Questions — CAIE (7659 questions)

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CAIE P2 2013 June Q4
4 The polynomial \(a x ^ { 3 } - 5 x ^ { 2 } + b x + 9\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 3 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 8 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2013 June Q5
5 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } , \quad y = 4 t \mathrm { e } ^ { t }$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ( t + 1 ) } { \mathrm { e } ^ { t } }\).
  2. Find the equation of the normal to the curve at the point where \(t = 0\).
CAIE P2 2013 June Q6
6
  1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 4 x - 2$$ where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.9\).
  3. Show that this root also satisfies the equation $$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
  4. Use the iterative formula \(x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2013 June Q7
7
  1. Express \(5 \sin 2 \theta + 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. Hence
  2. solve the equation $$5 \sin 2 \theta + 2 \cos 2 \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\),
  3. determine the least value of \(\frac { 1 } { ( 10 \sin 2 \theta + 4 \cos 2 \theta ) ^ { 2 } }\) as \(\theta\) varies.
CAIE P2 2018 June Q1
1 Solve the inequality \(| 3 x - 2 | < | x + 5 |\).
CAIE P2 2018 June Q2
2 A curve has equation \(y = 3 \ln ( 2 x + 9 ) - 2 \ln x\).
  1. Find the \(x\)-coordinate of the stationary point.
  2. Determine whether the stationary point is a maximum or minimum point.
CAIE P2 2018 June Q3
3
  1. Find the quotient when $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 13$$ is divided by \(x ^ { 2 } + 6\) and show that the remainder is 1 .
  2. Show that the equation $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 12 = 0$$ has no real roots.
CAIE P2 2018 June Q4
4
  1. Solve the equation \(2 \ln ( 2 x ) - \ln ( x + 3 ) = 4 \ln 2\).
  2. Hence solve the equation $$2 \ln \left( 2 ^ { u + 1 } \right) - \ln \left( 2 ^ { u } + 3 \right) = 4 \ln 2$$ giving the value of \(u\) correct to 4 significant figures.
CAIE P2 2018 June Q5
5 A curve has equation $$y ^ { 3 } \sin 2 x + 4 y = 8$$ Find the equation of the tangent to the curve at the point where it crosses the \(y\)-axis.
CAIE P2 2018 June Q6
6 It is given that \(\int _ { 0 } ^ { a } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 2 } x } \right) ^ { 2 } \mathrm {~d} x = 10\), where \(a\) is a positive constant.
  1. Show that \(a = 2 \ln \left( \frac { 15 - a } { 4 + \mathrm { e } ^ { \frac { 1 } { 2 } a } } \right)\).
  2. Use the equation in part (i) to show by calculation that \(1.5 < a < 1.6\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2018 June Q7
7
  1. Show that \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) \equiv \sec ^ { 2 } x\).
  2. Solve the equation \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = \tan x + 21\) for \(0 < x < \pi\), giving your answers correct to 3 significant figures.
  3. Find \(\int \left[ 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) - 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) \cos ( 4 y + 2 ) \right] \mathrm { d } y\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S2 2020 June Q1
1 The masses, in grams, of plums of a certain type have the distribution \(\mathrm { N } \left( 40.4,5.2 ^ { 2 } \right)\). The plums are packed in bags, with each bag containing 6 randomly chosen plums. If the total weight of the plums in a bag is less than 220 g the bag is rejected. Find the percentage of bags that are rejected.
CAIE S2 2020 June Q2
2 A shop obtains apples from a certain farm. It has been found that 5\% of apples from this farm are Grade A. Following a change in growing conditions at the farm, the shop management plan to carry out a hypothesis test to find out whether the proportion of Grade A apples has increased. They select 25 apples at random. If the number of Grade A apples is more than 3 they will conclude that the proportion has increased.
  1. State suitable null and alternative hypotheses for the test.
  2. Find the probability of a Type I error.
    In fact 2 of the 25 apples were Grade A .
  3. Which of the errors, Type I or Type II, is possible? Justify your answer.
CAIE S2 2020 June Q3
3 In the data-entry department of a certain firm, it is known that \(0.12 \%\) of data items are entered incorrectly, and that these errors occur randomly and independently.
  1. A random sample of 3600 data items is chosen. The number of these data items that are incorrectly entered is denoted by \(X\).
    1. State the distribution of \(X\), including the values of any parameters.
    2. State an appropriate approximating distribution for \(X\), including the values of any parameters. Justify your choice of approximating distribution.
    3. Use your approximating distribution to find \(\mathrm { P } ( X > 2 )\).
  2. Another large random sample of \(n\) data items is chosen. The probability that the sample contains no data items that are entered incorrectly is more than 0.1 . Use an approximating distribution to find the largest possible value of \(n\).
CAIE S2 2020 June Q4
4 The score on one spin of a 5 -sided spinner is denoted by the random variable \(X\) with probability distribution as shown in the table.
\(x\)01234
\(\mathrm { P } ( X = x )\)0.10.20.40.20.1
  1. Show that \(\operatorname { Var } ( X ) = 1.2\).
    The spinner is spun 200 times. The score on each spin is noted and the mean, \(\bar { X }\), of the 200 scores is found.
  2. Given that \(\mathrm { P } ( \bar { X } > a ) = 0.1\), find the value of \(a\).
  3. Explain whether it was necessary to use the Central Limit theorem in your answer to part (b).
  4. Johann has another, similar, spinner. He suspects that it is biased so that the mean score is less than 2 . He spins his spinner 200 times and finds that the mean of the 200 scores is 1.86 . Given that the variance of the score on one spin of this spinner is also 1.2 , test Johann's suspicion at the 5\% significance level.
CAIE S2 2020 June Q5
5
  1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\).
    1. State the values that \(X\) can take.
      It is given that \(\mathrm { P } ( X = 1 ) = 3 \times \mathrm { P } ( X = 0 )\).
    2. Find \(\lambda\).
    3. Find \(\mathrm { P } ( 4 \leqslant X \leqslant 6 )\).
  2. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\) where \(\mu\) is large. Using a suitable approximating distribution, it is found that \(\mathrm { P } ( Y < 46 ) = 0.0668\), correct to 4 decimal places. Find \(\mu\).
CAIE S2 2020 June Q6
6 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 1 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.
  1. Show that \(k = \frac { a } { a - 1 }\).
  2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
  3. Find the 60th percentile of \(X\) in terms of \(a\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S2 2020 June Q1
1 A random sample of 100 values of a variable \(X\) is taken. These values are summarised below. $$n = 100 \quad \Sigma x = 1556 \quad \Sigma x ^ { 2 } = 29004$$ Calculate unbiased estimates of the population mean and variance of \(X\).
CAIE S2 2020 June Q2
2 Each day at the gym, Sarah completes three runs. The distances, in metres, that she completes in the three runs have the independent distributions \(W \sim \mathrm {~N} ( 1520,450 ) , X \sim \mathrm {~N} ( 2250,720 )\) and \(Y \sim \mathrm {~N} ( 3860,1050 )\). Find the probability that, on a particular day, \(Y\) is less than the total of \(W\) and \(X\).
CAIE S2 2020 June Q3
3 The number of customers who visit a particular shop between 9.00 am and 10.00 am has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 5.2. Following some new advertising, the manager wishes to test whether the value of \(\lambda\) has increased. He chooses a random sample of 20 days and finds that the total number of customers who visited the shop between 9.00 am and 10.00 am on those days is 125 . Use an approximating distribution to test at the \(2.5 \%\) significance level whether the value of \(\lambda\) has increased.
CAIE S2 2020 June Q4
4 The random variable \(A\) has the distribution \(\operatorname { Po } ( 1.5 ) . A _ { 1 }\) and \(A _ { 2 }\) are independent values of \(A\).
  1. Find \(\mathrm { P } \left( A _ { 1 } + A _ { 2 } < 2 \right)\).
  2. Given that \(A _ { 1 } + A _ { 2 } < 2\), find \(\mathrm { P } \left( A _ { 1 } = 1 \right)\).
  3. Give a reason why \(A _ { 1 } - A _ { 2 }\) cannot have a Poisson distribution.
CAIE S2 2020 June Q5
5 Sunita has a six-sided die with faces marked \(1,2,3,4,5,6\). The probability that the die shows a six on any throw is \(p\). Sunita throws the die 500 times and finds that it shows a six 70 times.
  1. Calculate an approximate \(99 \%\) confidence interval for \(p\).
  2. Sunita believes that the die is fair. Use your answer to part (a) to comment on her belief.
  3. Sunita uses the result of her 500 throws to calculate an \(\alpha \%\) confidence interval for \(p\). This interval has width 0.04 . Find the value of \(\alpha\).
CAIE S2 2020 June Q6
6 The length, \(X\) centimetres, of worms of a certain type is modelled by the probability density function $$f ( x ) = \begin{cases} \frac { 6 } { 125 } ( 10 - x ) ( x - 5 ) & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$
  1. State the value of \(\mathrm { E } ( X )\).
  2. Find \(\operatorname { Var } ( X )\).
  3. Two worms of this type are chosen at random. Find the probability that exactly one of them has length less than 6 cm .
CAIE S2 2020 June Q7
7 A market researcher is investigating the length of time that customers spend at an information desk. He plans to choose a sample of 50 customers on a particular day.
  1. He considers choosing the first 50 customers who visit the information desk. Explain why this method is unsuitable.
    The actual lengths of time, in minutes, that customers spend at the information desk may be assumed to have mean \(\mu\) and variance 4.8. The researcher knows that in the past the value of \(\mu\) was 6.0. He wishes to test, at the \(2 \%\) significance level, whether this is still true. He chooses a random sample of 50 customers and notes how long they each spend at the information desk.
  2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
  3. Given that the mean time spent at the information desk by the 50 customers is 6.8 minutes, carry out the test.
  4. Give a reason why it was necessary to use the Central Limit theorem in your answer to part (c).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S2 2021 June Q1
1 The number of goals scored by a team in a match is independent of other matches, and is denoted by the random variable \(X\), which has a Poisson distribution with mean 1.36. A supporter offers to make a donation of \(\\) 5$ to the team for each goal that they score in the next 10 matches. Find the expectation and standard deviation of the amount that the supporter will pay.