Questions — OCR (4628 questions)

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OCR C4 2009 January Q5
8 marks Standard +0.3
5
  1. Show that the substitution \(u = \sqrt { x }\) transforms \(\int \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x\) to \(\int \frac { 2 } { u ( 1 + u ) } \mathrm { d } u\).
  2. Hence find the exact value of \(\int _ { 1 } ^ { 9 } \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x\).
OCR C4 2009 January Q6
9 marks Moderate -0.3
6 A curve has parametric equations $$x = t ^ { 2 } - 6 t + 4 , \quad y = t - 3 .$$ Find
  1. the coordinates of the point where the curve meets the \(x\)-axis,
  2. the equation of the curve in cartesian form, giving your answer in a simple form without brackets,
  3. 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.
OCR C4 2009 January Q7
10 marks Standard +0.3
7
  1. Show that the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 4 \\ - 2 \end{array} \right)\) meets the line passing through ( \(9,7,5\) ) and ( \(7,8,2\) ), and find the point of intersection of these lines.
  2. Find the acute angle between these lines.
OCR C4 2009 January Q8
12 marks Standard +0.3
8 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } = 6 x y\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Show that the point \(\left( 2 ^ { \frac { 4 } { 3 } } , 2 ^ { \frac { 5 } { 3 } } \right)\) lies on the curve and that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at this point.
  3. The point \(( a , a )\), where \(a > 0\), lies on the curve. Find the value of \(a\) and the gradient of the curve at this point.
OCR C4 2009 January Q9
11 marks Standard +0.3
9 A liquid is being heated in an oven maintained at a constant temperature of \(160 ^ { \circ } \mathrm { C }\). It may be assumed that the rate of increase of the temperature of the liquid at any particular time \(t\) minutes is proportional to \(160 - \theta\), where \(\theta ^ { \circ } \mathrm { C }\) is the temperature of the liquid at that time.
  1. Write down a differential equation connecting \(\theta\) and \(t\). When the liquid was placed in the oven, its temperature was \(20 ^ { \circ } \mathrm { C }\) and 5 minutes later its temperature had risen to \(65 ^ { \circ } \mathrm { C }\).
  2. Find the temperature of the liquid, correct to the nearest degree, after another 5 minutes. 4
OCR C4 2010 January Q1
4 marks Moderate -0.3
1 Find the quotient and the remainder when \(x ^ { 4 } + 11 x ^ { 3 } + 28 x ^ { 2 } + 3 x + 1\) is divided by \(x ^ { 2 } + 5 x + 2\).
OCR C4 2010 January Q2
6 marks Standard +0.3
2 Points \(A , B\) and \(C\) have position vectors \(- 5 \mathbf { i } - 10 \mathbf { j } + 12 \mathbf { k } , \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } + 6 \mathbf { j } + p \mathbf { k }\) respectively, where \(p\) is a constant.
  1. Given that angle \(A B C = 90 ^ { \circ }\), find the value of \(p\).
  2. Given instead that \(A B C\) is a straight line, find the value of \(p\).
OCR C4 2010 January Q3
5 marks Moderate -0.3
3 By expressing \(\cos 2 x\) in terms of \(\cos x\), find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \cos 2 x } { \cos ^ { 2 } x } \mathrm {~d} x\).
OCR C4 2010 January Q4
6 marks Moderate -0.3
4 Use the substitution \(u = 2 + \ln t\) to find the exact value of $$\int _ { 1 } ^ { \mathrm { e } } \frac { 1 } { t ( 2 + \ln t ) ^ { 2 } } \mathrm {~d} t$$
OCR C4 2010 January Q5
7 marks Moderate -0.3
5
  1. Expand \(( 1 + x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
  2. (a) Hence, or otherwise, expand \(( 8 + 16 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
    (b) State the set of values of \(x\) for which the expansion in part (ii) (a) is valid.
OCR C4 2010 January Q6
6 marks Standard +0.3
6 A curve has parametric equations $$x = 9 t - \ln ( 9 t ) , \quad y = t ^ { 3 } - \ln \left( t ^ { 3 } \right)$$ Show that there is only one value of \(t\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) and state that value.
OCR C4 2010 January Q7
8 marks Standard +0.3
7 Find the equation of the normal to the curve \(x ^ { 3 } + 2 x ^ { 2 } y = y ^ { 3 } + 15\) at the point \(( 2,1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C4 2010 January Q8
7 marks Standard +0.3
8
  1. State the derivative of \(\mathrm { e } ^ { \cos x }\).
  2. Hence use integration by parts to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos x \sin x \mathrm { e } ^ { \cos x } \mathrm {~d} x$$
OCR C4 2010 January Q9
10 marks Standard +0.3
9 The equation of a straight line \(l\) is \(\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) . O\) is the origin.
  1. The point \(P\) on \(l\) is given by \(t = 1\). Calculate the acute angle between \(O P\) and \(l\).
  2. Find the position vector of the point \(Q\) on \(l\) such that \(O Q\) is perpendicular to \(l\).
  3. Find the length of \(O Q\).
OCR C4 2010 January Q10
13 marks Standard +0.3
10
  1. Express \(\frac { 1 } { ( 3 - x ) ( 6 - x ) }\) in partial fractions.
  2. In a chemical reaction, the amount \(x\) grams of a substance at time \(t\) seconds is related to the rate at which \(x\) is changing by the equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 3 - x ) ( 6 - x )$$ where \(k\) is a constant. When \(t = 0 , x = 0\) and when \(t = 1 , x = 1\).
    (a) Show that \(k = \frac { 1 } { 3 } \ln \frac { 5 } { 4 }\).
    (b) Find the value of \(x\) when \(t = 2\).
OCR C4 2011 January Q1
6 marks Moderate -0.3
1
  1. Expand \(( 1 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) as far as the term in \(x ^ { 2 }\).
  2. Hence expand \(\left( 1 - 2 y + 4 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(y\) as far as the term in \(y ^ { 2 }\).
OCR C4 2011 January Q2
7 marks Moderate -0.3
2
  1. Express \(\frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } }\) in the form \(\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }\), where \(A\) and \(B\) are constants.
  2. Hence find the exact value of \(\int _ { 4 } ^ { 5 } \frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x\).
OCR C4 2011 January Q3
8 marks Standard +0.3
3
  1. Show that the derivative of \(\sec x\) can be written as \(\sec x \tan x\).
  2. Find \(\int \frac { \tan x } { \sqrt { 1 + \cos 2 x } } \mathrm {~d} x\).
OCR C4 2011 January Q4
7 marks Moderate -0.3
4 A curve has parametric equations $$x = 2 + t ^ { 2 } , \quad y = 4 t$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the normal at the point where \(t = 4\), giving your answer in the form \(y = m x + c\).
  3. Find a cartesian equation of the curve.
OCR C4 2011 January Q5
9 marks Standard +0.3
5 In this question, \(I\) denotes the definite integral \(\int _ { 2 } ^ { 5 } \frac { 5 - x } { 2 + \sqrt { x - 1 } } \mathrm {~d} x\). The value of \(I\) is to be found using two different methods.
  1. Show that the substitution \(u = \sqrt { x - 1 }\) transforms \(I\) to \(\int _ { 1 } ^ { 2 } \left( 4 u - 2 u ^ { 2 } \right) \mathrm { d } u\) and hence find the exact value of \(I\).
  2. (a) Simplify \(( 2 + \sqrt { x - 1 } ) ( 2 - \sqrt { x - 1 } )\).
    (b) By first multiplying the numerator and denominator of \(\frac { 5 - x } { 2 + \sqrt { x - 1 } }\) by \(2 - \sqrt { x - 1 }\), find the exact value of \(I\).
OCR C4 2011 January Q6
10 marks Standard +0.3
6 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 3 \\ 0 \\ - 2 \end{array} \right) + s \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right)\). The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 0 \\ 1 \\ - 2 \end{array} \right)\).
  1. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
  2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
  3. One of the numbers in the equation of line \(l _ { 1 }\) is changed so that the equation becomes \(\mathbf { r } = \left( \begin{array} { l } 3 \\ 0 \\ a \end{array} \right) + s \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right)\). Given that \(l _ { 1 }\) and \(l _ { 2 }\) now intersect, find \(a\).
OCR C4 2011 January Q7
7 marks Standard +0.8
7 Show that \(\int _ { 0 } ^ { \pi } \left( x ^ { 2 } + 5 x + 7 \right) \sin x \mathrm {~d} x = \pi ^ { 2 } + 5 \pi + 10\).
OCR C4 2011 January Q8
8 marks Standard +0.3
8 The points \(P\) and \(Q\) lie on the curve with equation $$2 x ^ { 2 } - 5 x y + y ^ { 2 } + 9 = 0$$ The tangents to the curve at \(P\) and \(Q\) are parallel, each having gradient \(\frac { 3 } { 8 }\).
  1. Show that the \(x\) - and \(y\)-coordinates of \(P\) and \(Q\) are such that \(x = 2 y\).
  2. Hence find the coordinates of \(P\) and \(Q\).
OCR C4 2011 January Q9
10 marks Standard +0.3
9 Paraffin is stored in a tank with a horizontal base. At time \(t\) minutes, the depth of paraffin in the tank is \(x \mathrm {~cm}\). When \(t = 0 , x = 72\). There is a tap in the side of the tank through which the paraffin can flow. When the tap is opened, the flow of the paraffin is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 4 ( x - 8 ) ^ { \frac { 1 } { 3 } }$$
  1. How long does it take for the level of paraffin to fall from a depth of 72 cm to a depth of 35 cm ?
  2. The tank is filled again to its original depth of 72 cm of paraffin and the tap is then opened. The paraffin flows out until it stops. How long does this take?
OCR C4 2012 January Q1
3 marks Moderate -0.8
1 When the polynomial \(\mathrm { f } ( x )\) is divided by \(x ^ { 2 } + 1\), the quotient is \(x ^ { 2 } + 4 x + 2\) and the remainder is \(x - 1\). Find \(\mathrm { f } ( x )\), simplifying your answer.