Questions C2 (1410 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 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 \(R\).
AQA C2 2011 January Q7
  1. Given that \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find an equation of the tangent at the point on the curve \(C\) where \(x = 1\).
  3. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
    1. Find \(\int \left( x + 3 + \frac { 8 } { x ^ { 4 } } \right) \mathrm { d } x\).
    2. Hence find the area of the region bounded by the curve \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).
  5. The curve \(C\) is translated by \(\left[ \begin{array} { l } 0
    k \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the \(x\)-axis is a tangent to the curve \(y = \mathrm { f } ( x )\), state the value of the constant \(k\).
    (1 mark)
AQA C2 2011 June Q6
  1. The area of the shaded region is given by \(\int _ { 0 } ^ { 2 } \sin x \mathrm {~d} x\), where \(x\) is in radians. Use the trapezium rule with five ordinates (four strips) to find an approximate value for the area of the shaded region, giving your answer to three significant figures.
  2. Describe the geometrical transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = 2 \sin x\).
  3. Use a trigonometrical identity to solve the equation $$2 \sin x = \cos x$$ in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your solutions in radians to three significant figures.
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.
OCR C2 2007 January Q9
  1. Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures.
  2. There are 39 tonnes of coal available. An engineer wishes to calculate \(N\), the total number of trips possible. Show that \(N\) satisfies the inequality $$1.02 ^ { N } \leqslant 1.52$$
  3. Hence, by using logarithms, find the greatest number of trips possible.
OCR MEI C2 2006 January Q11
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find, in exact form, the range of values of \(x\) for which \(x ^ { 3 } - 6 x + 2\) is a decreasing function.
  3. Find the equation of the tangent to the curve at the point \(( - 1,7 )\). Find also the coordinates of the point where this tangent crosses the curve again.
OCR MEI C2 2011 January Q11
  1. Use calculus to find \(\int _ { 1 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } - x + 3 \right) \mathrm { d } x\) and state what this represents.
  2. Find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\) is a decreasing function.
OCR MEI C2 2009 June Q10
  1. On the insert, complete the table and plot \(h\) against \(\log _ { 10 } t\), drawing by eye a line of best fit.
  2. Use your graph to find an equation for \(h\) in terms of \(\log _ { 10 } t\) for this model.
  3. Find the height of the tree at age 100 years, as predicted by this model.
  4. Find the age of the tree when it reaches a height of 29 m , according to this model.
  5. Comment on the suitability of the model when the tree is very young.
OCR MEI C2 2013 June Q5
  1. On the copy of Fig. 5, draw by eye a tangent to the curve at the point where \(x = 2\). Hence find an estimate of the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
  2. Calculate the \(y\)-values on the curve when \(x = 1.8\) and \(x = 2.2\). Hence calculate another approximation to the gradient of \(y = 2 ^ { x }\) when \(x = 2\).
    \(6 S\) is the sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\).
  3. Another geometric progression has first term \(2 a\) and common ratio \(r\). Express the sum to infinity of this progression in terms of \(S\).
  4. A third geometric progression has first term \(a\) and common ratio \(r ^ { 2 }\). Express, in its simplest form, the sum to infinity of this progression in terms of \(S\) and \(r\).
OCR MEI C2 Q11
  1. The speed-time graph on the insert sheet provides the axes and the first two points plotted. Plot the remainder of these points and join them with a smooth curve. The area between this curve and the \(t\)-axis represents the distance travelled by the car in this time.
  2. Using the trapezium rule with 6 values of \(t\) estimate the area under the curve to give the distance travelled. Illustrate on your graph the area found.
  3. John's teacher suggests that the equation of the curve could be \(v = 6 t - \frac { 1 } { 2 } t ^ { 2 }\). Find, by calculus, the area between this curve and the \(t\) axis.
  4. Plot this curve on your graph. Comment on whether the estimates obtained in parts (ii) and (iii) are overestimates or underestimates. 12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions \(2 x\) metres horizontally by \(y\) metres vertically. The top is a semicircle of radius \(x\) metres. The perimeter of the window is 10 metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-4_428_433_1638_766} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure}
  5. Express \(y\) as a function of \(x\).
  6. Find the total area, \(A \mathrm {~m} ^ { 2 }\), in terms of \(x\) and \(y\). Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
  7. Prove that for the maximum value of \(A\), \(y = x\) exactly.
    \section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
    Insert sheet for question 11}
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.
Edexcel C2 Q6
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ).
    1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
    2. Hence, or otherwise, solve the equation $$2 x ^ { 3 } + 3 x ^ { 2 } - 6 x - 8 = 0 ,$$ giving your answers to 2 decimal places where appropriate.
OCR C2 Q5
  1. Find the number of sit-ups that Habib will do in the fifth week.
  2. Show that he will do a total of 1512 sit-ups during the first eight weeks. In the \(n\)th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
  3. Find the value of \(n\).
OCR C2 Q7
  1. Show that the common difference is 5 .
  2. Find the 12th term. \end{enumerate} Another arithmetic sequence has first term -12 and common difference 7 .
    Given that the sums of the first \(n\) terms of these two sequences are equal,
  3. find the value of \(n\).