Questions — OCR (4619 questions)

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OCR C4 2005 June Q7
10 marks Standard +0.8
7 A curve is given parametrically by the equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { t }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
  2. Show that the equation of the tangent at the point \(P \left( 4 , - \frac { 1 } { 2 } \right)\) is $$x - 16 y = 12$$
  3. Find the value of the parameter at the point where the tangent at \(P\) meets the curve again.
OCR C4 2005 June Q8
11 marks Standard +0.3
8
  1. Given that \(\frac { 3 x + 4 } { ( 1 + x ) ( 2 + x ) ^ { 2 } } \equiv \frac { A } { 1 + x } + \frac { B } { 2 + x } + \frac { C } { ( 2 + x ) ^ { 2 } }\), find \(A , B\) and \(C\).
  2. Hence or otherwise expand \(\frac { 3 x + 4 } { ( 1 + x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
  3. State the set of values of \(x\) for which the expansion in part (ii) is valid.
OCR C4 2005 June Q9
13 marks Standard +0.3
9 Newton's law of cooling states that the rate at which the temperature of an object is falling at any instant is proportional to the difference between the temperature of the object and the temperature of its surroundings at that instant. A container of hot liquid is placed in a room which has a constant temperature of \(20 ^ { \circ } \mathrm { C }\). At time \(t\) minutes later, the temperature of the liquid is \(\theta ^ { \circ } \mathrm { C }\).
  1. Explain how the information above leads to the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 ) ,$$ where \(k\) is a positive constant.
  2. The liquid is initially at a temperature of \(100 ^ { \circ } \mathrm { C }\). It takes 5 minutes for the liquid to cool from \(100 ^ { \circ } \mathrm { C }\) to \(68 ^ { \circ } \mathrm { C }\). Show that $$\theta = 20 + 80 \mathrm { e } ^ { - \left( \frac { 1 } { 5 } \ln \frac { 5 } { 3 } \right) t }$$
  3. Calculate how much longer it takes for the liquid to cool by a further \(32 ^ { \circ } \mathrm { C }\).
OCR C4 2006 June Q1
4 marks Moderate -0.3
1 Find the gradient of the curve \(4 x ^ { 2 } + 2 x y + y ^ { 2 } = 12\) at the point \(( 1,2 )\).
OCR C4 2006 June Q2
7 marks Standard +0.3
2
  1. Expand \(( 1 - 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
  2. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { ( 1 + 2 x ) ^ { 2 } } { ( 1 - 3 x ) ^ { 2 } }\) in ascending powers of \(x\).
OCR C4 2006 June Q3
8 marks Moderate -0.3
3
  1. Express \(\frac { 3 - 2 x } { x ( 3 - x ) }\) in partial fractions.
  2. Show that \(\int _ { 1 } ^ { 2 } \frac { 3 - 2 x } { x ( 3 - x ) } \mathrm { d } x = 0\).
  3. What does the result of part (ii) indicate about the graph of \(y = \frac { 3 - 2 x } { x ( 3 - x ) }\) between \(x = 1\) and \(x = 2\) ?
OCR C4 2006 June Q4
8 marks Moderate -0.3
4 The position vectors of three points \(A , B\) and \(C\) relative to an origin \(O\) are given respectively by and $$\begin{aligned} & \overrightarrow { O A } = 7 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k } \\ & \overrightarrow { O B } = 4 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k } \\ & \overrightarrow { O C } = 5 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } \end{aligned}$$
  1. Find the angle between \(A B\) and \(A C\).
  2. Find the area of triangle \(A B C\).
OCR C4 2006 June Q5
8 marks Moderate -0.3
5 A forest is burning so that, \(t\) hours after the start of the fire, the area burnt is \(A\) hectares. It is given that, at any instant, the rate at which this area is increasing is proportional to \(A ^ { 2 }\).
  1. Write down a differential equation which models this situation.
  2. After 1 hour, 1000 hectares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt.
  3. Show that the substitution \(u = \mathrm { e } ^ { x } + 1\) transforms \(\int \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x\) to \(\int \frac { u - 1 } { u } \mathrm {~d} u\).
  4. Hence show that \(\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x = \mathrm { e } - 1 - \ln \left( \frac { \mathrm { e } + 1 } { 2 } \right)\).
OCR C4 2006 June Q7
8 marks Standard +0.3
7 Two lines have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } + \lambda ( 3 \mathbf { i } + \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) ,$$ where \(a\) is a constant.
  1. Given that the lines are skew, find the value that \(a\) cannot take.
  2. Given instead that the lines intersect, find the point of intersection.
OCR C4 2006 June Q8
9 marks Standard +0.3
8
  1. Show that \(\int \cos ^ { 2 } 6 x \mathrm {~d} x = \frac { 1 } { 2 } x + \frac { 1 } { 24 } \sin 12 x + c\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } x \cos ^ { 2 } 6 x \mathrm {~d} x\).
OCR C4 2006 June Q9
12 marks Standard +0.3
9 A curve is given parametrically by the equations $$x = 4 \cos t , \quad y = 3 \sin t$$ where \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Show that the equation of the tangent at the point \(P\), where \(t = p\), is $$3 x \cos p + 4 y \sin p = 12$$
  3. The tangent at \(P\) meets the \(x\)-axis at \(R\) and the \(y\)-axis at \(S . O\) is the origin. Show that the area of triangle \(O R S\) is \(\frac { 12 } { \sin 2 p }\).
  4. Write down the least possible value of the area of triangle \(O R S\), and give the corresponding value of \(p\).
OCR C4 2007 June Q1
5 marks Moderate -0.3
1 The equation of a curve is \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 3 x + 1 } { ( x + 2 ) ( x - 3 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\mathrm { f } ^ { \prime } ( x )\) and deduce that the gradient of the curve is negative at all points on the curve.
OCR C4 2007 June Q2
6 marks Standard +0.3
2 Find the exact value of \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).
OCR C4 2007 June Q3
6 marks Moderate -0.3
3 Find the exact volume generated when the region enclosed between the \(x\)-axis and the portion of the curve \(y = \sin x\) between \(x = 0\) and \(x = \pi\) is rotated completely about the \(x\)-axis.
OCR C4 2007 June Q4
7 marks Moderate -0.3
4
  1. Expand \(( 2 + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), and state the set of values of \(x\) for which the expansion is valid.
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }\).
OCR C4 2007 June Q5
9 marks Moderate -0.3
5 A curve \(C\) has parametric equations $$x = \cos t , \quad y = 3 + 2 \cos 2 t , \quad \text { where } 0 \leqslant t \leqslant \pi$$
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and hence show that the gradient at any point on \(C\) cannot exceed 8 .
  2. Show that all points on \(C\) satisfy the cartesian equation \(y = 4 x ^ { 2 } + 1\).
  3. Sketch the curve \(y = 4 x ^ { 2 } + 1\) and indicate on your sketch the part which represents \(C\).
OCR C4 2007 June Q6
8 marks Standard +0.3
6 The equation of a curve is \(x ^ { 2 } + 3 x y + 4 y ^ { 2 } = 58\). Find the equation of the normal at the point \(( 2,3 )\) on the curve, giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C4 2007 June Q7
10 marks Moderate -0.3
7
  1. Find the quotient and the remainder when \(2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12\) is divided by \(x ^ { 2 } + 4\).
  2. Hence express \(\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 }\) in the form \(A x + B + \frac { C x + D } { x ^ { 2 } + 4 }\), where the values of the constants \(A , B , C\) and \(D\) are to be stated.
  3. Use the result of part (ii) to find the exact value of \(\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 } \mathrm {~d} x\).
OCR C4 2007 June Q8
10 marks Moderate -0.3
8 The height, \(h\) metres, of a shrub \(t\) years after planting is given by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 6 - h } { 20 }$$ A shrub is planted when its height is 1 m .
  1. Show by integration that \(t = 20 \ln \left( \frac { 5 } { 6 - h } \right)\).
  2. How long after planting will the shrub reach a height of 2 m ?
  3. Find the height of the shrub 10 years after planting.
  4. State the maximum possible height of the shrub.
OCR C4 2007 June Q9
11 marks Standard +0.3
9 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } : \mathbf { r } = ( 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) + s ( - 6 \mathbf { i } + 8 \mathbf { j } - 2 \mathbf { k } ) , \\ & L _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } - 8 \mathbf { j } ) + t ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) , \\ & L _ { 3 } : \mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + u ( 3 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$
  1. Calculate the acute angle between \(L _ { 1 }\) and \(L _ { 2 }\).
  2. Given that \(L _ { 1 }\) and \(L _ { 3 }\) are parallel, find the value of \(c\).
  3. Given instead that \(L _ { 2 }\) and \(L _ { 3 }\) intersect, find the value of \(c\). 4
OCR C4 2008 June Q1
6 marks Moderate -0.3
1
  1. Simplify \(\frac { \left( 2 x ^ { 2 } - 7 x - 4 \right) ( x + 1 ) } { \left( 3 x ^ { 2 } + x - 2 \right) ( x - 4 ) }\).
  2. Find the quotient and remainder when \(x ^ { 3 } + 2 x ^ { 2 } - 6 x - 5\) is divided by \(x ^ { 2 } + 4 x + 1\).
OCR C4 2008 June Q2
5 marks Standard +0.3
2 Find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } x ^ { 4 } \ln x \mathrm {~d} x\).
OCR C4 2008 June Q3
8 marks Standard +0.3
3 The equation of a curve is \(x ^ { 2 } y - x y ^ { 2 } = 2\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } - 2 x y } { x ^ { 2 } - 2 x y }\).
  2. (a) Show that, if \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\), then \(y = 2 x\).
    (b) Hence find the coordinates of the point on the curve where the tangent is parallel to the \(x\)-axis.
OCR C4 2008 June Q4
7 marks Standard +0.3
4 Relative to an origin \(O\), the points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively.
  1. Find a vector equation of the line passing through \(A\) and \(B\).
  2. Find the position vector of the point \(P\) on \(A B\) such that \(O P\) is perpendicular to \(A B\).
OCR C4 2008 June Q5
8 marks Standard +0.3
5
  1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
  2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).