Questions C2 (1550 questions)

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OCR MEI C2 2012 June Q10
14 marks Moderate -0.3
10
  1. Use calculus to find, correct to 1 decimal place, the coordinates of the turning points of the curve \(y = x ^ { 3 } - 5 x\). [You need not determine the nature of the turning points.]
  2. Find the coordinates of the points where the curve \(y = x ^ { 3 } - 5 x\) meets the axes and sketch the curve.
  3. Find the equation of the tangent to the curve \(y = x ^ { 3 } - 5 x\) at the point \(( 1 , - 4 )\). Show that, where this tangent meets the curve again, the \(x\)-coordinate satisfies the equation $$x ^ { 3 } - 3 x + 2 = 0$$ Hence find the \(x\)-coordinate of the point where this tangent meets the curve again.
OCR MEI C2 2012 June Q11
10 marks Standard +0.3
11 A geometric progression has first term \(a\) and common ratio \(r\). The second term is 6 and the sum to infinity is 25 .
  1. Write down two equations in \(a\) and \(r\). Show that one possible value of \(a\) is 10 and find the other possible value of \(a\). Write down the corresponding values of \(r\).
  2. Show that the ratio of the \(n\)th terms of the two geometric progressions found in part (i) can be written as \(2 ^ { n - 2 } : 3 ^ { n - 2 }\).
OCR MEI C2 2015 June Q1
5 marks Easy -1.8
1
  1. Differentiate \(12 \sqrt [ 3 ] { x }\).
  2. Integrate \(\frac { 6 } { x ^ { 3 } }\).
OCR MEI C2 2015 June Q2
3 marks Moderate -0.8
2 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { k + 1 } = \frac { 10 } { u _ { k } ^ { 2 } }\).
Calculate \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).
OCR MEI C2 2015 June Q3
5 marks Easy -1.2
3 An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression.
OCR MEI C2 2015 June Q4
4 marks Moderate -0.5
4 A sector of a circle has angle 1.5 radians and area \(27 \mathrm {~cm} ^ { 2 }\). Find the perimeter of the sector.
OCR MEI C2 2015 June Q5
5 marks Moderate -0.3
5 Use calculus to find the set of values of \(x\) for which \(x ^ { 3 } - 6 x\) is an increasing function.
OCR MEI C2 2015 June Q6
5 marks Moderate -0.8
6
  1. On the same axes, sketch the curves \(y = 3 ^ { x }\) and \(y = 3 ^ { 2 x }\), identifying clearly which is which.
  2. Given that \(3 ^ { 2 x } = 729\), find in either order the values of \(3 ^ { x }\) and \(x\).
OCR MEI C2 2015 June Q7
5 marks Moderate -0.3
7 Show that the equation \(\sin ^ { 2 } x = 3 \cos x - 2\) can be expressed as a quadratic equation in \(\cos x\) and hence solve the equation for values of \(x\) between 0 and \(2 \pi\).
OCR MEI C2 2015 June Q8
4 marks Moderate -0.8
8 Fig. 8 shows the graph of \(\log _ { 10 } y\) against \(\log _ { 10 } x\). It is a straight line passing through the points \(( 2,8 )\) and \(( 0,2 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-2_460_634_1868_717} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Find the equation relating \(\log _ { 10 } y\) and \(\log _ { 10 } x\) and hence find the equation relating \(y\) and \(x\).
OCR MEI C2 2015 June Q9
11 marks Moderate -0.8
9 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-3_253_1486_328_292} \captionsetup{labelformat=empty} \caption{Fig. 9.1}
\end{figure}
  1. Jean is designing a model aeroplane. Fig. 9.1 shows her first sketch of the wing's cross-section. Calculate angle A and the area of the cross-section.
  2. Jean then modifies her design for the wing. Fig. 9.2 shows the new cross-section, with 1 unit for each of \(x\) and \(y\) representing one centimetre. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5c7ac296-a911-451b-ad18-5ade3ac23e74-3_431_1682_970_194} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    \end{figure} Here are some of the coordinates that Jean used to draw the new cross-section.
    Upper surfaceLower surface
    \(x\)\(y\)\(x\)\(y\)
    0000
    41.454-0.85
    81.568-0.76
    121.2712-0.55
    161.0416-0.30
    200200
    Use the trapezium rule with trapezia of width 4 cm to calculate an estimate of the area of this cross-section.
OCR MEI C2 2015 June Q10
13 marks Standard +0.3
10 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x + 3\). The curve passes through the point ( 2,9 ).
  1. Find the equation of the tangent to the curve at the point \(( 2,9 )\).
  2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve.
  3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\). Write down the coordinates of the minimum point of the transformed curve.
OCR MEI C2 2015 June Q11
12 marks Standard +0.3
11 Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants.
Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.
  1. How many of Jill's descendants would there be in generation 8 ?
  2. How many of Jill's descendants would there be altogether in the first 15 generations?
  3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 2000003 } { \log _ { 10 } 3 } - 1 .$$ Hence find the least possible value of \(n\).
  4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? \section*{END OF QUESTION PAPER}
AQA C2 Q4
Standard +0.3
4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).
  1. Show that \(\theta = 0.430\) correct to three significant figures.
  2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
  3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-004_417_883_1436_557} Calculate, to two significant figures:
    1. the length of the \(\operatorname { arc } A D\);
    2. the area of the shaded region.
AQA C2 Q5
Moderate -0.3
5 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants. The first three terms of the sequence are given by $$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$
  1. Show that \(p = 0.6\) and find the value of \(q\).
  2. Find the value of \(u _ { 4 }\).
  3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).
AQA C2 Q6
Moderate -0.8
6
  1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
    1. \(y = 2 \sin x\);
    2. \(y = - \sin x\);
    3. \(\quad y = \sin \left( x - 30 ^ { \circ } \right)\).
  2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).
AQA C2 Q8
Standard +0.3
8 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(A ( 9,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-006_763_879_466_577} The curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
    2. Show that the equation of the tangent at \(A ( 9,0 )\) is \(2 y = 3 x - 27\).
    3. Hence find the coordinates of the point \(P\) where the two tangents meet.
  3. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x\).
  4. Calculate the area of the shaded region bounded by the curve and the tangents \(O P\) and \(A P\).
AQA C2 2005 January Q1
8 marks Moderate -0.8
1 A curve is defined for \(x > 0\) by the equation \(y = x + \frac { 2 } { x }\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence show that the gradient of the curve at the point \(P\) where \(x = 2\) is \(\frac { 1 } { 2 }\).
  2. Find an equation of the normal to the curve at this point \(P\).
AQA C2 2005 January Q2
10 marks Moderate -0.8
2 The diagram shows a triangle \(A B C\) and the arc \(A B\) of a circle whose centre is \(C\) and whose radius is 24 cm . \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-2_506_403_1187_781} The length of the side \(A B\) of the triangle is 32 cm . The size of the angle \(A C B\) is \(\theta\) radians.
  1. Show that \(\theta = 1.46\) correct to three significant figures.
  2. Calculate the length of the \(\operatorname { arc } A B\) to the nearest cm .
    1. Calculate the area of the sector \(A B C\) to the nearest \(\mathrm { cm } ^ { 2 }\).
    2. Hence calculate the area of the shaded segment to the nearest \(\mathrm { cm } ^ { 2 }\).
AQA C2 2005 January Q3
6 marks Moderate -0.3
3 An arithmetic series has fifth term 46 and twentieth term 181.
    1. Show that the common difference is 9 .
    2. Find the first term.
  2. Find the sum of the first 20 terms of the series.
  3. The \(n\)th term of the series is \(u _ { n }\). Given that the sum of the first 50 terms of the series is 11525 , find the value of $$\sum _ { n = 21 } ^ { 50 } u _ { n }$$
AQA C2 2005 January Q4
9 marks Moderate -0.8
4
  1. Write \(\sqrt { x }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
  2. Hence express \(\sqrt { x } ( x - 1 )\) in the form \(x ^ { p } - x ^ { q }\).
  3. Find \(\int \sqrt { x } ( x - 1 ) \mathrm { d } x\).
  4. Hence show that \(\int _ { 1 } ^ { 2 } \sqrt { x } ( x - 1 ) \mathrm { d } x = \frac { 4 } { 15 } ( \sqrt { 2 } + 1 )\).
AQA C2 2005 January Q5
7 marks Easy -1.2
5
  1. Given that $$\log _ { a } x = 3 \log _ { a } 6 - \log _ { a } 8$$ where \(a\) is a positive constant, show that \(x = 27\).
  2. Write down the value of:
    1. \(\quad \log _ { 4 } 1\);
    2. \(\log _ { 4 } 4\);
    3. \(\log _ { 4 } 2\);
    4. \(\quad \log _ { 4 } 8\).
AQA C2 2005 January Q6
10 marks Moderate -0.8
6
    1. Using the binomial expansion, or otherwise, express \(( 2 + x ) ^ { 3 }\) in the form \(8 + a x + b x ^ { 2 } + x ^ { 3 }\), where \(a\) and \(b\) are integers. (3 marks)
    2. Write down the expansion of \(( 2 - x ) ^ { 3 }\).
  2. Hence show that \(( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 } = 24 x + 2 x ^ { 3 }\).
  3. Hence show that the curve with equation $$y = ( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 }$$ has no stationary points.
AQA C2 2005 January Q7
11 marks Moderate -0.8
7 The diagram shows the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-4_518_906_1098_552}
  1. Write down the coordinates of the points \(A , B\) and \(C\) marked on the diagram.
  2. Describe the single geometrical transformation by which the curve with equation \(y = \cos 2 x\) can be obtained from the curve with equation \(y = \cos x\).
  3. Solve the equation $$\cos 2 x = 0.37$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). (No credit will be given for simply reading values from a graph.)
    (5 marks)
AQA C2 2005 January Q8
12 marks Moderate -0.8
8 The diagram shows a sketch of the curve with equation \(y = 3 ^ { x } + 1\). \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-5_535_1011_411_513} The curve intersects the \(y\)-axis at the point \(A\).
  1. Write down the \(y\)-coordinate of point \(A\).
    1. Use the trapezium rule with five ordinates (four strips) to find an approximation for \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\), giving your answer to three significant figures.
      (4 marks)
    2. By considering the graph of \(y = 3 ^ { x } + 1\), explain with the aid of a diagram whether your approximation will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\).
      (2 marks)
  3. The line \(y = 5\) intersects the curve \(y = 3 ^ { x } + 1\) at the point \(P\). By solving a suitable equation, find the \(x\)-coordinate of the point \(P\). Give your answer to four decimal places.
    (4 marks)
  4. The curve \(y = 3 ^ { x } + 1\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
    (1 mark)