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OCR C4 2015 June Q1
5 marks Moderate -0.8
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
6 marks Standard +0.3
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
6 marks Standard +0.3
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
5 marks Moderate -0.3
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
6 marks Standard +0.3
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
8 marks Standard +0.8
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 marks Standard +0.3
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 marks Standard +0.3
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
7 marks Standard +0.3
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
14 marks Standard +0.3
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
3 marks Moderate -0.5
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
5 marks Standard +0.3
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
5 marks Standard +0.3
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
5 marks Standard +0.3
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
6 marks Standard +0.3
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 marks Standard +0.3
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
6 marks Standard +0.3
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
9 marks Standard +0.3
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).
OCR C4 2016 June Q9
15 marks Standard +0.3
9 A curve has parametric equations \(x = 1 - \cos t , y = \sin t \sin 2 t\), for \(0 \leqslant t \leqslant \pi\).
  1. Find the coordinates of the points where the curve meets the \(x\)-axis.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \cos 2 t + 2 \cos ^ { 2 } t\). Hence find, in an exact form, the coordinates of the stationary points.
  3. Find the cartesian equation of the curve. Give your answer in the form \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a polynomial.
  4. Sketch the curve.
OCR C4 2016 June Q10
12 marks Standard +0.8
10
  1. Express \(\frac { 16 + 5 x - 2 x ^ { 2 } } { ( x + 1 ) ^ { 2 } ( x + 4 ) }\) in partial fractions.
  2. It is given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( 16 + 5 x - 2 x ^ { 2 } \right) y } { ( x + 1 ) ^ { 2 } ( x + 4 ) }$$ and that \(y = \frac { 1 } { 256 }\) when \(x = 0\). Find the exact value of \(y\) when \(x = 2\). Give your answer in the form \(A \mathrm { e } ^ { n }\).
OCR S1 2009 January Q1
8 marks Easy -1.2
1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table.
NumberProbability
00.7
10.2
20.1
The spinner is spun twice. The total of the two numbers on which it lands is denoted by \(X\).
  1. Show that \(\mathrm { P } ( X = 2 ) = 0.18\). The probability distribution of \(X\) is given in the table.
    \(x\)01234
    \(\mathrm { P } ( X = x )\)0.490.280.180.040.01
  2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 2009 January Q2
8 marks Moderate -0.8
2 The table shows the age, \(x\) years, and the mean diameter, \(y \mathrm {~cm}\), of the trunk of each of seven randomly selected trees of a certain species.
Age \(( x\) years \()\)11122028354551
Mean trunk diameter \(( y \mathrm {~cm} )\)12.216.026.439.239.651.360.6
$$\left[ n = 7 , \Sigma x = 202 , \Sigma y = 245.3 , \Sigma x ^ { 2 } = 7300 , \Sigma y ^ { 2 } = 10510.65 , \Sigma x y = 8736.9 . \right]$$
  1. (a) Use an appropriate formula to show that the gradient of the regression line of \(y\) on \(x\) is 1.13 , correct to 2 decimal places.
    (b) Find the equation of the regression line of \(y\) on \(x\).
  2. Use your equation to estimate the mean trunk diameter of a tree of this species with age
    (a) 30 years,
    (b) 100 years. It is given that the value of the product moment correlation coefficient for the data in the table is 0.988 , correct to 3 decimal places.
  3. Comment on the reliability of each of your two estimates.
OCR S1 2009 January Q3
10 marks Moderate -0.8
3 Erika is a birdwatcher. The probability that she will see a woodpecker on any given day is \(\frac { 1 } { 8 }\). It is assumed that this probability is unaffected by whether she has seen a woodpecker on any other day.
  1. Calculate the probability that Erika first sees a woodpecker
    (a) on the third day,
    (b) after the third day.
  2. Find the expectation of the number of days up to and including the first day on which she sees a woodpecker.
  3. Calculate the probability that she sees a woodpecker on exactly 2 days in the first 15 days.
OCR S1 2009 January Q4
7 marks Moderate -0.3
4 Three tutors each marked the coursework of five students. The marks are given in the table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)
Tutor 17367604839
Tutor 26250617665
Tutor 34250635471
  1. Calculate Spearman's rank correlation coefficient, \(r _ { \mathrm { s } }\), between the marks for tutors 1 and 2 .
  2. The values of \(r _ { \mathrm { s } }\) for the other pairs of tutors, are as follows. $$\begin{array} { c c } \text { Tutors } 1 \text { and 3: } & r _ { \mathrm { s } } = - 0.9 \\ \text { Tutors } 2 \text { and 3: } & r _ { \mathrm { s } } = 0.3 \end{array}$$ State which two tutors differ most widely in their judgements. Give your reason.
OCR S1 2009 January Q5
8 marks Easy -1.3
5 The stem-and-leaf diagram shows the masses, in grams, of 23 plums, measured correct to the nearest gram.
5567889
61235689
700245678
80
97
9
\(\quad\) Key \(: 6 \mid 2\) means 62
  1. Find the median and interquartile range of these masses.
  2. State one advantage of using the interquartile range rather than the standard deviation as a measure of the variation in these masses.
  3. State one advantage and one disadvantage of using a stem-and-leaf diagram rather than a box-and-whisker plot to represent data.
  4. James wished to calculate the mean and standard deviation of the given data. He first subtracted 5 from each of the digits to the left of the line in the stem-and-leaf diagram, giving the following.
    0567889
    11235689
    200245678
    30
    47
    The mean and standard deviation of the data in this diagram are 18.1 and 9.7 respectively, correct to 1 decimal place. Write down the mean and standard deviation of the data in the original diagram.