Questions C4 (1162 questions)

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OCR C4 2013 January Q5
5 The parametric equations of a curve are $$x = 2 + 3 \sin \theta \text { and } y = 1 - 2 \cos \theta \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$$
  1. Find the coordinates of the point on the curve where the gradient is \(\frac { 1 } { 2 }\).
  2. Find the cartesian equation of the curve.
OCR C4 2013 January Q6
6 Use the substitution \(u = 2 x + 1\) to evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 4 x - 1 } { ( 2 x + 1 ) ^ { 5 } } \mathrm {~d} x\).
OCR C4 2013 January Q7
7
  1. Given that \(y = \ln ( 1 + \sin x ) - \ln ( \cos x )\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \cos x }\).
  2. Using this result, evaluate \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sec x \mathrm {~d} x\), giving your answer as a single logarithm.
OCR C4 2013 January Q8
8 The points \(A ( 3,2,1 ) , B ( 5,4 , - 3 ) , C ( 3,17 , - 4 )\) and \(D ( 1,6,3 )\) form a quadrilateral \(A B C D\).
  1. Show that \(A B = A D\).
  2. Find a vector equation of the line through \(A\) and the mid-point of \(B D\).
  3. Show that \(C\) lies on the line found in part (ii).
  4. What type of quadrilateral is \(A B C D\) ?
OCR C4 2013 January Q9
9 The temperature of a freezer is \(- 20 ^ { \circ } \mathrm { C }\). A container of a liquid is placed in the freezer. The rate at which the temperature, \(\theta ^ { \circ } \mathrm { C }\), of a liquid decreases is proportional to the difference in temperature between the liquid and its surroundings. The situation is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta + 20 ) ,$$ where time \(t\) is in minutes and \(k\) is a positive constant.
  1. Express \(\theta\) in terms of \(t , k\) and an arbitrary constant. Initially the temperature of the liquid in the container is \(40 ^ { \circ } \mathrm { C }\) and, at this instant, the liquid is cooling at a rate of \(3 ^ { \circ } \mathrm { C }\) per minute. The liquid freezes at \(0 ^ { \circ } \mathrm { C }\).
  2. Find the value of \(k\) and find also the time it takes (to the nearest minute) for the liquid to freeze. The procedure is repeated on another occasion with a different liquid. The initial temperature of this liquid is \(90 ^ { \circ } \mathrm { C }\). After 19 minutes its temperature is \(0 ^ { \circ } \mathrm { C }\).
  3. Without any further calculation, explain what you can deduce about the value of \(k\) in this case.
OCR C4 2013 January Q10
10
  1. Use algebraic division to express \(\frac { x ^ { 3 } - 2 x ^ { 2 } - 4 x + 13 } { x ^ { 2 } - x - 6 }\) in the form \(A x + B + \frac { C x + D } { x ^ { 2 } - x - 6 }\), where \(A , B , C\) and \(D\) are constants.
  2. Hence find \(\int _ { 4 } ^ { 6 } \frac { x ^ { 3 } - 2 x ^ { 2 } - 4 x + 13 } { x ^ { 2 } - x - 6 } \mathrm {~d} x\), giving your answer in the form \(a + \ln b\).
OCR C4 2009 June Q1
1 Find the quotient and the remainder when \(3 x ^ { 4 } - x ^ { 3 } - 3 x ^ { 2 } - 14 x - 8\) is divided by \(x ^ { 2 } + x + 2\).
OCR C4 2009 June Q2
2 Use the substitution \(x = \tan \theta\) to find the exact value of $$\int _ { 1 } ^ { \sqrt { 3 } } \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } } \mathrm {~d} x$$
OCR C4 2009 June Q3
3
  1. Expand \(( a + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
  2. When \(( 1 - x ) ( a + x ) ^ { - 2 }\) is expanded, the coefficient of \(x ^ { 2 }\) is 0 . Find the value of \(a\).
OCR C4 2009 June Q4
4
  1. Differentiate \(\mathrm { e } ^ { x } ( \sin 2 x - 2 \cos 2 x )\), simplifying your answer.
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { e } ^ { x } \sin 2 x \mathrm {~d} x\).
OCR C4 2009 June Q5
5 A curve has parametric equations $$x = 2 t + t ^ { 2 } , \quad y = 2 t ^ { 2 } + t ^ { 3 }$$
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and find the gradient of the curve at the point \(( 3 , - 9 )\).
  2. By considering \(\frac { y } { x }\), find a cartesian equation of the curve, giving your answer in a form not involving fractions.
OCR C4 2009 June Q6
6 The expression \(\frac { 4 x } { ( x - 5 ) ( x - 3 ) ^ { 2 } }\) is denoted by \(\mathrm { f } ( x )\).
  1. Express f \(( x )\) in the form \(\frac { A } { x - 5 } + \frac { B } { x - 3 } + \frac { C } { ( x - 3 ) ^ { 2 } }\), where \(A , B\) and \(C\) are constants.
  2. Hence find the exact value of \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).
OCR C4 2009 June Q8
8
  1. Given that \(14 x ^ { 2 } - 7 x y + y ^ { 2 } = 2\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 28 x - 7 y } { 7 x - 2 y }\).
  2. The points \(L\) and \(M\) on the curve \(14 x ^ { 2 } - 7 x y + y ^ { 2 } = 2\) each have \(x\)-coordinate 1 . The tangents to the curve at \(L\) and \(M\) meet at \(N\). Find the coordinates of \(N\).
OCR C4 2009 June Q9
9 A tank contains water which is heated by an electric water heater working under the action of a thermostat. The temperature of the water, \(\theta ^ { \circ } \mathrm { C }\), may be modelled as follows. When the water heater is first switched on, \(\theta = 40\). The heater causes the temperature to increase at a rate \(k _ { 1 } { } ^ { \circ } \mathrm { C }\) per second, where \(k _ { 1 }\) is a constant, until \(\theta = 60\). The heater then switches off.
  1. Write down, in terms of \(k _ { 1 }\), how long it takes for the temperature to increase from \(40 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\). The temperature of the water then immediately starts to decrease at a variable rate \(k _ { 2 } ( \theta - 20 ) ^ { \circ } \mathrm { C }\) per second, where \(k _ { 2 }\) is a constant, until \(\theta = 40\).
  2. Write down a differential equation to represent the situation as the temperature is decreasing.
  3. Find the total length of time for the temperature to increase from \(40 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\) and then decrease to \(40 ^ { \circ } \mathrm { C }\). Give your answer in terms of \(k _ { 1 }\) and \(k _ { 2 }\). 4
OCR C4 2010 June Q1
1 Expand \(( 1 + 3 x ) ^ { - \frac { 5 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
OCR C4 2010 June Q2
2 Given that \(y = \frac { \cos x } { 1 - \sin x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying your answer.
OCR C4 2010 June Q3
3 Express \(\frac { x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( x - 2 ) }\) in partial fractions.
OCR C4 2010 June Q4
4 Use the substitution \(u = \sqrt { x + 2 }\) to find the exact value of $$\int _ { - 1 } ^ { 7 } \frac { x ^ { 2 } } { \sqrt { x + 2 } } \mathrm {~d} x$$
OCR C4 2010 June Q5
5 Find the coordinates of the two stationary points on the curve with equation $$x ^ { 2 } + 4 x y + 2 y ^ { 2 } + 18 = 0$$
OCR C4 2010 June Q6
6 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + a \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - \mathbf { k } + s ( 2 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } )$$ respectively, where \(t\) and \(s\) are parameters and \(a\) is a constant.
  1. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular, find the value of \(a\).
  2. Given instead that \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find
    (a) the value of \(a\),
    (b) the angle between the lines.
OCR C4 2010 June Q7
7 The parametric equations of a curve are \(x = \frac { t + 2 } { t + 1 } , y = \frac { 2 } { t + 3 }\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 0\).
  2. Find the cartesian equation of the curve, giving your answer in a form not involving fractions.
OCR C4 2010 June Q8
8
  1. Find the quotient and the remainder when \(x ^ { 2 } - 5 x + 6\) is divided by \(x - 1\).
  2. (a) Find the general solution of the differential equation $$\left( \frac { x - 1 } { x ^ { 2 } - 5 x + 6 } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = y - 5 .$$ (b) Given that \(y = 7\) when \(x = 8\), find \(y\) when \(x = 6\).
OCR C4 2010 June Q9
9
  1. Find \(\int ( x + \cos 2 x ) ^ { 2 } \mathrm {~d} x\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{80f94db1-39be-46f5-896e-277c93cbe4b8-3_538_935_383_646} The diagram shows the part of the curve \(y = x + \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The shaded region bounded by the curve, the axes and the line \(x = \frac { 1 } { 2 } \pi\) is rotated completely about the \(x\)-axis to form a solid of revolution of volume \(V\). Find \(V\), giving your answer in an exact form.
OCR C4 2011 June Q1
1 Simplify \(\frac { x ^ { 4 } - 10 x ^ { 2 } + 9 } { \left( x ^ { 2 } - 2 x - 3 \right) \left( x ^ { 2 } + 8 x + 15 \right) }\).
OCR C4 2011 June Q2
2 Find the unit vector in the direction of \(\left( \begin{array} { c } 2
- 3
\sqrt { 12 } \end{array} \right)\).