Questions C4 (1219 questions)

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Edexcel C4 Q1
8 marks Standard +0.3
  1. A curve has the equation
$$x ^ { 3 } + 2 x y - y ^ { 2 } + 24 = 0$$ Show that the normal to the curve at the point \(( 2 , - 4 )\) has the equation \(y = 3 x - 10\). (8)
Edexcel C4 Q2
9 marks Standard +0.3
2. (a) Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
(b) State the set of values of \(x\) for which your expansion is valid.
(c) Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.
Edexcel C4 Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-04_560_1059_146_406} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \ln ( 2 + \cos x ) , 0 \leq x \leq \pi\).
  1. Complete the table below for points on the curve, giving the \(y\) values to 4 decimal places.
  2. Giving your answers to 3 decimal places, find estimates for the area of the region bounded by the curve and the coordinate axes using the trapezium rule with
    1. 1 strip,
    2. 2 strips,
    3. 4 strips.
  3. Making your reasoning clear, suggest a value to 2 decimal places for the actual area of the region bounded by the curve and the coordinate axes.
    \(x\)0\(\frac { \pi } { 4 }\)\(\frac { \pi } { 2 }\)\(\frac { 3 \pi } { 4 }\)\(\pi\)
    \(y\)1.09860
    1. continued
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-06_563_983_146_379} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Edexcel C4 Q5
11 marks Standard +0.3
5. Relative to a fixed origin, the points \(A , B\) and \(C\) have position vectors ( \(2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }\) ), \(( 5 \mathbf { i } - 4 \mathbf { j } )\) and \(( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )\) respectively.
  1. Show that \(A , B\) and \(C\) all lie on a single straight line.
  2. Write down the ratio \(A B : B C\) The point \(D\) has position vector \(( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\).
  3. Show that \(A D\) is perpendicular to \(B D\).
  4. Find the exact area of triangle \(A B D\).
    5. continued
Edexcel C4 Q6
11 marks Standard +0.3
6. (a) Use the substitution \(x = 2 \sin u\) to evaluate $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$$ (b) Use integration by parts to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x d x$$ 6. continued
Edexcel C4 Q7
15 marks Standard +0.8
7. When a plague of locusts attacks a wheat crop, the proportion of the crop destroyed after \(t\) hours is denoted by \(x\). In a model, it is assumed that the rate at which the crop is destroyed is proportional to \(x ( 1 - x )\). A plague of locusts is discovered in a wheat crop when one quarter of the crop has been destroyed. Given that the rate of destruction at this instant is such that if it remained constant, the crop would be completely destroyed in a further six hours,
  1. show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 } x ( 1 - x )\),
  2. find the percentage of the crop destroyed three hours after the plague of locusts is first discovered.
    7. continued
    7. continued
Edexcel C4 Q1
6 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{80bef9d4-b84c-4d3a-a093-67a466c6d1b9-02_615_791_146_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \frac { 3 x + 1 } { \sqrt { x } } , x > 0\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Find the volume of the solid formed when the shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer in the form \(\pi ( a + \ln b )\), where \(a\) and \(b\) are integers.
□
Edexcel C4 Q2
7 marks Standard +0.3
2. (a) Expand \(( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(b) Hence, or otherwise, show that for small \(x\), $$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$
Edexcel C4 Q3
11 marks Standard +0.3
3. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that $$\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = p - \ln q$$ where \(p\) is rational and \(q\) is an integer.
    3. continued
Edexcel C4 Q4
11 marks Standard +0.3
4. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7 \\ 0 \\ - 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5 \\ 4 \\ - 2 \end{array} \right) \end{aligned}$$
Edexcel C4 Q5
12 marks Moderate -0.8

& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right)
Edexcel C4 Q7
15 marks Moderate -0.5
7
0
- 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5
4
- 2 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right) + \mu \left( \begin{array} { c } - 5
Edexcel C4 Q14
Standard +0.3
14
2 \end{array} \right) , $$ and\\ where \(a\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.\\ Given that the two lines intersect,
  1. find the position vector of their point of intersection,
  2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
  3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.\\ 4. continued\\ 5. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$
  4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
  5. Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
  6. Find the coordinates of \(Q\).\\ 5. continued\\ 6. The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).
  7. Write down a differential equation connecting \(N\) and \(t\). Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
  8. Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant. Given also that the number of bacteria present doubles every six hours,
  9. find the value of \(k\),
  10. find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. of ten, giving your answer to the nearest minute.\\ 6. continued\\ 7. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 } .$$
  11. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
  12. find a cartesian equation for the curve.
  13. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
  14. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
    7. continued
    7. continued
Edexcel C4 2013 January Q5
15 marks Moderate -0.3
  1. Show that \(A\) has coordinates \(( 0,3 )\).
  2. Find the \(x\) coordinate of the point \(B\).
  3. Find an equation of the normal to \(C\) at the point \(A\). The region \(R\), as shown shaded in Figure 2, is bounded by the curve \(C\), the line \(x = - 1\) and the \(x\)-axis.
  4. Use integration to find the exact area of \(R\).
AQA C4 2014 June Q7
9 marks Moderate -0.3
    1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence find the exact value of the gradient of the curve at \(A\).
  2. The normal at \(A\) crosses the \(y\)-axis at the point \(B\). Find the exact value of the \(y\)-coordinate of \(B\).
    [0pt] [2 marks]
OCR C4 2008 January Q7
8 marks Standard +0.3
  1. Given that $$A ( \sin \theta + \cos \theta ) + B ( \cos \theta - \sin \theta ) \equiv 4 \sin \theta$$ find the values of the constants \(A\) and \(B\).
  2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 4 \sin \theta } { \sin \theta + \cos \theta } \mathrm { d } \theta$$ giving your answer in the form \(a \pi - \ln b\).
OCR C4 2008 June Q7
8 marks Moderate -0.3
  1. Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
  2. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\).
OCR C4 2009 June Q7
9 marks Moderate -0.3
  1. The vector \(\mathbf { u } = \frac { 3 } { 13 } \mathbf { i } + b \mathbf { j } + c \mathbf { k }\) is perpendicular to the vector \(4 \mathbf { i } + \mathbf { k }\) and to the vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find the values of \(b\) and \(c\), and show that \(\mathbf { u }\) is a unit vector.
  2. Calculate, to the nearest degree, the angle between the vectors \(4 \mathbf { i } + \mathbf { k }\) and \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\).
OCR MEI C4 2010 June Q5
8 marks Standard +0.3
  1. Verify that \(\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
  2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
  3. Find the coordinates of the point where the pipeline meets the layer of rock.
  4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer. 8 Part of the track of a roller-coaster is modelled by a curve with the parametric equations $$x = 2 \theta - \sin \theta , \quad y = 4 \cos \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ This is shown in Fig. 8. B is a minimum point, and BC is vertical. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-4_602_1447_488_351} \caption{Fig. 8}
    \end{figure}
  5. Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is \(( \pi - 1 ) : ( \pi + 1 )\).
  6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Find the gradient of the track at A .
  7. Show that, when the gradient of the track is \(1 , \theta\) satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
  8. Express \(\cos \theta - 4 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\). Hence solve the equation \(\cos \theta - 4 \sin \theta = 2\) for \(0 \leqslant \theta \leqslant 2 \pi\). {www.ocr.org.uk}) after the live examination series.
    If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
    OCR is part of the \section*{ADVANCED GCE
    MATHEMATICS (MEI)} 4754B
    Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{Candidates answer on the Question Paper} OCR Supplied Materials:
    • Insert (inserted)
    • MEI Examination Formulae and Tables (MF2)
    \section*{Other Materials Required:}
    • Rough paper
    • Scientific or graphical calculator
    Wednesday 9 June 2010 Afternoon \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-5_264_456_881_1361} 1 The train journey from Swansea to London is 307 km and that by road is 300 km . Carry out the calculations performed on the First Great Western website to estimate how much lower the carbon dioxide emissions are when travelling by rail rather than road.
    2 The equation of the curve in Fig. 3 is $$y = \frac { 1 } { 10 ^ { 4 } } \left( x ^ { 3 } - 100 x ^ { 2 } - 10000 x + 2100100 \right)$$ Calculate the speed at which the car has its lowest carbon dioxide emissions and the value of its emissions at that speed.
    [0pt] [An answer obtained from the graph will be given no marks.]
    3
  9. In line 109 the carbon dioxide emissions for a particular train journey from Exeter to London are estimated to be 3.7 tonnes. Obtain this figure.
  10. The text then goes on to state that the emissions per extra passenger on this journey are less than \(\frac { 1 } { 2 } \mathrm {~kg}\). Justify this figure.
  11. \(\_\_\_\_\)
  12. \(\_\_\_\_\) 4 The daily number of trains, \(n\), on a line in another country may be modelled by the function defined below, where \(P\) is the annual number of passengers. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 } \\ & n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 } \\ & n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 } \\ & n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 } \\ & n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 } \\ & \ldots \text { and so on } \ldots \end{aligned}$$
  13. Sketch the graph of \(n\) against \(P\).
  14. Describe, in words, the relationship between the daily number of trains and the annual number of passengers.
  15. \includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
  16. \(\_\_\_\_\)
Edexcel C4 Q4
11 marks Challenging +1.2
  1. Show that the volume of the solid formed is \(\frac { 1 } { 4 } \pi ( \pi + 2 )\).
  2. Find a cartesian equation for the curve.
OCR MEI C4 Q6
4 marks Moderate -0.8
6 Use the Insert provided for this question. The graph of \(y = \tan x\) is given on the Insert.
On this graph sketch the graph of \(y = \operatorname { cotx }\).
Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes.
AQA C4 Q5
10 marks Standard +0.3
5
    1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
    2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
  2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
  3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
  4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
AQA C4 2006 January Q1
8 marks Moderate -0.8
1
  1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2\).
    1. Find f(1).
    2. Show that \(\mathrm { f } ( - 2 ) = 0\).
    3. Hence, or otherwise, show that $$\frac { ( x - 1 ) ( x + 2 ) } { 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 } = \frac { 1 } { a x + b }$$ where \(a\) and \(b\) are integers.
  2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(( 3 x - 1 )\), the remainder is 2 . Find the value of \(d\).
AQA C4 2006 January Q2
11 marks Moderate -0.3
2 A curve is defined by the parametric equations $$x = 3 - 4 t \quad y = 1 + \frac { 2 } { t }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the tangent to the curve at the point where \(t = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Verify that the cartesian equation of the curve can be written as $$( x - 3 ) ( y - 1 ) + 8 = 0$$
AQA C4 2006 January Q3
6 marks Moderate -0.3
3 It is given that \(3 \cos \theta - 2 \sin \theta = R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  1. Find the value of \(R\).
  2. Show that \(\alpha \approx 33.7 ^ { \circ }\).
  3. Hence write down the maximum value of \(3 \cos \theta - 2 \sin \theta\) and find a positive value of \(\theta\) at which this maximum value occurs.