Questions — OCR (4619 questions)

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OCR S2 2008 January Q1
1 The random variable \(T\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( T > 80 ) = 0.05\) and \(\mathrm { P } ( T > 50 ) = 0.75\). Find the values of \(\mu\) and \(\sigma\).
OCR S2 2008 January Q2
2 A village has a population of 600 people. A sample of 12 people is obtained as follows. A list of all 600 people is obtained and a three-digit number, between 001 and 600 inclusive, is allocated to each name in alphabetical order. Twelve three-digit random numbers, between 001 and 600 inclusive, are obtained and the people whose names correspond to those numbers are chosen.
  1. Find the probability that all 12 of the numbers chosen are 500 or less.
  2. When the selection has been made, it is found that all of the numbers chosen are 500 or less. One of the people in the village says, "The sampling method must have been biased." Comment on this statement.
OCR S2 2008 January Q3
3 The random variable \(G\) has the distribution \(\operatorname { Po } ( \lambda )\). A test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 4.5\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda \neq 4.5\), based on a single observation of \(G\). The critical region for the test is \(G \leqslant 1\) and \(G \geqslant 9\).
  1. Find the significance level of the test.
  2. Given that \(\lambda = 5.5\), calculate the probability that the test results in a Type II error.
OCR S2 2008 January Q4
4 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The results of 40 independent observations of \(Y\) are summarised by $$\Sigma y = 3296.0 , \quad \Sigma y ^ { 2 } = 286800.40$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Use your answers to part (i) to estimate the probability that a single random observation of \(Y\) will be less than 60.0.
  3. Explain whether it is necessary to know that \(Y\) is normally distributed in answering part (i) of this question.
OCR S2 2008 January Q5
5 Over a long period the number of visitors per week to a stately home was known to have the distribution \(\mathrm { N } \left( 500,100 ^ { 2 } \right)\). After higher car parking charges were introduced, a sample of four randomly chosen weeks gave a mean number of visitors per week of 435 . You should assume that the number of visitors per week is still normally distributed with variance \(100 ^ { 2 }\).
  1. Test, at the \(10 \%\) significance level, whether there is evidence that the mean number of visitors per week has fallen.
  2. Explain why it is necessary to assume that the distribution of the number of visitors per week (after the introduction of higher charges) is normal in order to carry out the test.
OCR S2 2008 January Q6
6 The number of house sales per week handled by an estate agent is modelled by the distribution \(\operatorname { Po } ( 3 )\).
  1. Find the probability that, in one randomly chosen week, the number of sales handled is
    (a) greater than 4 ,
    (b) exactly 4 .
  2. Use a suitable approximation to the Poisson distribution to find the probability that, in a year consisting of 50 working weeks, the estate agent handles more than 165 house sales.
  3. One of the conditions needed for the use of a Poisson model to be valid is that house sales are independent of one another.
    (a) Explain, in non-technical language, what you understand by this condition.
    (b) State another condition that is needed.
OCR S2 2008 January Q7
7 A continuous random variable \(X _ { 1 }\) has probability density function given by $$f ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 2 }\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Find \(\mathrm { E } \left( X _ { 1 } \right)\) and \(\operatorname { Var } \left( X _ { 1 } \right)\).
  4. Sketch the graph of \(y = \mathrm { f } ( x - 1 )\).
  5. The continuous random variable \(X _ { 2 }\) has probability density function \(\mathrm { f } ( x - 1 )\) for all \(x\). Write down the values of \(\mathrm { E } \left( X _ { 2 } \right)\) and \(\operatorname { Var } \left( X _ { 2 } \right)\).
OCR S2 2008 January Q8
4 marks
8 Consultations are taking place as to whether a site currently in use as a car park should be developed as a shopping mall. An agency acting on behalf of a firm of developers claims that at least \(65 \%\) of the local population are in favour of the development. In a survey of a random sample of 12 members of the local population, 6 are in favour of the development.
  1. Carry out a test, at the \(10 \%\) significance level, to determine whether the result of the survey is consistent with the claim of the agency.
  2. A local residents' group claims that no more than \(35 \%\) of the local population are in favour of the development. Without further calculations, state with a reason what can be said about the claim of the local residents' group.
  3. A test is carried out, at the \(15 \%\) significance level, of the agency's claim. The test is based on a random sample of size \(2 n\), and exactly \(n\) of the sample are in favour of the development. Find the smallest possible value of \(n\) for which the outcome of the test is to reject the agency's claim.
    [0pt] [4] 4
OCR C1 Q1
  1. Solve the equation
$$9 ^ { x } = 3 ^ { x + 2 } .$$
OCR C1 Q2
  1. The straight line \(l\) has the equation \(x - 5 y = 7\).
The straight line \(m\) is perpendicular to \(l\) and passes through the point \(( - 4,1 )\).
Find an equation for \(m\) in the form \(y = m x + c\).
OCR C1 Q3
3.
\includegraphics[max width=\textwidth, alt={}, center]{4fec0924-d727-4d4f-81e6-918e1ccfedbd-1_330_1230_829_386} The diagram shows the rectangles \(A B C D\) and \(E F G H\) which are similar.
Given that \(A B = ( 3 - \sqrt { 5 } ) \mathrm { cm } , A D = \sqrt { 5 } \mathrm {~cm}\) and \(E F = ( 1 + \sqrt { 5 } ) \mathrm { cm }\), find the length \(E H\) in cm, giving your answer in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are integers.
OCR C1 Q4
4. (i) Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
(ii) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$
OCR C1 Q5
  1. (i) Solve the inequality
$$x ^ { 2 } + 3 x > 10 .$$ (ii) Find the set of values of \(x\) which satisfy both of the following inequalities: $$\begin{aligned} & 3 x - 2 < x + 3
& x ^ { 2 } + 3 x > 10 \end{aligned}$$
OCR C1 Q6
6. $$f ( x ) = 4 x ^ { 2 } + 12 x + 9 .$$
  1. Determine the number of real roots that exist for the equation \(\mathrm { f } ( x ) = 0\).
  2. Solve the equation \(\mathrm { f } ( x ) = 8\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.
OCR C1 Q7
7. The circle \(C\) has centre \(( - 1,6 )\) and radius \(2 \sqrt { 5 }\).
  1. Find an equation for \(C\). The line \(y = 3 x - 1\) intersects \(C\) at the points \(A\) and \(B\).
  2. Find the \(x\)-coordinates of \(A\) and \(B\).
  3. Show that \(A B = 2 \sqrt { 10 }\).
OCR C1 Q8
8. $$f ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0 .$$
  1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
  2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
  3. Determine whether the turning point is a maximum or minimum point.
OCR C1 Q9
9. (i) Find an equation for the tangent to the curve \(y = x ^ { 2 } + 2\) at the point \(( 1,3 )\) in the form \(y = m x + c\).
(ii) Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers.
(iii) Describe fully the transformation that maps the graph of \(y = x ^ { 2 } + 2\) onto the graph of \(y = x ^ { 2 } - 6 x + 11\).
(iv) Use your answers to parts (i) and (iii) to deduce an equation for the tangent to the curve \(y = x ^ { 2 } - 6 x + 11\) at the point with \(x\)-coordinate 4.
OCR C1 Q10
10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 2 ) ^ { 3 }$$
  1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
  2. Find \(\mathrm { f } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
  3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
  4. Show that \(m\) has the equation \(y = 3 x + 8\).
OCR C1 Q1
  1. Solve the equation
$$x ^ { 2 } - 4 x - 8 = 0$$ giving your answers in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.
OCR C1 Q2
2. The curve \(C\) has the equation $$y = x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
Given that the minimum point of \(C\) has coordinates \(( - 2,5 )\), find the values of \(a\) and \(b\).
OCR C1 Q3
3. (i) Solve the simultaneous equations $$\begin{aligned} & y = x ^ { 2 } - 6 x + 7
& y = 2 x - 9 \end{aligned}$$ (ii) Hence, describe the geometrical relationship between the curve \(y = x ^ { 2 } - 6 x + 7\) and the straight line \(y = 2 x - 9\).
OCR C1 Q4
4. (i) Evaluate $$\left( 36 ^ { \frac { 1 } { 2 } } + 16 ^ { \frac { 1 } { 4 } } \right) ^ { \frac { 1 } { 3 } }$$ (ii) Solve the equation $$3 x ^ { - \frac { 1 } { 2 } } - 4 = 0 .$$
OCR C1 Q5
  1. (i) Sketch on the same diagram the curve with equation \(y = ( x - 2 ) ^ { 2 }\) and the straight line with equation \(y = 2 x - 1\).
Label on your sketch the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the set of values of \(x\) for which $$( x - 2 ) ^ { 2 } > 2 x - 1$$
OCR C1 Q6
  1. (i) Given that \(y = x ^ { \frac { 1 } { 3 } }\), show that the equation
$$2 x ^ { \frac { 1 } { 3 } } + 3 x ^ { - \frac { 1 } { 3 } } = 7$$ can be rewritten as $$2 y ^ { 2 } - 7 y + 3 = 0 .$$ (ii) Hence, solve the equation $$2 x ^ { \frac { 1 } { 3 } } + 3 x ^ { - \frac { 1 } { 3 } } = 7$$
OCR C1 Q7
  1. Given that
$$y = \sqrt { x } - \frac { 4 } { \sqrt { x } }$$
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
  3. show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = 0 .$$