Questions C4 (1219 questions)

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OCR MEI C4 Q6
4 marks Challenging +1.2
6 A sequence is defined by $$a _ { n + 1 } = 2 a _ { n } + 3 a _ { n - 1 } \quad \text { with } a _ { 1 } = 1 \text { and } a _ { 2 } = 1 .$$ Using the method on page 5, show that the value to which the ratio of successive terms converges is 3 .
[0pt] [4]
OCR MEI C4 Q8
Standard +0.3
8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{070e9904-12b9-4458-b8f2-60c89b31b828-093_1013_1399_488_372} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)
  1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
  2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). Show that the vector \(2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
  3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
  4. Write down a vector equation of the line RS. Calculate the coordinates of S.
OCR MEI C4 2005 June Q2
6 marks Moderate -0.5
2 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.
OCR MEI C4 2005 June Q3
4 marks Easy -1.2
3 Solve the equation $$\sec ^ { 2 } \theta = 4 , \quad 0 \leqslant \theta \leqslant \pi ,$$ giving your answers in terms of \(\pi\).
OCR MEI C4 2005 June Q4
5 marks Standard +0.3
4 Fig. 4 shows a sketch of the region enclosed by the curve \(\sqrt { 1 + \mathrm { e } ^ { - 2 x } }\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7a1123f8-53cd-4b24-bec6-8c3bccc22653-3_517_755_1576_649} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Find the volume of the solid generated when this region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in an exact form.
OCR MEI C4 2005 June Q5
7 marks Moderate -0.3
5 Solve the equation \(2 \cos 2 x = 1 + \cos x\), for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
OCR MEI C4 2005 June Q6
8 marks Standard +0.3
6 A curve has cartesian equation \(y ^ { 2 } - x ^ { 2 } = 4\).
  1. Verify that $$x = t - \frac { 1 } { t } , \quad y = t + \frac { 1 } { t } ,$$ are parametric equations of the curve.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - 1 ) ( t + 1 ) } { t ^ { 2 } + 1 }\). Hence find the coordinates of the stationary points of the curve. Section B (36 marks)
OCR MEI C4 2005 June Q7
18 marks Standard +0.3
7 In a chemical process, the mass \(M\) grams of a chemical at time \(t\) minutes is modelled by the differential equation $$\frac { \mathrm { d } M } { \mathrm {~d} t } = \frac { M } { t \left( 1 + t ^ { 2 } \right) }$$
  1. Find \(\int \frac { t } { 1 + t ^ { 2 } } \mathrm {~d} t\).
  2. Find constants \(A , B\) and \(C\) such that $$\frac { 1 } { t \left( 1 + t ^ { 2 } \right) } = \frac { A } { t } + \frac { B t + C } { 1 + t ^ { 2 } } .$$
  3. Use integration, together with your results in parts (i) and (ii), to show that $$M = \frac { K t } { \sqrt { 1 + t ^ { 2 } } } ,$$ where \(K\) is a constant.
  4. When \(t = 1 , M = 25\). Calculate \(K\). What is the mass of the chemical in the long term?
AQA C4 2011 January Q1
6 marks Moderate -0.3
1
  1. Express \(2 \sin x + 5 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
    1. Write down the maximum value of \(2 \sin x + 5 \cos x\).
    2. Find the value of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) at which this maximum occurs, giving the value of \(x\) to the nearest \(0.1 ^ { \circ }\).
AQA C4 2011 January Q2
10 marks Moderate -0.3
2
  1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } - x - 2\).
    1. Use the Factor Theorem to show that \(3 x + 1\) is a factor of \(\mathrm { f } ( x )\).
    2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
    3. Simplify \(\frac { 9 x ^ { 3 } + 21 x ^ { 2 } + 6 x } { \mathrm { f } ( x ) }\).
  2. When the polynomial \(9 x ^ { 3 } + p x ^ { 2 } - x - 2\) is divided by \(3 x - 2\), the remainder is - 4 . Find the value of the constant \(p\).
AQA C4 2011 January Q3
12 marks Standard +0.3
3
  1. Express \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) in the form \(\frac { A } { 1 + x } + \frac { B } { 3 + 5 x }\), where \(A\) and \(B\) are integers.
  2. Hence, or otherwise, find the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) up to and including the term in \(x ^ { 2 }\).
  3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) is valid.
    (2 marks)
AQA C4 2011 January Q4
6 marks Standard +0.3
4 A curve is defined by the parametric equations $$x = 3 \mathrm { e } ^ { t } , \quad y = \mathrm { e } ^ { 2 t } - \mathrm { e } ^ { - 2 t }$$
    1. Find the gradient of the curve at the point where \(t = 0\).
    2. Find an equation of the tangent to the curve at the point where \(t = 0\).
  2. Show that the cartesian equation of the curve can be written in the form $$y = \frac { x ^ { 2 } } { k } - \frac { k } { x ^ { 2 } }$$ where \(k\) is an integer.
AQA C4 2011 January Q5
7 marks Moderate -0.3
5 A model for the radioactive decay of a form of iodine is given by $$m = m _ { 0 } 2 ^ { - \frac { 1 } { 8 } t }$$ The mass of the iodine after \(t\) days is \(m\) grams. Its initial mass is \(m _ { 0 }\) grams.
  1. Use the given model to find the mass that remains after 10 grams of this form of iodine have decayed for 14 days, giving your answer to the nearest gram.
  2. A mass of \(m _ { 0 }\) grams of this form of iodine decays to \(\frac { m _ { 0 } } { 16 }\) grams in \(d\) days. Find the value of \(d\).
  3. After \(n\) days, a mass of this form of iodine has decayed to less than \(1 \%\) of its initial mass. Find the minimum integer value of \(n\).
AQA C4 2011 January Q6
10 marks Standard +0.3
6
    1. Given that \(\tan 2 x + \tan x = 0\), show that \(\tan x = 0\) or \(\tan ^ { 2 } x = 3\).
    2. Hence find all solutions of \(\tan 2 x + \tan x = 0\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
      (l mark)
    1. Given that \(\cos x \neq 0\), show that the equation $$\sin 2 x = \cos x \cos 2 x$$ can be written in the form $$2 \sin ^ { 2 } x + 2 \sin x - 1 = 0$$
    2. Show that all solutions of the equation \(2 \sin ^ { 2 } x + 2 \sin x - 1 = 0\) are given by \(\sin x = \frac { \sqrt { 3 } - 1 } { p }\), where \(p\) is an integer.
AQA C4 2011 January Q7
10 marks Moderate -0.3
7
    1. Solve the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)\) to find \(x\) in terms of \(t\).
    2. Given that \(x = 1\) when \(t = 0\), show that the solution can be written as $$x = ( a - \cos b t ) ^ { 2 }$$ where \(a\) and \(b\) are constants to be found.
  2. The height, \(x\) metres, above the ground of a car in a fairground ride at time \(t\) seconds is modelled by the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)\). The car is 1 metre above the ground when \(t = 0\).
    1. Find the greatest height above the ground reached by the car during the ride.
    2. Find the value of \(t\) when the car is first 5 metres above the ground, giving your answer to one decimal place.
AQA C4 2011 January Q8
14 marks Standard +0.3
8 The coordinates of the points \(A\) and \(B\) are \(( 3 , - 2,4 )\) and \(( 6,0,3 )\) respectively.
The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ - 1 \\ 3 \end{array} \right]\).
    1. Find the vector \(\overrightarrow { A B }\).
    2. Calculate the acute angle between \(\overrightarrow { A B }\) and the line \(l _ { 1 }\), giving your answer to the nearest \(0.1 ^ { \circ }\).
  2. The point \(D\) lies on \(l _ { 1 }\) where \(\lambda = 2\). The line \(l _ { 2 }\) passes through \(D\) and is parallel to \(A B\).
    1. Find a vector equation of line \(l _ { 2 }\) with parameter \(\mu\).
    2. The diagram shows a symmetrical trapezium \(A B C D\), with angle \(D A B\) equal to angle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{5fe2527a-33da-4076-b3fa-4cab545336ec-9_620_675_1197_726} The point \(C\) lies on line \(l _ { 2 }\). The length of \(A D\) is equal to the length of \(B C\). Find the coordinates of \(C\).
AQA C4 2012 January Q1
11 marks Standard +0.3
1
  1. Express \(\frac { 2 x + 3 } { 4 x ^ { 2 } - 1 }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { 2 x + 1 }\), where \(A\) and \(B\) are integers. (3 marks)
  2. Express \(\frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 }\) in the form \(C x + \frac { D ( 2 x + 3 ) } { 4 x ^ { 2 } - 1 }\), where \(C\) and \(D\) are integers.
    (3 marks)
  3. Evaluate \(\int _ { 1 } ^ { 2 } \frac { 12 x ^ { 3 } - 7 x - 6 } { 4 x ^ { 2 } - 1 } \mathrm {~d} x\), giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational numbers.
    (5 marks)
AQA C4 2012 January Q2
6 marks Moderate -0.3
2 Angle \(\alpha\) is acute and \(\cos \alpha = \frac { 3 } { 5 }\). Angle \(\beta\) is obtuse and \(\sin \beta = \frac { 1 } { 2 }\).
    1. Find the value of \(\tan \alpha\) as a fraction.
      (1 mark)
    2. Find the value of \(\tan \beta\) in surd form.
  2. Hence show that \(\tan ( \alpha + \beta ) = \frac { m \sqrt { 3 } - n } { n \sqrt { 3 } + m }\), where \(m\) and \(n\) are integers.
    (3 marks)
AQA C4 2012 January Q3
7 marks Moderate -0.3
3
  1. Find the binomial expansion of \(( 1 + 6 x ) ^ { \frac { 2 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
    (2 marks)
  2. Find the binomial expansion of \(( 8 + 6 x ) ^ { \frac { 2 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
    (3 marks)
  3. Use your answer from part (b) to find an estimate for \(\sqrt [ 3 ] { 100 }\) in the form \(\frac { a } { b }\), where \(a\) and \(b\) are integers.
    (2 marks)
AQA C4 2012 January Q4
9 marks Standard +0.3
4 A scientist is testing models for the growth and decay of colonies of bacteria. For a particular colony, which is growing, the model is \(P = A \mathrm { e } ^ { \frac { 1 } { 8 } t }\), where \(P\) is the number of bacteria after a time \(t\) minutes and \(A\) is a constant.
  1. This growing colony consists initially of 500 bacteria. Calculate the number of bacteria, according to the model, after one hour. Give your answer to the nearest thousand.
  2. For a second colony, which is decaying, the model is \(Q = 500000 \mathrm { e } ^ { - \frac { 1 } { 8 } t }\), where \(Q\) is the number of bacteria after a time \(t\) minutes. Initially, the growing colony has 500 bacteria and, at the same time, the decaying colony has 500000 bacteria.
    1. Find the time at which the populations of the two colonies will be equal, giving your answer to the nearest 0.1 of a minute.
    2. The population of the growing colony will exceed that of the decaying colony by 45000 bacteria at time \(T\) minutes. Show that $$\left( \mathrm { e } ^ { \frac { 1 } { 8 } T } \right) ^ { 2 } - 90 \mathrm { e } ^ { \frac { 1 } { 8 } T } - 1000 = 0$$ and hence find the value of \(T\), giving your answer to one decimal place.
      (4 marks)
AQA C4 2012 January Q5
11 marks Moderate -0.3
5 A curve is defined by the parametric equations $$x = 8 t ^ { 2 } - t , \quad y = \frac { 3 } { t }$$
  1. Show that the cartesian equation of the curve can be written as \(x y ^ { 2 } + 3 y = k\), stating the value of the integer \(k\).
    (2 marks)
    1. Find an equation of the tangent to the curve at the point \(P\), where \(t = \frac { 1 } { 4 }\).
    2. Verify that the tangent at \(P\) intersects the curve when \(x = \frac { 3 } { 2 }\).
AQA C4 2012 January Q6
10 marks Standard +0.3
6
  1. Use the Factor Theorem to show that \(4 x - 3\) is a factor of $$16 x ^ { 3 } + 11 x - 15$$
  2. Given that \(x = \cos \theta\), show that the equation $$27 \cos \theta \cos 2 \theta + 19 \sin \theta \sin 2 \theta - 15 = 0$$ can be written in the form $$16 x ^ { 3 } + 11 x - 15 = 0$$
  3. Hence show that the only solutions of the equation $$27 \cos \theta \cos 2 \theta + 19 \sin \theta \sin 2 \theta - 15 = 0$$ are given by \(\cos \theta = \frac { 3 } { 4 }\).
AQA C4 2012 January Q7
9 marks Standard +0.3
7 Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } x \sin 3 x$$ given that \(y = 1\) when \(x = \frac { \pi } { 6 }\). Give your answer in the form \(y = \frac { 9 } { \mathrm { f } ( x ) }\).
AQA C4 2012 January Q8
12 marks Standard +0.3
8 The points \(A\) and \(B\) have coordinates \(( 4 , - 2,3 )\) and \(( 2,0 , - 1 )\) respectively. The line \(l\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 4 \\ - 2 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 5 \\ - 2 \end{array} \right]\).
    1. Find the vector \(\overrightarrow { A B }\).
    2. Find the acute angle between \(A B\) and the line \(l\), giving your answer to the nearest degree.
  2. The point \(C\) lies on the line \(l\) such that the angle \(A B C\) is a right angle. Given that \(A B C D\) is a rectangle, find the coordinates of the point \(D\).
AQA C4 2013 January Q1
7 marks Moderate -0.3
1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 7\).
  1. Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x + 1 )\).
    (2 marks)
  2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + d\), where \(d\) is a constant.
    1. Given that \(( 2 x + 1 )\) is a factor of \(\mathrm { g } ( x )\), show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 4\).
      (1 mark)
    2. Given that \(\mathrm { g } ( x )\) can be written as \(\mathrm { g } ( x ) = ( 2 x + 1 ) \left( x ^ { 2 } + a \right)\), where \(a\) is an integer, express \(\mathrm { g } ( x )\) as a product of three linear factors.
    3. Hence, or otherwise, show that \(\frac { \mathrm { g } ( x ) } { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x } = p + \frac { q } { x }\), where \(p\) and \(q\) are integers.
      (3 marks)