Questions — OCR C4 (310 questions)

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OCR C4 2014 June Q4
4 Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 - 2 \sin ^ { 2 } x } { 1 + 2 \sin x \cos x } \mathrm {~d} x = \frac { 1 } { 2 } \ln 2\).
OCR C4 2014 June Q5
5 The equations of three lines are as follows. $$\begin{array} { l l } \text { Line } A : & \mathbf { r } = \mathbf { i } + 4 \mathbf { j } + \mathbf { k } + s ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )
\text { Line } B : & \mathbf { r } = 2 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } )
\text { Line } C : & \mathbf { r } = - \mathbf { i } + 19 \mathbf { j } + 15 \mathbf { k } + u ( 2 \mathbf { i } - 4 \mathbf { j } - 4 \mathbf { k } ) \end{array}$$
  1. Show that lines \(A\) and \(B\) are skew.
  2. Determine, giving reasons, the geometrical relationship between lines \(A\) and \(C\).
OCR C4 2014 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{02e31b5d-10dd-42b1-885a-6db610d788c3-2_570_1191_1509_420} The diagram shows the curve with equation \(x ^ { 2 } + y ^ { 3 } - 8 x - 12 y = 4\). At each of the points \(P\) and \(Q\) the tangent to the curve is parallel to the \(y\)-axis. Find the coordinates of \(P\) and \(Q\).
OCR C4 2014 June Q7
7 A curve has parametric equations $$x = 2 \sin t , \quad y = \cos 2 t + 2 \sin t$$ for \(- \frac { 1 } { 2 } \pi \leqslant t \leqslant \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - 2 \sin t\) and hence find the coordinates of the stationary point.
  2. Find the cartesian equation of the curve.
  3. State the set of values that \(x\) can take and hence sketch the curve.
OCR C4 2014 June Q8
8
  1. Use division to show that \(\frac { t ^ { 3 } } { t + 2 } \equiv t ^ { 2 } - 2 t + 4 - \frac { 8 } { t + 2 }\).
  2. Find \(\int _ { 1 } ^ { 2 } 6 t ^ { 2 } \ln ( t + 2 ) \mathrm { d } t\). Give your answer in the form \(A + B \ln 3 + C \ln 4\).
OCR C4 2014 June Q9
9 Express \(\frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\) in partial fractions and hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } } \frac { 2 + x ^ { 2 } } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \frac { 3 } { 2 } + \frac { 1 } { 3 }\).
OCR C4 2014 June Q10
10 A container in the shape of an inverted cone of radius 3 metres and vertical height 4.5 metres is initially filled with liquid fertiliser. This fertiliser is released through a hole in the bottom of the container at a rate of \(0.01 \mathrm {~m} ^ { 3 }\) per second. At time \(t\) seconds the fertiliser remaining in the container forms an inverted cone of height \(h\) metres.
[0pt] [The volume of a cone is \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  1. Show that \(h ^ { 2 } \frac { \mathrm {~d} h } { \mathrm {~d} t } = - \frac { 9 } { 400 \pi }\).
  2. Express \(h\) in terms of \(t\).
  3. Find the time it takes to empty the container, giving your answer to the nearest minute.
OCR C4 2015 June Q1
1
  1. Express \(\frac { 2 } { 3 - x } + \frac { 3 } { 1 + x }\) as a single fraction in its simplest form.
  2. Hence express \(\left( \frac { 2 } { 3 - x } + \frac { 3 } { 1 + x } \right) \times \frac { x ^ { 2 } + 8 x - 33 } { 121 - x ^ { 2 } }\) as a single fraction in its lowest terms.
OCR C4 2015 June Q2
2 A triangle has vertices at \(A ( 1,1,3 ) , B ( 5,9 , - 5 )\) and \(C ( 6,5 , - 4 ) . P\) is the point on \(A B\) such that \(A P : P B = 3 : 1\).
  1. Show that \(\overrightarrow { C P }\) is perpendicular to \(\overrightarrow { A B }\).
  2. Find the area of the triangle \(A B C\).
OCR C4 2015 June Q3
3 The equation of a curve is \(y = \mathrm { e } ^ { 2 x } \cos x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any stationary points for which \(- \pi \leqslant x \leqslant \pi\). Give your answers correct to 3 significant figures.
OCR C4 2015 June Q4
4
  1. Find the first three terms in the binomial expansion of \(( 8 - 9 x ) ^ { \frac { 2 } { 3 } }\) in ascending powers of \(x\).
  2. State the set of values of \(x\) for which this expansion is valid.
OCR C4 2015 June Q5
5 By first using the substitution \(t = \sqrt { x + 1 }\), find \(\int \mathrm { e } ^ { 2 \sqrt { x + 1 } } \mathrm {~d} x\).
OCR C4 2015 June Q6
6
  1. Use the quotient rule to show that the derivative of \(\frac { \cos x } { \sin x }\) is \(\frac { - 1 } { \sin ^ { 2 } x }\).
  2. Show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \frac { \sqrt { 1 + \cos 2 x } } { \sin x \sin 2 x } \mathrm {~d} x = \frac { 1 } { 2 } ( \sqrt { 6 } - \sqrt { 2 } )\).
OCR C4 2015 June Q7
7 A curve has equation \(( x + y ) ^ { 2 } = x y ^ { 2 }\). Find the gradient of the curve at the point where \(x = 1\).
OCR C4 2015 June Q8
8 In the year 2000 the population density, \(P\), of a village was 100 people per \(\mathrm { km } ^ { 2 }\), and was increasing at the rate of 1 person per \(\mathrm { km } ^ { 2 }\) per year. The rate of increase of the population density is thought to be inversely proportional to the size of the population density. The time in years after the year 2000 is denoted by \(t\).
  1. Write down a differential equation to model this situation, and solve it to express \(P\) in terms of \(t\).
  2. In 2008 the population density of the village was 108 people per \(\mathrm { km } ^ { 2 }\) and in 2013 it was 128 people per \(\mathrm { km } ^ { 2 }\). Determine how well the model fits these figures.
OCR C4 2015 June Q9
9 Two lines have equations $$\mathbf { r } = 3 \mathbf { i } + 5 \mathbf { j } - \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \text { and } \mathbf { r } = 4 \mathbf { i } + 10 \mathbf { j } + 19 \mathbf { k } + \mu ( \mathbf { i } - \mathbf { j } + \alpha \mathbf { k } ) ,$$ where \(\alpha\) is a constant.
Find the value of \(\alpha\) in each of the following cases.
  1. The lines intersect at the point (7,7,1).
  2. The angle between their directions is \(60 ^ { \circ }\).
OCR C4 2015 June Q10
10
  1. Express \(\frac { x + 8 } { x ( x + 2 ) }\) in partial fractions.
  2. By first using division, express \(\frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }\) in the form \(P + \frac { Q } { x } + \frac { R } { x + 2 }\). A curve has parametric equations \(x = \frac { 2 t } { 1 - t } , y = 3 t + \frac { 4 } { t }\).
  3. Show that the cartesian equation of the curve is \(y = \frac { 7 x ^ { 2 } + 16 x + 16 } { x ( x + 2 ) }\).
  4. Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). Give your answer in the form \(L + M \ln 2 + N \ln 3\).
OCR C4 2016 June Q1
1 Find the quotient and the remainder when \(4 x ^ { 3 } + 8 x ^ { 2 } - 5 x + 12\) is divided by \(2 x ^ { 2 } + 1\).
OCR C4 2016 June Q2
2 Use integration to find the exact value of \(\int _ { \frac { 1 } { 16 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( 9 - 6 \cos ^ { 2 } 4 x \right) \mathrm { d } x\).
OCR C4 2016 June Q3
3 Given that \(y \sin 2 x + \frac { 1 } { x } + y ^ { 2 } = 5\), find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 2016 June Q4
4 Find the exact value of \(\int _ { 1 } ^ { 8 } \frac { 1 } { \sqrt [ 3 ] { x } } \ln x \mathrm {~d} x\), giving your answer in the form \(A \ln 2 + B\), where \(A\) and \(B\) are constants to be found.
OCR C4 2016 June Q5
5 The vector equations of two lines are as follows. $$L : \mathbf { r } = \left( \begin{array} { l } 1
4
5 \end{array} \right) + s \left( \begin{array} { c } 2
- 1
3 \end{array} \right) \quad M : \mathbf { r } = \left( \begin{array} { c } 3
2
- 5 \end{array} \right) + t \left( \begin{array} { c } 5
- 3
1 \end{array} \right)$$
  1. Show that the lines \(L\) and \(M\) meet, and find the coordinates of the point of intersection.
  2. Show that the line \(L\) can also be represented by the equation \(\mathbf { r } = \left( \begin{array} { c } 7
    1
    14 \end{array} \right) + u \left( \begin{array} { c } - 4
    2
    - 6 \end{array} \right)\).
OCR C4 2016 June Q6
6 Use the substitution \(u = x ^ { 2 } - 2\) to find \(\int \frac { 6 x ^ { 3 } + 4 x } { \sqrt { x ^ { 2 } - 2 } } \mathrm {~d} x\).
OCR C4 2016 June Q7
7 Given that the binomial expansion of \(( 1 + k x ) ^ { n }\) is \(1 - 6 x + 30 x ^ { 2 } + \ldots\), find the values of \(n\) and \(k\). State the set of values of \(x\) for which this expansion is valid.
OCR C4 2016 June Q8
8 The points \(A\) and \(B\) have position vectors relative to the origin \(O\) given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 \sin \alpha
2 \cos \alpha
- 1 \end{array} \right) \text { and } \overrightarrow { O B } = \left( \begin{array} { c } 2 \cos \alpha
4 \sin \alpha
3 \end{array} \right)$$ where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). It is given that \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are perpendicular.
  1. Calculate the two possible values of \(\alpha\).
  2. Calculate the area of triangle \(O A B\) for the smaller value of \(\alpha\) from part (i).