Use the graphs to solve the simultaneous equations 2 – 2y = x and 2y = 3x – 22.
Use these graphs to solve the simultaneous equations
2 - 2y = x
2y = 3x - 22
2 - 2y = x
2y = 3x - 22
The graph of y = 1x - 2 is drawn on the grid for -2 ≤ x ≤ 6.
(a) There are no values of x for which 1x - 2 = k.
Find the value of k. [1 mark]
Find the value of k. [1 mark]
k =
Worked Solution & Answer
(a)
The horizontal asymptote is y = 0, so 1/(x − 2) never equals 0 ⇒ k = 0. B1
Answer (a): k = 0
Find the coordinates of the midpoint of PQ where P = (–3, –2) and Q = (2, 4).
Find the coordinates of the midpoint of PQ.
Worked Solution & Answer
Here is the graph of y = x2 - 2x - 2
(a) Write down the coordinates of the turning point on the graph of y = x2 - 2x - 2 [1 mark]
(b) Write down an estimate for one of the roots of x2 - 2x - 2 = 0 [1 mark]
Worked Solution & Answer
(a)
Minimum at (1, –3) B1
Answer (a): (1, –3)
(b)
x-intercepts at x ≈ –0.7 and x ≈ 2.7 B1
Answer (b): –0.7 or 2.7
Phil sells ties.
He increases the original price of each tie by 10% to £13.20
A month later he announces a sale: 10% OFF ALL TIES.
He increases the original price of each tie by 10% to £13.20
A month later he announces a sale: 10% OFF ALL TIES.
Phil says,
“The ties will be back to their original price, because each change was by 10%”
Is he correct?
Tick a box: [ ] Yes [ ] No
Show working to support your answer.
“The ties will be back to their original price, because each change was by 10%”
Is he correct?
Tick a box: [ ] Yes [ ] No
Show working to support your answer.
Worked Solution & Answer
Original price = £13.20 ÷ 1.10 = £12.00 M1
Sale price = £13.20 × 0.90 = £11.88 M1
Since £11.88 ≠ £12.00, Phil is incorrect (tick No). A1
Sale price = £13.20 × 0.90 = £11.88 M1
Since £11.88 ≠ £12.00, Phil is incorrect (tick No). A1
Answer: No, sale price is £11.88 whereas original price was £12.00
The table gives information about the population and the total amount of money, in dollars, spent on healthcare for two countries in 2016
Work out how much more was spent per person on healthcare in Luxembourg than in Austria.
Give your answer correct to the nearest whole number.
Give your answer correct to the nearest whole number.
dollars
Worked Solution & Answer
Luxembourg per person = (3.7 × 10⁹) ÷ (6.3 × 10⁵) = $5873.0158.. M1
Austria per person = (4.2 × 10¹⁰) ÷ (8.7 × 10⁶) = $4827.5862.. M1
Difference = 5873.0158 – 4827.5862 = 1045.429... → 1045 dollars. A1
Austria per person = (4.2 × 10¹⁰) ÷ (8.7 × 10⁶) = $4827.5862.. M1
Difference = 5873.0158 – 4827.5862 = 1045.429... → 1045 dollars. A1
Answer: 1045 dollars
(a) Ben makes m subject of p = m/3 + 5: p – 5 = m/3; 3 × p – 5 = m; m = 3p – 5. What mistake has Ben made? (1)
(b) Factorise fully 2x³y + 4xy² (2)
(b) Factorise fully 2x³y + 4xy² (2)
(a) What mistake has Ben made? [1 mark]
(b) Factorise fully 2x3 y + 4xy2 [2 marks]
Worked Solution & Answer
(a)
When multiplying both sides by 3, the entire left hand side must be multiplied: 3(p – 5) = 3p – 15, not 3p – 5 B1
Answer (a): He didn't multiply the –5 by 3 (should be 3(p – 5) or 3p – 15)
(b)
Common factor is 2xy: 2xy(x² + 2y) M1 A1
Answer (b): 2xy(x2 + 2y)
Rectangle with sides (x + 2) cm and (x – 5) cm has area 120 cm². Work out x. [4 marks]
The area of the rectangle is 120 cm2
Work out the value of x.
Work out the value of x.
x =
Worked Solution & Answer
(The area of a rectangle is length × width)
(x + 2)(x – 5) = 120 ⇒ x² – 3x – 10 = 120 ⇒ x² – 3x – 130 = 0. M3
Factorise: (x – 13)(x + 10) = 0 ⇒ x = 13 (length > 0). A1
(x + 2)(x – 5) = 120 ⇒ x² – 3x – 10 = 120 ⇒ x² – 3x – 130 = 0. M3
Factorise: (x – 13)(x + 10) = 0 ⇒ x = 13 (length > 0). A1
Answer: x = 13
A pattern is made from four identical rectangles.
The sides of the rectangles are parallel to the axes.
The sides of the rectangles are parallel to the axes.
Point A has coordinates (3, 4)
Point B has coordinates (11, 20)
Point C is marked on the diagram.
Work out the coordinates of C.
You must show all your working.
Point B has coordinates (11, 20)
Point C is marked on the diagram.
Work out the coordinates of C.
You must show all your working.
Worked Solution & Answer
Let length = L and width = W M2
Horizontal change from A(3,4) to B(11,20): 2L – W = 11 – 3 = 8. Vertical change from A(3,4) to B(11,20): 2L + W = 20 – 4 = 16 M2
Adding gives 4L = 24 ⇒ L = 6, W = 4. Therefore C = (3 + L, 4 + W) = (3 + 6, 4 + 4) = (9, 8). A1
Horizontal change from A(3,4) to B(11,20): 2L – W = 11 – 3 = 8. Vertical change from A(3,4) to B(11,20): 2L + W = 20 – 4 = 16 M2
Adding gives 4L = 24 ⇒ L = 6, W = 4. Therefore C = (3 + L, 4 + W) = (3 + 6, 4 + 4) = (9, 8). A1
Answer: (9, 8)
(a) Factorise y² − 2y − 48 (2) (b) Write down the inequality shown on the number line (1) (c) Solve 7w + 6 > 12w + 14 (3)
(a) Factorise y2 - 2y - 48 [2 marks]
(b) Write down the inequality shown on the number line [1 mark]
(c) Solve the inequality 7w + 6 > 12w + 14 [3 marks]
Worked Solution & Answer
(a)
Numbers multiplying to −48 and adding to −2 are +6 and −8 M1
⇒ (y + 6)(y − 8) A1
⇒ (y + 6)(y − 8) A1
Answer (a): (y + 6)(y - 8)
(b)
Filled circle at 3 pointing left ⇒ x ≤ 3 B1
Answer (b): x ≤ 3
(c)
6 − 14 > 12w − 7w M1
⇒ −8 > 5w M1
⇒ w < −8/5 A1
⇒ −8 > 5w M1
⇒ w < −8/5 A1
Answer (c): w < -85