2021 HKDSE Mathematics Practice Paper | DSE Mock

Make \( y \) the subject of the formula \( \frac{3x - 2y}{5 + y} = 4a \).

Simplify \( \frac{(u^3 v^{-2})^4}{u^{-5} v^3} \) and express your answer with positive indices.

Let \( f(x) = 2x^3 + ax^2 - 7x + b \), where \( a \) and \( b \) are constants. When \( f(x) \) is divided by \( x - 1 \), the remainder is \( -6 \). It is given that \( x + 2 \) is a factor of \( f(x) \). Find the values of \( a \) and \( b \).

Find the range of values of \( k \) such that the quadratic equation \( x^2 - 2kx + (3k - 2) = 0 \) has no real roots.

It is given that \( z \) is partly constant and partly varies directly as \( x^2 \). When \( x = 2 \), \( z = 14 \); and when \( x = 5 \), \( z = 77 \). Find the value of \( z \) when \( x = -3 \).

Solve the compound inequality \( 3(x + 2) > 5x - 4 \) and \( \frac{3 - x}{2} \le x + 3 \). Hence, write down the number of integers satisfying both inequalities.

The 3rd term and the 8th term of an arithmetic sequence are \( 11 \) and \( 31 \) 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.

The coordinates of the points \( A \) and \( B \) are \( (2, 5) \) and \( (6, -3) \) respectively. Let \( L \) be the perpendicular bisector of \( AB \). Find the equation of \( L \).

The stem-and-leaf diagram below shows the distribution of the test scores of 15 students in a class:

Stem (tens) | Leaf (units)
5 | 2, 4, 7
6 | 1, 3, 3, 5, 8
7 | 0, 2, 4, 6
8 | 1, 5, 9

Find the median, range, and standard deviation of the distribution. (Correct the standard deviation to 2 decimal places.)

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

The cost of manufacturing a solid metal cylinder, \(\$C\), is the sum of two parts. One part is constant, and the other part varies jointly as the square of the base radius, \(r\text{ cm}\), and the height, \(h\text{ cm}\). When \(r = 3\) and \(h = 5\), the cost is \(\$130\). When \(r = 4\) and \(h = 10\), the cost is \(\$360\). (a) Find the cost of manufacturing a cylinder with base radius \(5\text{ cm}\) and height \(8\text{ cm}\). (b) If the height of a cylinder is doubled and its base radius is halved, find the percentage change in the part of the cost that varies.

Let \(C\) be the circle \(x^2 + y^2 - 12x - 4y + 15 = 0\). (a) Find the coordinates of the center and the radius of \(C\). (b) A line \(L\) passing through \(P(2, 1)\) is parallel to the line \(3x - 4y + 5 = 0\). (i) Find the equation of \(L\). (ii) Determine whether \(L\) intersects \(C\). Explain your answer.

The 2nd term and the 5th term of a geometric sequence are \(12\) and \(96\) respectively. (a) Find the first term and the common ratio of the sequence. (b) Let \(G_n\) be the \(n\)-th term of the geometric sequence. (i) Find the least value of \(n\) such that \(\sum_{i=1}^n G_i > 10^5\). (ii) If \(A_n = \log_2 (G_n)\) for any positive integer \(n\), prove that \(A_1, A_2, A_3, \dots\) is an arithmetic sequence.

The stem-and-leaf diagram below shows the distribution of the test scores (in marks) of a class of 20 students: \[ \begin{array}{r|l} \text{Stem (tens)} & \text{Leaf (units)} \\ \hline 4 & 2\quad 5\quad 5\quad 8 \\ 5 & 0\quad 3\quad 4\quad 4\quad 7\quad 8 \\ 6 & 1\quad 1\quad 1\quad 5\quad 5 \\ 7 & 3\quad 4\quad 5\quad 7 \\ 8 & 2 \end{array} \] (a) Find the median, the range, and the interquartile range of the distribution. (b) Two students who were absent from the test took the make-up test later. Their scores were \(x\) and \(y\). After including their scores, the mean of the distribution remains unchanged, but the range of the distribution increases by 6 marks. (i) Find the mean of the original 20 scores. (ii) Find the values of \(x\) and \(y\).

(a) The 2nd term of a geometric sequence \(G\) is \(12\) and its sum to infinity is \(64\). Find the two possible values of the common ratio of \(G\). (4 marks)
(b) Suppose \(G\) has the larger common ratio of the two values found in (a). Let \(S_k\) be the sum of the first \(k\) terms of \(G\). Find the least value of \(k\) such that \(S_k > 47.9\). (3 marks)

Let the equation of the circle \(C\) be \(x^2 + y^2 - 10x - 24y + 144 = 0\).
(a) Find the center and the radius of \(C\). (2 marks)
(b) Find the equations of the two tangents from the origin \(O(0,0)\) to \(C\). (3 marks)
(c) Let \(P\) and \(R\) be the points of contact of the two tangents from \(O\) to \(C\) respectively, and let \(Q\) be the center of \(C\). Find the area of the quadrilateral \(OPQR\). (2 marks)

A bag contains 4 red balls, 3 blue balls, and 3 yellow balls.
(a) If 3 balls are randomly drawn from the bag at the same time, find the probability that the 3 balls drawn are of different colors. (3 marks)
(b) In a game, 3 balls are randomly drawn from the bag at the same time. If the 3 balls are of different colors, a player wins 20 tokens. If exactly 2 of the balls are of the same color, the player wins 5 tokens. Otherwise, the player loses 10 tokens. Find the expected number of tokens won by a player in a game. (4 marks)

In the figure (not shown), \(A\), \(B\), and \(C\) are three points on a horizontal ground such that \(AB = 8\text{ m}\), \(BC = 7\text{ m}\), and \(\angle ABC = 60^\circ\). \(TA\) is a vertical flagpole standing at \(A\). The angle of elevation of the top of the flagpole \(T\) from \(B\) is \(30^\circ\).
(a) Find the height of the flagpole \(TA\). (2 marks)
(b) Find the distance between \(T\) and \(C\). (3 marks)
(c) Find the angle of elevation of \(T\) from \(C\). (2 marks)

The mean and the standard deviation of the test scores of a class of 20 students are 65 marks and 8 marks respectively.
(a) Find the sum of the test scores of these 20 students, and the sum of the squares of the test scores. (3 marks)
(b) It is found that there are two clerical errors. The scores of two students, which were recorded as 50 and 80, should actually be 55 and 75 respectively.
(i) Find the correct mean of the test scores of the class.
(ii) Find the correct standard deviation of the test scores of the class. (4 marks)

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - 5x + 1 = 0\), find the value of \(\alpha^3 + \beta^3\).

Let \(P(x) = 2x^3 + ax^2 + bx - 6\), where \(a\) and \(b\) are constants. When \(P(x)\) is divided by \(x-1\) and \(x+2\), the remainders are \(-6\) and \(-24\) respectively. Find the remainder when \(P(x)\) is divided by \(2x-1\).

Solve the equation \(3^{2x+1} - 10 \cdot 3^x + 3 = 0\).

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 \(19\%\), find the percentage change of \(z\).

Let \(S_n\) be the sum of the first \(n\) terms of an arithmetic sequence. If \(S_n = 2n^2 + 5n\) for all positive integers \(n\), find the 10th term of the sequence.

Find the range of values of \(k\) such that the inequality \(x^2 + kx + (k+3) > 0\) is satisfied for all real values of \(x\).

Simplify \(\frac{\sin(180^\circ - \theta)\cos(90^\circ + \theta)}{\tan(360^\circ - \theta)}\).

The equation of a circle \(C\) is \(x^2 + y^2 - 6x + 8y + k = 0\), where \(k\) is a constant. If the straight line \(3x - 4y + 5 = 0\) is tangent to \(C\), find the value of \(k\).

A committee of 5 members is to be selected from 6 men and 4 women. If the committee must contain at least 2 women, how many different committees can be formed?

The mean and the standard deviation of a set of 10 numbers are 20 and 4 respectively. If a new number 20 is added to the set, find the mean and the standard deviation of the new set of numbers.

Let \(p(x) = ax^3 + bx^2 - 11x - 6\). If \(x-2\) and \(2x+1\) are factors of \(p(x)\), find the remainder when \(p(x)\) is divided by \(x-1\).

If \(\alpha\) and \(\beta\) (where \(\alpha \neq \beta\)) are the real roots of the quadratic equation \(x^2 - 2(k-1)x + k^2 - 5k = 0\), and \(\alpha^2 + \beta^2 = 28\), find the value of \(k\).

If \(\log_9 x - \log_3 y = 1\), which of the following must be true?

The 3rd term and the 6th term of a geometric sequence are 12 and 96 respectively. Find the sum of the first 10 terms of the sequence.

Find the number of non-negative integers \(x\) satisfying the system of inequalities: \(\frac{3x - 5}{2} < 2x + 1\) and \(4x - 7 \le 2(x + 3)\).

The equation of a circle \(C\) is \(x^2 + y^2 - 8x + 6y - 11 = 0\). Which of the following statements is/are true?

I. The coordinates of the center of \(C\) are \((4, -3)\).
II. The radius of \(C\) is 6.
III. The point \((1, 2)\) lies inside \(C\).

For \(0^\circ \le \theta < 360^\circ\), how many roots does the equation \(3 \sin^2 \theta - 5 \cos \theta - 1 = 0\) have?

Let \(A\) and \(B\) be the points \((2, 5)\) and \((8, -3)\) respectively. If \(P\) is a moving point in the rectangular coordinate plane such that \(AP \perp BP\), find the equation of the locus of \(P\).

A bag contains 4 red balls, 5 blue balls and 3 yellow balls. If 3 balls are randomly drawn from the bag one by one without replacement, find the probability that at least 2 blue balls are drawn.

The mean and the standard deviation of a set of data are 48 and 8 respectively. If each datum in the set is multiplied by \(-3\) and then 10 is added to each resulting value, find the new mean and the new standard deviation.

Let \( f(x) = 2x^3 + ax^2 + bx - 5 \). When \( f(x) \) is divided by \( x-2 \), the remainder is \( 21 \). When \( f(x) \) is divided by \( x+1 \), the remainder is \( -9 \). Find the remainder when \( f(x) \) is divided by \( x-1 \).

Let the circle \( C \) be \( x^2 + y^2 - 6x + 8y + k = 0 \). If the line \( 3x - 4y + 5 = 0 \) is tangent to the circle \( C \), find the value of \( k \).

Simplify \( \frac{\sin(360^\circ - \theta)\cos(90^\circ - \theta)}{\sin(180^\circ + \theta)\tan(180^\circ - \theta)} \).

If \( \log_4 x - \log_{16} y = 1 \), express \( y \) in terms of \( x \).

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 \( z \).

Let \( \alpha \) and \( \beta \) be the real roots of the quadratic equation \( 2x^2 - 6x + 3 = 0 \). Find the value of \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \).

In a geometric sequence, the 2nd term is \( 12 \) and the 5th term is \( 324 \). Find the sum of the first 6 terms of the sequence.

If \( (x, y) \) is a point in the region bounded by \( x + y \le 6 \), \( 2x - y \ge 0 \), and \( y \ge 1 \), find the maximum value of \( 3x + 2y \).

A set of data \( x_1, x_2, \dots, x_{10} \) has a mean of \( 40 \) and a standard deviation of \( 6 \). If each datum \( x_i \) is replaced by \( y_i = 3 - 2x_i \) for \( i = 1, 2, \dots, 10 \), find the mean and the standard deviation of the new set of data \( y_1, y_2, \dots, y_{10} \).

A committee of 5 members is to be selected from 6 teachers and 5 students. If the committee must contain at least 3 teachers, how many different committees can be formed?

The graph shows the linear relation between \(\log_5 y\) and \(\log_5 x\). The intercept on the horizontal axis is \(3\) and the intercept on the vertical axis is \(-2\). Which of the following is true?

In a geometric sequence, the sum of the first two terms is \(8\), and the sum to infinity is \(9\). Find the sum of all possible values of the first term.

In the figure, \(ABCD\) is a regular tetrahedron with side length \(6\). Let \(M\) be the midpoint of \(AD\) and \(N\) be the midpoint of \(BC\). Find the length of \(MN\).

A team of 5 representatives is to be selected from 6 boys and 5 girls. If the team must include at least 2 boys and at least 2 girls, how many different teams can be formed?

Let \(C\) be the circle \(x^2 + y^2 - 8x - 8y + 24 = 0\). If a straight line \(L\) passes through the origin \(O(0,0)\) and is tangent to the circle \(C\) at point \(P\), find the length of \(OP\).

Let \(x_1, x_2, \dots, x_{20}\) be a set of 20 data with mean \(m\) and variance \(v\). If each data is multiplied by \(-3\) and then \(5\) is added to it to form a new set of data, let the new mean and new variance be \(m'\) and \(v'\) respectively. Which of the following is/are correct?

I. \(m' = 5 - 3m\)
II. \(v' = 9v\)
III. The standard deviation of the new set is \(3\) times the standard deviation of the original set.

A box contains 4 red balls and 6 blue balls. A boy randomly draws balls from the box one by one without replacement until he gets a red ball. Find the probability that he needs at least 3 draws.

Let \(x\) and \(y\) be non-negative real numbers satisfying the system of inequalities:
$$\begin{cases} 2x + y \le 12 \\ x + 3y \le 11 \end{cases}$$
Find the maximum value of \(P = 3x + 4y\).

Let \(z = \frac{a+i}{1-2i}\), where \(a\) is a real number and \(i^2 = -1$. If the real part of \)z\) is equal to its imaginary part, find the value of \(a\).

The coordinates of the points \(A\) and \(B\) are \((2, 6)\) and \((8, -2)\) respectively. If \(P(x, y)\) is a moving point in the coordinate plane such that \(\angle APB = 90^\circ\), find the equation of the locus of \(P\).

The figure shows the linear relation between \(\log_3 x\) and \(\log_9 y\). The intercepts of the straight line on the horizontal axis and the vertical axis are \(4\) and \(2\) respectively. Which of the following must be true?

Let \(x\) be a constant. If the first three terms of a geometric sequence are \(x + 3\), \(x\) and \(x - 2\) respectively, find the sum to infinity of the sequence.

If the straight line \(3x - 4y + k = 0\) (where \(k\) is a constant) intersects the circle \(x^2 + y^2 - 4x + 6y - 12 = 0\) at two distinct points, find the range of values of \(k\).

A committee of 4 members is to be selected from 6 boys and 5 girls. If the committee must contain at least one boy and at least one girl, how many different committees can be formed?

Find the maximum value of \(\frac{12}{3 - \cos^2 \theta - 2\sin \theta}\), where \(0^\circ \le \theta < 360^\circ\).