2023 HKDSE Mathematics Practice Paper | DSE Mock

Let \( p(x) = 3x^3 - hx^2 - 5x + 12 \), where \( h \) is a constant. If \( p(x) \) is divisible by \( 3x - 4 \), find the value of \( h \).

It is given that \( y \) is partly constant and partly varies directly as \( x^2 \). When \( x = 2 \), \( y = 18 \); and when \( x = 3 \), \( y = 33 \). Find the value of \( y \) when \( x = 4 \).

The mean and the standard deviation of a set of 7 numbers are 12 and 4 respectively. If a number 12 is added to the set, find the new standard deviation of the set. (Leave your answer in surd form.)

Let \(f(x) = 2x^3 - kx^2 - 13x + 6\), where \(k\) is a constant. It is given that \(2x - 1\) is a factor of \(f(x)\).
(a) Find the value of \(k\).
(b) Factorize \(f(x)\) completely.

It is given that \(z\) is the sum of two parts, one part is a constant and the other part varies inversely as \(y^2\). When \(y = 2\), \(z = 11\); and when \(y = 4\), \(z = 5\).
(a) Express \(z\) in terms of \(y\).
(b) Find the value of \(z\) when \(y = \frac{1}{2}\).

In a rectangular coordinate system, the coordinates of the points \(A\) and \(B\) are \((-2, 1)\) and \((6, 7)\) respectively. Let \(C\) be the circle with \(AB\) as a diameter.
(a) Find the equation of \(C\).
(b) Find the equation of the tangent to \(C\) at \(B\).

The stem-and-leaf diagram below shows the distribution of the weights (in kg) of 15 students.

$$\begin{array}{r|l}
\text{Stem (tens)} & \text{Leaf (units)} \\
\hline
4 & 2 \quad 5 \quad 5 \quad 8 \\
5 & 1 \quad 1 \quad 3 \quad 4 \quad 6 \quad 7 \quad 9 \\
6 & 0 \quad 2 \quad 5 \quad 8
\end{array}$$
Key: \(4 \mid 2\) means \(42\) kg.

(a) Find the median, the range, and the interquartile range of the distribution.
(b) If a student of weight 53 kg leaves the group, find the change in the median.

Let \( f(x) = 2x^3 + kx^2 - 13x - 6 \), where \( k \) is a constant. It is given that \( x - 2 \) is a factor of \( f(x) \).
(a) Find the value of \( k \). (2 marks)
(b) Solve the equation \( f(x) = 0 \). (3 marks)

It is given that \( y \) is the sum of two parts, one part varies directly as \( x^2 \) and the other part is a constant. When \( x = 2 \), \( y = 13 \); when \( x = 4 \), \( y = 37 \).
(a) Find the formula for \( y \) in terms of \( x \). (3 marks)
(b) If \( y = 55 \), find the value(s) of \( x \). (2 marks)

The manufacturing cost of a custom crystal trophy, \(\$C\), is the sum of two parts. One part is constant, and the other part varies directly as the square of the height \(h\text{ cm}\) of the trophy and inversely as its thickness \(t\text{ mm}\). When the height is \(10\text{ cm}\) and the thickness is \(4\text{ mm}\), the manufacturing cost is \(\$350\). When the height is \(6\text{ cm}\) and the thickness is \(3\text{ mm}\), the manufacturing cost is \(\$220\). (a) Find the manufacturing cost of a trophy with height \(12\text{ cm}\) and thickness \(5\text{ mm}\). (4 marks) (b) If the manufacturing cost of a trophy of thickness \(8\text{ mm}\) is \(\$600\), find the height of this trophy. (2 marks)

Let \(f(x) = 2x^3 + ax^2 + bx + 6\), where \(a\) and \(b\) are constants. When \(f(x)\) is divided by \(x - 2\), the remainder is \(0\). When \(f(x)\) is divided by \(x + 1\), the remainder is \(9\). (a) Find the values of \(a\) and \(b\). (4 marks) (b) Someone claims that all the real roots of the equation \(f(x) = 0\) are rational numbers. Do you agree? Explain your answer. (3 marks)

The equation of the circle \(C\) is \(x^2 + y^2 - 8x - 6y = 0\). (a) Find the coordinates of the center and the radius of \(C\). (2 marks) (b) Let \(L\) be the straight line \(3x - 4y + c = 0\), where \(c\) is a constant. If \(L\) intersects \(C\) at two points \(A\) and \(B\) such that the length of the chord \(AB\) is \(8\), find the possible values of \(c\). (5 marks)

Let \(V\) be the volume of a solid. It is known that \(V\) is the sum of two parts, one part varies directly as the square of its height \(h\), and the other part varies directly as the cube of its height \(h\). When \(h = 2\), \(V = 36\); when \(h = 3\), \(V = 99\). (a) Find the volume of the solid when its height is \(4\). (4 marks) (b) If the height of the solid decreases from \(2\) to \(1.5\), find the percentage decrease in its volume. (3 marks)

Let \(C\) be a circle passing through \(P(2, 8)\) and \(Q(8, 2)\). It is given that the center of \(C\) lies on the straight line \(L: 2x + 3y - 15 = 0\). (a) Find the equation of \(C\). (4 marks) (b) Let \(R\) be a point such that \(PR\) is a tangent to \(C\) at \(P\). If the coordinates of \(R\) are \((k, k+10)\), find \(k\). (4 marks)

A bag contains 5 red balls, 4 green balls and 3 blue balls.

(a) If 3 balls are randomly selected from the bag at the same time, find the probability that at least two balls of the same color are selected.
(2 marks)

(b) If 4 balls are randomly selected from the bag one by one with replacement, find the probability that the color of the selected balls changes exactly once.
(2 marks)

Let \(C\) be a circle passing through \(A(2, 8)\) and \(B(6, 0)\). It is given that the center of \(C\) lies on the straight line \(L: x - y - 1 = 0\).

(a) Find the equation of \(C\). (3 marks)
(b) Find the equations of the tangent lines to \(C\) which are parallel to \(L\). (2 marks)

(a) The circle \(C\) passes through the points \(A(0, 8)\) and \(B(6, 0)\), and its center lies on the straight line \(L: x - y - 1 = 0\). Find the equation of \(C\). (4 marks)

(b) A straight line \(L_1\) is parallel to \(L\) and is tangent to \(C\). Find the \(y\)-intercept(s) of \(L_1\). (2 marks)

Let \(C\) be a circle passing through \(A(6, 0)\) and \(B(-1, -7)\). It is given that the center of \(C\) lies on the straight line \(L: 2x - y - 7 = 0\).

(a) Find the equation of \(C\). (4 marks)

(b) Suppose another straight line \(L_2: 3x - 4y + k = 0\) is tangent to \(C\), where \(k\) is a constant. Find the possible values of \(k\). (4 marks)

Let \(C\) be the circle \(x^2 + y^2 - 12x - 16y + 64 = 0\).

(a) Find the center \(G\) and the radius of \(C\). (2 marks)

(b) Let \(P\) be the point \((14, 14)\).
(i) Show that \(P\) lies outside \(C\). (1 mark)
(ii) Find the equations of the two tangents from \(P\) to \(C\). (4 marks)

(c) The two tangents touch \(C\) at points \(Q_1\) and \(Q_2\).
(i) Find the area of the quadrilateral \(GQ_1PQ_2\). (2 marks)
(ii) Let \(C'\) be the circumcircle of the triangle \(GQ_1Q_2\). Find the equation of \(C'\). (3 marks)

Let \(x_1, x_2, \dots, x_n\) be a set of data with range \(R\), interquartile range \(I\), and variance \(V\). If each datum is multiplied by \(-3\) and then \(5\) is added to it, find the new range, new interquartile range, and new variance of the data.

Consider two groups of students, Group A and Group B, each having 40 students. The statistical data of their test scores are shown below: Group A: Minimum = 20, First quartile = 45, Median = 60, Third quartile = 75, Maximum = 95. Group B: Minimum = 30, First quartile = 50, Median = 65, Third quartile = 70, Maximum = 90. Which of the following must be true? I. There are more students in Group B than in Group A who scored 70 or above. II. The range of scores in Group A is greater than that in Group B. III. The interquartile range of scores in Group A is greater than that in Group B.

The mean and the standard deviation of a set of 10 numbers are 15 and 4 respectively. If two numbers, 11 and 19, are added to the set, find the standard deviation of the new set of 12 numbers.

The equation of a circle \(C\) is \(x^2 + y^2 - 4x + 6y - 12 = 0\). A straight line \(L\) passes through the center of \(C\) and is perpendicular to the line \(3x - 4y + 5 = 0\). Find the equation of \(L\).

If the straight line \(3x - 4y + k = 0\) is tangent to the circle \(x^2 + y^2 - 2x - 2y - 7 = 0\), find the possible values of \(k\).

Let \(A(1, 2)\) and \(B(5, -6)\) be two points. If \(AB\) is a diameter of a circle \(C\), which of the following is/are true? I. The equation of \(C\) is \(x^2 + y^2 - 6x + 4y - 7 = 0\). II. The origin lies inside \(C\). III. The line \(y = 2x - 8\) passes through the center of \(C\).

Let \(P(x) = x^3 + ax^2 + bx - 6\), where \(a\) and \(b\) are constants. When \(P(x)\) is divided by \(x - 1\), the remainder is \(-8\). It is given that \(x + 1\) is a factor of \(P(x)\). Find the remainder when \(P(x)\) is divided by \(x - 3\).

Find the Least Common Multiple (LCM) of \(12x^2 y^3 z\), \(18x^3 y (z - 1)^2\) and \(8x y^2 (z - 1)\).

It is given that \(z\) varies directly as \(x^2\) and inversely as \(\sqrt{y}\). If \(x\) is increased by 20% and \(y\) is decreased by 36%, find the percentage change in \(z\).

It is given that \(u\) is the sum of two parts, where one part varies directly as \(v\) and the other part varies directly as \(v^2\). When \(v = 2\), \(u = 10\); when \(v = 3\), \(u = 21\). Find the value of \(u\) when \(v = 5\).

When the polynomial \(f(x)\) is divided by \(x-2\), the remainder is \(5\). When \(f(x)\) is divided by \(2x+1\), the remainder is \(-5\). Find the remainder when \(f(x)\) is divided by \(2x^2-3x-2\).

It is given that \(z\) varies directly as \(x^2\) and inversely as \(\sqrt{y\)}\. If \(x\) is decreased by \(10\%\) and \(y\) is increased by \(44\%\), find the percentage change in \(z\).

A circle \(C\) passes through \(P(0, 8)\) and \(Q(6, 0)\). If the center of \(C\) lies on the line \(x + y - 7 = 0\), find the equation of \(C\).

A set of 10 data has mean 50 and standard deviation 8. If two new data, 42 and 58, are added to the set, find the new standard deviation of the set of 12 data.

If \(3x^3 + ax^2 + bx - 12\) is divisible by \(x^2 - x - 6\), find the value of \(a - b\).

Let \(P(k, 1)\) be a point, where \(k\) is a constant. The length of the tangent from \(P\) to the circle \(x^2 + y^2 - 4x + 6y - 12 = 0\) is \(4\). Find the possible value(s) of \(k\).

The stem-and-leaf diagram below shows the distribution of the weekly pocket money (in dollars) of a group of students.
\(\begin{array}{r|l} \text{Stem (tens)} & \text{Leaf (units)} \\ \hline 4 & 2\ \ 5\ \ 5\ \ 8 \\ 5 & 0\ \ 3\ \ 3\ \ 3\ \ 7\ \ 9 \\ 6 & 1\ \ 4\ \ 4\ \ 8 \\ 7 & 2\ \ 5 \end{array}\)
Which of the following must be true?
I. The range is 33.
II. The interquartile range is 16.
III. The mode of the distribution is 53.

If the variance of a set of 8 numbers \(x_1, x_2, \dots, x_8\) is \(12\), find the variance of the 8 numbers \(3 - 2x_1, 3 - 2x_2, \dots, 3 - 2x_8\).

The equation of a circle \(C\) is \(x^2 + y^2 - 8x + 10y + 5 = 0\). If the line \(L: 3x - 4y + k = 0\) is a tangent to the circle \(C\), find the possible values of \(k\).

A group of 20 boys and 30 girls sat for a test. The mean score of the boys is 65 with a standard deviation of 8. The mean score of the girls is 75 with a standard deviation of 8. Find the standard deviation of the test scores of the 50 students combined.

The standard deviation of a set of data \(x_1, x_2, \dots, x_{40}\) is \(4\). If \(y_i = 5 - 3x_i\) for \(i = 1, 2, \dots, 40\), find the variance of \(y_1, y_2, \dots, y_{40}\).

The mean of the eleven numbers \(14, 15, 15, 16, 17, 18, 18, 19, 20, 22\) and \(x\) is \(18\). Find the range of these eleven numbers.

In a school, the mean score and standard deviation of a Mathematics exam are \(64\) marks and \(12\) marks respectively. The mean score and standard deviation of an English exam are \(56\) marks and \(8\) marks respectively. Mary gets \(76\) marks in Mathematics and \(66\) marks in English. John gets \(70\) marks in Mathematics and his standard score in English is equal to Mary's standard score in Mathematics. Which of the following statements must be true? I. Mary performs better in English than in Mathematics relative to other students. II. John's score in the English exam is \(64\). III. John's standard score in English is \(1.25\).

Let \(C\) be the circle \(x^2 + y^2 - 4x + 6y + k = 0\). If the straight line \(3x - 4y + 2 = 0\) is tangent to \(C\), find the value of \(k\).

A circle passes through the origin \(O\) and its center is \((3, 4)\). Find the equation of the tangent to the circle at \(O\).

The equation of the circle \(C\) is \(x^2 + y^2 - 6x - 2y - 15 = 0\). The equation of the straight line \(L\) is \(3x + 4y - 28 = 0\). Find the length of the chord intercepted by \(C\) on \(L\).

Let \(P(x) = 2x^3 + ax^2 + bx - 6\). When \(P(x)\) is divided by \(x-1\), the remainder is \(-4\). When \(P(x)\) is divided by \(x+2\), the remainder is \(-10\). Find the remainder when \(P(x)\) is divided by \(2x-1\).

Find the least common multiple (LCM) of \(12a^2b^3c\), \(18ab^4d^2\) and \(8a^3c^2\).

It is given that \(z\) is the sum of two parts, one part varies as \(x\) and the other part varies inversely as \(y\). When \(x=2\) and \(y=3\), \(z=10\). When \(x=3\) and \(y=1\), \(z=21\). Find the value of \(z\) when \(x=4\) and \(y=2\).

Suppose \(u\) varies directly as \(v^2\) and inversely as \(\sqrt{w}\). If \(v\) is increased by \(20\%\) and \(w\) is decreased by \(36\%\), find the percentage change in \(u\).

Let \(k\) be a constant. If the line \(x - 2y + k = 0\) intersects the circle \(x^2 + y^2 - 4x - 6y - 12 = 0\) at two distinct points \(A\) and \(B\) such that the midpoint of \(AB\) is \((1, y_0)\), find the value of \(k\).

A circle \(C\) passes through the point \((1, 8)\) and is tangent to both the \(x\)-axis and the \(y\)-axis. Find the sum of the possible radii of \(C\).

Let \(P(x)\) be a polynomial. When \(P(x)\) is divided by \(x - 1\), the remainder is \(3\). When \(P(x)\) is divided by \(x + 2\), the remainder is \(-3\). Find the remainder when \((x+1)P(x)\) is divided by \(x^2 + x - 2\).

Find the least common multiple (LCM) of \(3x^2 - 12\), \(x^2 - 4x + 4\), and \(2x^2 - 4x\).

It is given that \(z\) varies directly as \(x^2\) and inversely as \(\sqrt{y}\). If \(x\) is decreased by \(30\%\) and \(y\) is increased by \(96\%\), find the percentage change in \(z\).

It is given that \(z\) is the sum of two parts, one part varies directly as \(x\) and the other part varies directly as \(y^2\). When \(x = 2\) and \(y = 3\), \(z = 22\). When \(x = 3\) and \(y = -2\), \(z = 17\). Find the value of \(z\) when \(x = -1\) and \(y = 4\).

Let \(\{x_1, x_2, \dots, x_{20}\}\) be a set of data with mean \(m\) and standard deviation \(s\). If a new data set is formed by replacing each \(x_i\) with \(y_i = 3 - 2x_i\) for \(i=1, 2, \dots, 20\), which of the following must be true?
I. The mean of the new data set is \(3 - 2m\).
II. The standard deviation of the new data set is \(3 - 2s\).
III. The variance of the new data set is \(4s^2\).

In a mathematics test, the mean score is \(64\) marks and the standard deviation is \(8\) marks. The standard score of Alan in the test is \(1.5\). If the test scores of all students are linearly adjusted such that the new mean score is \(70\) marks and the new standard score of Alan is \(1.2\) with his actual mark unchanged, find the new standard deviation of the test.

A committee consisting of \(4\) boys and \(4\) girls is to be formed from \(7\) boys and \(6\) girls. If two particular girls, Mary and Susan, cannot be both selected, how many different committees can be formed?

Find the number of roots of the equation \(4 \cos^2 x + 5 \sin x - 5 = 0\) in the interval \(0 \le x \le 2\pi\).

Let \(x_1, x_2, \dots, x_n\) (where \(n > 1\)) be a set of data. Let \(\bar{x}\) and \(v\) (where \(v > 0\)) be the mean and the variance of the data set respectively. If a new datum \(x_{n+1} = \bar{x}\) is added to the data set, let \(\bar{y}\) and \(u\) be the mean and the variance of the new data set \(\{x_1, x_2, \dots, x_n, x_{n+1}\}\) respectively. Which of the following must be true? I. \(\bar{y} = \bar{x}\) II. \(u < v\) III. The standard deviation of the new data set is \(\sqrt{\frac{n}{n+1}}\) times the standard deviation of the original data set.

Let \(C\) be the circle \(x^2 + y^2 - 6x - 2y + 5 = 0\). If the straight line \(L: y = kx\) intersects \(C\) at two points \(A\) and \(B\) such that the length of the chord \(AB\) is \(4\), find the possible value(s) of \(k\).

Let \(p(x) = 2x^3 - kx^2 + hx - 6\), where \(k\) and \(h\) are constants. It is given that \(2x - 3\) is a factor of \(p(x)\). When \(p(x)\) is divided by \(x + 1\), the remainder is \(-15\). Let \(\alpha, \beta\) and \(\gamma\) be the roots of the equation \(p(x) = 0\). Find the value of \(\alpha^2 \beta \gamma + \alpha \beta^2 \gamma + \alpha \beta \gamma^2\).

The graph of \(\log_4 y\) against \(\log_2 x\) is a straight line. The intercept on the vertical axis is \(3\) and the slope of the line is \(-2\). Which of the following relations between \(x\) and \(y\) is correct?

Let \(a_1, a_2, a_3, \dots\) be a geometric sequence with common ratio \(r\), where \(0 < r < 1\). If \(a_1 = 16\) and \(a_2 + a_4 = \frac{10}{3} a_3\), find the sum to infinity of the sequence.