Questions — OCR (4619 questions)

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OCR S2 2007 June Q1
1 A random sample of observations of a random variable \(X\) is summarised by $$n = 100 , \quad \Sigma x = 4830.0 , \quad \Sigma x ^ { 2 } = 249 \text { 509.16. }$$
  1. Obtain unbiased estimates of the mean and variance of \(X\).
  2. The sample mean of 100 observations of \(X\) is denoted by \(\bar { X }\). Explain whether you would need any further information about the distribution of \(X\) in order to estimate \(\mathrm { P } ( \bar { X } > 60 )\). [You should not attempt to carry out the calculation.]
OCR S2 2007 June Q2
2 It is given that on average one car in forty is yellow. Using a suitable approximation, find the probability that, in a random sample of 130 cars, exactly 4 are yellow.
OCR S2 2007 June Q3
3 The proportion of adults in a large village who support a proposal to build a bypass is denoted by \(p\). A random sample of size 20 is selected from the adults in the village, and the members of the sample are asked whether or not they support the proposal.
  1. Name the probability distribution that would be used in a hypothesis test for the value of \(p\).
  2. State the properties of a random sample that explain why the distribution in part (i) is likely to be a good model.
    \(4 X\) is a continuous random variable.
OCR S2 2007 June Q5
5 The number of system failures per month in a large network is a random variable with the distribution \(\operatorname { Po } ( \lambda )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 2.5\) is carried out by counting \(R\), the number of system failures in a period of 6 months. The result of the test is that \(\mathrm { H } _ { 0 }\) is rejected if \(R > 23\) but is not rejected if \(R \leqslant 23\).
  1. State the alternative hypothesis.
  2. Find the significance level of the test.
  3. Given that \(\mathrm { P } ( R > 23 ) < 0.1\), use tables to find the largest possible actual value of \(\lambda\). You should show the values of any relevant probabilities.
OCR S2 2007 June Q6
6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter \(e\) appears \(19 \%\) of the time. A certain encoded message of 20 letters contains one letter \(e\).
  1. Using an exact binomial distribution, test at the \(10 \%\) significance level whether there is evidence that the proportion of the letter \(e\) in the language from which this message is a sample is less than in German, i.e., less than \(19 \%\).
  2. Give a reason why a binomial distribution might not be an appropriate model in this context.
OCR S2 2007 June Q7
7 Two continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}
T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases} \end{array}$$
  1. Sketch on the same axes the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). [You should not use graph paper or attempt to plot points exactly.]
  2. Explain in everyday terms the difference between the two random variables.
  3. Find the value of \(t\) such that \(\mathrm { P } ( T > t ) = 0.2\).
OCR S2 2007 June Q8
8 A random variable \(Y\) is normally distributed with mean \(\mu\) and variance 12.25. Two statisticians carry out significance tests of the hypotheses \(\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0\).
  1. Statistician \(A\) uses the mean \(\bar { Y }\) of a sample of size 23, and the critical region for his test is \(\bar { Y } > 64.20\). Find the significance level for \(A\) 's test.
  2. Statistician \(B\) uses the mean of a sample of size 50 and a significance level of \(5 \%\).
    (a) Find the critical region for \(B\) 's test.
    (b) Given that \(\mu = 65.0\), find the probability that \(B\) 's test results in a Type II error.
  3. Given that, when \(\mu = 65.0\), the probability that \(A\) 's test results in a Type II error is 0.1365 , state with a reason which test is better.
OCR S2 2007 June Q9
9
  1. The random variable \(G\) has the distribution \(\mathrm { B } ( n , 0.75 )\). Find the set of values of \(n\) for which the distribution of \(G\) can be well approximated by a normal distribution.
  2. The random variable \(H\) has the distribution \(\mathrm { B } ( n , p )\). It is given that, using a normal approximation, \(\mathrm { P } ( H \geqslant 71 ) = 0.0401\) and \(\mathrm { P } ( H \leqslant 46 ) = 0.0122\).
    1. Find the mean and standard deviation of the approximating normal distribution.
    2. Hence find the values of \(n\) and \(p\). 4
OCR C1 Q1
  1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(k \sqrt { 2 }\).
  2. Find the coordinates of the stationary point of the curve with equation
$$y = x + \frac { 4 } { x ^ { 2 } } .$$
OCR C1 Q3
3.
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The diagram shows the curve with equation \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). Show that \(a = - 5\) and find the values of \(b\) and \(c\).
OCR C1 Q4
4. The curve \(C\) has the equation \(y = ( x - a ) ^ { 2 }\) where \(a\) is a constant. Given that $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 x - 6 ,$$
  1. find the value of \(a\),
  2. describe fully a single transformation that would map \(C\) onto the graph of \(y = x ^ { 2 }\).
OCR C1 Q5
5. The straight line \(l _ { 1 }\) has the equation \(3 x - y = 0\). The straight line \(l _ { 2 }\) has the equation \(x + 2 y - 4 = 0\).
  1. Sketch \(l _ { 1 }\) and \(l _ { 2 }\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes.
  2. Find, as exact fractions, the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
OCR C1 Q6
6. (a) Given that \(y = 2 ^ { x }\), find expressions in terms of \(y\) for
  1. \(2 ^ { x + 2 }\),
  2. \(2 ^ { 3 - x }\).
    (b) Show that using the substitution \(y = 2 ^ { x }\), the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$ can be rewritten as $$4 y ^ { 2 } - 33 y + 8 = 0$$ (c) Hence solve the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$
OCR C1 Q7
  1. The point \(A\) has coordinates ( 4,6 ).
Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR C1 Q8
8. (i) Express \(3 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\).
(ii) Sketch the curve with equation \(y = 3 x ^ { 2 } - 12 x + 11\), showing the coordinates of the minimum point of the curve. Given that the curve \(y = 3 x ^ { 2 } - 12 x + 11\) crosses the \(x\)-axis at the points \(A\) and \(B\),
(iii) find the length \(A B\) in the form \(k \sqrt { 3 }\).
OCR C1 Q9
9. A curve has the equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x\).
  1. Show that the curve only crosses the \(x\)-axis at one point. The point \(P\) on the curve has coordinates \(( 3,3 )\).
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
  3. Show that triangle \(O Q R\), where \(O\) is the origin, has area \(28 \frac { 1 } { 8 }\).
OCR C1 Q1
  1. Solve the inequality
$$4 ( x - 2 ) < 2 x + 5$$
OCR C1 Q2
2. $$f ( x ) = 2 - x - x ^ { 3 } .$$ Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
OCR C1 Q3
3. (i) Solve the equation $$y ^ { 2 } + 8 = 9 y .$$ (ii) Hence solve the equation $$x ^ { 3 } + 8 = 9 x ^ { \frac { 3 } { 2 } } .$$
OCR C1 Q4
  1. Given that
$$y = \frac { x ^ { 4 } - 3 } { 2 x ^ { 2 } } ,$$
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 4 } - 9 } { x ^ { 4 } }\).
OCR C1 Q5
5. Find the pairs of values \(( x , y )\) which satisfy the simultaneous equations $$\begin{aligned} & 3 x ^ { 2 } + y ^ { 2 } = 21
& 5 x + y = 7 \end{aligned}$$
OCR C1 Q6
  1. (i) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
    (ii) Find the value of \(x\) such that
$$\frac { 1 + x } { x } = \sqrt { 3 } ,$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.
OCR C1 Q7
7. The straight line \(l\) passes through the point \(P ( - 3,6 )\) and the point \(Q ( 1 , - 4 )\).
  1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The straight line \(m\) has the equation \(2 x + k y + 7 = 0\), where \(k\) is a constant.
    Given that \(l\) and \(m\) are perpendicular,
  2. find the value of \(k\).
OCR C1 Q8
8. (i) Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
(ii) Sketch the graph of \(y = \frac { 3 } { x }\) and write down the equations of any asymptotes.
(iii) Find the values of the constant \(c\) for which the straight line \(y = c - 3 x\) is a tangent to the curve \(y = \frac { 3 } { x }\).
OCR C1 Q9
9. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0$$
  1. Find the coordinates of the centre of \(C\).
  2. Find the radius of \(C\).
  3. Sketch \(C\). Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
  4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).