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CAIE Further Paper 4 2023 November Q3
8 marks Standard +0.3
3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10 of these people, the LDL blood cholesterol levels were measured at the beginning and the end of the three-month period. The results, measured in suitable units, are as follows.
\cline { 2 - 12 } \multicolumn{1}{c|}{}Person\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
\multirow{2}{*}{
Cholesterol
level
}		Beginning72841209010213564758088
\cline { 2 - 12 }End64761059210511567757584
  1. Test, at the \(2.5 \%\) significance level, whether there is evidence that the population mean LDL blood cholesterol level has reduced by more than 2 units after the three-month period.
  2. State any assumption that you have made in part (a).
    \includegraphics[max width=\textwidth, alt={}, center]{b6635fbc-3c9d-4f93-b51a-b1cbd71ddbb1-06_399_1383_269_324} As shown in the diagram, the continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 2 \\ \frac { k } { x ^ { 2 } } + c & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(m , k\) and \(c\) are constants.
CAIE Further Paper 4 2024 November Q1
6 marks Standard +0.3
1 Ellie is investigating the heights of two types of beech tree, \(A\) and \(B\), in a certain region. She has chosen a random sample of 60 beech trees of type \(A\) in the region, recorded their heights, \(x \mathrm {~m}\), and calculated unbiased estimates for the population mean and population variance as 35.6 m and \(4.95 \mathrm {~m} ^ { 2 }\) respectively. Ellie also chooses a random sample of 50 beech trees of type \(B\) in the region and records their heights, \(y \mathrm {~m}\). Her results are summarised as follows. $$\sum y = 1654 \quad \sum y ^ { 2 } = 54850$$ Find a \(95 \%\) confidence interval for the difference between the population mean heights of type \(A\) and type \(B\) beech trees in the region.
CAIE Further Paper 4 2024 November Q2
9 marks Standard +0.3
2 A school with a large number of students is updating its logo. Each student has designed a new logo and two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random sample of 12 students are shown in the following table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)
Teacher 1363840362234454448352830
Teacher 2384232413241425036444241
One of the students claims that Teacher 2 is awarding higher marks than Teacher 1.
  1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the claim.
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-04_2720_38_109_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-05_2717_29_105_22} It was later discovered that Teacher 1 had entered her mark for student \(C\) incorrectly. Her intended mark was 24 not 40 . This was corrected.
  2. Determine whether this correction affects the conclusion of the test carried out in part (a).
CAIE Further Paper 4 2024 November Q3
8 marks Standard +0.3
3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).
Number of calls0123456 or more
Observed frequency101835211141
Expected frequency14.95728.41826.9971.322
  1. Complete the table.
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable.
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-07_2726_35_97_20}
CAIE Further Paper 4 2024 November Q4
10 marks Standard +0.3
4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 3 } & 0 \leqslant x < 1 , \\ k ( 5 - x ) & 1 \leqslant x \leqslant 5 , \\ 0 & \text { otherwise } , \end{cases}$$ where \(k\) is a constant.
  1. Sketch the graph of f.
  2. Show that \(k = \frac { 4 } { 33 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-09_2725_35_99_20}
  3. Find the cumulative distribution function of \(X\).
  4. Find the median value of \(X\).
CAIE Further Paper 4 2024 November Q5
9 marks Standard +0.8
5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.
  1. Find the probability generating function of \(X\).
    Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
    ΝΙΟθW SΙΗΙ ΝΙ ΞιΙΥΜ ιΟΝ Ο0\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_446_37_674_2013}\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_444_37_1245_2013}\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_441_33_1816_2013}\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_443_33_2387_2013}
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-11_2726_35_97_20}
  3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).
CAIE Further Paper 4 2024 November Q6
8 marks Standard +0.8
6 Ansal is investigating the wingspans of Monarch butterflies in two different regions, \(X\) and \(Y\). He takes a random sample of 8 Monarch butterflies from region \(X\) and records their wingspans, \(x \mathrm {~cm}\). His results are as follows. $$\begin{array} { l l l l l l l l } 8.2 & 7.0 & 7.3 & 8.8 & 7.8 & 8.5 & 9.2 & 7.4 \end{array}$$ Ansal also takes a random sample of 9 Monarch butterflies from region \(Y\) and records their wingspans, \(y \mathrm {~cm}\). His results are summarised as follows. $$\sum y = 71.10 \quad \sum y ^ { 2 } = 567.13$$ Ansal suspects that the mean wingspan of Monarch butterflies from region \(X\) is greater than the mean wingspan of Monarch butterflies from region \(Y\). It is known that the wingspans of Monarch butterflies in regions \(X\) and \(Y\) are normally distributed with equal population variances. Test, at the 10\% significance level, whether Ansal's suspicion is supported by the data.
\includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-12_2717_35_109_2012}
\includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-13_2726_35_97_20}
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-14_2715_33_109_2012}
CAIE Further Paper 4 2024 November Q1
4 marks Standard +0.3
1 A scientist is investigating the lengths of the leaves of a certain type of plant. The scientist assumes that the lengths of the leaves of this type of plant are normally distributed. He measures the lengths, \(x \mathrm {~cm}\), of the leaves of a random sample of 8 plants of this type. His results are as follows.
\(\begin{array} { l l l l l l l l } 3.5 & 4.2 & 3.8 & 5.2 & 2.9 & 3.7 & 4.1 & 3.2 \end{array}\) Find a \(90 \%\) confidence interval for the population mean length of leaves of this type of plant.
CAIE Further Paper 4 2024 November Q2
8 marks Challenging +1.2
2 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 5 } + p t + q t ^ { 2 }$$ where \(p\) and \(q\) are constants.
  1. Given that \(\mathrm { E } ( X ) = 1.1\), find the numerical value of \(\operatorname { Var } ( X )\).
    \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-04_2714_38_109_2010} The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t )\) given by $$\mathrm { G } _ { Y } ( t ) = \frac { 2 } { 3 } t \left( 1 + \frac { 1 } { 2 } t ^ { 2 } \right)$$ The random variable \(Z\) is the sum of independent observations of \(X\) and \(Y\).
  2. Find the probability generating function of \(Z\).
  3. Find \(\mathrm { P } ( Z > 2 )\).
  4. State the most probable value of \(Z\).
CAIE Further Paper 4 2024 November Q3
10 marks Standard +0.3
3 Rosie sows 5 seeds in each of 150 plant pots. The number of seeds that germinate is recorded for each pot. The results are summarised in the following table.
Number of seeds that germinate012345
Number of pots12404335164
Rosie suggests that the number of seeds that germinate follows the binomial distribution \(\mathrm { B } ( 5 , p )\).
  1. Use Rosie's results to show that \(p = 0.42\).
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether the distribution \(\mathrm { B } ( 5,0.42 )\) is a good fit for the data.
    \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-06_2720_38_109_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-07_2726_35_97_20}
CAIE Further Paper 4 2024 November Q4
10 marks Standard +0.8
4 The random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } ( x - 1 ) ^ { 2 } & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$
  1. Find the cumulative distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = ( X - 1 ) ^ { 4 }\).
  2. Find the probability density function of \(Y\).
    \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-09_2725_35_99_20}
  3. Find the median value of \(Y\).
  4. Find \(\mathrm { E } ( Y )\).
CAIE Further Paper 4 2024 November Q5
9 marks Challenging +1.2
5 Dev owns a small company which produces bottles of juice. He uses two machines, \(X\) and \(Y\), to fill empty bottles with juice. Dev is investigating the volumes of juice in the bottles. He chooses a random sample of 35 bottles filled by machine \(X\) and a random sample of 60 bottles filled by machine \(Y\). The volumes of juice, \(x\) and \(y\) respectively, measured in suitable units, are summarised by $$\sum x = 30.8 , \quad \sum x ^ { 2 } = 29.0 , \quad \sum y = 62.4 , \quad \sum y ^ { 2 } = 76.8 .$$ Dev claims that the mean volume of juice in bottles filled by machine \(Y\) is greater than the mean volume of juice in bottles filled by machine \(X\). A test at the \(\alpha \%\) significance level suggests that there is sufficient evidence to support Dev's claim. Find the set of possible values of \(\alpha\).
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-10_2717_33_109_2014}
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-11_2726_35_97_20}
CAIE Further Paper 4 2024 November Q6
9 marks Standard +0.3
6 A sports college keeps records of the times taken by students to run one lap of a running track. The population median time taken is 51.0 seconds. After a month of intensive training, a random sample of 22 new students run one lap of the track, giving times, in seconds, as follows.
51.352.053.449.249.351.152.247.2
53.048.549.450.350.851.649.152.3
51.852.447.948.950.651.9
It is claimed that the intensive training has led to a decrease in the median time taken to run one lap of the track. Carry out a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test whether there is sufficient evidence to support the claim.
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-13_2726_35_97_20}
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-14_2715_33_109_2012}
CAIE Further Paper 4 2024 November Q2
9 marks Standard +0.3
2 A school with a large number of students is updating its logo. Each student has designed a new logo and two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random sample of 12 students are shown in the following table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)
Teacher 1363840362234454448352830
Teacher 2384232413241425036444241
One of the students claims that Teacher 2 is awarding higher marks than Teacher 1.
  1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the claim.
    \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-04_2715_38_109_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-05_2716_29_107_22} It was later discovered that Teacher 1 had entered her mark for student \(C\) incorrectly. Her intended mark was 24 not 40 . This was corrected.
  2. Determine whether this correction affects the conclusion of the test carried out in part (a).
CAIE Further Paper 4 2024 November Q3
8 marks Standard +0.3
3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).
Number of calls0123456 or more
Observed frequency101835211141
Expected frequency14.95728.41826.9971.322
  1. Complete the table.
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable.
    \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-07_2726_35_97_20}
CAIE Further Paper 4 2024 November Q4
10 marks Standard +0.3
4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 3 } & 0 \leqslant x < 1 \\ k ( 5 - x ) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Sketch the graph of f.
  2. Show that \(k = \frac { 4 } { 33 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-09_2725_35_99_20}
  3. Find the cumulative distribution function of \(X\).
  4. Find the median value of \(X\).
CAIE Further Paper 4 2024 November Q5
9 marks Standard +0.8
5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.
  1. Find the probability generating function of \(X\).
    Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
    \includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_444_33_106_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_443_33_675_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_440_33_1247_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_441_33_1816_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_443_31_2385_2015}
  3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).
CAIE Further Paper 4 2024 November Q6
8 marks Standard +0.3
6 Ansal is investigating the wingspans of Monarch butterflies in two different regions, \(X\) and \(Y\). He takes a random sample of 8 Monarch butterflies from region \(X\) and records their wingspans, \(x \mathrm {~cm}\). His results are as follows. $$\begin{array} { l l l l l l l l } 8.2 & 7.0 & 7.3 & 8.8 & 7.8 & 8.5 & 9.2 & 7.4 \end{array}$$ Ansal also takes a random sample of 9 Monarch butterflies from region \(Y\) and records their wingspans, \(y \mathrm {~cm}\). His results are summarised as follows. $$\sum y = 71.10 \quad \sum y ^ { 2 } = 567.13$$ Ansal suspects that the mean wingspan of Monarch butterflies from region \(X\) is greater than the mean wingspan of Monarch butterflies from region \(Y\). It is known that the wingspans of Monarch butterflies in regions \(X\) and \(Y\) are normally distributed with equal population variances. Test, at the 10\% significance level, whether Ansal's suspicion is supported by the data.
\includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-12_2715_44_110_2006}
\includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-13_2726_35_97_20}
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-14_2714_38_109_2010}
CAIE P2 2024 November Q4
7 marks Moderate -0.3
4
  1. Sketch the graphs of \(y = 1 + \mathrm { e } ^ { 2 x }\) and \(y = | x - 4 |\) on the same diagram.
  2. The two graphs meet at the point \(P\) .
    Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 3 - x )\) .
    \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-06_2716_38_109_2012}
  3. Use an iterative formula, based on the equation in part (b), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Use an initial value of 0.45 and give the result of each iteration to 5 significant figures.
CAIE Further Paper 2 2024 November Q7
10 marks Challenging +1.8
7
  1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) .
    \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-15_2723_33_99_22}
  2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).
CAIE Further Paper 2 2024 November Q8
14 marks Hard +2.3
8
  1. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 7 }\) ,where \(z = \cos \theta + \mathrm { i } \sin \theta\) ,use de Moivre's theorem to show that $$\cos ^ { 7 } \theta = a \cos 7 \theta + b \cos 5 \theta + c \cos 3 \theta + d \cos \theta$$ where \(a , b , c\) and \(d\) are constants to be determined.
    Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { n } \theta \mathrm {~d} \theta\).
  2. Show that $$n I _ { n } = 2 ^ { - \frac { 1 } { 2 } n } + ( n - 1 ) I _ { n - 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-18_2718_42_107_2007}
  3. Using the results given in parts (a) and (b), find the exact value of \(I _ { 9 }\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2024 November Q7
10 marks Challenging +1.8
7
  1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) .
    \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-15_2723_33_99_22}
  2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).
CAIE P3 2012 June Q9
10 marks Standard +0.3
9 The lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } - \mathbf { k } )\) respectively, where \(a\) and \(b\) are constants.
  1. Given that \(l\) and \(m\) intersect, show that $$2 a - b = 4 .$$
  2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
  3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).
CAIE P3 2021 November Q9
11 marks Standard +0.3
9 Two lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + s ( 4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + t ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) respectively.
  1. Show that \(l\) and \(m\) are perpendicular.
  2. Show that \(l\) and \(m\) intersect and state the position vector of the point of intersection.
  3. Show that the length of the perpendicular from the origin to the line \(m\) is \(\frac { 1 } { 3 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q1
3 marks Moderate -0.5
1 Fird bet \(\mathbf { 6 }\) le \(\mathrm { s } \mathbf { 6 } x\) fo wh clB \(\left. 2 ^ { 3 x + 1 } \right) < \mathscr { G }\) in an wer ira simp ified \& ct fo m. [ \(\beta\)