2023 HKDSE Mathematics M1 (Calculus and Statistics) Practice Paper | DSE Mock

Let \(X\) be a discrete random variable with the following probability distribution:\n\n\(\begin{array}{c|c|c|c|c} x & 1 & 3 & 5 & 7 \\ \hline P(X=x) & p & q & 0.3 & p \end{array}\)\n\nwhere \(p\) and \(q\) are constants.\n\n(a) Given that \(E(X) = 4.1\), find \(p\) and \(q\).\n(b) Find \(Var(3 - 2X)\).

The number of accidents occurring at a busy intersection per week follows a Poisson distribution with a mean of 4.
Let \(\bar{X}\) be the average weekly number of accidents at this intersection recorded over a random sample of 100 weeks.

(a) Write down the mean and the variance of \(\bar{X}\).
(b) Using the Central Limit Theorem, find the probability that \(\bar{X}\) is between 3.7 and 4.3.

Placeholder

An electronics company imports components from three suppliers, \(A\), \(B\), and \(C\), with proportions \(40\%\), \(35\%\), and \(25\%\) respectively. The defective rates of components from \(A\), \(B\), and \(C\) are \(2\%\), \(3\%\), and \(5\%\) respectively. (a) Find the probability that a randomly selected component is defective. (b) Given that a randomly selected component is defective, find the probability that it was supplied by \(A\) or \(C\).

Let \(A\) and \(B\) be two events. Suppose that \(P(A) = 0.4\), \(P(B | A) = 0.3\), and \(P(A' \cap B') = 0.48\), where \(A'\) and \(B'\) are the complementary events of \(A\) and \(B\) respectively.\n\n(a) Find \(P(A \cap B)\).\n\n(b) Find \(P(B)\).\n\n(c) Are \(A\) and \(B\) independent? Explain your answer.

Let \(f(x) = (1 - 2x)^3 (1 + ax)^n\) for all real numbers \(x\), where \(a\) is a constant and \(n\) is a positive integer.
(a) Expand \(f(x)\) in ascending powers of \(x\) up to the term \(x^2\). (3 marks)
(b) It is given that the coefficient of \(x\) in the expansion of \(f(x)\) is \(2\), and \(f''(0) = -24\). Find the values of \(a\) and \(n\). (4 marks)

The number of bacteria in a culture, \( N \), is modeled by \[ N(t) = 500 + a \ln(bt + 1), \] where \( t \ge 0 \) is the time in hours since the observation began, and \( a \) and \( b \) are positive constants. It is given that when \( t = 2 \), \( N = 500 + 10 \ln 3 \) and the rate of change of the number of bacteria with respect to \( t \) is \( \frac{10}{3} \) per hour. (a) Find the values of \( a \) and \( b \). (4 marks) (b) Find the rate of change of the number of bacteria in the culture when \( t = 5 \). (2 marks)

Evaluate \(\int_{0}^{1} \frac{x^3}{\sqrt{1+3x^2}} \, dx\).

Consider the curve \(C: y = x e^{-x}\), where \(x \ge 0\).

(a) Find \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\). (2 marks)

(b) Use the trapezoidal rule with 4 subintervals to estimate the area of the region bounded by \(C\), the \(x\)-axis, and the line \(x=1\), correct to 4 decimal places. (3 marks)

(c) Determine whether the estimate in (b) is an over-estimate or an under-estimate. Explain your answer. (2 marks)

A manufacturer produces organic honey jars. The weight of honey in a jar, \(X\) grams, is assumed to follow a normal distribution.
(a) A random sample of 100 jars is selected. Let \(x_i\) (for \(i = 1, 2, \dots, 100\)) be the weight of honey (in grams) in the \(i\)-th jar in the sample. It is given that \(\sum x_i = 25200\) and \(\sum (x_i - \bar{x})^2 = 396\), where \(\bar{x}\) is the sample mean.
(i) Find unbiased estimates of the population mean and population variance of the weight of honey in a jar.
(ii) Construct a 95% confidence interval for the population mean of the weight of honey in a jar.
(5 marks)
(b) Suppose it is known that the weight of honey in a jar indeed follows a normal distribution with mean \(\mu = 252\) grams and standard deviation \(\sigma = 2\) grams. A jar of honey is classified as 'underfilled' if its weight is less than 248.5 grams.
(i) Find the probability that a randomly selected jar of honey is underfilled.
(3 marks)
(c) The honey jars are packed into boxes, each containing 12 jars.
(i) Find the probability that a box of honey contains at least 2 underfilled jars.
(ii) A box of honey is sent for inspection if it contains at least 2 underfilled jars. If 20 boxes are randomly selected one by one, find the probability that the 3rd box sent for inspection is the 8th box selected.
(5 marks)

A manufacturer produces organic honey jars. The weight of honey in a jar, \(X\) grams, is assumed to follow a normal distribution.
(a) A random sample of 100 jars is selected. Let \(x_i\) (for \(i = 1, 2, \dots, 100\)) be the weight of honey (in grams) in the \(i\)-th jar in the sample. It is given that \(\sum x_i = 25200\) and \(\sum (x_i - \bar{x})^2 = 396\), where \(\bar{x}\) is the sample mean.
(i) Find unbiased estimates of the population mean and population variance of the weight of honey in a jar.
(ii) Construct a 95% confidence interval for the population mean of the weight of honey in a jar.
(5 marks)
(b) Suppose it is known that the weight of honey in a jar indeed follows a normal distribution with mean \(\mu = 252\) grams and standard deviation \(\sigma = 2\) grams. A jar of honey is classified as 'underfilled' if its weight is less than 248.5 grams.
(i) Find the probability that a randomly selected jar of honey is underfilled.
(3 marks)
(c) The honey jars are packed into boxes, each containing 12 jars.
(i) Find the probability that a box of honey contains at least 2 underfilled jars.
(ii) A box of honey is sent for inspection if it contains at least 2 underfilled jars. If 20 boxes are randomly selected one by one, find the probability that the 3rd box sent for inspection is the 8th box selected.
(5 marks)

Suppose the number of customer inquiries arriving at a customer service desk per hour follows a Poisson distribution with a mean of 3.2. (a) (i) Find the probability that exactly 2 inquiries arrive at the customer service desk in a given hour. (ii) Find the probability that at least 3 inquiries arrive at the customer service desk in a given hour. (5 marks) (b) Suppose a day has 8 working hours, and the numbers of customer inquiries arriving in these hours are independent. Find the probability that there are at least 6 working hours in a day, each having at least 3 inquiries. (3 marks) (c) Suppose each customer inquiry is classified as either a "complaint" or a "general inquiry" with probabilities 0.25 and 0.75 respectively, independent of other inquiries. Given that exactly 4 inquiries arrive in a particular hour, find the probability that at least 2 of them are complaints. (4 marks)

A medical researcher models the concentration of a drug in the bloodstream of a patient, \(C(t)\) (in \(\text{mg/L}\)), \(t\) hours after injection by: \(C(t) = A(t+1)e^{-0.5t}\) for \(t \ge 0\), where \(A\) is a positive constant.

(a) Find the range of \(t\) for which \(C(t)\) is increasing, and the range of \(t\) for which \(C(t)\) is decreasing. Find the coordinates of the local maximum of \(C(t)\) in terms of \(A\). (3 marks)

(b) Find the coordinates of the point of inflection of the curve \(y = C(t)\) for \(t \ge 0\). (3 marks)

(c) Sketch the curve \(y = C(t)\) for \(t \ge 0\), showing the coordinates of the local maximum, the point of inflection, and the \(y\)-intercept. (2 marks)

(d) The concentration of a second drug, \(D(t)\) (in \(\text{mg/L}\)), is modeled by: \(D(t) = B t^2 e^{-0.5t}\) for \(t \ge 0\), where \(B\) is a positive constant. Suppose at the instant when the concentration of the first drug \(C(t)\) reaches its maximum, the concentration of the second drug \(D(t)\) is increasing at a rate of \(1.5 e^{-0.5} \text{ mg/L/hour}\).
(i) Find the value of \(B\).
(ii) The total concentration of the two drugs in the bloodstream is denoted by \(T(t) = C(t) + D(t)\). A researcher claims that the maximum total concentration occurs at \(t = 2\). Assuming \(A = 4\), determine whether this claim is correct. (4 marks)

During a laboratory experiment, the rate of change of the amount of pollutant \(P\) (in mg) in a water sample is modeled by
\[ \frac{dP}{dt} = \frac{160 t e^{-0.1 t^2}}{(3 + e^{-0.1 t^2})^2} \]
where \(t\) is the time in hours since the experiment started (\(t \ge 0\)). Initially, there are \(150\text{ mg}\) of pollutant in the water sample.

(a) Find \( \int \frac{t e^{-0.1 t^2}}{(3 + e^{-0.1 t^2})^2} dt \). (4 marks)

(b) (i) Using the result in (a), find \(P\) in terms of \(t\).
(ii) Find the amount of pollutant in the water sample after a very long time. (4 marks)

(c) Find the value of \( \frac{d^2P}{dt^2} \) at \(t = 5\). Hence, determine whether the rate of increase of the amount of pollutant is increasing or decreasing at \(t = 5\). Correct your numerical answers to 2 decimal places. (5 marks)