(p\) is a constant. These first 3 terms are \(1,36 x\) and \(q x ^ { 2 }\), where \(q\) is a constant.
(b) Find the value of \(p\) and the value of \(q\).
Edexcel C2 2006 January Q3
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{84b2d36b-c112-4d35-84a1-bc2b707f162d-04_675_792_287_568}
\end{figure} In Figure \(1 , A ( 4,0 )\) and \(B ( 3,5 )\) are the end points of a diameter of the circle \(C\). Find
  1. the exact length of \(A B\),
  2. the coordinates of the midpoint \(P\) of \(A B\),
  3. an equation for the circle \(C\).
Edexcel C2 2006 January Q4
  1. The first term of a geometric series is 120 . The sum to infinity of the series is 480 .
    1. Show that the common ratio, \(r\), is \(\frac { 3 } { 4 }\).
    2. Find, to 2 decimal places, the difference between the 5th and 6th term.
    3. Calculate the sum of the first 7 terms.
    The sum of the first \(n\) terms of the series is greater than 300 .
  2. Calculate the smallest possible value of \(n\).
Edexcel C2 2006 January Q5
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{84b2d36b-c112-4d35-84a1-bc2b707f162d-07_538_611_301_680}
\end{figure} In Figure \(2 O A B\) is a sector of a circle radius 5 m . The chord \(A B\) is 6 m long.
  1. Show that \(\cos A \hat { O } B = \frac { 7 } { 25 }\).
  2. Hence find the angle \(A \hat { O } B\) in radians, giving your answer to 3 decimal places.
  3. Calculate the area of the sector \(O A B\).
  4. Hence calculate the shaded area.
Edexcel C2 2006 January Q6
  1. The speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a train at time \(t\) seconds is given by
$$v = \sqrt { } \left( 1.2 ^ { t } - 1 \right) , \quad 0 \leqslant t \leqslant 30$$ The following table shows the speed of the train at 5 second intervals.
\(t\)051015202530
\(v\)01.222.286.11
  1. Complete the table, giving the values of \(v\) to 2 decimal places. The distance, \(s\) metres, travelled by the train in 30 seconds is given by $$s = \int _ { 0 } ^ { 30 } \sqrt { } \left( 1.2 ^ { t } - 1 \right) \mathrm { d } t$$
  2. Use the trapezium rule, with all the values from your table, to estimate the value of \(s\).
    (3)
Edexcel C2 2006 January Q7
7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Using the result from part (a), find the coordinates of the turning points of \(C\).
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  4. Hence, or otherwise, determine the nature of the turning points of \(C\).
Edexcel C2 2006 January Q8
  1. (a) Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which
$$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3$$ (b) Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which $$\tan ^ { 2 } \theta = 4$$
Edexcel C2 2006 January Q9
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{84b2d36b-c112-4d35-84a1-bc2b707f162d-14_545_922_312_497}
\end{figure} Figure 3 shows the shaded region \(R\) which is bounded by the curve \(y = - 2 x ^ { 2 } + 4 x\) and the line \(y = \frac { 3 } { 2 }\). The points \(A\) and \(B\) are the points of intersection of the line and the curve. Find
  1. the \(x\)-coordinates of the points \(A\) and \(B\),
  2. the exact area of \(R\).
Edexcel C2 2007 January Q1
1. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5$$ Find
  1. \(\mathrm { f } ^ { \prime \prime } ( x )\),
  2. \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).
Edexcel C2 2007 January Q2
2. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 - 2 x ) ^ { 5 }\). Give each term in its simplest form.
(b) If \(x\) is small, so that \(x ^ { 2 }\) and higher powers can be ignored, show that $$( 1 + x ) ( 1 - 2 x ) ^ { 5 } \approx 1 - 9 x$$ DU
Edexcel C2 2007 January Q3
3. The line joining the points \(( - 1,4 )\) and \(( 3,6 )\) is a diameter of the circle \(C\). Find an equation for \(C\).
Edexcel C2 2007 January Q4
4. Solve the equation $$5 ^ { x } = 17$$ giving your answer to 3 significant figures.
Edexcel C2 2007 January Q5
5. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + x - 6$$
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Factorise \(\mathrm { f } ( x )\) completely.
  3. Write down all the solutions to the equation $$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0$$
Edexcel C2 2007 January Q6
6. Find all the solutions, in the interval \(0 \leqslant x < 2 \pi\), of the equation $$2 \cos ^ { 2 } x + 1 = 5 \sin x$$ giving each solution in terms of \(\pi\).
Edexcel C2 2007 January Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{872356ab-68d3-43ee-8b76-650a2697d80e-08_1052_1116_351_413}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 5 )$$ Use calculus to find the total area of the finite region, shown shaded in Figure 1, that is between \(x = 0\) and \(x = 2\) and is bounded by \(C\), the \(x\)-axis and the line \(x = 2\).
(9)