HKDSE (Hong Kong) · Mathematics Compulsory Part

Number and Algebra

90 practice questions, each with the answer and an explanation.

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All 90 questions

Simplify $\frac{( 3 x ^{2} y ^{- 3} ) ^{2}}{9 x y ^{2}}$ and express your answer with positive indices.
The marked price of a watch is increased by $20%$ and then sold at a discount of $25%$ . What is the overall percentage change in the price of the watch?
Solve the simultaneous equations ${2 x + 3 y = 7 3 x - 2 y = 4$ . What is the value of $x + y$ ?
In an arithmetic sequence, the 4th term is $14$ and the 8th term is $30$ . Find the sum of the first $10$ terms of this sequence.
Solve the linear inequality $2 (x - 3) \leq 5 x + 3$ .
The sum to infinity of a convergent geometric series is $16$ , and the second term is $3$ . What are the possible values of the common ratio $r$ ?
A prize of $\$ 63$ is shared by Amy and Ben in the ratio $2:5$. How much does Ben receive?
A prize of $\$ 75$ is shared by Amy and Ben in the ratio $2:3$. How much does Ben receive?
A prize of $\$ 108$ is shared by Amy and Ben in the ratio $5:7$. How much does Ben receive?
A prize of $\$ 108$ is shared by Amy and Ben in the ratio $4:5$. How much does Ben receive?
Solve $7 x + 9 = 37$ .
Solve $2 x + 2 = 14$ .
Solve $5 x + 4 = 39$ .
Solve $5 x + 9 = 24$ .
Solve ${x + 4 y = 41 4 x + 4 y = 68$ . What is the value of $x y$ ?
Solve ${4 x + y = 27 2 x + 2 y = 24$ . What is the value of $x + y$ ?
Solve ${4 x + 4 y = 44 2 x + 4 y = 26$ . What is the value of $x + y$ ?
Solve ${4 x + 2 y = 46 3 x + 3 y = 45$ . What is the value of $x + y$ ?
Solve ${4 x + 4 y = 16 3 x + 2 y = 9$ . What is the value of $x - y$ ?
Solve ${4 x + y = 44 3 x + 3 y = 51$ . What is the value of $x - y$ ?
If $α$ and $β$ are the roots of $2 x^{2} - 2 x - 60 = 0$ , find $α \cdot β$ .
If $α$ and $β$ are the roots of $2 x^{2} + 6 x - 8 = 0$ , find $α \cdot β$ .
If $α$ and $β$ are the roots of $2 x^{2} + 2 x - 40 = 0$ , find $α \cdot β$ .
If $α$ and $β$ are the roots of $x^{2} + 6 x + 5 = 0$ , find $α \cdot β$ .
If $α$ and $β$ are the roots of $x^{2} + 3 x - 4 = 0$ , find $α + β$ .
What is the larger root of $x^{2} - 3 x + 2 = 0$ ?
What is the larger root of $x^{2} - 13 x + 42 = 0$ ?
What is the larger root of $x^{2} + 4 x - 12 = 0$ ?
What is the larger root of $x^{2} - x - 20 = 0$ ?
The equation $x^{2} + 6 x + k = 0$ has two equal real roots. Find $k$ .
The equation $x^{2} + 12 x + k = 0$ has two equal real roots. Find $k$ .
The equation $x^{2} + 8 x + k = 0$ has two equal real roots. Find $k$ .
The equation $x^{2} + 10 x + k = 0$ has two equal real roots. Find $k$ .
Simplify $\frac{5 ^{8} \times 5 ^{6}}{5 ^{1}}$ , leaving the answer in index form.
Simplify $\frac{3 ^{5} \times 3 ^{6}}{3 ^{2}}$ , leaving the answer in index form.
Simplify $\frac{5 ^{6} \times 5 ^{5}}{5 ^{1}}$ , leaving the answer in index form.
Simplify $\frac{5 ^{5} \times 5 ^{6}}{5 ^{3}}$ , leaving the answer in index form.
The first term of an arithmetic sequence is $8$ and the common difference is $2$ . Find the sum of the first 17 terms.
The first term of an arithmetic sequence is $6$ and the common difference is $7$ . Find the sum of the first 11 terms.
The first term of an arithmetic sequence is $8$ and the common difference is $4$ . Find the sum of the first 8 terms.
The first term of an arithmetic sequence is $5$ and the common difference is $6$ . Find the $9 t h$ term.
The first term of an arithmetic sequence is $4$ and the common difference is $6$ . Find the $11 t h$ term.
The first term of an arithmetic sequence is $2$ and the common difference is $3$ . Find the sum of the first 24 terms.
A geometric sequence has first term $4$ and common ratio $2$ . Find its term number 7.
A geometric sequence has first term $3$ and common ratio $3$ . Find its term number 7.
A geometric sequence has first term $5$ and common ratio $3$ . Find its term number 5.
A geometric sequence has first term $3$ and common ratio $2$ . Find its term number 7.
Find the greatest integer $x$ satisfying $3 x - 7 < 52$ .
Find the greatest integer $x$ satisfying $3 x - 13 < 37$ .
Find the greatest integer $x$ satisfying $6 x - 12 < 29$ .
Find the greatest integer $x$ satisfying $2 x - 13 < 22$ .
$y$ varies inversely as $x$ . When $x = 4$ , $y = 3$ . Find $y$ when $x = 6$ .
$y$ varies directly as $x$ . When $x = 4$ , $y = 3$ . Find $y$ when $x = 12$ .
$y$ varies inversely as $x$ . When $x = 2$ , $y = 3$ . Find $y$ when $x = 6$ .
$y$ varies directly as $x$ . When $x = 6$ , $y = 2$ . Find $y$ when $x = 9$ .
$y$ varies inversely as $x$ . When $x = 4$ , $y = 6$ . Find $y$ when $x = 8$ .
$y$ varies directly as $x$ . When $x = 6$ , $y = 3$ . Find $y$ when $x = 10$ .
A car travels from A to B at 60 km/h and returns along the same road at 120 km/h. Find the average speed for the whole journey.
A car travels from A to B at 20 km/h and returns along the same road at 30 km/h. Find the average speed for the whole journey.
A car travels from A to B at 50 km/h and returns along the same road at 75 km/h. Find the average speed for the whole journey.
A car travels from A to B at 30 km/h and returns along the same road at 60 km/h. Find the average speed for the whole journey.
A car travels from A to B at 40 km/h and returns along the same road at 80 km/h. Find the average speed for the whole journey.
A car travels from A to B at 40 km/h and returns along the same road at 60 km/h. Find the average speed for the whole journey.
If $f (x) = 2 x^{2} - 2 x - 5$ , find $f (- 2)$ .
If $f (x) = 3 x^{2} + 5 x + 3$ , find $f (- 1)$ .
If $f (x) = x^{2} - 3 x - 3$ , find $f (3)$ .
If $f (x) = 3 x^{2} - x + 5$ , find $f (2)$ .
If $f (x) = 4 x^{2} - 5 x + 1$ , find $f (4)$ .
Find the minimum value of $x^{2} - 16 x + 71$ .
Find the minimum value of $x^{2} + 10 x + 24$ .
Find the minimum value of $x^{2} + 12 x + 45$ .
Find the minimum value of $x^{2} + 14 x + 42$ .
Find the minimum value of $x^{2} - 16 x + 69$ .
Find the minimum value of $x^{2} + 10 x + 28$ .
Find the minimum value of $x^{2} - 12 x + 42$ .
Find the minimum value of $x^{2} + 16 x + 59$ .
Find the remainder when $x^{3} + 4 x^{2} - x - 6$ is divided by $x + 3$ .
Find the remainder when $3 x^{3} - 3 x^{2} + 4 x + 2$ is divided by $x + 3$ .
Find the remainder when $3 x^{3} + x^{2} - 3 x - 2$ is divided by $x + 3$ .
Find the remainder when $x^{3} + x^{2} + 2 x + 5$ is divided by $x - 1$ .
Find the remainder when $2 x^{3} - x^{2} + 3 x - 1$ is divided by $x + 2$ .
Find the remainder when $x^{3} + 3 x^{2} - 2 x + 3$ is divided by $x - 1$ .
Find the remainder when $2 x^{3} - x^{2} - 4 x + 3$ is divided by $x + 2$ .
Find the remainder when $3 x^{3} + 2 x^{2} + 2 x + 2$ is divided by $x - 1$ .
Factorise $x^{2} - 2 x - 8$ .
Expand $(x + 4)^{2}$ .
Expand $(x + 6)^{2}$ .
Factorise $x^{2} - 2 x - 48$ .
Factorise $x^{2} - 4 x - 32$ .
Factorise $x^{2} - x - 56$ .

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