2021 HKDSE Mathematics M1 (Calculus and Statistics) Practice Paper | DSE Mock

In the expansion of \((1 + ax)^4 (1 - x)^n\), where \(a\) is a constant and \(n\) is a positive integer, the coefficient of \(x\) is \(-1\) and the coefficient of \(x^2\) is \(-12\). Find the values of \(a\) and \(n\).

The number of customers arriving at a checkout counter in a supermarket follows a Poisson distribution with a mean of 4 per 10-minute interval.
(a) Find the probability that exactly 3 customers arrive at the counter in a certain 10-minute interval.
(b) Find the probability that at least 2 customers arrive at the counter in a certain 5-minute interval.
(c) Given that at least 2 customers arrive at the counter in a certain 5-minute interval, find the probability that at most 4 customers arrive at the counter in that interval.

Let \(f(x) = x e^{-x}\).
(a) Use the trapezoidal rule with 4 subintervals to estimate \(\int_0^2 f(x) dx\).
(b) Determine whether the estimate in (a) is an over-estimate or an under-estimate. Explain your answer.

The lifetimes of a certain brand of light bulbs follow a normal distribution with mean \(\mu\) hours and standard deviation \(\sigma\) hours. It is known that 6.68% of the light bulbs have a lifetime of less than 850 hours, and 11.51% of the light bulbs have a lifetime of more than 1120 hours.
(a) Find \(\mu\) and \(\sigma\).
(b) A batch of 5 light bulbs is selected at random. Find the probability that at least 1 of them has a lifetime of more than 1120 hours.

The derivative of a curve is given by \(\frac{dy}{dx} = x \sqrt{2x^2 + 1}\). If the curve passes through the point \((2, 6)\), find the equation of the curve.

A company produces light switches, and the probability of a switch being defective is \(p\). It is known that in a randomly selected batch of 40 light switches, the expected number of defective switches is 2.4.
(a) Find the value of \(p\).
(b) Find the variance of the number of defective switches in a batch of 40.
(c) If a batch of 15 light switches is randomly selected, find the probability that:
    (i) exactly 1 switch is defective;
    (ii) at least 2 switches are defective.

A factory has two machines, A and B, producing the same component. Machine A produces 60% of the components and Machine B produces 40% of the components. The defective rates of Machine A and Machine B are 2% 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 produced by Machine A.

A rectangular storage container with an open top is to have a volume of \(36\text{ m}^3\). The length of its base is twice its width. Let \(x\) metres be the width of the base.
(a) Show that the total surface area of the container, \(A\text{ m}^2\), is given by \(A = 2x^2 + \frac{108}{x}\).
(b) Find the value of \(x\) that minimizes the total surface area.

Consider a drug concentration model where the concentration \( C(t) \) of the drug in the bloodstream \( t \) hours after injection is given by \( C(t) = 8 t^2 e^{-0.5t} \) for \( t \ge 0 \).

(a) Find the exact interval of \( t \) for which the concentration is increasing. (3 marks)

(b) Find the maximum concentration of the drug and the time at which it occurs. (3 marks)

(c) Find the value of \( C''(t) \) and hence find the exact coordinates of the point(s) of inflexion of the curve \( y = C(t) \) for \( t > 0 \). (5 marks)

(d) Describe the behavior of the rate of change of concentration of the drug as \( t \to \infty \). (2 marks)

The number of complaints received by a customer service center follows a Poisson distribution with a mean of 1.8 per hour.

(a) Find the probability that the center receives:
(i) exactly 2 complaints in a given hour.
(ii) at least 3 complaints in a given hour.
(4 marks)

(b) The center is open for 8 hours a day.
(i) Find the probability that there are at least 6 hours in a day in which the center receives at least 1 complaint.
(ii) If the center receives at least 1 complaint in a day (defined as the entire 8-hour period), find the probability that the total number of complaints received in that day is at most 10. (8 marks)

Let \( f(x) = \frac{4}{e^x + 1} \) for \( x \ge 0 \).

(a) Using the substitution \( u = e^x \), find the exact value of \( \int_{0}^{2} f(x) dx \). (4 marks)

(b) (i) Using the trapezoidal rule with 4 sub-intervals, estimate the value of \( \int_{0}^{2} f(x) dx \). (3 marks)
(ii) Determine whether the estimate in (b)(i) is an over-estimate or an under-estimate. Explain your answer. (3 marks)

(c) Show that \( \int_{0}^{2} \frac{4}{e^x+1} dx + \int_{0}^{2} \frac{4e^x}{e^x+1} dx = 8 \). Hence, find the exact value of \( \int_{0}^{2} \frac{4e^x}{e^x+1} dx \). (3 marks)

The weights of packages of a certain brand of coffee powder are normally distributed with a mean of 250 grams and a standard deviation of 4 grams.

(a) Find the probability that a randomly selected package of coffee powder weighs less than 245 grams. (3 marks)

(b) The manufacturer offers a refund if a package weighs less than \( w \) grams. If only 1.5% of the packages qualify for a refund, find the value of \( w \) correct to 1 decimal place. (3 marks)

(c) Now, a box contains 6 packages of coffee powder. The box is considered "underweight" if at least 2 packages in the box weigh less than 245 grams.
(i) Find the probability that a randomly selected box is underweight.
(ii) If a box is NOT underweight, find the probability that there is exactly 1 package in the box that weighs less than 245 grams. (6 marks)