2021 HKDSE Mathematics M2 (Algebra and Calculus) Practice Paper | DSE Mock

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

In the expansion of \((1+ax)^n\), where \(n\) is a positive integer and \(a\) is a non-zero constant, the coefficients of \(x\) and \(x^2\) are \(12\) and \(60\) respectively. (a) Find the values of \(a\) and \(n\). (b) Find the coefficient of \(x^3\) in the expansion of \((2-x)(1+ax)^n\).

Solve the equation \(\cos 3\theta + \cos \theta = \cos 2\theta\) for \(0 \le \theta \le \pi\).

Let \(f(x) = \sqrt{2x+3}\) for \(x > -\frac{3}{2}\). Find \(f'(x)\) from first principles.

The volume of a spherical balloon is increasing at a constant rate of \(12\pi \text{ cm}^3\text{s}^{-1}\). Let \(V \text{ cm}^3\), \(A \text{ cm}^2\) and \(r \text{ cm}\) be the volume, surface area and radius of the balloon at time \(t\) seconds respectively. (a) Find the rate of change of the radius when the radius is \(3 \text{ cm}\). (b) Find the rate of change of the surface area when the radius is \(3 \text{ cm}\).

Find \(\int_{0}^{1} x^2 e^{2x} \, \mathrm{d}x\).

Consider the system of linear equations in \(x, y, z\): \((E): \begin{cases} x + y + z = 3 \\ x + 2y + 3z = 4 \\ x + 3y + az = b \end{cases}\), where \(a, b \in \mathbb{R}\). (a) Find the range of values of \(a\) for which \((E)\) has a unique solution. (b) Suppose \(a = 5\). Find the value of \(b\) for which \((E)\) is consistent, and solve \((E)\) in this case.

Let \(A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\). (a) Find \(A^{-1}\). (b) Find the \(2 \times 2\) matrix \(X\) such that \(A X A = B\).

Consider the system of linear equations in real variables $x, y, z$:
\( \begin{cases} x + 2y - z = 1 \\ 2x + (a+3)y + 3z = a + 3 \\ 3x + 6y + (a^2-4)z = a^2 + 2a \end{cases} \) where \(a\) is a real constant.

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

(b) Suppose \(a = 1\).
(i) Solve the system.
(ii) If \((x, y, z)\) is a real solution of the system, find the minimum value of \(x^2 + y^2\). (5 marks)

(c) Suppose \(a = -1\).
(i) Show that the system is inconsistent.
(ii) If the third equation is replaced by \(3x + 6y - 3z = k\), find the value of \(k\) such that the system is consistent. (4 marks)

(a) Prove that \(\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx\). (2 marks)

(b) (i) Using (a), show that \(\int_0^{\pi/2} \frac{\sin^3 x}{\sin x + \cos x} \, dx = \int_0^{\pi/2} \frac{\cos^3 x}{\sin x + \cos x} \, dx\).
(ii) Hence, evaluate \(\int_0^{\pi/2} \frac{\sin^3 x}{\sin x + \cos x} \, dx\). (5 marks)

(c) Using the substitution \(t = \tan \frac{x}{2}\), evaluate \(\int_0^{\pi/2} \frac{1}{\sin x + \cos x + 1} \, dx\). (6 marks)

Let \(O\) be the origin. The coordinates of points \(A\), \(B\) and \(C\) are \((2, 1, 0)\), \((0, 3, 2)\) and \((1, 0, 4)\) respectively.

(a) Find \(\vec{AB} \times \vec{AC}\). Hence, find the area of triangle \(ABC\). (4 marks)

(b) Let \(D(k, 2, -1)\) be a point, where \(k\) is a constant.
(i) Find the volume of the tetrahedron \(ABCD\) in terms of \(k\).
(ii) If the volume of the tetrahedron \(ABCD\) is \(5\), find the possible values of \(k\). (4 marks)

(c) Let \(k = -\frac{6}{5}\).
(i) Find the unit normal vector of the plane \(ABC\) which makes an obtuse angle with the positive z-axis.
(ii) Hence, find the shortest distance from \(D\) to the plane \(ABC\). (4 marks)

Let \(f(x) = \frac{x^2 - 3x}{x - 4}\) for all real numbers \(x \neq 4\). Let \(C\) be the curve \(y = f(x)\).

(a) Find the vertical asymptote(s) and oblique asymptote(s) of \(C\). (3 marks)

(b) Find the coordinate(s) of all local maximum and local minimum point(s) of \(C\). (4 marks)

(c) Find the range of values of \(x\) for which the curve \(C\) is concave upward. (2 marks)

(d) Sketch \(C\), showing the asymptotes and the stationary points with their coordinates. (4 marks)