Questions — OCR C4 (310 questions)

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OCR C4 2011 January Q4
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
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
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 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 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
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
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.
OCR C4 2012 January Q2
2
  1. Find, in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), an equation of the line \(l\) through the points ( \(4,2,7\) ) and ( \(5 , - 4 , - 1\) ).
  2. Find the acute angle between the line \(l\) and a line in the direction of the vector \(\left( \begin{array} { l } 1
    2
    3 \end{array} \right)\).
OCR C4 2012 January Q3
3 The equation of a curve \(C\) is \(( x + 3 ) ( y + 4 ) = x ^ { 2 } + y ^ { 2 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. The line \(2 y = x + 3\) meets \(C\) at two points. What can be said about the tangents to \(C\) at these points? Justify your answer.
  3. Find the equation of the tangent at the point ( 6,0 ), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C4 2012 January Q4
4
  1. Expand \(( 1 - 4 x ) ^ { \frac { 1 } { 4 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  2. The term of lowest degree in the expansion of $$( 1 + a x ) \left( 1 + b x ^ { 2 } \right) ^ { 7 } - ( 1 - 4 x ) ^ { \frac { 1 } { 4 } }$$ in ascending powers of \(x\) is the term in \(x ^ { 3 }\). Find the values of the constants \(a\) and \(b\).
OCR C4 2012 January Q5
5 Use the substitution \(u = \cos x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \cos ^ { 2 } x d x$$
OCR C4 2012 January Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{cf154c94-6248-4dda-91e8-61349cc10482-3_606_846_251_614} The diagram shows the curves \(y = \cos x\) and \(y = \sin x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The region \(R\) is bounded by the curves and the \(x\)-axis. Find the volume of the solid of revolution formed when \(R\) is rotated completely about the \(x\)-axis, giving your answer in terms of \(\pi\).
OCR C4 2012 January Q7
3 marks
7 The equation of a straight line \(l\) is $$\mathbf { r } = \left( \begin{array} { l } 1
0
2 \end{array} \right) + t \left( \begin{array} { r } 1
- 1
0 \end{array} \right) .$$ \(O\) is the origin.
  1. Find the position vector of the point \(P\) on \(l\) such that \(O P\) is perpendicular to \(l\).
  2. A point \(Q\) on \(l\) is such that the length of \(O Q\) is 3 units. Find the two possible position vectors of \(Q\). [3]
OCR C4 2012 January Q8
8 A curve is defined by the parametric equations $$x = \sin ^ { 2 } \theta , \quad y = 4 \sin \theta - \sin ^ { 3 } \theta ,$$ where \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 - 3 \sin ^ { 2 } \theta } { 2 \sin \theta }\).
  2. Find the coordinates of the point on the curve at which the gradient is 2 .
  3. Show that the curve has no stationary points.
  4. Find a cartesian equation of the curve, giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR C4 2012 January Q9
9 Find the exact value of \(\int _ { 0 } ^ { 1 } \left( x ^ { 2 } + 1 \right) \mathrm { e } ^ { 2 x } \mathrm {~d} x\).
OCR C4 2012 January Q10
10
  1. Write down the derivative of \(\sqrt { y ^ { 2 } + 1 }\) with respect to \(y\).
  2. Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( x - 1 ) \sqrt { y ^ { 2 } + 1 } } { x y }\) and that \(y = \sqrt { \mathrm { e } ^ { 2 } - 2 \mathrm { e } }\) when \(x = \mathrm { e }\),
    find a relationship between \(x\) and \(y\).
OCR C4 2013 January Q1
1 Find \(\int x \cos 3 x \mathrm {~d} x\).
OCR C4 2013 January Q2
2 Find the first three terms in the expansion of \(( 9 - 16 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\), and state the set of values for which this expansion is valid.
OCR C4 2013 January Q3
3 The equation of a curve is \(x y ^ { 2 } = x ^ { 2 } + 1\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\), and hence find the coordinates of the stationary points on the curve.
OCR C4 2013 January Q4
4 The equations of two lines are $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \text { and } \mathbf { r } = 6 \mathbf { i } + 8 \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } ) .$$
  1. Show that these lines meet, and find the coordinates of the point of intersection.
  2. Find the acute angle between these lines.
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.