2024 HKDSE Mathematics Practice Paper | DSE Mock

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.

The coordinates of the center of the circle \( C \) are \( (4, -3) \). The straight line \( L: 3x - 4y + 1 = 0 \) is tangent to \( C \).
(a) Find the equation of \( C \).
(b) Determine whether the origin \( O(0,0) \) lies inside, outside, or on the circle \( C \).

Solve the equation \( 2\cos^2 \theta + 5\sin \theta - 4 = 0 \), where \( 0^\circ \le \theta \le 360^\circ \).

A set of data consists of seven positive integers: \( 3, 5, 6, 8, 9, x, y \), where \( x \le y \). It is given that the mean and the range of this set of data are 7 and 9 respectively.
(a) Find the values of \( x \) and \( y \).
(b) Find the standard deviation of the set of data. (Give your answer correct to 2 decimal places.)

Consider the quadratic equation \( x^2 - 2(k - 1)x + (2k + 1) = 0 \), where \( k \) is a constant. It is given that the equation has real roots.
(a) Find the range of values of \( k \).
(b) If \( k \) is the smallest positive integer satisfying the condition in (a), solve the quadratic equation.

Let \( g(x) = ax^3 + bx^2 - 5x - 2 \), where \( a \) and \( b \) are constants. When \( g(x) \) is divided by \( x - 1 \) and \( x + 2 \), the remainders are \( 6 \) and \( -12 \) respectively.
(a) Find the values of \( a \) and \( b \).
(b) Find the remainder when \( g(x) \) is divided by \( x^2 + x - 2 \).

It is given that \( y \) is partly constant and partly varies inversely as \( x^2 \). When \( x = 1 \), \( y = 8 \); and when \( x = 2 \), \( y = 5 \).
(a) Find \( y \) in terms of \( x \).
(b) If \( y = 4.25 \), find the value(s) of \( x \).

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 least value of \( n \) such that the sum of the first \( n \) terms of the sequence is greater than 500.

In \( \triangle ABC \), \( AB = 8\text{ cm} \), \( AC = 5\text{ cm} \), and \( \angle BAC = 60^\circ \).
(a) Find the length of \( BC \).
(b) Find the area of \( \triangle ABC \). (Leave your answer in surd form.)

Let \(p(x) = 3x^3 + ax^2 + bx - 12\), where \(a\) and \(b\) 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 \(-15\).

(a) Find the values of \(a\) and \(b\). (4 marks)

(b) Someone claims that all the roots of the equation \(p(x) = 0\) are real numbers. Do you agree? Explain your answer. (3 marks)

Let the circle \(C\) have the equation \(x^2 + y^2 - 8x + 2y - 8 = 0\). Let \(L\) be the straight line \(3x - 4y + k = 0\), where \(k\) is a constant.

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

(b) If \(L\) is tangent to \(C\) and the \(y\)-intercept of \(L\) is positive, find the value of \(k\). (3 marks)

(c) Find the coordinates of the point of contact between \(L\) and \(C\). (2 marks)

The stem-and-leaf diagram below shows the distribution of the weekly pocket money (in HKD) of a group of 15 students:

Stem (tens) | Leaf (units)
1 | 5, 5, 8, 9
2 | 0, 2, 5, \(a\), 8
3 | 1, 1, 6
4 | 0, 2, \(b\)

where \(5 \le a \le 8\) and \(2 \le b \le 9\). It is given that the range and the median of the distribution are \$32 and \$25 respectively.

(a) Find the values of \(a\) and \(b\). (3 marks)

(b) If two more students with weekly pocket money of \$25 and \$43 join the group, find the change in the interquartile range of the distribution. (4 marks)

In the figure, \(A\), \(B\) and \(C\) are three points on a horizontal ground. \(B\) is due East of \(A\). The bearing of \(C\) from \(A\) is \(N50^\circ E\), and the bearing of \(C\) from \(B\) is \(N30^\circ W\). The distance between \(A\) and \(B\) is \(120\text{ m}\).

(a) Find the distance between \(A\) and \(C\). (3 marks)

(b) A vertical pole \(CP\) of height \(h\text{ m}\) is erected at \(C\). If the angle of elevation of \(P\) from \(A\) is \(35^\circ\), find:
(i) the value of \(h\),
(ii) the angle of elevation of \(P\) from \(B\).
(4 marks)
(Give your answers correct to 3 significant figures.)

The 1st, 2nd, and 5th terms of an arithmetic sequence \(A(n)\) are the 1st, 2nd, and 3rd terms of a geometric sequence \(G(n)\) respectively. It is given that the first term of both sequences is \(4\), and the common ratio of \(G(n)\) is not equal to \(1\).

(a) Find the common difference of \(A(n)\) and the common ratio of \(G(n)\). (4 marks)

(b) Let \(S_n\) be the sum of the first \(n\) terms of \(A(n)\). Find the least value of \(n\) such that \(S_n > 5000\). (3 marks)

Let \(P(x) = 2x^3 + ax^2 + bx - 12\), where \(a\) and \(b\) 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 \(-21\).

(a) Find the values of \(a\) and \(b\). (3 marks)

(b) Someone claims that all the roots of the equation \(P(x) = 0\) are real numbers. Do you agree? Explain your answer. (4 marks)

There are 15 distinct numbers in a set \(S\). The mean of \(S\) is 48 and the standard deviation of \(S\) is 8.

(a) Find the sum of the numbers in \(S\) and the sum of squares of the numbers in \(S\). (3 marks)

(b) Two additional numbers, 36 and 60, are added to the set \(S\) to form a new set \(S'\).
(i) Find the mean of \(S'\).
(ii) Determine whether the standard deviation of \(S'\) is greater than, equal to, or less than the standard deviation of \(S\). Explain your answer. (4 marks)

Consider the circle \(C: x^2 + y^2 - 10x - 6y + 9 = 0\). Let \(L\) be the straight line \(4x - 3y + k = 0\), where \(k\) is a constant.

(a) Find the range of values of \(k\) such that \(L\) intersects \(C\) at two distinct points. (3 marks)

(b) Suppose \(k = -1\) and \(L\) intersects \(C\) at two points \(A\) and \(B\). Let \(G\) be the center of \(C\). Find the area of \(\triangle GAB\). (4 marks)

In the figure, \(ABCD\) is a triangular pyramid. The base \(BCD\) is a horizontal right-angled triangle with \(\angle BCD = 90^\circ\). It is given that \(BC = 12\text{ cm}\), \(CD = 5\text{ cm}\) and \(AB\) is perpendicular to the horizontal plane \(BCD\) with \(AB = 9\text{ cm}\).

(a) Find the length of \(AD\) correct to 3 significant figures. (2 marks)

(b) Find the angle between the plane \(ACD\) and the horizontal plane \(BCD\) correct to 3 significant figures. (2 marks)

(c) Find the angle between the line \(AD\) and the plane \(ABC\) correct to 3 significant figures. (3 marks)

Let \(F(x) = 2x^3 - 5x^2 - 4x + 3\) and \(G(x) = 2x^2 + kx - 3\), where \(k\) is a constant.

(a) Suppose \(F(x)\) and \(G(x)\) have a common linear factor \(2x - 1\).
(i) Find the value of \(k\).
(ii) Hence, find the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of \(F(x)\) and \(G(x)\) in factorised form. (5 marks)

(b) Solve the equation \(\frac{G(x)}{F(x)} = \frac{2}{x+1}\). (2 marks)