2022 HKDSE Mathematics M1 (Calculus and Statistics) Practice Paper | DSE Mock

In the expansion of \(\left(1 + \frac{x}{k}\right)^n\) in ascending powers of \(x\), where \(n\) is a positive integer and \(k\) is a non-zero real number, the coefficient of \(x\) is \(2\) and the coefficient of \(x^2\) is \(\frac{3}{2}\). Find the values of \(n\) and \(k\).

Box \(X\) contains 3 red balls and 2 blue balls. Box \(Y\) contains 2 red balls and 4 blue balls. A fair die is rolled. If the result is 1 or 2, a ball is randomly drawn from Box \(X\); otherwise, a ball is randomly drawn from Box \(Y\). (a) Find the probability that a red ball is drawn. (b) Given that a red ball is drawn, find the probability that the die rolled was 1 or 2.

In the expansion of \((1 + ax)^n (1 - 3x)^2\), where \(n\) is a positive integer and \(a\) is a non-zero constant, the coefficient of \(x\) is \(4\) and the coefficient of \(x^2\) is \(-11\). Find the values of \(a\) and \(n\).

Consider the curve \(C: y = \frac{e^{2x}}{x-1}\) for \(x > 1\). (a) Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\). (2 marks) (b) Find the coordinates of the local minimum point of \(C\). (4 marks)

In the expansion of \((1 + kx)^n (1 - 2x)^5\), where \(n\) is a positive integer and \(k\) is a constant, the coefficients of \(x\) and \(x^2\) are \(-2\) and \(-16\) respectively. Find the values of \(n\) and \(k\).

A software system contains code written by two programmers, Alice and Bob. Alice wrote \(60\%\) of the code, and Bob wrote the remaining \(40\%\). It is known that \(2\%\) of the lines of code written by Alice contain errors, while \(5\%\) of the lines of code written by Bob contain errors. A line of code is randomly selected. (a) Find the probability that the selected line of code contains an error. (b) Given that the selected line of code contains an error, find the probability that it was written by Bob.

The profit \(P\) (in million dollars) of a startup company \(t\) years after its establishment is modeled by \(P(t) = 12 t^2 e^{-0.5t} + 5\) for \(t \ge 0\). (a) Find \(\frac{\mathrm{d}P}{\mathrm{d}t}\). (b) Determine the value of \(t\) at which the profit is maximum. Hence, find the maximum profit of the company, leaving your answer in terms of \(e\).

Let \(C\) be a curve passing through the point \((0, 4)\). It is given that the slope of the tangent to \(C\) at any point \((x, y)\) is given by \(\frac{\mathrm{d}y}{\mathrm{d}x} = x \sqrt{2x^2 + 9}\). Find the equation of \(C\).

The rate of change of the mass of a certain chemical substance \(X\) in a chemical reaction tank, in grams per week, is modeled by \( \frac{dX}{dt} = \frac{80 \ln(t+1)}{(t+1)^2} \) where \(t\) (\(t \ge 0\)) is the number of weeks elapsed since the reaction started.\n(a) Find the value of \(t\) at which the rate of change of the mass of chemical \(X\) is maximum. (4 marks)\n(b) (i) Using integration by parts, find \( \int \frac{\ln(t+1)}{(t+1)^2} dt \). (3 marks)\n(ii) It is given that the initial mass of chemical \(X\) in the tank is 100 grams. Find the mass of chemical \(X\) in the tank after 5 weeks, correct to 2 decimal places. (4 marks)

The concentration of a pollutant in a water sample, \(C(t)\) (in ppm), at \(t\) hours after a chemical treatment is initiated, is modeled by\n\(C(t) = \frac{a t + b}{t^2 + 3}\) for \(t \ge 0\),\nwhere \(a\) and \(b\) are constants.\nIt is given that the initial concentration of the pollutant is \(4\text{ ppm}\), and the rate of change of the concentration of the pollutant is \(0\text{ ppm/hour}\) when \(t = 1\).\n\n(a) Find the values of \(a\) and \(b\). (4 marks)\n\n(b) Show that the concentration of the pollutant is decreasing for \(t > 1\). (2 marks)\n\n(c) Find the rate of change of the concentration of the pollutant when \(t = 3\). (2 marks)\n\n(d) Let \(C''(t)\) be the second derivative of \(C(t)\).\n(i) Find \(C''(t)\).\n(ii) A researcher claims that the rate of decrease of the concentration of the pollutant is the fastest when \(t = 3\). Do you agree? Explain your answer. (4 marks)

A company introduces a new product. The sales rate \( S(t) \) (in thousands of units per month) of the product at time \( t \) months (where \( t \ge 0 \)) is modeled by
\( S(t) = \frac{a \ln(t+1) + b}{t+1} \),
where \( a \) and \( b \) are positive constants.

(a) Show that the maximum value of \( S(t) \) occurs at \( t = e^{1 - \frac{b}{a}} - 1 \). (4 marks)

(b) It is given that the maximum sales rate of the product is \( 12 \) thousand units per month, and this maximum occurs at \( t = 1 \) month.
(i) Find the values of \( a \) and \( b \).
(ii) Hence, find the sales rate at the start of the product launch (i.e. at \( t = 0 \)). (4 marks)

(c) The total sales of the product over the first \( T \) months (in thousands of units) is given by \( I(T) = \int_0^T S(t) dt \).
(i) Using the substitution \( u = \ln(t+1) \), find \( I(T) \) in terms of \( T \).
(ii) The marketing department claims that the total sales of the product will exceed 50 thousand units within the first 5 months. Do you agree? Explain your answer. (5 marks)

A chemical factory releases pollutants into a lake. Let \(A(t)\) (in \(\text{kg}\)) be the total amount of pollutant in the lake at time \(t\) days, where \(t \ge 0\). Initially, there are \(20\text{ kg}\) of pollutant in the lake. The rate of change of the amount of pollutant in the lake is modeled by:
$$\frac{dA}{dt} = \begin{cases} t \sqrt{4-t} & \text{for } 0 \le t \le 4 \\\\ \frac{k \ln(t-3)}{(t-3)^2} & \text{for } t > 4 \end{cases}$$
where \(k\) is a positive constant.

(a) (i) Find \(\int_0^4 t \sqrt{4-t} dt\).
(ii) Hence, find the amount of pollutant in the lake at \(t=4\).
(4 marks)

(b) Assume that \(A(t)\) is continuous at \(t=4\).
(i) Using the substitution \(u = t-3\), find \(\int \frac{\ln(t-3)}{(t-3)^2} dt\).
(ii) It is given that as \(t \to \infty\), the total amount of pollutant in the lake approaches a maximum level of \(40\text{ kg}\). Show that \(k = \frac{172}{15}\).
(7 marks)

(c) Let \(k = \frac{172}{15}\).
(i) Using the trapezoidal rule with 2 subintervals, estimate the increase in the amount of pollutant in the lake from \(t=6\) to \(t=10\).
(ii) Determine whether the estimate in (c)(i) is an overestimate or an underestimate. Explain your answer.
(3 marks)