Questions — OCR (4907 questions)

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OCR C1 Q7
9 marks Standard +0.3
7. A circle has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).
  1. Find the length of the diameter of the circle.
  2. Find an equation for the circle.
  3. Show that the line \(y = 2 x - 3\) is a tangent to the circle and find the coordinates of the point of contact.
OCR C1 Q8
9 marks Moderate -0.8
8. (i) Sketch the graphs of \(y = 2 x ^ { 4 }\) and \(y = 2 \sqrt { x } , x \geq 0\) on the same diagram and write down the coordinates of the point where they intersect.
(ii) Describe fully the transformation that maps the graph of \(y = 2 \sqrt { x }\) onto the graph of \(y = 2 \sqrt { x - 3 }\).
(iii) Find and simplify the equation of the graph obtained when the graph of \(y = 2 x ^ { 4 }\) is stretched by a factor of 2 in the \(x\)-direction, about the \(y\)-axis.
OCR C1 Q9
11 marks Moderate -0.3
9. The straight line \(l _ { 1 }\) passes through the point \(A ( - 2,5 )\) and the point \(B ( 4,1 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The straight line \(l _ { 2 }\) passes through \(B\) and is perpendicular to \(l _ { 1 }\).
  2. Find an equation for \(l _ { 2 }\). Given that \(l _ { 2 }\) meets the \(y\)-axis at the point \(C\),
  3. show that triangle \(A B C\) is isosceles.
OCR C1 Q10
13 marks Standard +0.3
10. \includegraphics[max width=\textwidth, alt={}, center]{6ef55dbd-f18d-4264-b80c-d181473ca7b3-3_531_786_246_523} The diagram shows an open-topped cylindrical container made from cardboard. The cylinder is of height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). Given that the area of card used to make the container is \(192 \pi \mathrm {~cm} ^ { 2 }\),
  1. show that the capacity of the container, \(\mathrm { V } \mathrm { cm } ^ { 3 }\), is given by $$V = 96 \pi r - \frac { 1 } { 2 } \pi r ^ { 3 } .$$
  2. Find the value of \(r\) for which \(V\) is stationary.
  3. Find the corresponding value of \(V\) in terms of \(\pi\).
  4. Determine whether this is a maximum or a minimum value of \(V\).
OCR S2 2005 June Q1
4 marks Easy -1.8
1 It is desired to obtain a random sample of 15 pupils from a large school. One pupil suggests listing all the pupils in the school in alphabetical order and choosing the first 15 names on the list.
  1. Explain why this method is unsatisfactory.
  2. Suggest a better method.
OCR S2 2005 June Q2
4 marks Moderate -0.3
2 A continuous random variable has a normal distribution with mean 25.0 and standard deviation \(\sigma\). The probability that any one observation of the random variable is greater than 20,0 is 0.75 . Find the value of \(\sigma\).
OCR S2 2005 June Q3
8 marks Standard +0.3
3
  1. The random variable \(X\) has a \(\mathrm { B } ( 60,0.02 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( X \leqslant 2 )\).
  2. The random variable \(Y\) has a \(\operatorname { Po } ( 30 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( Y \leqslant 38 )\).
OCR S2 2005 June Q4
9 marks Moderate -0.3
4 The height of sweet pea plants grown in a nursery is a random variable. A random sample of 50 plants is measured and is found to have a mean height 1.72 m and variance \(0.0967 \mathrm {~m} ^ { 2 }\).
  1. Calculate an unbiased estimate for the population variance of the heights of sweet pea plants.
  2. Hence test, at the \(10 \%\) significance level, whether the mean height of sweet pea plants grown by the nursery is 1.8 m , stating your hypotheses clearly.
OCR S2 2005 June Q5
11 marks Moderate -0.3
5 The random variable \(W\) has the distribution \(\mathbf { B } ( 30 , p )\).
  1. Use the exact binomial distribution to calculate \(\mathbf { P } ( W = 10 )\) when \(p = 0.4\).
  2. Find the range of values of \(p\) for which you would expect that a normal distribution could be used as an approximation to the distribution of \(W\).
  3. Use a normal approximation to calculate \(\mathrm { P } ( W = 10 )\) when \(p = 0.4\).
OCR S2 2005 June Q6
11 marks Standard +0.3
6 A factory makes chocolates of different types. The proportion of milk chocolates made on any day is denoted by \(p\). It is desired to test the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.8\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p < 0.8\). The test consists of choosing a random sample of 25 chocolates. \(\mathrm { H } _ { 0 }\) is rejected if the number of milk chocolates is \(k\) or fewer. The test is carried out at a significance level as close to \(5 \%\) as possible.
  1. Use tables to find the value of \(k\), giving the values of any relevant probabilities.
  2. The test is carried out 20 times, and each time the value of \(p\) is 0.8 . Each of the tests is independent of all the others. State the expected number of times that the test will result in rejection of the null hypothesis.
  3. The test is carried out once. If in fact the value of \(p\) is 0.6 , find the probability of rejecting \(\mathrm { H } _ { 0 }\).
  4. The test is carried out twice. Each time the value of \(p\) is equally likely to be 0.8 or 0.6 . Find the probability that exactly one of the two tests results in rejection of the null hypothesis.
OCR S2 2005 June Q7
13 marks Standard +0.3
7 The continuous random variable \(X\) has the probability density function shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b69b1fe8-790d-4727-a892-8ab2ade08962-3_364_766_1229_699}
  1. Find the value of the constant \(k\).
  2. Write down the mean of \(X\), and use integration to find the variance of \(X\).
  3. Three observations of \(X\) are made. Find the probability that \(X < 9\) for all three observations.
  4. The mean of 32 observations of \(X\) is denoted by \(\bar { X }\). State the approximate distribution of \(\bar { X }\), giving its mean and variance. \section*{[Question 8 is printed overleaf.]}
OCR S2 2005 June Q8
12 marks Standard +0.3
8 In excavating an archaeological site, Roman coins are found scattered throughout the site.
  1. State two assumptions needed to model the number of coins found per square metre of the site by a Poisson distribution. Assume now that the number of coins found per square metre of the site can be modelled by a Poisson distribution with mean \(\lambda\).
  2. Given that \(\lambda = 0.75\), calculate the probability that exactly 3 coins are found in a region of the site of area \(7.20 \mathrm {~m} ^ { 2 }\). A test is carried out, at the \(5 \%\) significance level, of the null hypothesis \(\lambda = 0.75\), against the alternative hypothesis \(\lambda > 0.75\), in Region LVI which has area \(4 \mathrm {~m} ^ { 2 }\).
  3. Determine the smallest number of coins that, if found in Region LVI, would lead to rejection of the null hypothesis, stating also the values of any relevant probabilities.
  4. Given that, in fact, \(\lambda = 1.2\) in Region LVI, find the probability that the test results in a Type II error.
OCR C1 Q1
4 marks Moderate -0.3
  1. Solve the inequality
$$x ( 2 x + 1 ) \leq 6 .$$
OCR C1 Q2
4 marks Easy -1.2
  1. Differentiate with respect to \(x\)
$$3 x ^ { 2 } - \sqrt { x } + \frac { 1 } { 2 x }$$
OCR C1 Q3
6 marks Moderate -0.3
  1. The straight line \(l\) has the equation \(x - 2 y = 12\) and meets the coordinate axes at the points \(A\) and \(B\).
Find the distance of the mid-point of \(A B\) from the origin, giving your answer in the form \(k \sqrt { 5 }\).
OCR C1 Q4
6 marks Moderate -0.3
4. (i) By completing the square, find in terms of the constant \(k\) the roots of the equation $$x ^ { 2 } + 2 k x + 4 = 0 .$$ (ii) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$
OCR C1 Q5
6 marks Moderate -0.8
  1. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2 .
Find an equation for the tangent to the curve at \(A\).
OCR C1 Q6
7 marks Moderate -0.3
6. $$f ( x ) = x ^ { \frac { 3 } { 2 } } - 8 x ^ { - \frac { 1 } { 2 } }$$
  1. Evaluate \(\mathrm { f } ( 3 )\), giving your answer in its simplest form with a rational denominator.
  2. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(k \sqrt { 2 }\).
OCR C1 Q7
7 marks Standard +0.3
7. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 7 = 0 \\ & x ^ { 2 } + 2 x y - y ^ { 2 } = 7 \end{aligned}$$
OCR C1 Q8
9 marks Standard +0.3
8.
\includegraphics[max width=\textwidth, alt={}]{4fa65854-801c-4a93-866e-796c000a649f-2_675_689_251_495}
The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x - 18 y + 73 = 0\) and the straight line with equation \(y = 2 x - 3\).
  1. Find the coordinates of the centre and the radius of the circle.
  2. Find the coordinates of the point on the line which is closest to the circle.
OCR C1 Q9
10 marks Standard +0.3
9. \(f ( x ) = 2 x ^ { 2 } + 3 x - 2\).
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) crosses the coordinate axes. When the graph of \(y = \mathrm { f } ( x )\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.
  4. Find the values of \(a , b\) and \(c\).
OCR C1 Q10
13 marks Moderate -0.3
10. The curve with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).
  1. Sketch the curve, showing the coordinates of \(A\) and \(B\).
  2. Show that the tangent to the curve at \(A\) has the equation $$x + y = 2$$ Given that the curve is stationary at the points \(B\) and \(C\),
  3. find the exact coordinates of \(C\).
OCR S2 2006 June Q1
6 marks Moderate -0.5
1 Calculate the variance of the continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 37 } x ^ { 2 } & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
OCR S2 2006 June Q2
7 marks Standard +0.3
2
  1. The random variable \(R\) has the distribution \(\mathrm { B } ( 6 , p )\). A random observation of \(R\) is found to be 6. Carry out a \(5 \%\) significance test of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.45\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p \neq 0.45\), showing all necessary details of your calculation.
  2. The random variable \(S\) has the distribution \(\mathrm { B } ( n , p ) . \mathrm { H } _ { 0 }\) and \(\mathrm { H } _ { 1 }\) are as in part (i). A random observation of \(S\) is found to be 1 . Use tables to find the largest value of \(n\) for which \(\mathrm { H } _ { 0 }\) is not rejected. Show the values of any relevant probabilities.
OCR S2 2006 June Q3
8 marks Standard +0.3
3 The continuous random variable \(T\) has mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( T < 140 ) = 0.01\) and \(\mathrm { P } ( T < 300 ) = 0.8\).
  1. Assuming that \(T\) is normally distributed, calculate the values of \(\mu\) and \(\sigma\). In fact, \(T\) represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about \(10 \%\) of runners took longer than 400 minutes.
  2. State with a reason whether the mean of \(T\) would be higher than, equal to, or lower than the value calculated in part (i).