2023 HKDSE Mathematics M2 (Algebra and Calculus) Practice Paper | DSE Mock

In the expansion of \((1 + ax)^n\), the coefficient of \(x\) is \(-16\) and the coefficient of \(x^2\) is \(120\), where \(n\) is a positive integer and \(a\) is a non-zero constant. (a) Find the values of \(a\) and \(n\). (b) Find the coefficient of \(x^3\) in the expansion.

Prove, from first principles, that \(\frac{d}{dx}\sqrt{5-2x} = -\frac{1}{\sqrt{5-2x}}\).

Prove by mathematical induction that \(\sum_{r=1}^n \frac{1}{(2r-1)(2r+1)} = \frac{n}{2n+1}\) for all positive integers \(n\).

Let \(A = \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix}\). (a) Find \(A^2\). (b) Prove by mathematical induction that \(A^n = \begin{pmatrix} 2n+1 & -4n \\ n & 1-2n \end{pmatrix}\) for all positive integers \(n\).

A vessel in the shape of an inverted right circular cone of height \(12\text{ cm}\) and base radius \(6\text{ cm}\) is placed vertex downwards. Water is poured into the vessel at a constant rate of \(3\pi\text{ cm}^3\text{s}^{-1}\). Find the rate of increase of the depth of water when the depth of water is \(4\text{ cm}\).

(a) Show that \( \int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} dx = \frac{\pi}{2} \int_0^{\pi} \frac{\sin x}{1 + \cos^2 x} dx \). (b) Hence, evaluate \( \int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} dx \).

Consider the system of linear equations in \(x, y, z\): \((E): \begin{cases} x + y + z = 2 \\ 2x + 3y + kz = 5 \\ x + 2y + 3z = 3 \end{cases}\) where \(k \in \mathbb{R}\). (a) Find the value of \(k\) for which \((E)\) has infinitely many solutions. (b) Solve \((E)\) for the value of \(k\) found in (a).

Let \(\vec{u} = \vec{i} + 2\vec{j} - \vec{k}\), \(\vec{v} = 2\vec{i} - \vec{j} + 3\vec{k}\), and \(\vec{w} = 3\vec{i} + \lambda\vec{j} + 2\vec{k}\) be three vectors, where \(\lambda \in \mathbb{R}\). If the volume of the parallelepiped spanned by \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) is \(15\), find the possible values of \(\lambda\).

Consider the system of linear equations in real variables \(x, y, z\):
\( (E): \begin{cases} x + ay + z = 1 \\ ax + y + (a-1)z = a \\ 2x + 2ay + az = b \end{cases} \) where \(a, b\) are real constants.

(a) Find the range of values of \(a\) for which \( (E) \) has a unique solution. (3 marks)

(b) Suppose \(a = 2\).
(i) Find the value of \(b\) for which \( (E) \) is consistent.
(ii) Solve \( (E) \) under the condition in (b)(i). (5 marks)

(c) Suppose \(a = 1\).
(i) Find the value of \(b\) for which \( (E) \) is consistent.
(ii) Under the condition in (c)(i), is it possible to find a real constant \(c\) such that the system of equations
\( (F): \begin{cases} x + y + z = 1 \\ x + y = 1 \\ 2x + 2y + z + c(x-y) = b \end{cases} \)
has infinitely many solutions? If yes, find \(c\); if no, explain briefly. (5 marks)

(a) Prove by mathematical induction that for all positive integers \(n\),
\( \sin \theta +
\sin 3\theta + \dots + \sin(2n-1)\theta = \frac{\sin^2 n\theta}{\sin\theta} \) (where \( \sin \theta \neq 0 \)). (5 marks)

(b) (i) Using (a), show that for any positive integer \(n > 1\),
\( \frac{\sin^2 n\theta}{\sin\theta} - \frac{\sin^2 (n-1)\theta}{\sin\theta} = \sin(2n-1)\theta \). (2 marks)

(ii) Evaluate \( \int_{\pi/6}^{\pi/3} \frac{\sin^2 3\theta - \sin^2 2\theta}{\sin\theta} d\theta \). (5 marks)

Let \( C \) be the curve \( y = f(x) \), where \( f(x) = \frac{x^2 - 3x + 6}{x-1} \) for \( x \neq 1 \).

(a) Find the coordinates of the local maximum point and local minimum point of \( C \). (4 marks)

(b) Find the asymptote(s) of \( C \). (3 marks)

(c) Find the range of values of \( x \) such that the curve \( C \) is concave upward, and the range of values of \( x \) such that \( C \) is concave downward. (2 marks)

(d) Sketch \( C \), showing its asymptotes and turning points. (3 marks)

Let \( O \) be the origin. Three points \( A(2, 1, -1) \), \( B(3, -1, 2) \), and \( C(1, 2, k) \) are given, where \( k \) is a real constant.

(a) Find \( \overrightarrow{AB} \times \overrightarrow{AC} \) in terms of \( k \). (3 marks)

(b) Suppose the area of triangle \( ABC \) is \( \frac{3\sqrt{6}}{2} \).
(i) Find the two possible values of \( k \).
(ii) For the integer value of \( k \) found in (b)(i), find the equation of the plane \( \Pi \) passing through \( A \), \( B \), and \( C \). (6 marks)

(c) A fourth point \( D(1, -1, 4) \) is given.
For the integer value of \( k \) found in (b)(i), find:
(i) the volume of the tetrahedron \( ABCD \).
(ii) the shortest distance from \( D \) to the plane \( \Pi \). (4 marks)