Questions — OCR (4619 questions)

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OCR S2 2005 June Q8
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
  1. Solve the inequality
$$x ( 2 x + 1 ) \leq 6 .$$
OCR C1 Q2
  1. Differentiate with respect to \(x\)
$$3 x ^ { 2 } - \sqrt { x } + \frac { 1 } { 2 x }$$
OCR C1 Q3
  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
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
  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
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. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 7 = 0
& x ^ { 2 } + 2 x y - y ^ { 2 } = 7 \end{aligned}$$
OCR C1 Q8
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
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
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
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
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
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).
OCR S2 2006 June Q4
4
  1. Explain briefly what is meant by a random sample. Random numbers are used to select, with replacement, a sample of size \(n\) from a population numbered 000, 001, 002, ..., 799.
  2. If \(n = 6\), find the probability that exactly 4 of the selected sample have numbers less than 500 .
  3. If \(n = 60\), use a suitable approximation to calculate the probability that at least 40 of the selected sample have numbers less than 500 .
OCR S2 2006 June Q5
5 An airline has 300 seats available on a flight to Australia. It is known from experience that on average only \(99 \%\) of those who have booked seats actually arrive to take the flight, the remaining \(1 \%\) being called 'no-shows'. The airline therefore sells more than 300 seats. If more than 300 passengers then arrive, the flight is over-booked. Assume that the number of no-show passengers can be modelled by a binomial distribution.
  1. If the airline sells 303 seats, state a suitable distribution for the number of no-show passengers, and state a suitable approximation to this distribution, giving the values of any parameters. Using the distribution and approximation in part (i),
  2. show that the probability that the flight is over-booked is 0.4165 , correct to 4 decimal places,
  3. find the largest number of seats that can be sold for the probability that the flight is over-booked to be less than 0.2.
OCR S2 2006 June Q6
6 Customers arrive at a post office at a constant average rate of 0.4 per minute.
  1. State an assumption needed to model the number of customers arriving in a given time interval by a Poisson distribution. Assuming that the use of a Poisson distribution is justified,
  2. find the probability that more than 2 customers arrive in a randomly chosen 1 -minute interval,
  3. use a suitable approximation to calculate the probability that more than 55 customers arrive in a given two-hour interval,
  4. calculate the smallest time for which the probability that no customers arrive in that time is less than 0.02 , giving your answer to the nearest second.
OCR S2 2006 June Q7
7 Three independent researchers, \(A , B\) and \(C\), carry out significance tests on the power consumption of a manufacturer's domestic heaters. The power consumption, \(X\) watts, is a normally distributed random variable with mean \(\mu\) and standard deviation 60. Each researcher tests the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 4000\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu > 4000\). Researcher \(A\) uses a sample of size 50 and a significance level of \(5 \%\).
  1. Find the critical region for this test, giving your answer correct to 4 significant figures. In fact the value of \(\mu\) is 4020 .
  2. Calculate the probability that Researcher \(A\) makes a Type II error.
  3. Researcher \(B\) uses a sample bigger than 50 and a significance level of \(5 \%\). Explain whether the probability that Researcher \(B\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
  4. Researcher \(C\) uses a sample of size 50 and a significance level bigger than \(5 \%\). Explain whether the probability that Researcher \(C\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
  5. State with a reason whether it is necessary to use the Central Limit Theorem at any point in this question.
OCR C1 Q1
  1. Evaluate \(49 ^ { \frac { 1 } { 2 } } + 8 ^ { \frac { 2 } { 3 } }\).
  2. Solve the equation
$$3 x - \frac { 5 } { x } = 2 .$$
OCR C1 Q3
  1. Find the set of values of \(x\) for which
    1. \(6 x - 11 > x + 4\),
    2. \(x ^ { 2 } - 6 x - 16 < 0\).
    3. (i) Sketch on the same diagram the graphs of \(y = ( x - 1 ) ^ { 2 } ( x - 5 )\) and \(y = 8 - 2 x\).
    Label on your diagram the coordinates of any points where each graph meets the coordinate axes.
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$( x - 1 ) ^ { 2 } ( x - 5 ) = 8 - 2 x$$
  3. State the integer, \(n\), such that $$n < \alpha < n + 1 .$$
OCR C1 Q5
5. $$f ( x ) = x ^ { 2 } - 10 x + 17$$
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. State the coordinates of the minimum point of the curve \(y = \mathrm { f } ( x )\).
  3. Deduce the coordinates of the minimum point of each of the following curves:
    1. \(y = \mathrm { f } ( x ) + 4\),
    2. \(y = \mathrm { f } ( 2 x )\).
OCR C1 Q6
6. The points \(P , Q\) and \(R\) have coordinates (-5, 2), (-3, 8) and (9, 4) respectively.
  1. Show that \(\angle P Q R = 90 ^ { \circ }\). Given that \(P , Q\) and \(R\) all lie on a circle,
  2. find the coordinates of the centre of the circle,
  3. show that the equation of the circle can be written in the form $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y = k$$ where \(k\) is an integer to be found.
OCR C1 Q7
7. The straight line \(l _ { 1 }\) has gradient \(\frac { 3 } { 2 }\) and passes through the point \(A ( 5,3 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\). The straight line \(l _ { 2 }\) has the equation \(3 x - 4 y + 3 = 0\) and intersects \(l _ { 1 }\) at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find the coordinates of the mid-point of \(A B\).
  4. Show that the straight line parallel to \(l _ { 2 }\) which passes through the mid-point of \(A B\) also passes through the origin.
OCR C1 Q8
8.
\includegraphics[max width=\textwidth, alt={}, center]{b7078372-d0d3-4563-818d-637260be5efc-3_592_727_251_493} The diagram shows the curve with equation \(y = 2 + 3 x - x ^ { 2 }\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.
  1. Find an equation for \(l\). The line \(m\) is the normal to the curve at the point \(B\).
    Given that \(l\) and \(m\) are parallel,
  2. find the coordinates of \(B\).
OCR C1 Q9
9. The curve \(C\) has the equation $$y = 3 - x ^ { \frac { 1 } { 2 } } - 2 x ^ { - \frac { 1 } { 2 } } , \quad x > 0 .$$
  1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis.
  2. Find the exact coordinates of the stationary point of \(C\).
  3. Determine the nature of the stationary point.
  4. Sketch the curve \(C\).