Which of the following equations shows p inversely proportional to q?
A) p = 7q
B) p = 7/q
C) p = q + 7
D) p = q2/7
(Total for Question 1 is 1 mark)
2
Which of the following equations shows y directly proportional to x3?
A) y = k/x3
B) y = 3x
C) y = kx3
D) y = x3 + k
(Total for Question 2 is 1 mark)
3
y is directly proportional to x. Given that y = 20 when x = 4, state the value of the constant of proportionality k.
(Total for Question 3 is 1 mark)
4
m is inversely proportional to n. Given that m = 6 when n = 5, state the value of the constant of proportionality k.
(Total for Question 4 is 1 mark)
5
y is directly proportional to x2. Given that y = 48 when x = 4, state the value of the constant of proportionality k.
(Total for Question 5 is 1 mark)
6
p is inversely proportional to q2. Given that p = 5 when q = 2, state the value of the constant of proportionality k.
(Total for Question 6 is 1 mark)
7
y is directly proportional to x. When x = 9, y = 54. Find the value of y when x = 13.
(Total for Question 7 is 2 marks)
8
m is inversely proportional to n. When n = 6, m = 15. Find the value of m when n = 10.
(Total for Question 8 is 2 marks)
9
y is directly proportional to √x. When x = 25, y = 40. Find the value of y when x = 64.
(Total for Question 9 is 2 marks)
10
The time, h hours, taken to paint a fence is inversely proportional to the number of painters, p, working on it. Using 3 painters takes 8 hours. Find how long it would take using 4 painters, assuming all painters work at the same constant rate.
(Total for Question 10 is 2 marks)
11
y is directly proportional to x3. When x = 2, y = 48. Find the value of x when y = 750.
(Total for Question 11 is 2 marks)
12
z is directly proportional to x and inversely proportional to y. When x = 6 and y = 3, z = 8. Find the value of z when x = 9 and y = 2.
(Total for Question 12 is 2 marks)
13
y is directly proportional to x2. Find the percentage increase in y when x is increased by 20%.
(Total for Question 13 is 2 marks)
14
The energy, E joules, stored in a stretched spring is directly proportional to the square of its extension, x cm. When x = 4, E = 48. Find the value of E when x = 7.
(Total for Question 14 is 3 marks)
15
The force, F newtons, needed to tow a boat at a steady speed is directly proportional to the square of the speed, v m/s. A speed of 4 m/s requires a force of 20 N. Find the force needed to tow the boat at a speed of 10 m/s.
(Total for Question 15 is 3 marks)
16
p is directly proportional to q3. Show that if q is halved, p is divided by 8.
(Total for Question 16 is 3 marks)
17
y is directly proportional to x. When x = 4, y = 20. Find, in exact surd form, the positive value of x for which y = x2 - 9.
(Total for Question 17 is 4 marks)
18
y is directly proportional to x2. When x = 3, y = 36. Separately, y is inversely proportional to w, and when x = 3 (so y = 36), it is also known that w = 6 at this point. Find the value of w when x = 6.
(Total for Question 18 is 3 marks)
19
The volume, V cm3, of a certain type of resin block is directly proportional to the square of its width, w cm, and directly proportional to its depth, d cm. A block with width 3 cm and depth 8 cm has volume 216 cm3. Find the depth of a block with width 6 cm and volume 756 cm3.
(Total for Question 19 is 4 marks)
Mark scheme · A8D Direct and Inverse Proportion: Fluency and Exam Drill
Question 1
B1 B cao
Answer: B) p = 7/q
Question 2
B1 C cao
Answer: C) y = kx3
Question 3
B1 5 cao
Answer: k = 5
Question 4
B1 30 cao
Answer: k = 30
Question 5
B1 3 cao
Answer: k = 3
Question 6
B1 20 cao
Answer: k = 20
Question 7
M1 finds the constant of proportionality, k = 6 (from 54 = 9k)
A1 y = 78 cao
Answer: y = 78
Question 8
M1 finds the constant of proportionality, k = 90 (from k = 6 x 15)
A1 m = 9 cao
Answer: m = 9
Question 9
M1 finds k = 8 (from 40 = k √25, i.e. 40 = 5k)
A1 y = 64 cao
Answer: y = 64
Question 10
M1 finds k = 24 (from k = 3 x 8)
A1 h = 6 hours cao
Answer: h = 6 hours
Question 11
M1 finds k = 6 (from 48 = k(2)3, i.e. 48 = 8k) and sets up 750 = 6x3
A1 x = 5 cao
Answer: x = 5
Question 12
M1 finds k = 4 (from 8 = k(6)/3, i.e. 8 = 2k) and sets up z = 4(9)/2
A1 z = 18 cao
Answer: z = 18
Question 13
M1 lets the original y = kx2, and finds the new y when x is increased by 20%: k(1.2x)2 = 1.44kx2 = 1.44y
A1 44% cao
Answer: y increases by 44%
Question 14
M1 finds k = 3 (from 48 = k(4)2, i.e. 48 = 16k)
M1 substitutes x = 7 into E = 3x2
A1 E = 147 cao
Answer: E = 147
Question 15
M1 finds k = 1.25 (from 20 = k(4)2, i.e. 20 = 16k)
M1 substitutes v = 10 into F = 1.25v2
A1 F = 125 cao
Answer: F = 125 N
Question 16
M1 lets the original p = kq3
M1 finds the new p when q is halved: k(q/2)3 = kq3/8 = p/8
A1 correct conclusion: p is divided by 8, as required
Answer: p is divided by 8 (shown)
Question 17
M1 finds k = 5 (from 20 = 4k), so y = 5x
M1 sets up 5x = x2 - 9 and rearranges to x2 - 5x - 9 = 0
M1 applies the quadratic formula with a = 1, b = -5, c = -9, giving x = (5 ± √25 + 36)/2
A1 x = (5 + √61)/2 cao (exact surd form, rejecting the negative root)
Answer: x = (5 + √61)/2
Question 18
M1 finds the first constant, k1 = 4 (from 36 = k1(3)2), so y = 4x2, and finds y at x = 6: y = 4(6)2 = 144
M1 finds the second constant, k2 = 216 (from k2 = 36 x 6), so y = 216/w
A1 w = 1.5 cao
Answer: w = 1.5
Question 19
M1 finds k = 3 (from 216 = k(3)2(8), i.e. 216 = 72k)
A1 V = 3w2 d cao
M1 substitutes w = 6, V = 756 into 756 = 3(6)2 d and rearranges to d = 756/108