Questions — SPS (1106 questions)

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SPS SPS FM Statistics 2022 January Q5
5. In a large population of hens, the weight of a hen is normally distributed with mean \(\mu \mathrm { kg }\) and standard deviation \(\sigma \mathrm { kg }\). A random sample of 100 hens is taken from the population. The mean weight for the sample is denoted \(\bar { X }\).
a. State the distribution of \(\bar { X }\) giving its mean and variance. The sample values are summarised by \(\sum x = 199.8\) and \(\sum x ^ { 2 } = 407.8\) where \(x \mathrm {~kg}\) is the weight of a hen.
b. Find an unbiased estimate for \(\mu\).
c. Find an unbiased estimate for \(\sigma ^ { 2 }\).
d. Find a \(90 \%\) confidence interval for \(\mu\). It is found that \(\sigma = 0.27\). It is decided to test, at the \(1 \%\) level of significance, the null hypothesis \(\mu = 1.95\) against the alternative hypothesis \(\mu > 1.95\).
e. Find the \(p\)-value for the test.
f. Write down the conclusion reached.
g. Explain whether or not the central limit theorem was required in part e.
SPS SPS FM Statistics 2022 January Q6
6. The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematical examinations each year can be modelled by a Poisson distribution with a mean of 3 .
a. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations.
b. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with mean of 7 . Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A-grades in their Mathematics and English examinations.
c. Lowkey School is given a performance rating, \(P = 2 X + 3 Y\), based on the number of A-grades achieved in Mathematics and English. Find: $$\begin{array} { l l } \text { i. } & \mathrm { E } ( P )
\text { ii. } & \operatorname { Var } ( P ) \end{array}$$ d. What assumption did you make in answering part (b)? Did you need this assumption to answer part (c)? Justify your answers.
SPS SPS FM Statistics 2022 January Q7
7. The continuous random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c l } 0 & x < 1
\frac { 4 } { x ^ { 5 } } & x \geq 1 \end{array} \right.$$ a. Find the cumulative distribution function, \(F ( x )\), of \(X\).
b. Find the interquartile range of \(X\).
c. Show that the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), is given by $$g ( y ) = \left\{ \begin{array} { c l } 2 y & 0 < y \leq 1
0 & \text { otherwise } \end{array} \right.$$ d. Find the value of \(a\) for which \(\mathrm { E } \left( \frac { 1 } { X ^ { 2 } } \right) = a \mathrm { E } \left( X ^ { 2 } \right)\).
SPS SPS SM Statistics 2022 January Q1
  1. Each day Anna drives to work.
  • R is the event that it is raining.
  • \(L\) is the event that Anna arrives at work late.
You are given that \(\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25\) and \(\mathrm { P } ( R \cup L ) = 0.41\).
i. Draw a Venn diagram showing the events \(R\) and \(L\). Fill in the probability corresponding to each of the four regions of your diagram.
ii. Determine whether the events \(R\) and \(L\) are independent.
iii. Find \(\mathrm { P } ( L \mid R )\)
SPS SPS SM Statistics 2022 January Q2
2. The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{252d5094-6cdd-4379-bcd5-ca6a5cc48c7a-3_956_1497_1361_269}
i. Use the diagram to estimate the median and interquartile range of the data.
ii. Use your answers to part (i) to show that there are very few, if any, outliers in the sample. Below is the frequency table for these data:
Temperature
\(( t\) degrees Celsius \()\)
\(3.0 \leq t \leq 3.4\)\(3.4 < t \leq 3.8\)\(3.8 < t \leq 4.2\)\(4.2 < t \leq 4.6\)\(4.6 < t \leq 5.0\)
Frequency2512524315750
iii. Use the table to calculate estimates for the mean and standard deviation.
iv. The temperatures are converted from degrees Celsius to degrees Fahrenheit using the formula \(F = 1.8 C + 32\). Hence estimate the mean and standard deviation of the temperatures in degrees Fahrenheit.
SPS SPS SM Statistics 2022 January Q3
3. The weights of Braeburn apples on display in a supermarket, measured in grams, are Normally distributed with mean 210.5 and standard deviation 15.2.
i. Find the probability that a randomly selected apple weighs at least 220 grams.
ii. 80 apples are selected at random.
a) Find the probability that more than 18 of these apples weigh at least 220 grams.
b) Find the expectation and standard deviation for the number of apples that weigh at least 220 grams.
c) State a suitable approximating distribution, including any parameters, for the number of apples that weigh at least 220 grams.
d) Explain why this approximating distribution is suitable. The supermarket also sells Cox's Orange Pippin apples. The weights of these apples, measured in grams, are Normally distributed with mean 185 and standard deviation \(\sigma\).
iii. Given that \(10 \%\) of randomly selected Cox's Orange Pippin apples weigh less than 170 grams, calculate the value of \(\sigma\).
SPS SPS FM 2021 November Q1
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
The roots of the equation $$x ^ { 3 } - 8 x ^ { 2 } + 28 x - 32 = 0$$ are \(\alpha , \beta\) and \(\gamma\). Without solving the equation, find the value of $$( \alpha + 2 ) ( \beta + 2 ) ( \gamma + 2 )$$
SPS SPS FM 2021 November Q2
3 marks
  1. The equation of a curve in polar coordinates is
$$r = 11 + 9 \sec \theta$$ Show that a cartesian equation of the curve is $$( x - 9 ) \sqrt { x ^ { 2 } + y ^ { 2 } } = 11 x$$ [3 marks]
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SPS SPS FM 2021 November Q3
3. The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$| z - 6 i | = 2 | z - 3 |$$ Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle.
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SPS SPS FM 2021 November Q4
4 marks
4. Prove that $$\sum _ { r = 1 } ^ { n } 18 \left( r ^ { 2 } - 4 \right) = n \left( 6 n ^ { 2 } + 9 n - 69 \right) .$$ [4 marks]
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SPS SPS FM 2021 November Q5
4 marks
5. Use a trigonometrical substitution to show that $$\int _ { 0 } ^ { 2 } \frac { 1 } { \left( 16 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x = \frac { 1 } { 16 \sqrt { 3 } }$$ [4 marks]
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SPS SPS FM 2021 November Q6
6. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int _ { 1 } ^ { \infty } \frac { 1 } { \cosh \mathrm { u } } \mathrm { du }$$ giving your answer in an exact form.
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SPS SPS FM 2021 November Q7
11 marks
7. The curve with equation $$y = - x + \tanh ( 36 x ) , \quad x \geq 0$$ has a maximum turning point \(A\).
  1. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\).
  2. Show that the \(y\)-coordinate of \(A\) is $$\frac { \sqrt { 35 } } { 6 } - \frac { 1 } { 36 } \ln ( 6 + \sqrt { 35 } )$$ [BLANK PAGE] The function \(f\) is defined by \(f ( x ) = ( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\).
  3. Find \(\mathrm { f } ^ { \prime \prime \prime } ( \mathrm { x } )\) (i.e. the third derivative of \(f\) ) showing all your intermediate steps. Hence, find the Maclaurin series for \(f ( x )\) up to and including the \(x ^ { 3 }\) term.
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  4. Use the expansion of \(e ^ { x }\) together with the result in part (a) to show that, up to and including the \(x ^ { 3 }\) term, $$e ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } = 1 + 2 x + x ^ { 2 } + k x ^ { 3 }$$ where \(k\) is a rational number to be found.
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SPS SPS FM 2021 November Q9
9. (a) Show that $$\frac { 1 } { 9 r - 4 } - \frac { 1 } { 9 r + 5 } = \frac { 9 } { ( 9 r - 4 ) ( 9 r + 5 ) }$$ (b) Hence use the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 9 r - 4 ) ( 9 r + 5 ) }$$ [BLANK PAGE]
SPS SPS FM 2021 November Q10
4 marks
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6278666a-d95f-461c-ab81-742c8faae1d5-24_517_1596_331_278} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a closed curve \(C\) with equation $$r = 3 \sqrt { \cos ( 2 \theta ) } , \quad \text { where } - \frac { \pi } { 4 } < \theta \leq \frac { \pi } { 4 } , \quad \frac { 3 \pi } { 4 } < \theta \leq \frac { 5 \pi } { 4 }$$ The lines \(P Q , S R , P S\) and \(Q R\) are tangents to \(C\), where \(P Q\) and \(S R\) are parallel to the initial line and \(P S\) and \(Q R\) are perpendicular to the initial line. The point \(O\) is the pole.
  1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1.
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  2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1.
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SPS SPS SM 2021 November Q1
  1. Find \(\frac { d y } { d x }\) for the following functions, simplifying your answers as far as possible.
    i) \(y = \cos x - 2 \sin 2 x\)
    ii) \(y = \frac { 1 } { 2 } x ^ { 4 } + 2 x ^ { 4 } \ln x\)
    iii) \(y = \frac { 2 e ^ { 3 x } - 1 } { 3 e ^ { 3 x } - 1 }\)
a. Express \(\frac { 5 x + 7 } { ( x + 3 ) ( x + 1 ) ^ { 2 } }\) in partial fractions. In this question you must show all of your algebraic steps clearly. The function \(f ( x ) = \frac { 2 - 6 x + 5 x ^ { 2 } } { x ^ { 2 } ( 1 - 2 x ) }\) can be written in the form; $$f ( x ) = \frac { - 2 } { x } + \frac { 2 } { x ^ { 2 } } + \frac { 1 } { 1 - 2 x }$$ b. Hence find the exact value of \(\int _ { 2 } ^ { 3 } \frac { 2 - 6 x + 5 x ^ { 2 } } { x ^ { 2 } ( 1 - 2 x ) } d x\)
SPS SPS SM 2021 November Q3
3. In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = ( 1 - 3 x ) ( 3 - x ) ^ { 3 }$$
SPS SPS SM 2021 November Q4
  1. Find the equation of the normal to the curve \(y = 4 \ln ( 2 x - 3 )\) at the point where the curve crosses the \(x\) axis. Give your answer in the form \(a x + b y + k = 0\) where \(a > 0\).
i) Write \(\log _ { 16 } y - \log _ { 16 } x\) as a single logarithm.
ii) Solve the simultaneous equations, giving your answers in an exact form. $$\begin{gathered} \log _ { 3 } y = \log _ { 3 } ( 9 - 6 x ) + 1
\log _ { 16 } y - \log _ { 16 } x = \frac { 1 } { 4 } \end{gathered}$$
SPS SPS SM 2021 November Q6
6. a. Prove the following trigonometric identities. You must show all of your algebraic steps clearly. $$( \cos x + \sin x ) ( \operatorname { cosec } x - \sec x ) \equiv 2 \cot 2 x$$ b. Solve the following equation for \(x\) in the interval \(0 \leq x \leq \pi\). Giving your answers in terms of \(\pi\). $$\sin \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { 2 } \sin \left( 2 x - \frac { \pi } { 6 } \right)$$
SPS SPS SM 2021 November Q7
  1. The diagram below represents the graph of the function \(y = ( 2 x - 5 ) ^ { 4 } - 1\)
    \includegraphics[max width=\textwidth, alt={}, center]{1650b28f-be4e-4600-89ca-67c2d3026c5b-10_784_657_233_694}
    a. Find the intersections of this graph with the \(x\) axis.
    b. Hence find the exact value of the area bounded by the curve and the \(x\) axis.
a. Express \(2 \sqrt { 3 } \cos 2 x - 6 \sin 2 x\) in the form \(R \cos ( 2 x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
b. Hence
i. Solve the equation \(2 \sqrt { 3 } \cos 2 x - 6 \sin 2 x = 6\) for \(0 \leq x \leq 2 \pi\) Giving your answers in terms of \(\pi\).
c. It can be shown that \(y = 9 \sin 2 x + 4 \cos 2 x\) can be written as \(y = \sqrt { 97 } \sin \left( 2 x + 24.0 ^ { \circ } \right)\)
i. State the transformations in the order of occurrence which transform the curve \(y = 9 \sin 2 x + 4 \cos 2 x\) to the curve \(y = \sin x\)
ii. Find the exact maximum and minimum values of the function; $$f ( x ) = \frac { 1 } { 11 - 9 \sin 2 x - 4 \cos 2 x }$$
SPS SPS SM 2021 November Q9
9. a.
i. Show that \(\cos ^ { 2 } x \equiv \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos 2 x\)
ii. Hence find \(\int 2 \cos ^ { 2 } 4 x d x\)
b. Find \(\int \sin ^ { 3 } x d x\)
SPS SPS SM 2021 November Q10
10.
a. The parametric equations of a curve are \(x = \theta \cos \theta\) and \(y = \sin \theta\) Find the gradient of the curve at the point for which \(\theta = \pi\)
b. A curve is defined parametrically by the equations; $$x = \cos \theta \quad y = \left( \frac { \sin \theta } { 2 } \right) \left( \sin \frac { \theta } { 2 } \right)$$ Show that the cartesian equation of the curve can be written as \(y ^ { 2 } = \frac { 1 } { 8 } ( 1 - x ) ^ { 2 } ( 1 + x )\)
SPS SPS FM 2022 October Q1
  1. a) Rationalise the denominator for \(\frac { \sqrt { 8 } + 2 } { 5 - \sqrt { 2 } }\)
    b) Solve
$$( \sqrt { 2 } ) ^ { x + 1 } = \frac { 1 } { 4 ^ { 4 - 3 x } }$$ [BLANK PAGE]
SPS SPS FM 2022 October Q2
2. Given that $$f ( x ) = \ln x , x > 0$$ Sketch on separate axes the graphs of
i) \(y = f ( x )\)
ii) \(\quad y = f ( x - 4 )\) Show on each diagram, the point where the graph meets or crosses the \(x\)-axis. In each case, state the equation of the asymptote.
SPS SPS FM 2022 October Q3
3. The first term of a geometric series is 120 . The sum to infinity of the series is 480 .
a) Show that the common ratio, \(r\), is \(\frac { 3 } { 4 }\) The sum of the first n terms of the series is greater than 300 .
b) Calculate the smallest possible value of n
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