Algebra

What does the algebraic term ab mean?

• Ab subtracted from a
• Ba added to b
• Ca multiplied by b✓
• Da divided by b

Answer: In algebra, writing two letters next to each other always means multiplication, so ab is short for a × b. Addition would be written a + b, and division would need a fraction bar or a ÷ sign. Subtraction would be written a − b, not ab.

In algebra, what does the term 3y represent?

• Ay shared equally between three
• By raised to the power of three
• Cthree subtracted from y
• Dthree lots of y added together✓

Answer: A number written directly next to a letter is a coefficient, so 3y means 3 × y, which is the same as y + y + y. Raising to a power would be written y³, not 3y. Subtracting or dividing would need a minus sign or a fraction, neither of which appear here.

What does a² mean in algebra?

• Aa multiplied by itself✓
• Ba multiplied by 2
• Ca plus 2
• D2 multiplied by a

Answer: A small raised 2 is an index, so a² means a × a, the letter multiplied by itself. Multiplying a by 2 would be written 2a, not a². Adding 2 to a would be written a + 2, giving a added to the number 2.

Priya writes a fraction with a on top and b on the bottom to show a division. Which calculation does this fraction represent?

• Aa subtracted from b
• Bb divided by a
• Ca multiplied by b
• Da divided by b✓

Answer: A fraction bar always means the top is divided by the bottom, so a over b represents a ÷ b. Swapping the order to b divided by a gives a different value unless a equals b. Multiplication and subtraction would need different symbols, not a fraction bar.

A rectangular playground has perimeter P = 2(l + w). Using length l = 7 m and width w = 4 m, what is the perimeter?

• A18 m
• B22 m✓
• C15 m
• D11 m

Answer: Substituting the values gives P = 2(7 + 4) = 2 × 11 = 22 m. Forgetting to double the bracket gives 11 m. Doubling only the length before adding the width gives 18 m, and doubling only the width gives 15 m.

A market stall charges C = 3n + 5, where n is the number of items bought and C is the total cost in pounds including delivery. What is the cost for 6 items?

• A£23✓
• B£33
• C£48
• D£18

Answer: Substituting n = 6 gives C = 3 × 6 + 5 = 18 + 5 = £23. Adding the 5 before multiplying by 3 gives 3 × 11 = £33. Forgetting to add the delivery fee gives just £18, and multiplying the whole formula by 6 instead of substituting gives £48.

The area of a triangle is found using A = ½bh. A sail has base b = 8 m and height h = 5 m. What is its area?

• A40 m²
• B9 m²
• C13 m²
• D20 m²✓

Answer: Substituting gives A = ½ × 8 × 5 = ½ × 40 = 20 m². Forgetting to halve the product gives 40 m². Adding the base and height instead of multiplying gives 13 m², and halving the base but then adding the height instead of multiplying gives 9 m².

Which word correctly describes 3x + 5?

• Aan expression✓
• Ba formula
• Can equation
• Dan inequality

Answer: A collection of terms with no equals sign, like 3x + 5, is called an expression. An equation must contain an equals sign linking two expressions, and an inequality uses a symbol such as < or >. A formula is a special equation that shows the relationship between two or more quantities.

What kind of mathematical statement is x + 4 = 9?

• Aan equation✓
• Ban expression
• Ca sequence
• Dan inequality

Answer: A statement containing an equals sign that is true for a specific value of x, like x + 4 = 9, is an equation. An expression such as x + 4 has no equals sign and cannot be solved on its own. An inequality uses a symbol such as < or > instead of =, and a sequence is a list of numbers following a pattern.

What kind of mathematical statement is x < 7?

• Aa formula
• Ban equation
• Can expression
• Dan inequality✓

Answer: A statement comparing two values with a symbol such as < instead of an equals sign is called an inequality. An equation must use an equals sign, and an expression such as x − 7 has no comparison symbol at all. A formula shows a fixed relationship between quantities, such as A = lw.

In the expression 4x + 8, what is the highest common factor of the two terms?

• Ax
• B4✓
• C2
• D8

Answer: Both 4x and 8 divide exactly by 4, and no larger number divides both, so the highest common factor is 4. 2 divides both terms too but is not the highest such number. 8 does not divide 4x exactly for every value of x, and x is not a factor of the constant term 8.

Simplify 3x + 5x − 2x.

• A6x✓
• B8x
• C10x
• D−4x

Answer: Collecting the like terms gives 3x + 5x − 2x = 6x. Ignoring the minus sign and adding all three coefficients gives 10x. Subtracting the first term instead of adding it gives −4x, and adding only the first two terms while forgetting the last one gives 8x.

Expand 4(x + 3).

• A4x + 7
• B4x + 3
• C4x + 12✓
• Dx + 12

Answer: Every term inside the bracket must be multiplied by 4, giving 4 × x + 4 × 3 = 4x + 12. Multiplying only the x and leaving the 3 unchanged gives 4x + 3. Forgetting to multiply the x term gives x + 12, and adding 4 and 3 instead of multiplying gives 4x + 7.

Factorise 6x + 9 fully.

• A3(2x + 9)
• B3(2x + 3)✓
• C6(x + 9)
• D2(3x + 3)

Answer: The highest common factor of 6x and 9 is 3, so dividing both terms by 3 gives 3(2x + 3). Taking out 6 instead of the correct common factor leaves the second term wrong, giving 6(x + 9). Taking out 3 correctly but forgetting to divide the second term gives 3(2x + 9), and taking out 2, which is not a factor of 9, gives the invalid factorisation 2(3x + 3).

Expand and simplify (x + 2)(x + 5).

• Ax² + 7x + 10✓
• Bx² + 7x + 7
• Cx² + 10x + 7
• Dx² + 3x + 10

Answer: Multiplying each term in the first bracket by each term in the second gives x² + 5x + 2x + 10, which simplifies to x² + 7x + 10. Swapping the coefficient and the constant gives x² + 10x + 7. Adding 2 and 5 correctly but forgetting to multiply them for the constant term gives x² + 7x + 7, and subtracting the constants instead of multiplying them gives x² + 3x + 10.

Rearrange y = x + 7 to make x the subject.

• Ax = 7 − y
• Bx = y − 7✓
• Cx = 7y
• Dx = y + 7

Answer: Subtracting 7 from both sides of y = x + 7 isolates x, giving x = y − 7. Adding 7 instead of subtracting gives x = y + 7. Reversing the order of the subtraction gives x = 7 − y, which is only correct when y equals half of 7, and multiplying instead of subtracting gives x = 7y.

The formula v = u + at gives the final speed of an object. Rearrange it to make a the subject.

• Aa = (v − u) / t✓
• Ba = (v − u) × t
• Ca = (v + u) / t
• Da = v − u − t

Answer: Subtracting u from both sides gives v − u = at, and dividing both sides by t isolates a, giving a = (v − u) / t. Adding u instead of subtracting gives a = (v + u) / t. Multiplying by t instead of dividing gives a = (v − u) × t, and subtracting t as if it were a separate term gives a = v − u − t.

The perimeter of a rectangle is P = 2(l + w). Rearrange it to make l the subject.

• Al = P − w
• Bl = P/2 − w✓
• Cl = 2P − w
• Dl = P/2 + w

Answer: Dividing both sides by 2 first gives P/2 = l + w, and subtracting w from both sides gives l = P/2 − w. Adding w instead of subtracting gives l = P/2 + w. Forgetting to divide P by 2 gives l = P − w, and multiplying P by 2 instead of dividing gives l = 2P − w.

Solve 3x + 4 = 19.

• Ax = 45
• Bx = 5✓
• Cx = 23
• Dx = 6.33

Answer: Subtracting 4 from both sides gives 3x = 15, and dividing by 3 gives x = 5. Dividing 19 by 3 without first subtracting the 4 gives x = 6.33. Adding 4 instead of subtracting and stopping there gives x = 23, and multiplying both sides by 3 instead of dividing gives x = 45.

Solve 5x − 3 = 2x + 12.

• Ax = 5✓
• Bx = 2.14
• Cx = 4
• Dx = −5

Answer: Subtracting 2x and adding 3 to both sides gives 3x = 15, so x = 5. Forgetting the −3 term and solving 5x = 2x + 12 gives 3x = 12, so x = 4. Making a sign error to get 3x = −15 gives x = −5, and adding the x terms instead of subtracting them gives 7x = 15, so x = 2.14.

Solve x/4 + 2 = 7.

• Ax = 20✓
• Bx = 36
• Cx = 1.25
• Dx = 28

Answer: Subtracting 2 from both sides gives x/4 = 5, and multiplying both sides by 4 gives x = 20. Skipping the subtraction and multiplying 7 by 4 directly gives x = 28. Adding 2 instead of subtracting gives x/4 = 9, so x = 36, and dividing 5 by 4 instead of multiplying gives x = 1.25.

Solve 2(x + 3) = 16.

• Ax = 11
• Bx = 8
• Cx = 6.5
• Dx = 5✓

Answer: Dividing both sides by 2 gives x + 3 = 8, and subtracting 3 gives x = 5. Expanding the bracket incorrectly as 2x + 3 = 16 gives x = 6.5. Dividing by 2 correctly but forgetting to subtract 3 leaves x = 8, and adding 3 instead of subtracting gives x = 11.

In which quadrant does the point (−3, 5) lie?

• Athe second quadrant✓
• Bthe third quadrant
• Cthe first quadrant
• Dthe fourth quadrant

Answer: The first quadrant has both coordinates positive, the second has a negative x and a positive y, the third has both negative, and the fourth has a positive x and a negative y. Since (−3, 5) has a negative x-coordinate and a positive y-coordinate, it lies in the second quadrant, not the first, third or fourth.

In which quadrant does the point (4, −2) lie?

• Athe first quadrant
• Bthe third quadrant
• Cthe fourth quadrant✓
• Dthe second quadrant

Answer: The fourth quadrant contains points with a positive x-coordinate and a negative y-coordinate, which matches (4, −2). The first quadrant needs both coordinates positive, the second needs a negative x with a positive y, and the third needs both coordinates negative.

The point (2, 3) is reflected in the x-axis. What are the coordinates of its image?

• A(2, −3)✓
• B(−2, 3)
• C(−2, −3)
• D(3, 2)

Answer: Reflecting in the x-axis keeps the x-coordinate the same and changes the sign of the y-coordinate, giving (2, −3). Changing the sign of the x-coordinate instead describes a reflection in the y-axis, giving (−2, 3). Changing both signs describes a rotation of 180°, giving (−2, −3), and swapping the coordinates gives (3, 2), which is unrelated to either reflection.

Which of these points lies on the line y = 2x + 1?

• A(3, 5)
• B(2, 7)
• C(3, 6)
• D(3, 7)✓

Answer: Substituting x = 3 into y = 2x + 1 gives y = 2 × 3 + 1 = 7, so (3, 7) lies on the line. Forgetting to add the 1 gives (3, 6). Subtracting 1 instead of adding it gives (3, 5), and (2, 7) simply swaps which value belongs to x and which to the calculated y.

For the graph of y = x², what is the value of y when x = −3?

• A−6
• B−9
• C9✓
• D6

Answer: Squaring a negative number gives a positive result, so y = (−3)² = 9. Forgetting that a negative squared is positive gives −9. Doubling x instead of squaring it, while keeping the negative sign, gives −6, and doubling x while dropping the sign gives 6.

What is the gradient of the line y = 3x − 4?

• A4
• B−4
• C3✓
• D−3

Answer: In the form y = mx + c, the coefficient of x gives the gradient, so the gradient of y = 3x − 4 is 3. The constant term −4 is the y-intercept, not the gradient. Reading the sign from the wrong term gives −3, and reversing the sign of the intercept gives 4.

What is the y-intercept of the line y = −2x + 5?

• A5✓
• B−2
• C2
• D−5

Answer: In the form y = mx + c, the constant term c gives the y-intercept, so the y-intercept of y = −2x + 5 is 5. The coefficient −2 is the gradient, not the intercept. Dropping the negative sign from the gradient gives 2, and reversing the sign of the intercept gives −5.

A line has gradient 4 and passes through (0, −1). What is its equation?

• Ay = 4x − 1✓
• By = x − 1
• Cy = 4x + 1
• Dy = −4x − 1

Answer: The line crosses the y-axis at (0, −1), so c = −1, and with gradient m = 4 the equation is y = 4x − 1. Reversing the sign of the intercept gives y = 4x + 1. Reversing the sign of the gradient gives y = −4x − 1, and using a gradient of 1 instead of 4 gives y = x − 1.

A taxi firm's cost graph follows C = 5h + 20, where h is hours hired and C is the cost in pounds. Reading from the graph, what does it show for a 3-hour hire?

• A£65
• B£75
• C£15
• D£35✓

Answer: Substituting h = 3 gives C = 5 × 3 + 20 = 15 + 20 = £35. Leaving out the fixed £20 charge gives just £15. Swapping the roles of the rate and the fixed charge, as if C = 20h + 5, gives £65, and multiplying the whole bracket (5 + 20) by 3 gives £75.

A car park's piecewise charge graph shows the first hour free, then £2 for each hour after that. Reading from the graph, what is the charge for 3 hours?

• A£2
• B£4✓
• C£6
• D£8

Answer: Only the hours after the first free one are charged, so 2 chargeable hours at £2 each gives 2 × £2 = £4. Charging for all 3 hours at £2 each, ignoring the free hour, gives £6. Charging for only 1 hour after the free one gives £2, and doubling the correct answer by mistake gives £8.

The graph of y = 2ˣ models a bacteria colony doubling each hour. Reading from the graph, what value does it show at x = 3?

• A6
• B9
• C8✓
• D16

Answer: Evaluating gives y = 2³ = 2 × 2 × 2 = 8. Multiplying the base by the power instead of using repeated multiplication gives 2 × 3 = 6. Confusing the graph with y = 3² gives 9, and continuing the doubling for one extra hour gives 16.

The graph of y = 12/x is drawn. Reading from the graph, what value does it show at x = 3?

• A9
• B36
• C3
• D4✓

Answer: Substituting x = 3 gives y = 12 ÷ 3 = 4. Subtracting instead of dividing gives 12 − 3 = 9. Multiplying instead of dividing gives 12 × 3 = 36, and reading the x-value itself as the answer gives 3.

A sequence begins 5, 9, 13, 17, … following a term-to-term rule. What is the next term?

• A20
• B21✓
• C19
• D22

Answer: Each term increases by 4, so adding 4 to 17 gives 21. Adding only 3 gives 20, adding 5 gives 22, and adding 4 but then subtracting 2 by mistake gives 19.

A sequence has the position-to-term rule nth term = 3n + 2. What is the 5th term?

• A20
• B13
• C17✓
• D15

Answer: Substituting n = 5 gives 3 × 5 + 2 = 15 + 2 = 17. Forgetting to add the 2 gives 15. Adding 2 before multiplying by 3 gives 3 × 7 = 20, and subtracting 2 instead of adding gives 13.

A sequence begins 50, 44, 38, 32, … following a term-to-term rule. What is the next term?

• A24
• B26✓
• C27
• D38

Answer: Each term decreases by 6, so subtracting 6 from 32 gives 26. Subtracting only 5 gives 27, subtracting 8 gives 24, and repeating the previous term instead of continuing the pattern gives 38.

A sequence begins 7, 10, 13, 16, … What is its nth term rule?

• A3n + 3
• Bn + 6
• C3n + 4✓
• D3n + 7

Answer: The common difference is 3, so the rule starts with 3n. Testing n = 1 gives 3 + 4 = 7, which matches the first term, so the rule is 3n + 4. Using the first term as the constant without adjusting it gives 3n + 7, using a difference of 1 instead of 3 gives n + 6, and testing n = 1 with the wrong constant gives 3n + 3, which produces 6 rather than 7.

A savings sequence begins £4, £9, £14, £19, … increasing by the same amount each week. What is the 20th term?

• A£95
• B£100
• C£94
• D£99✓

Answer: The common difference is 5, so the nth term is 5n − 1, and the 20th term is 5 × 20 − 1 = £99. Forgetting to subtract 1 at the end gives £100. Using the wrong constant of −5 gives 5 × 20 − 5 = £95, and multiplying by 19 instead of 20 before subtracting 1 gives £94.

Which of these sequences is geometric?

• A2, 6, 10, 14, …
• B2, 8, 14, 20, …
• C2, 6, 18, 54, …✓
• D2, 4, 6, 8, …

Answer: A geometric sequence has a constant multiplier between terms, and in 2, 6, 18, 54 each term is 3 times the one before, so it is geometric. The sequence 2, 6, 10, 14 increases by adding 4 each time, making it arithmetic rather than geometric. The sequences 2, 4, 6, 8 and 2, 8, 14, 20 also increase by a constant addition, 2 and 6 respectively, so both are arithmetic too.