2025 HKDSE Mathematics Practice Paper | DSE Mock

Let \(f(x) = 2x^3 + ax^2 + bx - 6\), where \(a\) and \(b\) are constants. When \(f(x)\) is divided by \(x-1\), the remainder is \(-6\). When \(f(x)\) is divided by \(2x+1\), the remainder is \(-9\). Find the values of \(a\) and \(b\).

Let \(g(x) = 2x^3 + kx^2 - 13x + 6\), where \(k\) is a constant. It is given that \(x-2\) is a factor of \(g(x)\).
(a) Find the value of \(k\).
(b) Factorize \(g(x)\) completely.

Let \(y = f(x)\) be the graph of \(y = x^2 - 4x - 5\). The graph of \(y = f(x)\) is translated horizontally to the right by 3 units to form the graph \(y = g(x)\). Then, the graph of \(y = g(x)\) is reflected with respect to the \(x\)-axis to form the graph of \(y = h(x)\). Find the equation of the graph \(y = h(x)\), expressing your answer in the form \(y = ax^2 + bx + c\).

Let \(f(x) = -2x^2 + 12x - 10\).
(a) Find the coordinates of the vertex of the graph of \(y = f(x)\).
(b) Find the range of values of \(k\) such that the straight line \(y = k\) intersects the graph of \(y = f(x)\) at two distinct points.

The equation of the circle \(C\) is \(x^2 + y^2 - 6x + 8y + 9 = 0\).
(a) Find the coordinates of the center and the radius of \(C\).
(b) Let \(O\) be the origin. Find the length of the tangent from \(O\) to \(C\).

A circle \(C\) has its center at \(P(-3, 2)\) and passes through the point \(Q(1, 5)\).
(a) Find the equation of \(C\).
(b) Determine whether the point \(R(-1, -1)\) lies inside, outside, or on the circle \(C\). Explain your answer.

Consider a set of nine numbers: \(11, 13, 15, 15, 17, 19, 21, x, y\), where \(x\) and \(y\) are real numbers and \(x \le y\). It is given that the mean of the nine numbers is 17 and the range is 12. Find the values of \(x\) and \(y\).

A set of 8 data is given as follows: \(4, 6, 7, 8, 9, 10, 11, a\). It is given that the mean of the data is 8.
(a) Find the value of \(a\).
(b) Find the standard deviation of the data, correct to 2 decimal places.

The 3rd term and the 8th term of an arithmetic sequence are 13 and 33 respectively.
(a) Find the first term and the common difference of the sequence.
(b) Find the sum of the first 20 terms of the sequence.

(a) The circle \(C\) passes through \(P(0, 0)\) and its center is \(G(3, 4)\). Find the equation of \(C\) and the equation of the tangent \(L\) to \(C\) at \(P\). (4 marks)
(b) Another circle \(C'\) is obtained by translating \(C\) horizontally to the left by \(d\) units, where \(d > 0\). If \(C'\) is tangent to \(L\), find the value of \(d\). (3 marks)

Let \(f(x) = 2x^3 + ax^2 + bx - 12\), where \(a\) and \(b\) are constants. It is given that \(x - 2\) is a factor of \(f(x)\). When \(f(x)\) is divided by \(x + 1\), the remainder is \(-15\).
(a) Find the values of \(a\) and \(b\). (3 marks)
(b) Someone claims that all the roots of the equation \(f(x) = 0\) are real numbers. Do you agree? Explain your answer. (4 marks)

The stem-and-leaf diagram below shows the distribution of the hourly wages (in HK$) of 15 employees in a shop:
$$\begin{array}{r|l}
\text{Stem (tens)} & \text{Leaf (units)} \\
\hline
2 & 2,\, 5,\, 5,\, 8 \\
3 & 0,\, 3,\, 3,\, 5,\, 7,\, 8 \\
4 & 2,\, 4,\, 4 \\
5 & 0,\, 4
\end{array}$$
(a) Find the mean, the median, and the interquartile range of the distribution. (3 marks)
(b) Two more employees, whose hourly wages are \(W_1\) and \(W_2\) respectively (where \(W_1 \le W_2\)), join the shop.
(i) If the mean of the hourly wages of the 17 employees remains unchanged, find the value of \(W_1 + W_2\).
(ii) If the range of the hourly wages increases by 6, and the median remains unchanged, write down a pair of possible values of \(W_1\) and \(W_2\). (4 marks)

Let \(A\) be an arithmetic sequence with first term \(a\) and common difference \(d\), where \(d \neq 0\). The 1st term, the 3rd term, and the 9th term of \(A\) form a geometric sequence \(G\) in that order.
(a) Show that \(a = d\). (3 marks)
(b) If the sum of the first 10 terms of \(A\) is 110, find:
(i) the first term and the common ratio of \(G\);
(ii) the sum of the first 8 terms of \(G\). (4 marks)

Let \(f(x) = x^2 - 4x + 3\). Let the graph of \(y = f(x)\) be denoted by \(U\).
(a) By completing the square or otherwise, find the coordinates of the vertex of \(U\). (2 marks)
(b) Let \(V\) be the graph obtained by translating \(U\) vertically downwards by 5 units, and then reflecting the resulting graph with respect to the \(x\)-axis.
(i) Find the equation of \(V\).
(ii) Let the vertex of \(V\) be \(W\). If \(V\) cuts the \(x\)-axis at points \(A\) and \(B\), find the area of triangle \(WAB\). (5 marks)

The equation of the circle \(C\) is \(x^2 + y^2 - 4x - 12y + 31 = 0\). (a) Find the coordinates of the center \(G\) and the radius \(r\) of \(C\). (b) A straight line \(L\) passes through \(P(2, 1)\) with slope \(m\). (i) If \(L\) is tangent to \(C\), find the two possible values of \(m\). (ii) Let \(A\) and \(B\) be the points of contact of the two tangents from \(P\) to \(C\). Find the area of the quadrilateral \(PAGB\).

A set of test scores of 10 students has a mean of 64 and a standard deviation of 8. (a) Find the sum of the scores and the sum of the squares of the scores of these 10 students. (b) Two more students with scores \(x\) and \(y\) join the group (where \(x \le y\)). (i) If the mean of the 12 students remains unchanged, write down an equation relating \(x\) and \(y\). (ii) If the standard deviation of the 12 students is also 8, find the values of \(x\) and \(y\).

Let \(f(x) = ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The vertex of the graph of \(y = f(x)\) is \(V(3, -4)\), and the graph cuts the y-axis at \(P(0, 5)\). (a) Find the values of \(a\), \(b\) and \(c\). (b) The graph of \(y = f(x)\) is translated horizontally to the left by \(h\) units and then reflected along the x-axis to become the graph of \(y = g(x)\). The vertex of \(y = g(x)\) lies on the y-axis. (i) Find the value of \(h\). (ii) Write down the expression of \(g(x)\). (iii) If the graph of \(y = g(x)\) cuts the straight line \(y = k\) at two distinct points \(A\) and \(B\) such that the length of \(AB\) is 6, find the value of \(k\).

Let \(a_1, a_2, a_3, \dots\) be an arithmetic sequence with common difference \(d\). Let \(b_n = 2^{a_n}\) for all positive integers \(n\). (a) Show that \(b_1, b_2, b_3, \dots\) is a geometric sequence and express its common ratio in terms of \(d\). (b) It is given that \(b_1 b_2 b_3 = 512\) and \(b_1 + b_2 + b_3 = 28\). (i) Find the two possible values of \(d\). (ii) If \(d > 0\), find the minimum value of \(n\) such that the sum of the first \(n\) terms of the sequence \(a_n\) exceeds 2000.

Let \(P(x) = 2x^3 - 5x^2 + kx - 3\), where \(k\) is a constant. It is given that \(2x - 3\) is a factor of \(P(x)\). (a) Find the value of \(k\). (b) Someone claims that all the roots of the equation \(P(x) = 0\) are real numbers. Do you agree? Explain your answer.