Practice Paper

This paper has $150$ questions in seven sections (A to G), with the same kinds of question as the final paper. The numbers and situations are new, so you practise the method.

How to use your $2$ hours:

• 1First do the $48$ core questions, marked ★. They cover every type of question. Spend about $15$ minutes per section.
• 2Mark each section with the answers as soon as you finish it.
• 3Use any time left for the other questions — start with the sections where you lost marks.

Work on paper and show every step. Stuck? Go back to the recap page for that topic.

A1–A8: evaluate. A9–A16: simplify using the laws of indices. A17–A22: solve for $x$. (Recap: Basic Concepts of Algebra)

A1. ★ Evaluate $5x+23x+6$ when $x=3$.

A2. Evaluate $9x-4+2x$ when $x=5$.

A3. Evaluate $6a+14a-9$ when $a=2.5$.

A4. Evaluate $3x^2+2x-1$ when $x=4$.

A5. ★ Evaluate $\frac{(3x^2\cdot 2y)-4}{2x-y}$ when $x=2,y=3$.

A6. Evaluate $\frac{(2x^2\cdot 5y)+6}{x+y}$ when $x=3,y=1$.

A7. Evaluate $\frac{x^2-y^2}{x-y}$ when $x=7,y=3$.

A8. Evaluate $\frac{5x-2y}{x+2y}$ when $x=4,y=2$.

A9. ★ Simplify $\frac{x^8}{x^2}$.

A10. Simplify $\frac{a^9}{a^4}$.

A11. Simplify $\frac{y^5\times y^3}{y^4}$.

A12. ★ Simplify $2x^3(5y)(3xy)$.

A13. Simplify $4a^{2}b(3a{b}^2)$.

A14. Simplify $(x^3{)}^4$.

A15. Simplify $\frac{12x^5{y}^3}{4x^{2}y}$.

A16. Simplify $x^{-2}\times x^5$, and write down the value of $7x^0$.

A17. ★ Solve $9x-4y=19$, given $y=2$.

A18. Solve $7x+3y=43$, given $y=5$.

A19. Solve $15x-2y=50$, given $y=5$.

A20. ★ Solve $\left[\frac{2x+4}{3y}\right]=18-4y$, given $y=3$.

A21. Solve $\left[\frac{2x-6}{4y}\right]=10-3y$, given $y=2$.

A22. Solve $3(x+2y)=5x-4$, given $y=3$.

B1–B8 use the sports survey below. B9–B11 use the café survey (in B9). B12–B17 use the sets given in B12. (Recap: Set Operations)

A survey asked $200$ students which sports they play — Football (F), Cricket (C) and Volleyball (V).

• $70$ play Football
• $65$ play Cricket
• $55$ play Volleyball
• $25$ play Football and Cricket
• $20$ play Football and Volleyball
• $15$ play Cricket and Volleyball
• $10$ play all three sports

B1. ★ Use the survey above. Draw a Venn diagram with three circles and fill in the number of students in every region.

B2. ★ How many students play none of the three sports?

B3. ★ How many play Football only?

B4. ★ How many play Football and Cricket, but not Volleyball?

B5. ★ How many play Cricket and Volleyball, but not Football?

B6. How many play exactly one sport?

B7. How many play exactly two sports?

B8. How many play at least one sport?

B9. A café asked $120$ customers what they drank this week: Tea (T), Coffee (K) and Juice (J). $50$ had tea, $45$ coffee and $40$ juice; $18$ had tea and coffee, $12$ tea and juice, $10$ coffee and juice, and $5$ had all three. How many customers had none of the three drinks?

B10. Using the café survey, how many had coffee only?

B11. Using the café survey, how many had tea and juice, but not coffee?

B12. ★ For questions $12-17$ use the sets $P=\{2,4,6,8\}$, $Q=\{\{2\},\{4\},\{6\},\{8\}\}$, $R=P\cup \{10\}$, $S=P\cup \{\varnothing \}$. Is $\{\}=\{\varnothing \}$? Explain.

B13. ★ Which of these are true? (a) $\varnothing \subseteq P$ (b) $\varnothing \subseteq Q$ (c) $\{2\}\subseteq P$ (d) $\{2\}\subseteq Q$

B14. ★ Write down $Q\cup R$ and $Q\cup S$.

B15. Write down $P\cap Q$.

B16. ★ Write down $Q\setminus P$, $R\setminus P$ and $S\setminus P$.

B17. Find $n(Q)$ and $n(S)$.

B18. ★ $n(A)=18$, $n(B)=24$ and $n(A\cup B)=35$. Find $n(A\cap B)$.

B19. $n(A)=30$, $n(B)=22$ and $n(A\cap B)=9$. Find $n(A\cup B)$.

B20. $n(A\cup B)=50$, $n(A)=28$ and $n(A\cap B)=12$. Find $n(B)$.

B21. $n(A)=8$, $n(B)=20$ and $n(A\cup B)=15$. Find $n(A\cap B)$. Is this data possible?

B22. $U=\{1,2,\dots ,10\}$, $A=\{1,2,3,4,5\}$, $B=\{4,5,6,7\}$. Find $A\cup B$, $A\cap B$, $A\setminus B$, $B\setminus A$ and $A^{\prime }$.

Show the order of each matrix and every multiplication step. (Recap: Matrices)

C1. ★ Find $A+B$ and $A-B$, where $A=\left[\begin{array}{cc}3& 5\\ 2& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}6& 1\\ 4& 7\end{array}\right]$.

C2. Find $A+B$ and $A-B$, where $A=\left[\begin{array}{ccc}5& -3& 2\\ 1& 4& -6\end{array}\right]$ and $B=\left[\begin{array}{ccc}2& 6& -1\\ -3& 2& 0\end{array}\right]$.

C3. Find $A+B$ and $A-B$, where $A=\left[\begin{array}{ccc}1& -2& 3\\ 0& 4& -1\\ 5& 2& 6\end{array}\right]$ and $B=\left[\begin{array}{ccc}2& 1& -1\\ 3& -2& 0\\ -4& 1& 2\end{array}\right]$.

C4. ★ Find $A+B$ and $A-B$, where $A=\left[\begin{array}{cc}4& -2\\ 1& 5\\ 6& -3\end{array}\right]$ and $B=\left[\begin{array}{ccc}2& 0& 1\\ 3& -1& 4\\ 5& 2& -2\end{array}\right]$.

C5. State the order of $A=\left[\begin{array}{ccc}1& 0& -2\\ 4& 3& 5\end{array}\right]$ and of $B=\left[\begin{array}{c}7\\ -1\\ 2\end{array}\right]$.

C6. ★ Find $2A$, where $A=\left[\begin{array}{cc}14& -6\\ 3& 11\end{array}\right]$.

C7. Find $3A$, where $A=\left[\begin{array}{ccc}2& -5& 0\\ 7& 1& -4\\ -3& 8& 6\end{array}\right]$.

C8. Find $-2A$, where $A=\left[\begin{array}{cc}4& -1\\ 0& 6\end{array}\right]$.

C9. Find $2A-B$, where $A=\left[\begin{array}{cc}3& 1\\ 2& 5\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 4\\ 6& 2\end{array}\right]$.

C10. ★ Find $AB$ and $BA$, where $A=\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}-2& 5\\ 1& 0\end{array}\right]$.

C11. ★ Find $AB$ and $BA$, where $A=\left[\begin{array}{ccc}1& -2& 4\\ 3& 0& -1\end{array}\right]$ and $B=\left[\begin{array}{ccc}2& 1& 0\\ -1& 3& 2\\ 4& 0& -3\end{array}\right]$.

C12. Find $AB$ and $BA$, where $A=\left[\begin{array}{ccc}2& -1& 3\\ 0& 4& 1\\ 5& 2& -2\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 3\\ -2& 0\\ 4& -1\end{array}\right]$.

C13. Find $AB$, where $A=\left[\begin{array}{cc}2& 3\\ 1& -4\end{array}\right]$ and $B=\left[\begin{array}{c}5\\ 2\end{array}\right]$.

C14. A is a $2\times 3$ matrix and B is a $3\times 4$ matrix. Is $AB$ possible? What is its order? Is $BA$ possible?

C15. ★ Find the determinant of $\left[\begin{array}{cc}5& 3\\ 2& 4\end{array}\right]$.

C16. Find the determinant of $\left[\begin{array}{cc}7& -2\\ 3& 6\end{array}\right]$.

C17. Find the determinant of $\left[\begin{array}{cc}6& 9\\ 4& 6\end{array}\right]$.

C18. ★ Find the determinant of $\left[\begin{array}{ccc}3& -2& 1\\ 1& 4& 0\\ 2& -1& 5\end{array}\right]$.

C19. Find the determinant of $\left[\begin{array}{ccc}1& 2& -1\\ -3& 5& 2\\ 2& 0& 4\end{array}\right]$.

C20. Find the determinant of $\left[\begin{array}{ccc}4& 0& 2\\ 1& -3& 5\\ 2& 1& 0\end{array}\right]$.

C21. Find $x$ if $\left|\begin{array}{cc}x& 3\\ 2& 5\end{array}\right|=4$.

C22. What is the $2\times 2$ identity matrix $I$, and what is $AI$ for any $2\times 2$ matrix $A$?

Use $\sqrt{3}=1.732$ and $\sqrt{2}=1.414$. Give lengths to $2$ decimal places where needed. D1–D3 use Figure $1$. (Recap: Trigonometry)

D1. ★ Use Figure $1$. Find the height AB.

D2. ★ Use Figure $1$. Find AD.

D3. Use Figure $1$. Find AC.

D4. In another right-angled triangle ABD (right angle at B), point C lies on DB. $\angle ACB=60°$, $\angle ADB=45°$ and $CB=10$ $m$. Find AB, DC and AD. ($\sqrt{3}=1.732$, $\sqrt{2}=1.414$)

D5. ★ Nimal stands $25$ $m$ from a flagpole. The angle of elevation of the top is $45$°, and his eye level is $1.6$ $m$ above the ground. How tall is the flagpole?

D6. ★ From $30$ $m$ away, the angle of elevation of the top of a tower is $60$°. Eye level is $1.5$ $m$. Find the height of the tower. ($\sqrt{3}=1.732$)

D7. A student $50$ $m$ from a building sees the top at an angle of elevation of $30$°. Eye level is $1.7$ $m$. Find the height of the building. ($\sqrt{3}=1.732$)

D8. From the top of a $40$ $m$ cliff, the angle of depression of a boat is $30$°. How far is the boat from the foot of the cliff? ($\sqrt{3}=1.732$)

D9. A kite string is $80$ $m$ long and makes an angle of $60$° with the ground. How high is the kite? ($\sqrt{3}=1.732$)

D10. A ladder makes an angle of $60$° with the ground. Its foot is $2.5$ $m$ from the wall. How long is the ladder?

D11. ★ A right-angled triangle PQR has the right angle at Q, $PQ=9$ cm and $QR=40$ cm. Find PR.

D12. A right-angled triangle PQR has the right angle at Q, $PQ=12$ cm and $QR=35$ cm. Find PR.

D13. In triangle XYZ, the right angle is at Y. $XZ=29$ cm and $XY=20$ cm. Find YZ.

D14. ★ In triangle XYZ, the right angle is at Y. $XZ=25$ cm and $XY=24$ cm. Find YZ.

D15. ★ A ladder $6.5$ $m$ long leans against a wall. Its foot is $2.5$ $m$ from the wall. How high up the wall does it reach?

D16. A $10$ $m$ ladder reaches up a building. Its foot is $6$ $m$ from the building. Find the height it reaches.

D17. A $4$ $m$ ladder stands with its foot $1$ $m$ from a wall. How high does it reach? Give your answer to $2$ decimal places.

D18. A rectangle is $16$ cm long and $12$ cm wide. Find the length of its diagonal.

D19. Is a triangle with sides $8$ cm, $15$ cm and $17$ cm right-angled? Show how you know.

D20. Write down the exact values of $\sin 30°$, $\cos 60°$, $\tan 45°$ and $\sin 60°$.

D21. A right-angled triangle has an angle θ. The side opposite θ is $5$ cm and the hypotenuse is $13$ cm. Find the adjacent side, then $\sin \theta$, $\cos \theta$ and $\tan \theta$.

D22. A ramp rises $3$ $m$ over a horizontal distance of $3$ $m$. What angle does it make with the ground?

Show the sorted list or a frequency table in every answer. (Recap: Descriptive Statistics)

E1. ★ $11$ students took a quiz. Their scores out of $100$ were: $74$, $58$, $58$, $91$, $42$, $63$, $74$, $85$, $58$, $67$, $80$. Find the mean, median, mode and range.

E2. The daily rainfall (mm) over $9$ days was: $15$, $22$, $18$, $22$, $30$, $12$, $25$, $22$, $19$. Find the mean, median, mode and range.

E3. $10$ workers' ages were: $34$, $41$, $29$, $34$, $50$, $27$, $38$, $45$, $34$, $48$. Find the mean, median, mode and range.

E4. $7$ students' shoe sizes were: $5$, $8$, $3$, $8$, $6$, $3$, $9$. Find the mean, median, mode and range.

E5. $4$ parcels weighed (kg): $12$, $15$, $18$, $21$. Find the mean, median, mode and range.

E6. The mean of $5$ numbers is $20$. Four of them are $18$, $25$, $14$ and $22$. Find the fifth number.

E7. ★ A survey of $30$ families recorded the number of children in each family:

$0$, $2$, $1$, $3$, $2$, $1$, $0$, $2$, $2$, $1$, $4$, $2$, $1$, $0$, $3$, $2$, $1$, $2$, $2$, $0$, $1$, $3$, $2$, $1$, $2$, $0$, $1$, $2$, $3$, $1$

Make a frequency distribution table.

E8. ★ Using the family survey, find the mean number of children.

E9. ★ Using the family survey, find the median.

E10. ★ Using the family survey, find the mode.

E11. Using the family survey, find the range, and the percentage of families with more than $2$ children.

E12. ★ Draw a bar chart of the family survey.

E13. This frequency table shows how many books students read in a month:

How many students were surveyed, and what is the mode?

E14. Using the books table, find the mean and median.

E15. The marks of $6$ students are $45$, $52$, $38$, $61$, $52$, $47$. A seventh student scores $90$. Which of the mean, median and mode changes the most?

E16. Which average is best for the incomes $30$ $000$, $32$ $000$, $35$ $000$, $31$ $000$ and $400$ $000$ (LKR)? Explain.

E17. Add $5$ to every value in a data set. What happens to the mean and to the range?

E18. The mean number of cars per household is $1.4$. Can this be right if no family owns $1.4$ cars? Explain.

E19. Why should a bar chart of discrete data (such as the number of children) have gaps between the bars?

E20. Two classes took the same test. Class X: mean $62$, range $15$. Class Y: mean $62$, range $48$. Compare the classes.

Give each probability as a fraction in its simplest form (and as a decimal where helpful). (Recap: Probability)

F1. ★ A letter is picked at random from the word PEPPER. Which letter is most likely to be picked?

• a) E
• b) P
• c) R
• d) None of the above

F2. A letter is picked at random from SUCCESS. Which letter is most likely?

• a) U
• b) C
• c) S
• d) E

F3. A letter is picked at random from CHEESE. Find $P(E)$.

F4. A letter is picked at random from EXAMINATION. Find the probability that it is a vowel.

F5. ★ A number is chosen at random from: $14$, $7$, $9$, $20$, $36$, $5$, $13$, $8$. What is the probability that it is even?

• a) $\frac{3}{8}$
• b) $\frac{1}{2}$
• c) $\frac{1}{8}$
• d) None of the above

F6. A number is chosen at random from: $3$, $10$, $15$, $22$, $7$, $18$, $11$, $9$, $4$, $5$. Find the probability that it is even.

F7. A whole number from $1$ to $20$ is chosen at random. Find the probability that it is a multiple of $3$.

F8. A whole number from $1$ to $10$ is chosen at random. Find the probability that it is prime.

F9. ★ One card is drawn at random from a standard pack of $52$ cards. Find the probability that it is a king.

F10. One card is drawn at random from a standard pack of $52$ cards. Find the probability that it is a red card.

F11. ★ One card is drawn at random from a standard pack of $52$ cards. Find the probability that it is a picture card (J, Q or K).

F12. One card is drawn at random from a standard pack of $52$ cards. Find the probability that it is a heart.

F13. One card is drawn at random from a standard pack of $52$ cards. Find the probability that it is a red ace.

F14. One card is drawn at random from a standard pack of $52$ cards. Find the probability that it is not a spade.

F15. One card is drawn at random from a standard pack of $52$ cards. Find the probability that it is an ace or a king.

F16. ★ A bag holds $400$ counters in four colours. A counter is taken at random, its colour noted, and it is put back. This is done $250$ times: red $90$, blue $60$, green $75$, yellow $25$. Find the (experimental) probability of drawing a red counter.

F17. Using the counter experiment, find the probability of drawing a green counter.

F18. ★ Using the counter experiment, find the probability of drawing a blue or a yellow counter.

F19. ★ Using the counter experiment, find the probability of not drawing a yellow counter.

F20. Using the counter experiment, estimate how many of the $400$ counters are red.

F21. A fair six-sided die is rolled. Find $P(\text{number greater than 4})$ and $P(\text{even number})$.

F22. Which of these cannot be a probability? $0$, $0.75$, $1.2$, $\frac{3}{5}$. Explain.

G1–G8 use Figure $2$. (Recap: Accuracy and Precision)

G1. ★ Which statement describes Target $1$ in Figure $2$?

• a) High accuracy and high precision
• b) Low accuracy and low precision
• c) High accuracy and low precision
• d) Low accuracy and high precision

G2. ★ Which statement describes Target $2$ in Figure $2$?

• a) High accuracy and high precision
• b) Low accuracy and low precision
• c) High accuracy and low precision
• d) Low accuracy and high precision

G3. ★ Which statement describes Target $3$ in Figure $2$?

• a) High accuracy and high precision
• b) Low accuracy and low precision
• c) High accuracy and low precision
• d) Low accuracy and high precision

G4. ★ Which statement describes Target $4$ in Figure $2$?

• a) High accuracy and high precision
• b) Low accuracy and low precision
• c) High accuracy and low precision
• d) Low accuracy and high precision

G5. Which statement describes Target $5$ in Figure $2$?

• a) High accuracy and high precision
• b) Low accuracy and low precision
• c) High accuracy and low precision
• d) Low accuracy and high precision

G6. Which statement describes Target $6$ in Figure $2$?

• a) High accuracy and high precision
• b) Low accuracy and low precision
• c) High accuracy and low precision
• d) Low accuracy and high precision

G7. Which statement describes Target $7$ in Figure $2$?

• a) High accuracy and high precision
• b) Low accuracy and low precision
• c) High accuracy and low precision
• d) Low accuracy and high precision

G8. Which statement describes Target $8$ in Figure $2$?

• a) High accuracy and high precision
• b) Low accuracy and low precision
• c) High accuracy and low precision
• d) Low accuracy and high precision

G9. Which targets in Figure $2$ show both high accuracy and high precision?

G10. Which targets show shots clustered together but far from the centre? Name this combination.

G11. Which targets in Figure $2$ show low accuracy and low precision?

G12. Define accuracy and precision in one sentence each.

G13. ★ A $6.00$ kg mass is weighed three times: $5.02$ kg, $5.01$ kg, $5.02$ kg. Describe the scale.

G14. A thermometer measures water at exactly $100$ °C four times: $96$, $104$, $98$, $102$ °C. Describe it.

G15. A stopwatch records a $10.0$ s event as $10.1$, $9.9$, $10.0$, $10.0$ s. Describe it.

G16. A ruler with a worn end gives $12.4$, $12.4$, $12.5$ cm for a $12.0$ cm line. What kind of error is this, and how does it affect accuracy and precision?

G17. What kind of error causes readings to be scattered (low precision)?

G18. Can a set of results be precise but not accurate? Give an everyday example.

G19. How can you improve precision, and how can you improve accuracy?

G20. In a darts game, all of Amali's darts land in a tight group in the outer ring, far from the bullseye. Which describes her throws?

• a) High accuracy and low precision
• b) Low accuracy and high precision
• c) High accuracy and high precision
• d) Low accuracy and low precision