2022 HKDSE Mathematics Practice Paper | DSE Mock

Let \(P(x) = 3x^3 - kx^2 - 13x + 4\), where \(k\) is a constant. It is given that \(P(x)\) is divisible by \(3x - 1\). (a) Find the value of \(k\). (b) Find the remainder when \(P(x)\) is divided by \(x + 2\).

The coordinates of the point \(A\) are \((2, 6)\). The circle \(C\) has its center at the origin \(O\) and passes through \(A\). (a) Find the equation of \(C\). (b) Find the equation of the tangent to \(C\) at \(A\).

The heights (in \(\text{cm}\)) of 6 students are \(155\), \(158\), \(160\), \(162\), \(165\) and \(x\). It is given that the mean height of these students is \(161\text{ cm}\). (a) Find the value of \(x\). (b) Find the standard deviation of the heights of the 6 students, correct to 3 significant figures.

In \(\triangle ABC\), \(AB = 8\text{ cm}\), \(BC = 5\text{ cm}\) and \(\angle ABC = 120^\circ\). (a) Find the length of \(AC\), correct to 3 significant figures. (b) Find the area of \(\triangle ABC\) in surd form.

In an arithmetic sequence, the 3rd term is \(14\) and the 7th term is \(30\). (a) Find the first term and the common difference of the sequence. (b) Find the sum of the first \(20\) terms of the sequence.

It is given that \(z\) varies directly as \(x^2\) and inversely as \(\sqrt{y}\). When \(x = 3\) and \(y = 16\), \(z = 18\). (a) Express \(z\) in terms of \(x\) and \(y\). (b) If \(x\) is doubled and \(y\) is decreased by \(75\%\), find the percentage change in \(z\).

(a) Solve the compound inequality \(3x - 5 < 7x + 11\) and \(\frac{5 - 2x}{3} \ge x - 5\). (b) Write down the number of integers satisfying the compound inequality in (a).

(a) Solve the equation \(\log_2(x + 5) - \log_2(x - 1) = 2\). (b) Hence, solve the equation \(\log_2(2^y + 5) - \log_2(2^y - 1) = 2\), leaving your answer in exact form.

Let \(k\) be a constant. The quadratic equation \(x^2 + 2kx + (3k + 4) = 0\) has equal real roots. (a) Find the possible values of \(k\). (b) For the positive value of \(k\) obtained in (a), solve the quadratic equation.

Let \(P(x) = 2x^3 + ax^2 + bx - 6\), where \(a\) and \(b\) are constants. When \(P(x)\) is divided by \(x-2\), the remainder is \(0\). When \(P(x)\) is divided by \(x+1\), the remainder is \(-12\).
(a) Find \(a\) and \(b\). (4 marks)
(b) Let \(Q(x) = P(x) + k\). Someone claims that if \(k = 12\), then all roots of \(Q(x) = 0\) are real. Do you agree? Explain your answer. (3 marks)

The equation of the circle \(C\) is \(x^2 + y^2 - 8x - 6y + 20 = 0\).
(a) Find the coordinates of the center and the radius of \(C\). (2 marks)
(b) A straight line \(L\) with slope \(m > 1\) passes through the origin \(O(0,0)\) and is tangent to \(C\) at point \(P\).
(i) Find the value of \(m\).
(ii) Hence, find the coordinates of \(P\). (5 marks)

A set of 8 positive numbers has a mean of \(15\) and a standard deviation of \(4\).
(a) Find the sum of these 8 numbers, and the sum of the squares of these numbers. (3 marks)
(b) Two additional numbers, \(x\) and \(y\), are added to the set. It is given that \(x + y = 30\) and the new standard deviation of the 10 numbers is \(\sqrt{17.8}\). Find the values of \(x\) and \(y\). (4 marks)

In a quadrilateral \(ABCD\), \(AB = 10\text{ cm}\), \(\angle ABC = 120^\circ\), \(BC = 6\text{ cm}\), \(CD = 8\text{ cm}\), \(\angle CAD = 30^\circ\), and \(AD > CD\).
(a) Find the length of \(AC\). (3 marks)
(b) Find the two possible values of \(\angle ADC\), correct to 1 decimal place. (4 marks)

An arithmetic sequence has 3rd term \(15\) and 7th term \(39\).
(a) Find the first term and the common difference of the sequence. (2 marks)
(b) Let \(T_n\) be the sum of the first \(n\) terms of the arithmetic sequence. Another geometric sequence \(g_n\) has first term \(b\) and common ratio \(r > 0\). It is given that \(g_1 = T_2\) and \(g_3 = T_4\).
(i) Find \(b\) and \(r\).
(ii) Find the least value of \(n\) such that the sum of the first \(n\) terms of \(g_n\) exceeds \(10^6\). (5 marks)

Let \(P(x) = 2x^3 + px^2 + qx - 10\), where \(p\) and \(q\) are constants. It is given that \(x-2\) is a factor of \(P(x)\). When \(P(x)\) is divided by \(x+1\), the remainder is \(-9\). (a) Find \(p\) and \(q\). (3 marks) (b) Solve the equation \(P(x) = 0\). Show that only one of the roots is a real number. (4 marks)

Let \(C\) be the circle \(x^2 + y^2 - 10x + 16 = 0\). (a) Let \(L\) be a line passing through the origin \(O(0,0)\) with slope \(m\). If \(L\) is tangent to \(C\), find the two possible values of \(m\). (4 marks) (b) Let \(L_1\) and \(L_2\) be the two tangents to \(C\) in (a) with positive and negative slopes respectively. If \(L_1\) and \(L_2\) touch \(C\) at points \(A\) and \(B\) respectively, find the equation of the circle passing through \(O\), \(A\), and \(B\). (3 marks)

A set of 8 numbers has a mean of 15 and a standard deviation of 4. (a) Find the sum of these 8 numbers, and the sum of their squares. (3 marks) (b) Two numbers, 9 and 21, are removed from the set. Find the mean and the standard deviation of the remaining 6 numbers. (Give the standard deviation correct to 2 decimal places.) (4 marks)

In a tetrahedron \(VABC\), the base \(ABC\) is an equilateral triangle with side length \(12\text{ cm}\). \(V\) is vertically above the center \(O\) of the base \(ABC\), and the height \(VO\) is \(8\text{ cm}\). (a) Find the length of the slant edge \(VA\). (2 marks) (b) Let \(M\) be the midpoint of \(BC\). Find the angle between the face \(VBC\) and the base \(ABC\). (2 marks) (c) Find the angle between the line \(VA\) and the face \(VBC\). (3 marks) (Give your answers correct to 1 decimal place if necessary.)

The first, second and fifth terms of an arithmetic sequence with non-zero common difference \(d\) form the first three terms of a geometric sequence with common ratio \(r\). (a) Express \(d\) in terms of the first term \(a\), and find the value of \(r\). (3 marks) (b) It is given that the sum of the first \(n\) terms of the arithmetic sequence is \(S_n\). For another geometric sequence with first term \(a\) and common ratio \(\frac{1}{r}\), the sum to infinity is 12. (i) Find the value of \(a\). (ii) Find the least value of \(n\) such that \(S_n > 2024\). (4 marks)