2022 HKDSE Mathematics M2 (Algebra and Calculus) Practice Paper | DSE Mock

Prove by mathematical induction that \(\sum_{r=1}^n r(r+3) = \frac{n(n+1)(n+5)}{3}\) for all positive integers \(n\).

Given that the coefficient of the third term in the expansion of \((x^2 - \frac{2}{x})^n\) is \(144\), where \(n\) is a positive integer.
(a) Find the value of \(n\).
(b) Find the constant term in the expansion of \((1 + \frac{x^3}{4})(x^2 - \frac{2}{x})^n\).

Solve the equation \(\text{sin } 3\theta + \text{sin } \theta = \text{cos } \theta\) for \(0 \le \theta \le \pi\).

Find \(\frac{d}{dx} (\sqrt{2x+3})\) from first principles.

Find \(\int x^5 \cos(x^3) dx\).

Evaluate \(\int_{0}^{1} x^2 \sqrt{1 - x^2} dx\).

Let \(A = \begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\).
(a) Show that \(A^2 - 2A + I = 0\), where \(I\) is the \(2 \times 2\) identity matrix and \(0\) is the \(2 \times 2\) zero matrix.
(b) Using (a), express \(A^4\) and \(A^{-1}\) in the form \(\alpha A + \beta I\), where \(\alpha\) and \(\beta\) are real numbers.

Consider three vectors \(\mathbf{u} = \mathbf{i} + 2\mathbf{j} - \mathbf{k}\), \(\mathbf{v} = 2\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(\mathbf{w} = k\mathbf{i} + \mathbf{j} + 3\mathbf{k}\), where \(k\) is a constant.
(a) Find \(\mathbf{u} \times \mathbf{v}\).
(b) If the volume of the parallelepiped spanned by \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) is 15, find the possible values of \(k\).

Consider the curve \( C: y = f(x) = \frac{x^2 + a}{x - 1} \), where \( a > -1 \) is a constant and \( x \neq 1 \).
(a) Find the coordinates of the local maximum point and the local minimum point of \( C \) in terms of \( a \). (4 marks)
(b) If the distance between the two local extremum points of \( C \) is \( 2\sqrt{10} \), find the value of \( a \). (3 marks)
(c) Using the value of \( a \) obtained in (b),
    (i) find the equations of all asymptotes to \( C \); (2 marks)
    (ii) find the range of values of \( x \) for which the curve \( C \) is concave upward; (2 marks)
    (iii) sketch the curve \( C \). (1 mark)

Let \( M = \begin{pmatrix} 1 & p & 1 \\ p & 1 & 2 \\ 1 & 5 & 1 \end{pmatrix} \), where \( p \) is a real constant.
(a) Find the values of \( p \) for which \( M \) is singular. (3 marks)
(b) Suppose \( p = 3 \). Find the inverse of \( M \). (4 marks)
(c) Suppose \( p = 2 \ Bos\). Consider the system of linear equations
\( (S): \begin{cases} x + 2y + z = 4 \\ 2x + y + 2z = q \\ x + 5y + z = q + 2 \end{cases} \), where \( q \) is a real constant.
    (i) Find the value of \( q \) for which \( (S) \) is consistent. (3 marks)
    (ii) Solve \( (S) \) for the value of \( q \) obtained in (c)(i). (3 marks)

(a) Show that \( \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx \). (2 marks)
(b) Using (a), show that \( \int_{0}^{\pi/4} \ln(1 + \tan x) \, dx = \frac{\pi}{8} \ln 2 \). (5 marks)
(c) Evaluate \( \int_{0}^{\pi/4} \frac{x \sec^2 x}{1 + \tan x} \, dx \). (5 marks)

Consider the tetrahedron \( OABC \), where \( O \) is the origin. Let \( \mathbf{a} = \overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} - \mathbf{k} \), \( \mathbf{b} = \overrightarrow{OB} = 2\mathbf{i} - \mathbf{j} + \mathbf{k} \), and \( \mathbf{c} = \overrightarrow{OC} = \mathbf{i} + \mathbf{j} + z\mathbf{k} \), where \( z \) is a constant.
(a) Find \( \mathbf{a} \times \mathbf{b} \). (2 marks)
(b) If the volume of the tetrahedron \( OABC \) is \( \frac{7}{6} \), find the possible values of \( z \). (4 marks)
(c) Suppose \( z = 1 \).
    (i) Find the projection of \( \overrightarrow{OC} \) onto the plane \( OAB \). (4 marks)
    (ii) Find the shortest distance from \( C \) to the plane \( OAB \). (3 marks)