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More questions on this topic: IGCSE Algebra: Simplifying Basic Expressions (110)
13
| Answer | Mark | Mark scheme |
|---|---|---|
| \(n^3\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| 6 | M1 | for isolating \(k\) terms, eg \(5k = 26 + 4\) or \(5k = 30\) or \(k - \dfrac{4}{5} = \dfrac{26}{5}\) oe or \(26 + 4 \div 5\) |
| A1 | cao |
For M mark step must be carried out not just intention shown.
For example, if you see
\(5k - 4 = 26\)
\(+4 \qquad +4\)
award M1 for \(5k = r\) where \(r \gt 26\)
2 Simplify \(2 \times 3n\) (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(6n\) | B1 |
Allow \(n6\) but not \(6 \times n\) or \(6.n\)
1 Simplify \(e + e + e + e + e\) (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(5e\) | B1 | cao |
11
| Answer | Mark | Mark scheme |
|---|---|---|
| \(6xy\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(5d - 3e\) | M1 | for \(5d\) or \(-3e\) |
| A1 | for \(5d - 3e\) |
An answer of \(5d + -3e\) scores M1 A0
16
In a competition, a player gets
5 points for each game they win
2 points for each game they draw
0 points for each game they lose.
Amy wins \(x\) games and draws \(y\) games.
| Answer | Mark | Mark scheme |
|---|---|---|
| \(m^4\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(5x + 2y\) | M1 | for \(5x\) or \(2y\) or a linear expression in the form \(ax + by\) where \(a, b \gt 0\) |
| A1 | for \(5x + 2y\) oe |
Do not award M1 for \(-5x\) or \(-2y\)
13
\(T = x + 2y\)
\(x = 3\) and \(y = -4\)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(6cd\) | B1 | for \(6cd\) or \(6dc\) or \(cd6\) oe |
Inclusion of \(\times\) scores B0, eg \(6 \times cd\) oe
| Answer | Mark | Mark scheme |
|---|---|---|
| \(-5\) | M1 | for \(3 + 2 \times -4\ (= 3 - 8)\) |
| A1 | cao |
5 Complete the statement below to make it correct.
\[\ldots\ldots\ldots \times m = 2m\] (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| 2 | B1 | cao |
4 Simplify \(\quad 7a + a - 5a\) (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(3a\) | B1 | for \(3a\) |
Allow \(a3\)
13
| Answer | Mark | Mark scheme |
|---|---|---|
| \(20h^3\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(7y\) | B1 | cao |
3 Simplify \(\quad 3 \times a \times 4\) (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(12a\) | B1 |
13
| Answer | Mark | Mark scheme |
|---|---|---|
| \(5a\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(19 - 2b + 5c\) | M1 | for \(-2b\) or \(5c\) |
| A1 | for \(19 - 2b + 5c\) |
Accept the correct 3 terms in any order
The A mark is lost for any incorrect subsequent working, eg. \(17b + 5c\)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(2(4d - 3)\) | B1 | for \(2(4d - 3)\) oe |
Accept \((4d - 3)2\)
or \(2 \times (4d - 3)\) or \((4d - 3) \times 2\)
Condone missing final bracket,
eg \(2(4d - 3\)
11 Write \(\dfrac{(6x^5y^3)^2}{3x^2y^7 \times 4xy^{-3}}\) in the form \(ax^by^c\) where \(a\), \(b\) and \(c\) are integers. (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(3x^7y^2\) | M1 | for full evaluation of numerator or denominator with at least 2 of 3 terms correct in a product, eg \(36x^{10}y^6\) or \(12x^3y^4\) or full evaluation of \(\dfrac{6x^5y^3}{3x^2y^7}\) or \(\dfrac{6x^5y^3}{4xy^{-3}}\) with at least 2 of 3 terms correct in a product, eg \(2x^3y^{-4}\) or \(1.5x^4y^6\) |
| M1 | for correct evaluation of numerator and denominator, eg \(36x^{10}y^6\) and \(12x^3y^4\) or for full evaluation of numerator and denominator with no more than one error and a final answer of the form \(ax^by^c\) with two of \(a\), \(b\) and \(c\) correct or for correct evaluation of \(\dfrac{6x^5y^3}{3x^2y^7}\) and \(\dfrac{6x^5y^3}{4xy^{-3}}\) eg \(2x^3y^{-4}\) and \(1.5x^4y^6\) or for full evaluation of \(\dfrac{6x^5y^3}{3x^2y^7}\) and \(\dfrac{6x^5y^3}{4xy^{-3}}\) with no more than one error and a final answer of the form \(ax^by^c\) with two of \(a\), \(b\) and \(c\) correct | |
| A1 | for \(3x^7y^2\) oe |
Accept \(a = 3\), \(b = 7\), \(c = 2\)
4 Simplify \(\quad 3 \times 4t\) (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(12t\) | B1 | for \(12t\) |
3 Simplify \(\quad m + m + m + m\) (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(4m\) | B1 |
14
There are \(x\) sweets in a box.
There are \(y\) sweets in a packet.
| Answer | Mark | Mark scheme |
|---|---|---|
| \(7c + 6d\) | M1 | for \(7c\) or \(6d\) |
| A1 | for \(7c + 6d\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| 7 | M1 | for correct method to expand, eg \(5 \times 2m - 5 \times 6\), or divides both sides by 5 as a first step. |
| M1 | for correct method to isolate terms in \(m\), eg \(10m - 30 + 30 = 40 + 30\) | |
| A1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(3x + 2y\) | M1 | for \(3x\) or \(2y\) |
| A1 | cao |
Condone use of \(b\) and \(p\)
5 Simplify \(\quad 3 \times w \times 5 \times t\) (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(15tw\) | B1 | for \(15tw\) |
May be seen in different order
4 Simplify \(2m \times 3\) (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(6m\) | B1 | for \(6m\) |
2 Simplify \(e + e + e + e\) (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(4e\) | B1 | for \(4e\) oe |
\(e^4\) gets no marks, where the 4 is clearly a power
6
| Answer | Mark | Mark scheme |
|---|---|---|
| \(4ab\) | B1 |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(3x + 8\) | M1 | for method to collect terms eg \(3x\) or 8 |
| A1 | for \(3x + 8\) |
May be seen in working. Accept if no ambiguity.
4
| Answer | Mark | Mark scheme |
|---|---|---|
| \(4m\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(3p\) | B1 | cao |
18 Ben is \(n\) years old.
Chloe is twice as old as Ben.
Dan is five years younger than Ben.
The total of Ben’s age, Chloe’s age and Dan’s age is \(T\) years.
| \(3h + 2 = 14\) | |
| \(3a + 4b - 2c\) | |
| \(A = \pi r^2\) | |
| \(5m - 3m = 2m\) | |
| \(x + 7 \leqslant 12\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(T = 4n - 5\) | M1 | for \(2n\) or \(n - 5\) or \(T\) = a linear expression in \(n\) |
| M1 | for \(n + 2n + n - 5\) oe OR for \(T\) = an expression in \(n\) with 2 of 3 ages correct eg \(T = n + n^2 + n - 5\) | |
| A1 | for \(T = 4n - 5\) oe eg \(T = n + 2n + n - 5\) |
Allow variables other than \(n\)
Each age must be an expression in \(n\)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(5m - 3m = 2m\) | B1 | for \(5m - 3m = 2m\) indicated |
13
| Answer | Mark | Mark scheme |
|---|---|---|
| \(10ab\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(8x + y\) | M1 | for \(8x\) or \(y\) |
| A1 | for \(8x + y\) |
Accept \(1y\) for 1 or 2 marks
7
| Answer | Mark | Mark scheme |
|---|---|---|
| \(p^{10}\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(2x^4y^2\) | M1 | for any two of \(12 \div 6\ (= 2)\), \(x^{7-3}\ (= x^4)\), \(y^{3-1}\ (= y^2)\) in a single product or written as a fraction with complete and correct cancelling of at least two terms |
| A1 | cao |
9 Simplify \(4e + 6f + 7e - f\) (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(11e + 5f\) | M1 | for either \(11e\) or \(5f\) |
| A1 | for \(11e + 5f\) |
8
| Answer | Mark | Mark scheme |
|---|---|---|
| \(7ab\) | B1 | for \(7ab\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(y^3\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(\dfrac{e}{f}\) | M1 | for a correct first step, eg. numerator of \(e^3 \times f\) or denominator of \(e^2 \times f^2\) OR \(e \div f\) or \(e \times f^{-1}\) OR relevant crossings out for all the \(e\)’s and all the \(f\)’s |
| A1 | for \(\frac{e}{f}\) or \(ef^{-1}\) |
7
| Answer | Mark | Mark scheme |
|---|---|---|
| \(4m\) | B1 | cao |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(8np\) | B1 | cao |
6
| Answer | Mark | Mark scheme |
|---|---|---|
| \(12t\) | B1 | \(12t\) |
Accept \(t12\) but not \(12 \times t\) or \(t \times 12\)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(7a\) | B1 | \(7a\) |
Accept \(a7\) or \(7 \times a\) or \(a \times 7\)
Partial simplification of \(5a + 2a\) or \(8a - a\) does NOT get the mark
17
| expression | equation | formula | identity |
| inequality | term | factor | multiple |
| Answer | Mark | Notes |
|---|---|---|
| \(4(m + 3)\) | B1 | for \(4(m + 3)\) or \(2(2m + 6)\) |
| Answer | Mark | Notes |
|---|---|---|
| term, expression | B1 | for ‘term’ in the 1st space |
| B1 | for ‘expression’ in the 2nd space |
6 Here are five straight rods.
All measurements are in centimetres.
The total length of the five rods is \(L\) cm.
Find a formula for \(L\) in terms of \(a\).
Write your formula as simply as possible. (3)
| Answer | Mark | Notes |
|---|---|---|
| \(L = 5a + 3\) | M1 | for expression \(a - 1 + a + a + a + a + 4\) or \(L =\) an expression in \(a\) |
| M1 | for \(5a + 3\) or \(L = a + a + a - 1 + a + a + 4\) oe | |
| A1 | for \(L = 5a + 3\) |
3
| Answer | Mark | Notes |
|---|---|---|
| \(15fg\) | B1 | cao |
| Answer | Mark | Notes |
|---|---|---|
| \(t^2\) | B1 | cao |
| Answer | Mark | Notes |
|---|---|---|
| \(4n\) | B1 | cao |
2 Simplify \(y + 3y - 2y\) (1)
| Answer | Mark | Notes |
|---|---|---|
| \(2y\) | B1 | for \(2y\) |
3
| Answer | Mark | Notes |
|---|---|---|
| \(56ef\) | B1 | cao |
| Answer | Mark | Notes |
|---|---|---|
| 12.5 | B1 | oe |
1
| Answer | Mark | Notes |
|---|---|---|
| \(3p\) | B1 | cao |
| Answer | Mark | Notes |
|---|---|---|
| \(2m^3\) | B1 | cao |
| Answer | Mark | Notes |
|---|---|---|
| \(10 - 4c + 6d\) | M1 | for \(-4c\) or \(6d\) (accept \(+{-4c}\)) |
| A1 | for \(10 - 4c + 6d\) |
16
| Answer | Mark | Notes |
|---|---|---|
| \(7x\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(6y\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(8e - 2e^2\) | 2 | B2 or any correct two term factorised form (B1 for \(8e\) or \(-2e^2\)) |
8
| Answer | Mark | Notes |
|---|---|---|
| \(6ab - 7g\) | 2 | M1 for \(6ab\) or \(-7g\) A1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(3(2m - 3)\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(t^5\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(2xy(x + 2y)\) | 2 | B2 cao (B1 for \(xy(2x + 4y)\) or \(2x(xy + 2y^2)\) or \(2y(x^2 + 2xy)\) or \(2xy(\ldots)\) where \((\ldots)\) has two terms one of which is correct) |
| Working | Answer | Mark | Notes |
|---|---|---|---|
| \(w^2 - 5w - 5w + 25\) | \(w^2 - 10w + 25\) | 2 | M1 for all 4 terms (and no additional terms) correct with or without signs or 3 out of no more than 4 terms correct with signs A1 cao |
11
\(k = 5\)
\(m = 3\)
| Answer | Mark | Notes |
|---|---|---|
| \(3h\) | 1 | B1 |
| Answer | Mark | Notes |
|---|---|---|
| \(3pr\) | 1 | B1 |
| Answer | Mark | Notes |
|---|---|---|
| \(2x + 7y\) | 2 | M1 for \(2x\) or \(7y\) A1 for \(2x + 7y\) |
| Answer | Mark | Notes |
|---|---|---|
| 22 | 2 | M1 for correct substitution e.g. \(2 \times 5 + 4 \times 3\) A1 cao |
14
| Answer | Mark | Notes |
|---|---|---|
| \(5m\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(4pr\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(7x + 3y\) | 2 | B2 for \(7x + 3y\) oe (B1 for \(7x\) or \(3y\) seen) |
2
| Answer | Mark | Notes |
|---|---|---|
| \(30abc\) | 1 | B1 |
| Answer | Mark | Notes |
|---|---|---|
| \(3(y + 2)\) | 1 | B1 |
| Answer | Mark | Notes |
|---|---|---|
| \(x^2 - 3x\) | 1 | B1 |
10
| Answer | Mark | Notes |
|---|---|---|
| \(3t\) | 1 | B1 for \(3 \times t\) or \(3t\) |
| Answer | Mark | Notes |
|---|---|---|
| \(5ef\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(4x + 6y\) | 2 | M1 for \(4x\) or \(6y\) A1 cao |
8
| Answer | Mark | Notes |
|---|---|---|
| \(2g - 3h\) | 2 | M1 for \(2g\) or \(-3h\) A1 for \(2g - 3h\) or \(-3h + 2g\) |
| Answer | Mark | Notes |
|---|---|---|
| \(y(y - 2)\) | 1 | B1 |
| Answer | Mark | Notes |
|---|---|---|
| \(p^5\) | 2 | M1 for \(\dfrac{p^{3+4}}{p^2}\) \(\left(= \dfrac{p^7}{p^2}\right)\) or \(p^{3-2} \times p^4\) \((= p^1 \times p^4)\) or \(p^3 \times p^{4-2}\) \((= p^3 \times p^2)\) A1 cao |
5
| Answer | Mark | Notes |
|---|---|---|
| \(5e\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(7gh\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(a + 6d\) | 2 | M1 for \(a\) or \(6d\) A1 cao |
6
| Answer | Mark | Notes |
|---|---|---|
| \(3m\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(7e\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(15g\) | 1 | B1 cao |
3 \(f = 3g + 7h\)
| Answer | Mark | Notes |
|---|---|---|
| \(-1\) | 2 | M1 for \(3 \times -5 + 7 \times 2\) A1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(3(x + 2)\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(7y - 16\) | 2 | M1 for intention to expand a bracket eg \(5y - 10\) or \(2y - 6\) A1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(m^8\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(p^4\) | 1 | B1 cao |
12
\(a \times c \times 3\)
(1)\(p \times p \times p\)
(1)\(5x - 4y + 3x - 3y\)
(2)| Answer | Mark | Notes |
|---|---|---|
| \(3ac\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(p^3\) | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| \(8x - 7y\) | 2 | M1 for \(8x\) or \(\pm 7y\) A1 cao |
11
\(b + b + b + b\)
(1)\(8n - 3n\)
(1)\(3 \times c \times d\)
(1)\(3x + 7y + 2x - y\)
(2)| Answer | Mark | Notes |
|---|---|---|
| \(4b\) | 1 | B1 for \(4b\) oe |
| Answer | Mark | Notes |
|---|---|---|
| \(5n\) | 1 | B1 for \(5n\) cao |
| Answer | Mark | Notes |
|---|---|---|
| \(3cd\) | 1 | B1 for \(3cd\) oe algebraic simplified form |
| Answer | Mark | Notes |
|---|---|---|
| \(5x + 6y\) | 2 | M1 \(5x\) or \(6y\) A1 cao |
4
| Answer | Mark | Notes |
|---|---|---|
| \(5m\) | 1 | B1 for \(5m\) or \(5 \times m\) or \(m \times 5\) or \(m5\) |
| Answer | Mark | Notes |
|---|---|---|
| \(9p\) | 1 | B1 for \(9p\) or \(p9\) |
| Answer | Mark | Notes |
|---|---|---|
| \(4tw\) | 1 | B1 for \(4tw\) or \(4wt\) or \(tw4\) or \(wt4\) (condone \(4 \times tw\) and \(4 \times wt\)) |
9
| Answer | Mark | Notes |
|---|---|---|
| \(4a\) | 1 | B1 for \(4a\) oe as a single term |
| Answer | Mark | Notes |
|---|---|---|
| \(3cd\) | 1 | B1 for \(3cd\) oe as a single term |
| Answer | Mark | Notes |
|---|---|---|
| \(7ef\) | 1 | B1 for \(7ef\) oe as a single term |
| Answer | Mark | Notes |
|---|---|---|
| 3 | 1 | B1 cao |
| Answer | Mark | Notes |
|---|---|---|
| 2 | 2 | M1 for intention to subtract 7 from each side or divide each term by 5 or embedded method A1 cao |
7
| Answer | Mark | Notes |
|---|---|---|
| \(6x - 3y\) | 2 | M1 for an attempt to combine terms in \(x\) or terms in \(y\) correctly eg \(5x + x\) (\(= 6x\)), \(4y - 7y\) (\(= -3y\)) A1 for \(6x - 3y\) oe |
| Working | Answer | Mark | Notes |
|---|---|---|---|
| \(7x + 14 = 7\) or \(x + 2 = 1\) \(7x = -7\) | \(x = -1\) | 2 | M1 for correctly expanding the bracket or an attempt to divide both sides by 7 e.g. \(7x + 14\) or \(x + 2 = 7 \div 7\) oe A1 cao |
13
| Answer | Mark | Notes |
|---|---|---|
| \(3f\) | 1 | B1 for \(3f\) or \(f3\) or \(3 \times f\) or \(f \times 3\) |
| Answer | Mark | Notes |
|---|---|---|
| \(6m\) | 1 | B1 for \(6m\) or \(m6\) |
| Answer | Mark | Notes |
|---|---|---|
| \(4a + 5h\) | 2 | B2 for \(4a + 5h\) or \(5h + 4a\) (B1 for \(4a\) or \(5h\) or \(4a + 5h = 9ah\)) |
6
| Answer | Mark | Notes |
|---|---|---|
| \(3c\) | 1 | B1 \(3c\) oe |
| Answer | Mark | Notes |
|---|---|---|
| \(6ef\) | 1 | B1 \(6ef\) oe |
| Answer | Mark | Notes |
|---|---|---|
| \(7p + 5t\) | 2 | B2 for \(7p + 5t\) (B1 for either \(7p\) or \(5t\)) |
14 Match each description to the correct expression.
One has been done for you. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| All 4 correct matches \(2n\) \(n^2\) \(\dfrac{n}{2}\) \(n - 2\) | B4 | B1 for each correct match |
Accept any unambiguous indication
More than one line from a box on the left is incorrect for that box
11 Here are two expressions, A and B.
| A | B |
|---|---|
| \(7(x - 2) + 4x + 6\) | \(5(3x + 2) - 4x - 18\) |
Show that A and B are equivalent. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: expands brackets and simplifies | ||
| \(7x - 14\) or \(15x + 10\) | M1 | |
| \(7x - 14\ (+ 4x + 6)\) and \(11x - 8\) or \(15x + 10\ (- 4x - 18)\) and \(11x - 8\) | M1dep | A or B fully correct |
| \(7x - 14\ (+ 4x + 6)\) and \(11x - 8\) and \(15x + 10\ (- 4x - 18)\) and \(11x - 8\) | A1 | A and B fully correct SC1 \(11x - 8\) for one or both of A and B with no working seen |
| Alternative method 2: shows the \(x\) terms and constant terms are the same | ||
| Any one of \(7x + 4x = 11x\) or \(15x - 4x = 11x\) or \(-14 + 6 = -8\) or \(10 - 18 = -8\) | M1 | |
| Any two of \(7x + 4x = 11x\) or \(15x - 4x = 11x\) or \(-14 + 6 = -8\) or \(10 - 18 = -8\) | M1dep | |
| \(7x + 4x = 11x\) and \(15x - 4x = 11x\) and \(-14 + 6 = -8\) and \(10 - 18 = -8\) | A1 | SC1 \(11x - 8\) for one or both of A and B with no working seen |
| M1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| Use the scheme that favours the student | |
| Alt 2: \(7x + 4x = 11x\) and \(-14 + 6 = -8\) | M1M1 |
| Alt 2: \(7x + 4x = 11x\) and \(15x - 4x = 11x\) | M1M1 |
| Alt 2: \(11x\) only | M0 |
| Alt 2: \(-8\) only | M0 |
| Do not ignore subsequent incorrect work such as \(11x - 8 = 3\), which may be awarded up to M2 if working shown | |
| Substituting values into the two expressions | M0 |
4
| Answer | Mark | Comments |
|---|---|---|
| \(ab\) or \(ba\) | B1 |
Allow upper case A and/or B
Do not accept \(a \times b\) or \(b \times a\) or \(a^b\) or \(b^a\)
| Answer | Mark | Comments |
|---|---|---|
| \(3c\) | B1 |
Do not accept \(c3\) or \(3 \times c\) or \(c \times 3\)
| Answer | Mark | Comments |
|---|---|---|
| \(p^3\) | B1 |
Do not accept \(p3\)
| Answer | Mark | Comments |
|---|---|---|
| 1 | B1 |
Do not accept \(\dfrac{1}{1}\)
25 Here are three terms.
\(xy\) \(x^2\) \(5y^2\)
Alec multiplies two of these terms.
Work out the three possible fully simplified answers. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(x^3y\) or \(yx^3\) | B1 | |
| \(5xy^3\) or \(5y^3x\) | B1 | |
| \(5x^2y^2\) or \(5y^2x^2\) | B1 |
Mark the answer lines unless blank
Do not allow transcription errors
17 Match the algebra to the correct description.
One has been done for you. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| All correct | B3 | B2 for 2 or 3 correct B1 for 1 correct |
Two or more lines from one box on the left is choice so incorrect for that box
Accept any unambiguous indication
11 Here are three terms.
\(xy\) \(x^2\) \(5y^2\)
Alec multiplies two of these terms.
Work out the three possible fully simplified answers. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(x^3y\) or \(yx^3\) | B1 | |
| \(5xy^3\) or \(5y^3x\) | B1 | |
| \(5x^2y^2\) or \(5y^2x^2\) | B1 |
Mark the answer lines unless blank
Do not allow transcription errors
10
| Answer | Mark | Comments |
|---|---|---|
| \(6m + 11\) or \(11 + 6m\) | B2 | B1 \(6m\) or \((+)11\) |
| Do not ignore further work for B2 eg \(6m + 11 = 17m\) eg \(6m + 3 = 9m\) | B1 B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(3cd\) or \(3dc\) | B2 | B1 3 or \(cd\) or \(dc\) |
| \(cd3\) | B1 |
| Use of multiplication signs is max B1 eg \(3 \times cd\) eg \(c \times d\) | B1 B0 |
| \(6\dfrac{1}{2}cd\) | B1 |
| \(\dfrac{1}{2}c6d\) | B0 |
| \(\dfrac{1}{2}c3d\) | B0 |
2 Simplify fully \(\quad y + y + y\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(3y\) | B1 |
| \(3 \times y\) or \(y \times 3\) | B0 |
| \(y3\) | B0 |
23
Work out the next term. [1 mark]
Show that the third term is \(\quad 26 + 15\sqrt{3}\) [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{25}{16}\) or \(1\dfrac{9}{16}\) | B1 | oe with no surds or indices |
| Ignore an incorrect conversion of \(\dfrac{25}{16}\) to a mixed number | |
| \(\dfrac{5\sqrt{5}\sqrt{5}}{16}\) or \(\dfrac{5^2}{16}\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(4 + 2\sqrt{3} + 2\sqrt{3} + (\sqrt{3})^2\) or \(4 + 4\sqrt{3} + (\sqrt{3})^2\) or \(7 + 4\sqrt{3}\) | M1 | oe 4 terms with at least 3 correct or 3 terms with 2 correct including \(4\sqrt{3}\) terms may be seen in a grid |
| \(7 \times 2 + 7\sqrt{3} + 2 \times 4\sqrt{3} + 4\sqrt{3} \times \sqrt{3}\) or \(8 + 8\sqrt{3} + 6 + 4\sqrt{3} + 4 \times 3 + 3\sqrt{3}\) or \(14 + 7\sqrt{3} + 8\sqrt{3} + 12\) or \(8 + 4\sqrt{3} + 4\sqrt{3} + 6 + 4\sqrt{3} + 6 + 6 + 3\sqrt{3}\) | M1dep | oe full expansion with correct multiplication of their 2, 3 or 4 terms by \((2 + \sqrt{3})\) terms may be seen in a grid |
| \(8 + 4\sqrt{3} + 4\sqrt{3} + 6 + 4\sqrt{3} + 6 + 6 + 3\sqrt{3}\) and \(26 + 15\sqrt{3}\) or \(14 + 7\sqrt{3} + 8\sqrt{3} + 12\) and \(26 + 15\sqrt{3}\) | A1 | oe with full expansion terms may be seen in a grid condone \(15\sqrt{3} + 26\) |
Remember that the answer is given in the question
3 may be seen as \((\sqrt{3})^2\) for M1 only
Condone missing brackets if multiplications are correct
20
\[5x^3 + ax^2 + bx + c \equiv kx^3 + (2 - k)x^2 + (a^2 - 1)x + \dfrac{b}{2}\]Work out the values of \(a\), \(b\) and \(c\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(a = -3\) | B1 | |
| 8 or (their \(-3)^2 - 1\) correctly evaluated | B1ft | |
| 4 or their \(8 \div 2\) correctly evaluated | B1ft |
| \(a = -3 \quad b = -10 \quad c = -5\) | B1B0B1ft |
15 Complete these statements. [3 marks]
\(\ldots\ldots\ldots\) \(+\; 5x \;=\; 9x\)
\(y \;\times\) \(\ldots\ldots\ldots\) \(=\; y^2\)
\(\ldots\ldots\ldots\) \(-\; 2t \;=\; t\)
| Answer | Mark | Comments |
|---|---|---|
| \(4x\) | B1 | oe |
| \(y\) | B1 | oe |
| \(3t\) | B1 | oe |
4 \(3(x - 1) \equiv 3x - 3 \quad\) is an identity.
Tick one box. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| It is true for all values of \(x\) | B1 |
2
| Answer | Mark | Comments |
|---|---|---|
| \(5x\) | B1 |
| \(5 \times x\) or \(x \times 5\) or \(x5\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(10w\) | B1 |
| \(10 \times w\) or \(w \times 10\) or \(w10\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| 2 | B1 |
| \(\dfrac{2}{1}\) or \(2 \div 1\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(y^3\) | B1 |
| \(y^2 \times y\) or \(y \times y^2\) | B0 |
28 Here is the term-to-term rule for a sequence.
| Double the previous term and add 3 |
The first three terms of the sequence are \(\quad a + 1 \quad 2a + 5 \quad 4a + 13\)
Show that the sum of the first four terms is a multiple of 3 [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(8a + 29\) | B1 | oe eg \(2(4a + 13) + 3\) |
| \(15a + 48\) | B1ft | correct or ft B0 only their \(8a + 29\) must be in the form \(na + c\) where \(n \ne 0\) and \(c \ne 0\) implied by \(3(5a + 16)\) |
| \(3(5a + 16)\) or \(15 = 5 \times 3\) and \(48 = 16 \times 3\) | B1 | oe eg \(5a + 16\) so it divides by 3 |
| Ignore use of substitution as an attempt to show divisibility | |
| Ignore further non-contradictory statements | |
| Further simplification eg \(15a + 48 = 63\) which is \(21 \times 3\) | B1B1B0 |
| For the 1st B1 accept \(8a + 29\) embedded in a calculation for the sum of the first four terms eg \(a + 1 + 2a + 5 + 4a + 13 + 8a + 29\) | |
| For the 2nd B1 accept \(15a + 48\) embedded in a calculation to show divisibility eg \(\dfrac{15a + 48}{3} = 5a + 16\) | |
| For the 3rd B1 accept 15 is a multiple of 3 and 48 is a multiple of 3 | |
| \(8a + 29\) \(a + 2a + 4a + 8a = 15a \quad 1 + 5 + 13 + 29 = 48\) but \(15a + 48\) not seen \(15 = 5 \times 3\) and \(48 = 16 \times 3\) | B1 B0 B1 |
11
| Answer | Mark | Comments |
|---|---|---|
| \(10x + 4y - 6\) or \(2(5x + 2y - 3)\) | B3 | any order B2 two terms correct B1 one term correct |
| B1 may be awarded for correct work, with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| Further incorrect work after a B3 response is B2 eg1 \(10x + 4y - 6 = 8xy\) eg2 \(10x + 4y - 6\) and \(10x = -10\) eg3 \(10x + 4y - 6\) and \(2(5x + 2y - 6)\) | B2 |
| Further incorrect work after a B2 or B1 response is B1 eg1 \(10x + 4y + 6 = 20xy\) eg2 \(10x - 4y + 6\) and \(10x = -2\) eg3 \(10x + 4y + 6\) and \(2(5x + 2y + 6)\) | B1 |
| \(10x + 4y + 6\) and \(2(5x + 2y + 3)\) | B2 |
| \(10x\) and \(4y\) and \(-6\) | B2 |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{a^2}{2}\) | B1 | accept any indication |
9 Here is the term-to-term rule for a sequence.
| Double the previous term and add 3 |
The first three terms of the sequence are \(\quad a + 1 \quad 2a + 5 \quad 4a + 13\)
Show that the sum of the first four terms is a multiple of 3 [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(8a + 29\) | B1 | oe eg \(2(4a + 13) + 3\) |
| \(15a + 48\) | B1ft | correct or ft B0 only their \(8a + 29\) must be in the form \(na + c\) where \(n \ne 0\) and \(c \ne 0\) implied by \(3(5a + 16)\) |
| \(3(5a + 16)\) or \(15 = 5 \times 3\) and \(48 = 16 \times 3\) | B1 | oe eg \(5a + 16\) so it divides by 3 |
| Ignore use of substitution as an attempt to show divisibility | |
| Ignore further non-contradictory statements | |
| Further simplification eg \(15a + 48 = 63\) which is \(21 \times 3\) | B1B1B0 |
| For the 1st B1 accept \(8a + 29\) embedded in a calculation for the sum of the first four terms eg \(a + 1 + 2a + 5 + 4a + 13 + 8a + 29\) | |
| For the 2nd B1 accept \(15a + 48\) embedded in a calculation to show divisibility eg \(\dfrac{15a + 48}{3} = 5a + 16\) | |
| For the 3rd B1 accept 15 is a multiple of 3 and 48 is a multiple of 3 | |
| \(8a + 29\) \(a + 2a + 4a + 8a = 15a \quad 1 + 5 + 13 + 29 = 48\) but \(15a + 48\) not seen \(15 = 5 \times 3\) and \(48 = 16 \times 3\) | B1 B0 B1 |
4 Simplify \(\quad a + 3a \div a\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(a + 3\) or \(3 + a\) | B1 |
22 Expand and simplify fully \(\quad 5(3x + 4) - 2(x - 1)\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(13x + 22\) | B2 | B1 \(15x + 20\) or \(-2x + 2\) or \(13x + a\) or \(bx + 22\), where \(a\) and \(b\) can be any numbers |
| Do not ignore further working for B2 eg \(13x + 22 = 35x\) eg \(13x + 22,\ x = \dfrac{22}{13}\) | B1 B1 |
10
| Answer | Mark | Comments |
|---|---|---|
| \(p^3\) | B1 |
Accept \(1p^3\)
| Answer | Mark | Comments |
|---|---|---|
| \(2a + 11c\) | B2 | either order B1 \(2a\) or \(11c\) |
| Further incorrect work after a B2 response is B1 eg \(2a + 11c = 13ac\) | B1 |
| Further incorrect work after a B1 response is B1 eg \(3a + 11c = 14ac\) | B1 |
| \(a2 + 11c\) or \(2a + c11\) | B1 |
| \(a2\) or \(c11\) | B1 |
8
\[(a - 3)x^2 + 2b \equiv 5x^2 + 12\]Work out the values of \(a\) and \(b\). [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(a = 8\) and \(b = 6\) | B2 | B1 \(a - 3 = 5\) or \(a = 3 + 5\) or \(a = 8\) or \(2b = 12\) or \(b = 12 \div 2\) or \(b = 6\) SC1 \(a = 6\) and \(b = 8\) |
| Ignore working if B2 or B1 or SC1 seen | |
| \((a - 3)x^2 = 5x^2\) with no further correct work | B0 |
| For B1 do not allow embedded values eg \(2 \times 6 = 12\) | B0 |
2 Match each expression on the left with the simplified expression on the right.
One has been done for you. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| All 4 correct | B4 | B1 for each box correctly matched |
Connections do not have to be straight lines
The line from the second box on the left was given so do not count it
Two or more lines from one box on the left is choice so incorrect for that box
13 Emma tries to simplify \(\quad cd \times 2\)
Here is her method.
| \(c \times 2 = 2c\) \(d \times 2 = 2d\) \(2c \times 2d = 4cd\) |
What is wrong with her method? [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| Valid explanation referencing the multiplication by 2 twice | B1 | eg she has multiplied by 2 twice |
| She multiplied 2 by 2 but there was only one 2 to start with | B1 |
| \(2 \times 2\) should not be calculated | B1 |
| She doubled everything | B1 |
| There should only be one 2 | B1 |
| There should be a 2 | B0 |
| She’s adding up the 2s, whereas it should be \(cd \times 2 = 2cd\) | B0 |
| She multiplied by 4 (instead of 2) | B1 |
| She has 4 instead of 2 | B0 |
| The 4 is wrong | B0 |
| She should not have both \(2c\) and \(2d\) | B1 |
| She has multiplied (each of) \(c\) and \(d\) by 2 separately | B1 |
| She has multiplied (each of) \(c\) and \(d\) separately | B0 |
| She did \(2c \times 2d\) | B0 |
| Answer is \(2c + 2d\) | B0 |
| She shouldn’t separate the \(c\) and \(d\), it’s just \(2c\) | B0 |
| You don’t times each letter by 2 | B1 |
| She has multiplied each letter by 2 | B1 |
| She has multiplied each letter by 2, it should be \(2cd^2\) | B0 |
| It is \(c \times d \times 2\) not \(2c \times 2d\) | B1 |
| She shouldn’t do all that it is just \(cd \times 2 = 2cd\) | B0 |
| Answer is \(2cd\) | B0 |
| Her answer is wrong | B0 |
5 Simplify fully \(\quad 8a + 5b + 6a - 2b\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(14a + 3b\) or \(3b + 14a\) | B2 | B1 for \(14a\) or \((+)3b\) |
| \(14a + 3b\) followed by further work eg \(17ab\) | B1 |
| B1 response followed by further work eg \(2a + 3b = 5ab\) | B1 |
5
| Answer | Mark | Comments |
|---|---|---|
| \(d^2\) | B1 |
| Allow \(D^2\) | B1 |
| \(dd = d^2\) | B1 |
| \(dd\) | B0 |
| \(1d^2\) | B0 |
| \(d2\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| 1 or \(n^0\) | B1 |
| \(\dfrac{n}{n} = 1\) or \(\dfrac{n}{n} = n^0\) | B1 |
| \(\dfrac{n}{n}\) | B0 |
| \(\dfrac{1}{1}\) or \(1 \div 1\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(2t\) | B1 |
| Allow 2T | B1 |
| \(2 \times t = 2t\) | B1 |
| \(2 \times t\) | B0 |
| \(2^t\) | B0 |
| \(\dfrac{2t}{1}\) or \(\dfrac{2}{1}t\) | B0 |
4 Circle the correct statement. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(3x \equiv x + 2x\) | B1 |
18
\[12x^3 + 7x^2 + 3x - 10 \equiv 2(ax^3 + x^2 + 2x - 5) + x(bx + c)\]Work out the values of \(a\), \(b\) and \(c\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(2ax^3 + 2x^2 + 4x - 10\) and \(bx^2 + cx\) or \(2ax^3 + (2 + b)x^2 + (4 + c)x - 10\) or \(2a = 12\) or \(a = 6\) | M1 | oe correct expansions allow \(2ax^3 = 12x^3\) |
| \(2 + b = 7\) or \(b = 5\) or \(4 + c = 3\) or \(c = -1\) | M1 | oe equation |
| \(a = 6\) and \(b = 5\) and \(c = -1\) | A1 | SC2 any two of \(a = 6\) and \(b = 5\) and \(c = -1\) |
| One correct value without working only scores one mark eg \(a = 6\) and \(b = -5\) and \(c = 1\) with no working eg \(a = -6\) and \(b = 5\) and \(c = 1\) with no working | M1M0A0 M0M1A0 |
| Condone an incorrect \(-10\) in the expansion if A1 is awarded | M1M1A1 |
14 Match each expression on the left with one on the right.
One has been done for you. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| All 3 correct matches | B3 | B1 each correct match |
| Two different matches from left hand column is choice for that box | |
| Allow any unambiguous indication | |
 | B3 |
12
| \(8a\) | \(a^2 - b\) |
When \(\quad a = 25 \quad\) the expressions have the same value.
Work out the value of \(b\). [3 marks]
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(3x + 2y\) | B2 | either order B1 \(3x\) or \(2y\) |
| Further incorrect work after a B2 response is B1 | |
| eg \(3x + 2y\) followed by \(5xy\) | B1 |
| Further incorrect work after a B1 response is B1 | |
| eg \(15x + 2y\) followed by \(30xy\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(8 \times 25\) or 200 or \(25^2\) or 625 | M1 | oe |
| \(8 \times 25\) or 200 and \(25^2 - b\) or \(625 - b\) or \(25^2 - 8 \times 25\) or \(625 - 200\) | M1dep | oe may be seen in an equation |
| 425 | A1 |
| Embedded answer | M1M1A0 |
| Answer | Mark | Comments |
|---|---|---|
| \(3w + 5\) | B1 |
4 Simplify \(\quad 4 \times 2c\)
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(8c\) | B1 |
2 Here is an expression \(5a + 7b + 9c\)
Which is the second term?
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(7b\) | B1 |
19 Here are a triangle and a rectangle.
Not drawn accurately
\(a\) and \(b\) are positive numbers.
Which shape has the larger perimeter?
You must work out expressions for both perimeters. [3 marks]
Tick a box.
| Answer | Mark | Comments |
|---|---|---|
| \(5a + b + 4a + 7b + 2a + 3b\) or \(2(4a + 2b) + 2(a + 4b)\) | M1 | oe |
| \(11a + 11b\) or \(10a + 12b\) | A1 | oe |
| \(11a + 11b\) and \(10a + 12b\) and cannot tell | A1 | oe with no further incorrect working |
| Condone \(22ab\) after \(11a + 11b\) or \(10a + 12b\) for first A mark only | M1A1A0 |
| \(11a\) and \(11b\) or \(10a\) and \(12b\) implies M1 | M1A0 |
| \(5a + b = 6ab\) and \(4a + 7b = 11ab\) and \(2a + 3b = 5ab\) and \(6ab + 11ab + 5ab\) | M1A0 |
| \(6ab\) next to \(5a + b\) and \(11ab\) next to \(4a + 7b\) and \(5ab\) next to \(2a + 3b\) shown on diagram and \(6ab + 11ab + 5ab\) | M1A0 |
| \(5a + 4a + 2a = 15a\) and \(b + 7b + 3b = 12b\) and \(15a + 12b\) | M1A0 |
3 Circle the expression that means \(\quad\) half the value of \(x\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{x}{2}\) | B1 |
2 Circle the expression that is equal to \(\quad x + x + x - x + x\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(3x\) | B1 |
30 Expand and simplify fully \(\quad 4(2c + 3) - (5c - 1)\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(8c + 12\) or \(-5c + 1\) | M1 | may be seen in a grid implied by \(3c + 12 + 1\) or \(8c + 13 - 5c\) |
| \(3c + 13\) | A1 |
| Do not ignore further working eg \(3c + 13 = 16c\) eg \(3c + 13,\ c = \dfrac{-13}{3}\) | M1A0 M1A0 |
| \(8c + 12 - 5c - 1\) | M1 |
| \(8c + 3 - 5c + 1\) | M1 |
14
| Answer | Mark | Comments |
|---|---|---|
| \(6x = 13 + 11\) or \(6x = 24\) or \(\dfrac{24}{6}\) | M1 | oe eg \(-6x = -13 - 11\) or \(-6x = -24\) or \(\dfrac{-24}{-6}\) |
| 4 | A1 |
| Embedded answer, eg \(6 \times 4 - 11 = 13\) | M1A0 |
| 24 with no other working | M0A0 |
| Flow chart method, if 4 not given as the answer. \(x \to \times 6 \to -11 \to 13\) and \(13 \to +11 \to \div 6 \to x\) | M1A0 |
| Answer | Mark | Comments |
|---|---|---|
| \((2 \times 4a =)\ 8a\) | B1 | |
| \(\left(\dfrac{15a}{3} =\right) 5a\) | B1 | |
| \(13a + 2\) | B1ft | ft B1B0 or B0B1 for their \(8a +\) their \(5a + 9 - 7\) is in the form \(pa + q\) do not award with further incorrect work eg \(13a + 2 = 15a\) |
| \(13a + c\) could come from incorrect working eg \(8a + 4 + 9 + 5a - 7 = 13a + 16\) (their \(8a\) is \(8a + 4\)) eg \(8a + 4 + 9 + 5a - 7 = 13a + 6\) (their \(8a\) is \(8a + 4\)) eg \(8a + 9 + 5a - 7 = 13a + 16\) eg \(13a + 16\) (no other working) | B0B1B0ft B0B1B1ft B1B1B0ft B1B1B0ft |
| \(6a + 9 + 5a - 7 = 11a + 2\) | B0B1B1ft |
| \(8a + 9 + 12a - 7 = 20a + 2\) | B1B0B1ft |
| \(8a + 9 + 5 - 7 = 8a + 7\) | B1B0B1ft |
| \(8a + \dfrac{15a}{3} + 7\) | B1B0B0ft |
| \(6a + 9 + 12a - 7 = 18a + 2\) | B0B0B0ft |
| \(6a + 5a + 16 = 11a + 16\) | B0B1B0ft |
12 Match each expression on the left with one on the right.
One has been done for you. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| All 4 correct matches | B4 | B1 for each correct match |
 | B4 |
| Two different matches from left hand column is choice for that box | |
| Accept any unambiguous indication |
7 Match the algebra to the correct description.
One has been done for you. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Two correct matches | B2 | B1 one correct match |
| Do not accept two lines from an algebra box | |
 | B2 |
1 Which of these is a correct identity?
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(x + 4x \equiv 5x\) | B1 |
19 S and T are rectangles.
S has dimensions \((x + 2)\) and \(x\).
Some of these rectangles make the larger rectangle shown.
Not drawn accurately
Work out an expression for the perimeter of T.
Give your answer in its simplest form. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(4x + 12\) or \(2(2x + 6)\) or \(4(x + 3)\) | B3 | B2 correct expression for half the perimeter of T eg \(x + 2 + x + 2 + (x + 2 - x)\) \(x + 2 + x + 2 + 2\) \(2(x + 2) + (x + 2 - x)\) \(2(x + 2) + 2\) \(2x + 4 + (x + 2 - x)\) \(2x + 4 + 2\) \(2x + 6\) \(2(x + 3)\) or correct expression for the perimeter of T eg \(x + 2 + x + 2 + x + 2 + x + 2 + 2(x + 2 - x)\) \(x + 2 + x + 2 + x + 2 + x + 2 + 2 + 2\) \(2(x + 2 + x + 2) + 2(x + 2 - x)\) \(2(x + 2 + x + 2) + 2 \times 2\) \(2(2x + 4) + 2(x + 2 - x)\) \(2(2x + 4) + 2 \times 2\) \(4x + 8 + 4\) B1 simplified correct expression for the longer side of T \(2(x + 2)\) or \(2x + 4\) seen or simplified correct expression for the two longer sides of T \(4(x + 2)\) or \(2(2x + 4)\) or \(4x + 8\) seen SC1 \(8x + 12\) |
| Ignore further work with an incorrect attempt to factorise after \(4x + 12\) eg \(4x + 12\) and \(2(2x + 12)\) | B3 |
| Ignore further work with an incorrect attempt to expand after \(2(2x + 6)\) or \(4(x + 3)\) eg \(2(2x + 6)\) and \(4x + 6\) | B3 |
| Do not ignore further work with an incorrect attempt to simplify after \(4x + 12\) eg \(4x + 12\) and \(16x\) | B2 |
| Ignore further work with an incorrect attempt to simplify after a correct B2 or B1 expression | |
| Do not accept \(2x + 4\) seen as part of \(x^2 + 2x + 2x + 4\) for B1 | B0 |
17 Match each expression in Column P with the equivalent expression in Column Q.
One has been done for you. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| All 3 correct | B3 | B1 for each box in Column P correctly matched |
| Connections do not have to be straight lines | |
| Two or more lines from a box in Column P is choice so is incorrect for that box | |
 | B3 |
9
| Answer | Mark | Comments |
|---|---|---|
| \(p = m - 2\) or \(p = -2 + m\) | B1 |
| \(m - 2 = p\) or \(-2 + m = p\) | B1 |
| Answer without \(p =\) or \(= p\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(4x^2\) | B1 |
2 Circle the expression that can be written as \(3cd\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(3 \times c \times d\) | B1 |
1 Simplify \(\quad 8a - 3a + a\)
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(6a\) | B1 |
24
Which of these is equal to \(b\)?
Circle your answer. [1 mark]
Which of these is equal to \(d\)?
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(-a\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{1}{c}\) | B1 |
19 You are given that \(\quad 4a - 2b = 10\)
Write an expression in \(a\), \(b\) and \(c\) that is equal to 23
Give your answer in its simplest form.
You must show your working. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| 5 | B1 |
| Condone \(10 - 5 = 5\) | B1 |
| Condone \(x = 5\) | B1 |
| \(\dfrac{10}{2}\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(-10\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Unsimplified expression in \(a\), \(b\) and \(c\) which would evaluate to 23 | M1 | eg \(2(4a - 2b) + a + c\) or \(8a - 4b + a + c\) or \(11(a + c) - (4a - 2b)\) or \(11a + 11c - 4a + 2b\) |
| Simplified expression in \(a\), \(b\) and \(c\) which would evaluate to 23 | A1 | eg \(9a - 4b + c\) \(7a + 2b + 11c\) SC2 Values assigned to \(a\), \(b\) and \(c\) which satisfy original equations and expression given which has value 23 eg \(a = 3, b = 1, c = 0\) and \(7a + 2b + c\) |
| There are infinitely many correct solutions. Allow solutions where the coefficients are not integers if initial working is shown. eg \(3(4a - 2b) - \dfrac{7}{3}(a + c) = \dfrac{29}{3}a - 6b - \dfrac{7}{3}c\) | M1A1 |
| \(5a - 2b + c + 10 = 23\) | M1A1 |
| Condone ‘= 23’ after the expression | |
| Answer using only two variables eg \(2.3(4a - 2b)\) | M0A0 |
19
| Answer | Mark | Comments |
|---|---|---|
| \(2a^2 + 15a - 1\) | B3 | B2 \(2a^2 + 15a\) or \(2a^2 - 1\) or \(15a - 1\) B1 \(2a^2\) or \(15a\) or \(-1\) |
| \(2a + 15a - 1 = 17a - 1\) | B2 |
| \(2a^2 + 15a + -1\) | B2 |
| Do not ignore further incorrect algebraic simplification for B3 \(2a^2 + 15a - 1 = 17a - 1\) | B2 |
| Do not ignore further incorrect algebraic simplification for B2 \(2a + 15a - 1 = 17a - 1 = 16a\) \(2a^2 + 15a - 1 = 17a - 1 = 16a\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(4y(6y - 5)\) or \(-4y(5 - 6y)\) | B2 | B1 \(2y(12y - 10)\) or \(-2y(10 - 12y)\) or \(y(24y - 20)\) or \(-y(20 - 24y)\) or \(4(6y^2 - 5y)\) or \(-4(5y - 6y^2)\) or \(2(12y^2 - 10y)\) or \(-2(10y - 12y^2)\) |
| Ignore any ‘solutions’ seen eg \(4y(6y - 5)\) in working with 0 and \(\dfrac{5}{6}\) on answer line | B2 |
| Condone \(4y \times (6y - 5)\) | B2 |
| Condone \(y \times (24y - 20)\) | B1 |
| \((4y + 0)(6y - 5)\) | B1 |
| Do not ignore further incorrect algebraic simplification for B2 |
4 Circle the expression which means \(\quad x\) divided by \(y\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{x}{y}\) | B1 |
23 Two consecutive whole numbers are \(n\) and \(n + 1\)
Show that the answer must be an odd number. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(-1\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(n^2 + n\) or \(n + n^2\) | B1 |
| Accept \(1n^2 + 1n\) or \(1n^2 + n\) or \(n^2 + 1n\) etc… | B1 |
| Do not accept \(n \times n + n\) or \(n^2 + n1\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \((n + n + 1 =)\ 2n + 1\) and states that \(2n\) is even and states that even + 1 = odd or even + odd = odd | B2 | B1 \((n + n + 1 =)\ 2n + 1\) |
| Alternative method 2 | ||
| States that one of the numbers is even and the other is odd and states that even + odd = odd | B2 | B1 states that one of the numbers is even and the other is odd or states that even + odd = odd |
| Numerical examples with no other explanation | B0 |
| \(n + n + 1 = 2n + 1 = 3n\) | B0 |
3 Simplify \(\quad 16a^2 \div a + 3a \times 2\)
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(22a\) | B1 |
3 Simplify \(\quad 8x - 3 + 6x\)
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(14x - 3\) | B1 |
17 Simplify \(\qquad 7x - (3x - 2x)\)
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(6x\) | B1 |
15 \(3a + b = 7 \qquad\) and \(\qquad 6x + 8y = 40\)
Show that \(\quad 9a + 3b \quad\) has a greater value than \(\quad 3x + 4y\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(3 \times 7\) or 21 or \(40 \div 2\) or 20 | M1 | oe |
| 21 and 20 | A1 | |
| Alternative method 2 – works out and uses correct possible values for \(a\), \(b\), \(x\) and \(y\) | ||
| Substitute values into \(9a + 3b\) that satisfy \(3a + b = 7\) or substitute values into \(3x + 4y\) that satisfy \(6x + 8y = 40\) | M1 | eg \(a = 2\) and \(b = 1\) substituted into \(9a + 3b\) or \(x = 4\) and \(y = 2\) substituted into \(3x + 4y\) |
| 21 and 20 | A1 | Correct evaluation of their expressions with correct values for the letters |
| Beware 21 or 20 coming from wrong working | |
| Accept either of 21 or 20 seen if there is also an explanation that the other value is one more or one less (as appropriate) than the calculated one | M1A1 |
| Use the scheme that awards the better mark | |
| \(a = 3\) and \(b = -2\) then \(9 \times 3 + 3 \times -2\) or \(x = 0\) and \(y = 5\) then \(3 \times 0 + 4 \times 5\) | M1 |
7 \(e\) is 3 more than \(d\).
\(f\) is 5 less than \(d\).
Simplify your answer. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(d + 3\) or \(3 + d\) | B1 | must be seen in (a) |
| Condone \(e = d + 3\) or \(e = 3 + d\) | B1 |
| \(d = e - 3\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(d - 5\) | B1 | must be seen in (b) |
| Condone \(f = d - 5\) | B1 |
| \(d = f + 5\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| their \((d + 3) -\) their \((d - 5)\) or \(3 - -5\) or chooses values for \(d\), \(e\) and \(f\) with \(e\) 3 more than \(d\) and \(f\) 5 less than \(d\) and subtracts \(f\) from \(e\) or chooses values for \(e\) and \(f\) with \(e\) 8 more than \(f\) and subtracts \(f\) from \(e\) | M1 | oe eg \(d + 3 - d + 5\) or \(3 + d + 5 - d\) ft their expressions in (a) and (b) if both in terms of \(d\) and at least one has a numerical term may be implied by eg \(f = e - 8\) |
| 8 | A1ft | correct or ft their expressions in (a) and (b) if both in terms of \(d\) and at least one has a numerical term |
| 8 | M1A1 |
| \((d = 10,)\ e = 13\) and \(f = 5\) and \(13 - 5\) | M1 |
| Only condone missing brackets if recovered | |
| \(d + 3 - d - 5\) and no recovery | M0 |
| \(d + 3 - d - 5\) and answer 8 | M1A1 |
| \(d + 3\) in (a), \(5 - d\) in (b) and \(2d - 2\) in (c) | (B1B0)M1A1ft |
| \(3d\) in (a), \(d - 5\) in (b) and \(2d + 5\) in (c) | (B0B1)M1A1ft |
| \(3d\) in (a), \(-5d\) in (b) and \(8d\) in (c) | (B0B0)M0A0 |
3 Circle the expression that is equivalent to \(\quad 3a - a \times 4a + 2a\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(5a - 4a^2\) | B1 |
3 Simplify \(\quad 3 \times a \times 3 \times a\)
Circle your answer. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(9a^2\) | B1 |
1 Circle the expression that can be written as \(\quad 2y\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(y + y\) | B1 |
25 Circle the expression for the range of \(n\) consecutive integers. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(n - 1\) | B1 |
5
| Answer | Mark | Comments |
|---|---|---|
| \(y^2\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(4a + 11\) | B2 | B1 for each term |
| \(4a\) or 11 or \(4a + 11\) seen and answer eg \(15a\) | B1 |
| \(4a + 11\) seen and then ‘solves’ | B1 |
| 11 and –11 seen (without \(4a\) seen) | B0 |
5
| Answer | Mark | Comments |
|---|---|---|
| \(a^3 + 2b\) | B2 | B1 for \(a^3\) (+) or (+) \(2b\) |
| Do not accept \(2 \times b\) or \(b2\) for \(2b\) | |
| Do not accept \({}^3a\) for \(a^3\) | |
| Do not accept further working for B2 eg \(a^3 + 2b = a^32b\) | B1 |
| Do not accept further working for B1 eg \(3a + 2b = 5ab \quad\) or \(\quad a^3 \quad b^2 = a^3b^2\) | B0 |
| \(a^3 + b^2\) | B1 |
| \(3a + 2b\) | B1 |
| \(a^3 \quad 2b\) | B1 |
| \(a^3 \quad 2b = a^32b\) | B1 |
| \(a^3 \times 2b\) or \(a^32b\) without working for B1 | B0 |
| \(a^3 \times b^2\) or \(a^3b^2\) | B0 |
| \(3a \times 2b\) | B0 |
| \(3a - 2b\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(5x\) (+) 15 | B1 | Implied by correct answer |
| \(4x + 17\) | B2ft | B2ft their \(5x + 15\) in the form \(5x + b\) or \(ax + 15\), both their terms with correct ft in final answer B1ft \(4x\) or (+)17 B1ft their \(5x + 15\) in the form \(5x + b\) or \(ax + 15\), one of their terms with correct ft in final answer |
| ft \(4x\) or (+)17 or must use \(5x + b - x + 2\) or \(ax + 15 - x + 2\) | |
| \(4x + 17\) with no expansion seen | B1B2 |
| Ignore further working with an attempt to solve after their \(4x + 17\) eg \(4x + 17 = 0\) followed by \(x = -4.25\) | B1B2 |
| Do not ignore further working with an attempt to simplify after their \(4x + 17\) eg \(4x + 17\) followed by \(21x\) | B1B1 |
| \(5x + 15 - x + 2\) followed by \(4x + 15 = -2\) | B1B1 |
| \(5x + 3\) followed by \(4x + 5\) also \(5x - 15\) followed by \(4x - 13\) | B0B2ft |
| Ignore further working after \(5x + 15\) for first B1 eg \(5x + 15\) followed by \(20x\) and \(20x - x + 2\) followed by \(19x + 2\) | B1B0 |
| \(5x \quad 15\) | B1 |
| \(4x + k,\ k \ne 17\), with no expansion seen | B0B1ft |
| \(kx + 17,\ k \ne 4\), with no expansion seen | B0B1ft |
| \(5x + 15 - 5x + 10\) followed by 25 | B1B0 |
| \(5x + 3\) followed by \(4x + 1\) | B0B1ft |
| \(5x^2 + 15\) followed by \(5x^2 - x + 17\) | B0B1ft |
| \(5x + 3\) followed by \(4x + 1\) followed by \(5x\) | B0B0ft |
| \(5x + 3\) followed by \(6x + 1\) | B0B0ft |
| \(5x^2 + 3\) followed by \(5x^2 - x + 5\) | B0B0ft |
2 Circle the expression that is four times bigger than \(n\). [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(4n\) | B1 |
23
| \((x - 1)(x - 2) = 12\) | is an .......................... |
| \((x - 1)(x - 2) = x^2 - 3x + 2\) | is an .......................... |
| \(x^2 = 3x + 10\) | is an .......................... |
[1]
Find the equation of the line in the form \(y = mx + c\). [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Equation Identity Equation | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(y = 3x + 5\) as final answer | 3 | B2 for final answer \(3x + 5\) or for a correct equation but not in required form isw OR M2 for using \((2, 11)\) correctly in \(y = 3x + c\) oe or M1 for “\(y = 3x + c\)” or for \(y = 3x + k\) oe where \(k\) is any numerical value except 0 or 5 or for correctly substituting \((2, 11)\) in \(y = mx + c\) oe \((m \ne 3)\) | B2 for e.g. \((y - 11) = 3(x - 2)\) or \(y - 3x = 5\) or \(3x - y + 5 = 0\) as the answer B2 for \(y = 3x + 5\) seen in working and subsequently spoilt (excluding changing to a perpendicular gradient) e.g. \(11 = 3 \times 2 + c\) e.g. \(11 = \dfrac{1}{3} \times 2 + c\) |
10 Heidi is \(x\) years old.
Finley is 5 years younger than Heidi.
Zayn is 4 times Finley’s age.
Find an expression in terms of \(x\) for the total age of Heidi, Finley and Zayn.
Write your answer as simply as possible. [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(6x - 25\) final answer | 4 | B1 for \(x - 5\) B2FT for \(4x - 20\) or 4(their \((x - 5)\)) correctly expanded or B1FT for \(4(x - 5)\) or 4(their \((x - 5)\)) | accept e.g. \(1x - 5\) Their \(x - 5\) must be two term expression with one algebraic term e.g. \(5 - x\) |
9
\(4x - 3y + x + 7y\) [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(5x + 4y\) final answer | 2 | B1 for final answer \(5x + ky\) or \(kx + 4y\) or for correct answer seen then spoiled | Accept e.g. \(5x\) alone or \(4y\) alone for B1 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(10 - 5x\) final answer | 1 | ||
| (ii) \(2x^2 + 6x\) final answer | 2 | B1 for \(2x^2\) or \(6x\) in final answer or for correct answer seen then spoiled | |
7
\(6k - 3 = 177\) [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(a^7\) | 1 | ||
| (ii) \(20b^3\) as final answer | 2 | B1 for \(20b^k\) or \(kb^3\) \((k \ne 0)\) or correct answer seen and spoilt | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 30 as final answer | 2 | M1 for \(6k = 177 + 3\) or better or \(k - \dfrac{3}{6} = \dfrac{177}{6}\) or better or \(6 \times 30 - 3 = 177\) oe | Accept flow diagram |
5
| \((x - 1)(x - 2) = 12\) | is an |
| \((x - 1)(x - 2) = x^2 - 3x + 2\) | is an |
| \(x^2 = 3x + 10\) | is an |
Find the equation of the line in the form \(y = mx + c\). [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Equation Identity Equation | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(y = 3x + 5\) as final answer | 3 | B2 for final answer \(3x + 5\) or for a correct equation but not in required form isw | B2 for e.g. \((y - 11) = 3(x - 2)\) or \(y - 3x = 5\) or \(3x - y + 5 = 0\) as the answer B2 for \(y = 3x + 5\) seen in working and subsequently spoilt (excluding changing to a perpendicular gradient) |
| OR | |||
| M2 for using (2, 11) correctly in \(y = 3x + c\) oe or M1 for “\(y = 3x + c\)” or for \(y = 3x + k\) oe where \(k\) is any numerical value except 0 or 5 or for correctly substituting (2, 11) in \(y = mx + c\) oe \((m \ne 3)\) | e.g. \(11 = 3 \times 2 + c\) e.g. \(11 = \frac{1}{3} \times 2 + c\) | ||
20 Here is a question and an incorrect answer.
Explain why the answer is not correct. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Reason includes a4 should be written as \(4a\) oe or the 4 is written after the \(a\) oe and \(6 \times b\) should be written as \(6b\) oe or the 6 and \(b\) should not be separated oe | 2 | B1 for correct explanation for one term a4 should be written as \(4a\) oe or the 4 is written after the \(a\) oe or \(6 \times b\) should be written as \(6b\) oe or the 6 and b should not be separated oe or correct expression written \(4a + 6b\) | Must refer to error in each term for 2 marks Incorrect statements apply penalty 1 mark if 2 marks earned See appendix 3 |
APPENDIX 3 Question 20
Answers that imply the brackets have been multiplied incorrectly treat as incorrect statements
Do not accept for 2 marks \(4a + 6b\) alone without some explanation but allow for B1
| Response | Mark | |
|---|---|---|
| A | He didn’t put the 6 and the \(b\) together to show it is multiplied. He put the 4 after the \(a\) when it should be before (oe) | 2 |
| B | Because they separated the \(b\) from the 6 and put the a in front of the 4 | 2 |
| C | In algebra the 4 should come before the a so it is \(4a\), the multiplication sign is not needed so it should be \(6b\) \(4a + 6b\) should not be simplified so the answer is \(2a + 3b\) (penalty 1 after correct explanation for 2 marks then incorrect statement) | 1 |
| D | It is correct \(4a + 6b\) | B1 |
| E | It is not written correctly, it should be \(4a + 6b\) (B1 for \(4a + 6b\)) | B1 |
| F | The answer is \(4a + 6b\) | B1 |
| G | Because numbers have to be placed in front of letters BOD for term in \(a\) term in \(b\) not explained | B1 |
| H | Shows \(4a + 6b\) in working and then ‘because they can further simplify the answer by \(2(2a + 3b)\)’ | B1 |
| I | It should be \(4a + 6b\), the student has not timesed the 6 and the b together | B1 |
| J | It is \(6b\) not \(6 \times b\) – it has not been fully simplified | B1 |
| K | a4 is incorrect as 4 is the coefficient – it should be \(4a\) | B1 |
| L | The answer should be \(4a + 6b\), he did the \(2b\) calculation wrong he should have multiplied them together by 3 to get the answer | B1 |
| M | The number must be before the letter \(4a + 6b\) | B1 |
| N | \(4a + 6b\) written above answer lines and answer line blank | B1 |
| O | She has put 4 after the \(a\) and has only multiplied the number 2 by 3 without the \(b\) (one correct statement about \(a\)) | B1 |
| P | He has put \(4a\) as a power and \(6 \times b\) should be \(6b\), should be \(4a + 6b\) (B1 for \(4a + 6b\)) | B1 |
| Q | They should not be separating the 6 and the \(b\) it should be \(6b\) | B1 |
| R | The contents of the bracket were not expanded correctly it should be \(2 \times 3b = 6b\) (not explained error how term is written) | 0 |
| S | He has written \(a\)4 and \(6 \times b\) (further explanation about the error needed) | 0 |
10 Simplify.
\(8j + 4k - 10j + 7k\) [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(-2j + 11k\) final answer | 2 | B1 for \(-2j\) or \(11k\) in final answer or correct answer seen and spoilt | \(11k + -2j\) scores B1 |
8 Here is a question and an incorrect answer.
Explain why the answer is not correct. [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Reason includes \(a4\) should be written as \(4a\) oe or the 4 is written after the \(a\) oe and \(6 \times b\) should be written as \(6b\) oe or the 6 and \(b\) should not be separated oe | 2 | B1 for correct explanation for one term \(a4\) should be written as \(4a\) oe or the 4 is written after the \(a\) oe or \(6 \times b\) should be written as \(6b\) oe or the 6 and b should not be separated oe or correct expression written \(4a + 6b\) | Must refer to error in each term for 2 marks Incorrect statements apply penalty 1 mark if 2 marks earned See appendix 2 |
Answers that imply the brackets have been multiplied incorrectly treat as incorrect statements
Do not accept for 2 marks 4a + 6b alone without some explanation but allow for B1
| Response | Mark | |
|---|---|---|
| A | He didn’t put the 6 and the b together to show it is multiplied. He put the 4 after the a when it should be before (oe) | 2 |
| B | Because they separated the b from the 6 and put the a in front of the 4 | 2 |
| C | In algebra the 4 should come before the a so it is 4a, the multiplication sign is not needed so it should be 6b 4a + 6b should not be simplified so the answer is 2a + 3b (penalty 1 after correct explanation for 2 marks then incorrect statement) | 1 |
| D | It is correct 4a + 6b | B1 |
| E | It is not written correctly, it should be 4a + 6b (B1 for 4a + 6b) | B1 |
| F | The answer is 4a + 6b | B1 |
| G | Because numbers have to be placed in front of letters (BOD for term in a, term in b not explained) | B1 |
| H | Shows 4a + 6b in working and then ‘because they can further simplify the answer by 2(2a + 3b)’ | B1 |
| I | It should be 4a + 6b, the student has not timesed the 6 and the b together | B1 |
| J | It is 6b not 6 x b – it has not been fully simplified | B1 |
| K | a4 is incorrect as 4 is the coefficient – it should be 4a | B1 |
| L | The answer should be 4a + 6b, he did the 2b calculation wrong he should have multiplied them together by 3 to get the answer | B1 |
| M | The number must be before the letter 4a + 6b | B1 |
| N | 4a + 6b written above answer lines and answer line blank | B1 |
| O | She has put 4 after the a and has only multiplied the number 2 by 3 without the b (one correct statement about a) | B1 |
| P | He has put 4a as a power and 6 x b should be 6b, should be 4a + 6b (B1 for 4a + 6b) | B1 |
| Q | They should not be separating the 6 and the b it should be 6b | B1 |
| R | The contents of the bracket were not expanded correctly it should be 2 x 3b = 6b (not explained error how term is written) | 0 |
| S | He has written a4 and 6 × b (further explanation about the error needed) | 0 |
15 In this question all measurements are in centimetres.
The shaded shape is made by cutting a square from the corner of a rectangle.
The width of the rectangle is \(k\).
The length of the rectangle is \(2k\).
Each side of the square is \(g\).
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| The length is twice the width oe or The width is half the length | 1 | “Twice” or “double” or “half” is not enough unless it is clear that length = 2 × width Accept \(2k\) is double \(k\) oe or \(L = 2W\) oe Do not accept values or length = \(2 \times k\) Mark the best response as long as it is not contradictory or has an incorrect statement | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(2k^2 - g^2\) final answer | 2 | M1 for \(2k \times k - g \times g\) oe | Ignore units if included oe for M1 may be e.g. \(2k \times k - g^2\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(6k\) final answer | 3 | B2 for correct answer unsimplified or B2 for \(6k - 2g\) [\(+ g + g\)] or M2 for \(2k + k + 2k - g + k - g\) [\(+ g + g\)] oe or M1 for [height =] \(k - g\) or [length =] \(2k - g\) | Condone \(6k + 0\)[\(g\)] for 3 marks Accept in any order Accept e.g. \(2g\) for \(g + g\) Identified or seen on diagram in correct position |
| (ii) 10.4 nfww | 2 | M1 for their part (i) = 62.4 or \(\frac{62.4}{6}\) | Their part (i) must be algebraic in terms of \(k\) or \(k\) and \(g\) Their (a) can be rearranged Note: \(6k - 2g = 62.4\) scores M1 but does not score the second mark as from wrong working |
11 Choose a word from this list which best describes each statement.
Equation Expression Formula Identity Inequality Term
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Expression | 1 | Allow any clear intention e.g. arrow to expression | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Identity | 1 | Allow any clear intention e.g. arrow to identity | |
10 Simplify.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(2y + 11x\) | 2 | B1 for \(2y\) or \(11x\) in final answer | Accept \(11x + 2y\) for 2 marks Do not accept \(-2y\) or \(-11x\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(30ab\) or \(30ba\) | 1 | Not e.g. \(30 \times ab\), \(a30b\), \(b30a\), \(ab30\) | |
10
\(h \times h \times h \times h\) [1]
\(3fg + 12g\) [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(h^4\) final answer | 1 | Not h4 unless 4 clearly raised | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(3g(f + 4)\) final answer | 2 | B1 for \(3(fg + 4g)\) or \(g(3f + 12)\) | Allow e.g. \(1f\) and inclusion of \(\times\) Condone missing final bracket Correct answer seen and then spoilt B1 |
10 Eight consecutive numbers are written in ascending order in this grid, starting from the top and working left to right.
Show that for Kareem’s grid, the sum of the numbers in the top half of the grid is 16 less than the sum of the numbers in the bottom half of the grid. [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (9 + 10 + 11 + 12) – (5 + 6 + 7 + 8) = 16 | 1 | May be shown in stages Accept 42 – 26 = 16 May be explained via sum of differences of paired cells eg 4 × 4 from (9 – 5), (10 – 6) etc | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Eg. Top half: \(n + (n + 1) + (n + 2) + (n + 3) = 4n + 6\) Bottom half: \((n + 4) + (n + 5) + (n + 6) + (n + 7) = 4n + 22\) \((4n + 22) - (4n + 6) = 16\) | 5 | B2 consistent algebraic terms for consecutive numbers for the whole grid or B1 consistent algebraic terms for at least 3 consecutive numbers AND | eg. \(n\), \(n + 1\), \(n + 2\), \(n + 3\), \(n + 4\), \(n + 5\), \(n + 6\), \(n + 7\) For M2 and M1 FT expressions of form \(an + b\), \(b \ne 0\) eg. Top half: \(2n + (2n + 1) + (2n + 2) + (2n + 3) = 8n + 6\) Bottom half: \((2n + 4) + (2n + 5) + (2n + 6) + (2n + 7) = 8n + 22\) Accept unsimplified or simplified for M marks |
| M2 for algebraic sums for top half and bottom half of grid or M1 for algebraic sum for top half or bottom half of grid AND | |||
| A1dep for sum of bottom half – sum of top half = 16 shown algebraically or explained from correct working. | A1 is dep on B2M2 scored Condone \(\begin{array}{r} 4n + 22 \\ [-]\ 4n + 6 \\ \hline 16 \end{array}\) A0 for \(4n + 22 - 4n + 6\) or for just 22 – 6 or for use of \(2n\), \(2n + 1\), \(2n + 2\) etc as their algebraic terms | ||
| Alternative method for pairs of numbers B2 as above AND M2 for the difference of four pairs of algebraic terms from top and bottom calculated or M1 for the one pair of algebraic terms from top and bottom calculated AND A1dep for all four differences summed to 16 shown numerically or explained from correct working | eg. “difference between \(n\)+4 and \(n\) is 4” | ||
| If 0 scored, allow SC1 for a correct numerical or described example showing an overall difference of 16 | Must not be the numbers 5 to 12 | ||
10 Choose a word from this list which best describes each statement.
Equation Expression Formula Inequality Term
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Expression | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Equation | 1 | ||
8
\(p - 5 = -4\) [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(7a\) final answer | 1 | Accept \(7 \times a\) and \(a7\) and \(a \times 7\) | |
| (ii) \(-x + 4 - 3y\) final answer | 2 | B1 for \(-x\) seen or [\(+\)]\(4 - 3y\) seen in final answer | Accept answer in any order |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 1 | 1 | ||
3
\(3a^2 \times 4a^5\) [2]
\(4x^2 - 12x\) [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(12a^7\) final answer | 2 | B1 for answer \(ka^7\) or \(12a^k\) or for correct answer seen then spoiled | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(4x(x - 3)\) final answer | 2 | B1 for answer \(4(x^2 - 3x)\) or \(x(4x - 12)\) or \(2x(2x - 6)\) or for correct answer seen then spoiled | Condone omission of final bracket for 2 marks or B1 |
16 The diagram shows a quadrilateral and an equilateral triangle.
The perimeter of the quadrilateral is equal to the perimeter of the equilateral triangle.
Not to scale
Find an expression for the length of one side of the equilateral triangle.
Give your answer in terms of \(x\) in its simplest form. [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(7x + 3\) final answer | 4 | M2 for \(21x + 9\) isw or M1 for \(5x + 4 + x + 2 + 9x - 5 + 6x + 8\) M1 for their \((21x + 9) \div 3\) | Must be an algebraic expression in the form \(ax + b\) \(b \ne 0\) |
6
\(9 - 6y\) [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(12a\) final answer | 1 | Accept \(12 \times a\) or \(a12\) or \(a \times 12\) | |
| (ii) \(b^4\) final answer | 1 | Do not accept b4 but BOD if 4 appears raised above the base of b | |
| (iii) \(c^6\) final answer | 1 | Do not accept c6 but BOD if 6 appears raised above the base of c | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(3(3 - 2y)\) final answer | 1 | May be \(3 \times (3 - 2y)\) or \(-3(-3 + 2y)\) | |
9
\(4a - 12b\) [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(2r - 5t\) final answer | 2 | B1 for \(2r\) or \(-5t\) in final answer or correct answer seen and spoilt. \(2r + -5t\) scores B1 | |
| (ii) \(a^5\) | 1 | ||
| (iii) \(7b^4\) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(4(a - 3b)\) | 1 | Allow \(4(1a - 3b)\) and \(2(2a - 6b)\) | |
8
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(6a\) final answer | 1 | Condone poor algebra e.g. \(6 \times a\) and \(a \times 6\) but not \(a^6\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(\dfrac{1}{2}x^4\) or \(\dfrac{x^4}{2}\) or \(0.5x^4\) final answer | 2 | B1 for \(0.5x^k\), \(k \ne 0\) or \(gx^4\), \(g \ne 0\) as answer | Condone 1 before term in \(x\) Allow \(\dfrac{x^5}{2x}\) and \(\dfrac{2x^4}{4}\) for B1 |
7
\(t + 5t - 4t\) [1]
\(x^2 + 2x\) [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(2t\) final answer | 1 | Accept \(2 \times t\) Condone \(t2\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x(x + 2)\) final answer | 1 | Accept e.g. \(1x(x + 2)\), \((x + 0)(x + 2)\) Condone final bracket missing | |
11 Here are some algebraic statements.
\[v = u + at \qquad a + 2b \qquad 3(x + 2) = 3x + 6 \qquad 2y \lt x \qquad 2x = 5\]From the list above, write down an example of each of the following.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(a + 2b\) cao | 1 | Do not accept extras | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(2y \lt x\) cao | 1 | Do not accept extras | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(2x = 5\) cao | 1 | Do not accept extras | |
10
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(6c^2d^2\) cao, final answer | 2 | B1 for \(6c^2\) or \(d^2\) in final answer or correct answer seen and spoilt. | B1 for e.g. \(6c^2 \times d^2\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(7x(5 + x)\) final answer | 2 | B1 for \(7(5x + x^2)\) or \(x(35 + 7x)\) as final answer or correct answer seen and spoilt | Accept \(7x(5 + 1x)\) Condone final bracket missing |
8 Simplify. \(5t - 3u - t + 5u\) [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(4t + 2u\) final answer | 2 | B1 for \(4t\) or \(2u\) seen | |
15
[2]
[1]
[2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(2a + 3b\) final answer | 2 | B1 for \(2a\) or \(3b\) in answer If 0 scored, SC1 for correct answer seen then spoilt | Eg \(2a + 3b = 5ab\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(4x + 12\) final answer | 1 | ||
| (ii) \(x^2 + 3x - 10\) | 2 | Mark final answer B1 for 3 of \(x^2\), \(5x\), \(-2x\), \(-10\) | [+]\(3x\) counts as two terms For B1 could be in a grid |
13 Choose a word from this list that best describes each statement.
Identity Expression Formula Term Equation
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Equation | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Expression | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Identity | 1 | ||
10 Nine consecutive numbers are written on a 3-by-3 grid.
They are arranged, in ascending order, in a spiral as shown.
| 3 | 4 | 5 |
| 10 | 11 | 6 |
| 9 | 8 | 7 |
The total of the first column is \(3 + 10 + 9 = 22\).
Karen says
The total of the first column is one less than the total of the second column.
Show that this is correct for Karen’s grid. [1]
If any nine consecutive numbers are arranged in ascending order in this spiral on a 3-by-3 grid, the total of the first column will always be one less than the total of the second column.
Prove that Victor is correct. [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 4 + 11 + 8 = 23 seen | 1 | Accept written as a sum in a column | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| e.g. First column: \(n + (n + 7) + (n + 6) = 3n + 13\) Second column: \((n + 1) + (n + 8) + (n + 5) = 3n + 14\) \((3n + 14) - (3n + 13) = 1\) | 5 | B2 for consistent algebraic terms for at least first two columns of the grid or B1 for at least 3 algebraic terms for consecutive numbers seen AND M1 for algebraic sum of first or second column shown M1 for algebraic sum of first and second columns shown and correctly simplified A1 for sum of second column – sum of first column = 1 calculated or explained from correct working or M1 for difference of one pair of algebraic terms from first and second column shown M1 for differences of two further pairs of algebraic terms from first and second column, with all three pairs correctly simplified A1 for each difference found as +1 or –1 oe and summed/explained to a difference of +1. Correct algebra and reasoning throughout If 0 scored, allow SC1 for a correct numerical or descriptive example using either method and stating an overall difference of 1 | e.g. \(n\), \((n + 7)\), \((n + 6)\) and \((n + 1)\), \((n + 8)\), \((n + 5)\) e.g. \(n\), \((n + 1)\), \((n + 2)\) e.g. \(n + (n + 7) + (n + 6)\) or in column e.g. \(n + (n + 7) + (n + 6) = 3n + 13\) A1 for e.g. \(3n + 14\) and \(3n + 13\) and “second column is 1 more than the first” but A0 for e.g. \(3n + 14\) and \(3n + 13\) and “difference of 1” or for \((3n + 14) - 3n + 13 = 1\) e.g. “the difference between \(n + 1\) and \(n\) is 1” e.g. “\(n + 1\) is 1 more than \(n\)” Condone poor use of brackets for both M marks |
5 Simplify.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(2a\) | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(11g + 3f\) final answer | 2 | B1 for \(11g\) or \(3f\) in answer | |
3 Martina has answered some questions on algebra.
In each question, she has made an error.
Describe her error and give the correct answer to each problem.
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| She added the terms oe | 1 | In all 3 parts any incorrect statement treat as choice Allow correct descriptions of what Martina should have done in each part See AG | |
| \(2a^3\) | 1 | ||
| Response | Mark | |
|---|---|---|
| 1 | She did not multiply [the terms] | 1 |
| 2 | She added up all the a | 1 |
| 3 | She added [to] the 2a’s | 1BOD |
| 4 | She hasn’t multiplied the a’s | 1 |
| 5 | it is × not + | 1 |
| 6 | she would be right if they were plus signs and not times | 1 |
| 7 | Not 2a + a + a | 1 |
| 8 | a would equal 2a + a + a. When you times you add them 2a x a x a = 2a3 | 1 |
| 9 | She added a to the multiplication instead of using index laws | 0 |
| 10 | She has added the 2a’s then times by the 2a | 0 |
| 11 | She added the a’s to the 2 instead of multiplying them | 0 |
| 12 | She added the 2a to the a x a | 0 |
| 13 | 2a is different to a times a = a2 | 0 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| She divided the powers oe | 1 | See AG | |
| \(x^8\) | 1 | ||
| Response | Mark | |
|---|---|---|
| 1 | She divided the 10 and the 2/she divided the powers (must refer to ‘numbers’ or ‘indices’) | 1 |
| 2 | she done 10 ÷ 2 = 5 | 1 |
| 3 | She should have done 10 – 2 | 1 |
| 4 | She should take away the indices | 1 |
| 5 | She divided 10 by 2 instead of subtracting 2 | 1 |
| 6 | Laws of indices it should be taken away | 1 |
| 7 | Divided the numbers | 1 |
| 8 | She divided instead of taking away (‘indices’ implied by referring to division and subtraction) | 1 |
| 9 | She has used division | 0 |
| 10 | She divided it when it should be timesd | 0 |
| 11 | She didn’t use the laws of indices | 0 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| She squared (½ × 6 × 5) oe | 1 | See AG | |
| 75 | 1 | ||
| Response | Mark | |
|---|---|---|
| 1 | didn’t do the power first | 1 |
| 2 | She did 5 x 6 and then x ½ but the ² is near the 5² not all of it | 1 |
| 3 | She should have squared 5 first and then divide 6 by 2 and times them together | 1 |
| 4 | She didn’t do the 5² bit first | 1 |
| 5 | She didn’t square 5 and times 6 by 25 | 1 |
| 6 | She hasn’t squared the 5 | 1 |
| 7 | Doing 15² she needs to do 5² | 1 |
| 8 | She did 15² | 0 |
| 9 | She halved 30 when she meant to halve 6 earlier in the equation | 0 |
| 10 | She didn’t square the number 5 correctly nor did she times by 6 or halve it by half | 0 |
| 11 | She square after, when you square before | 0 |
| 12 | She halved 6 before multiplying by 5² | 0 |
| 13 | Didn’t use BODMAS | 0 |
| 14 | 6 x 25 = 150 | 0 |
13 Prove that the mean of any four consecutive even integers is an integer. [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(\dfrac{(2x) + (2x + 2) + (2x + 4) + (2x + 6)}{4}\) \(= \dfrac{8x + 12}{4}\) \(= 2x + 3\) which is an integer OR \((2x) + (2x + 2) + (2x + 4) + (2x + 6)\) \(= 8x + 12\) \(= 4(2x + 3)\) which is divisible by 4 oe | 4 | M1 for \(2x\), \(2x + 2\), \(2x + 4\) and \(2x + 6\) seen and M1 for adding their four terms in \(x\), e.g. \((2x) + (2x + 2) + (2x + 4) + (2x + 6)\) and M1 for their \((8x + 12) \div 4\) or better, condoning lack of brackets, or for \(4(2x + 3)\) and A1dep (dep on M0M1M1 or M1M1M1) for correct algebraic mean for their four terms and conclusion, e.g. \(2x + 3\) is an integer or \(4(2x + 3)\) which is divisible by 4 If 0 scored, allow SC1 for a numerical example with any 4 consecutive even integers with mean correctly calculated | Or equivalent algebraic representations of 4 consecutive even numbers. In this case, \(x\) does not need to be defined as being an integer. Using \(x\), \(x + 2\), \(x + 4\), \(x + 6\) oe does not score the first M mark unless \(x\) stated as even integer, but can score up to 3 marks for \((x) + (x + 2) + (x + 4) + (x + 6)\), their \((4x + 12) \div 4\) or better, or for \(4(x + 3)\) and the relevant conclusion Using \(x + 1\), \(x + 3\), \(x + 5\), \(x + 7\) oe does not score the first M mark unless \(x\) stated as odd integer but can score up to 3 marks similar to above. |
10
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(t + 8u\) final answer | 2 | M1 for [+1]\(t\) or [+]\(8u\) in final answer If 0 scored SC1 for correct answer seen then spoilt | Accept \(1t + 8u\), \(8u + 1t\) Condone capitals for 2 or 1 marks eg \(t + 8u = 9tu\) |
| (ii) \(12a^3\) final answer | 2 | B1 for \(12a^k\) or \(ka^3\) (\(k \ne 0\)) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(x = \sqrt{y + 1}\) final answer | 2 | M1 \(\sqrt{y + 1}\) final answer or for \(x^2 = y + 1\) or for correct FT step to answer after incorrect first step | Ignore \(\pm\) or – for 2 marks or M1 Condone \(\sqrt{y + 1} = x\) For M1 condone \(y + 1 = x^2\) eg \(x = \sqrt{y - 1}\), after \(x^2 = y - 1\) seen. |
16 The perimeter of the pentagon is equal to the perimeter of the square.
Not to scale
Find an expression for the length of one side of the square.
Give your answer in terms of \(x\) in its simplest form. [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(7x + 2\) final answer | 4 | B2 for \(28x + 8\) or B1 for \(28x + k\) or \(jx + 8\) or M1 for \(5x + 3 + 7x + 4 + 9x - 10 + 5x + 8 + 2x + 3\) AND M1 for their \((28x + 8) \div 4\) soi | \(j \ne 0\) B1 not from only one side e.g. \(5x + 8\) must be an algebraic expression in the form \(ax + b\) |
7
[2]
\(4a + 3c + 7a - 5c\) [2]
\(\dfrac{2x}{3} = 4\) [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Hollow circle at 3 only | 1 | No other blobs | |
| Line/arrow “pointing” right from 3 | 1 | Marks independent | Open line or arrow only and condone mark/blob over 8 or x If line, must reach approx. 8 Condone line/arrow starting closer to 3 than 4 |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(11a - 2c\) final answer | 2 | B1 for \(11a\) or \(-2c\) seen | Accept in any order \(11a + {-2c}\) scores 1 mark |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 6 | 2 | M1 for \(2x = 12\) or \(\dfrac{x}{3} = 2\) or \(\dfrac{x}{1.5} = 4\) | If T&I only correct answer scores Must be algebraic method for M1 Do not accept embedded answers |
13 In this question, assume all dimensions are in centimetres.
Jess and Pete have many rectangular tiles.
Each tile has length \(a + b\) and width \(2b\).
Not to scale
Not to scale
Factorise this expression fully. [2]
Draw a possible arrangement of tiles for Pete’s rectangle.
Write down expressions for the length and for the width of the rectangle. [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(6a + 10b\) or \(2(3a + 5b)\) final answer | 2 | M1 for \(6(a + b) + 2 \times 2b\) oe If 0 scored SC1 for \(3a + 5b\) as final answer | M1 for EG \(a + b + a + b + a + b + a + b + a + b + a + b + 2b + 2b\) or \(2 \times (3a + 3b + 2b)\) etc |
| (ii) \(6b(a + b)\) final answer | 2 | B1 for \(6(ab + b^2)\) or \(b(6a + 6b)\) or \(3(2ab + 2b^2)\) or \(3b(2a + 2b)\) or \(2(3ab + 3b^2)\) or \(2b(3a + 3b)\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 4 by 1 rectangle with \(4a + 4b\) and \(2b\) or 2 by 2 rectangle with \(2a + 2b\) and \(4b\) or 1 by 4 rectangle with \(a + b\) and \(8b\) stated or marked on rectangle | 5 | B4 for \(4a + 4b\) and \(2b\) or \(2a + 2b\) and \(4b\) or \(a + b\) and \(8b\) or B3 for rectangle drawn as (4 by 1) or (2 by 2) or (1 by 4) or B2 for one of \(2a + 2b\) or \(4a + 4b\) or \(4b\) or \(8b\) or B1 for any rectangle of 3 or more tiles drawn with \(a+b\) or \(2b\) marked on individual tiles | Accept unsimplified throughout Once correct expression(s) seen, ignore incorrect simplification to answer line In answer space or intended as final length and width Must clearly be answer May be in attempt to factorise EG \(4b(2a + b)\) Accept unsimplified EG \(a+b + a+b\) Only tiles that form the perimeter needed |
4
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(4x - 3y\) final answer | 2 | B1 for \(4x\) or \(-3y\) in final answer | \(4x + -3y\) scores B1 only |
| (ii) \(w^6\) final answer | 1 | ||
| (iii) \(15c^3d\) final answer | 1 | Accept \(15dc^3\) Do not accept eg \(15 \times c^3 \times d\) | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) 13 | 1 | ||
| (ii) 3 | 1 | ||
6
[2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(4p\) | 1 | ||
| (ii) \(5j - 2k\) | 2 | B1 for \(5j\) or \(-2k\) in final answer | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 144 | 2 | M1 for 120 or 24 or 10 \(\times\) 12 + 6 \(\times\) 4 | Not \(120h\) or \(24t\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(d = \dfrac{f - e}{7}\) oe nfww | 2 | M1 for correct first step or \(\dfrac{f - e}{7}\) | \(e + 7d = f\) or \(e - f = -7d\) oe |
10 A square is divided into three rectangles, A, B and C.
Not to scale
Rectangle A has length \(n\) cm and a width of 2 cm.
Rectangle C has length 4 cm.
Work out the value of \(n\). [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(4 - n\) final answer | 1 | Accept eg \(4 - x\) Ignore \(C =\) or \(w =\) etc Ignore any units given in all parts | |
| (ii) \(2n\) final answer | 1 | Accept \(n2\), \(2 \times n\), \(n \times 2\) Condone \(4(4 - n)\) Ignore \(A =\) etc | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(\dfrac{16}{6}\) or \(\dfrac{8}{3}\) or \(2\dfrac{4}{6}\) or \(2\dfrac{2}{3}\) or 2.6[6…] or 2.7 | 3 | M1 for \(4(4 - n)\) oe or \(4 \times 4 \div 3\) or better And M1 for their \(4(4 - n) = 2n\) or better or \(2n = \dfrac{16}{3}\) oe or \(4(4 - n) = \dfrac{16}{3}\) oe | Isw Accept eg \(5\dfrac{1}{3}\) or 5.33[…] for \(\dfrac{16}{3}\) M1M1 implied by eg \(5.33 \div 2\) |
No questions match these filters.