Which of the following equations shows y directly proportional to x?
A) y = x + 5
B) y = 3x
C) y = 5/x
D) y = x2
(Total for Question 1 is 1 mark)
2
Which of the following equations shows y inversely proportional to x2?
A) y = kx2
B) y = k/x2
C) y = kx
D) y = k/x
(Total for Question 2 is 1 mark)
3
y is directly proportional to x. When x = 6, y = 21. Find the value of y when x = 10.
(Total for Question 3 is 2 marks)
4
y is inversely proportional to x. When x = 5, y = 18. Find the value of y when x = 9.
(Total for Question 4 is 2 marks)
5
p is directly proportional to q2. When q = 4, p = 32. Find the value of p when q = 7.
(Total for Question 5 is 3 marks)
6
m is inversely proportional to n2. When n = 3, m = 8. Find the value of m when n = 6.
(Total for Question 6 is 3 marks)
7
y is directly proportional to √x. When x = 9, y = 15.
(a)Find a formula for y in terms of x.(2)
(b)Find the value of y when x = 25.(2)
(Total for Question 7 is 4 marks)
8
The time, T hours, taken to fill a water tank is inversely proportional to the number of pumps, n, used. Using 4 pumps takes 5 hours.
(a)Find a formula for T in terms of n.(2)
(b)Find the time taken to fill the tank using 8 pumps.(1)
(c)State one assumption needed for this model to be valid.(1)
(Total for Question 8 is 4 marks)
9
y is directly proportional to x3. When x = 2, y = 40.
(a)Find the value of y when x = 5.(3)
(b)Find the value of x when y = 1080.(2)
(Total for Question 9 is 5 marks)
10
The energy, E joules, stored in a stretched spring is directly proportional to the square of its extension, x cm. When x = 3, E = 27.
(a)Find a formula connecting E and x.(2)
(b)Find E when x = 5.(1)
(c)Find the extension when E = 108.(2)
(Total for Question 10 is 5 marks)
11
The force, F newtons, between two magnets is inversely proportional to the square of the distance, d cm, between them. When d = 2, F = 45.
(a)Find F in terms of d.(2)
(b)Find F when d = 6.(1)
(c)Find the distance when F = 20.(2)
(d)State what happens to F as d increases.(1)
(Total for Question 11 is 6 marks)
12
z is directly proportional to x and inversely proportional to y2. When x = 8 and y = 2, z = 12.
(a)Find a formula connecting x, y and z.(2)
(b)Find z when x = 10 and y = 5.(2)
(c)Find the value of y when x = 27 and z = 2.(2)
(Total for Question 12 is 6 marks)
13
y is directly proportional to x2. Show that if x is doubled, y increases by 300%.
(Total for Question 13 is 3 marks)
14
p is inversely proportional to q3. Show that if q is trebled, p is divided by 27.
(Total for Question 14 is 3 marks)
15
y is directly proportional to x. When x = 3, y = 12. Find, in exact surd form, the values of x for which y = x2 - 7.
(Total for Question 15 is 4 marks)
16
y is directly proportional to x2. When x = 2, y = 20.
(a)Find a formula for y in terms of x.(2)
(b)Separately, y is inversely proportional to w. When x = 2 (so y = 20), it is also known that w = 25 at this point. Find a formula for y in terms of w.(2)
(c)Hence find the value of w when x = 5.(3)
(Total for Question 16 is 7 marks)
17
The volume, V cm3, of a certain type of crystal is directly proportional to the square of its width, w cm, and directly proportional to its height, h cm. A crystal with width 4 cm and height 10 cm has volume 320 cm3.
(a)Find a formula for V in terms of w and h.(2)
(b)Find the volume of a crystal with width 5 cm and height 6 cm.(2)
(c)A crystal has height 8 cm and volume 144 cm3. Find its width, w cm.(2)
(Total for Question 17 is 6 marks)
18
y is directly proportional to x3 for x ≥ 0. The graph of y against x3 is a straight line through the origin. When x = 2, y = 56.
(a)State the gradient of the graph of y against x3.(1)
(b)Find the value of y when x = 1/2.(2)
(c)A second variable, z, is such that z is inversely proportional to x, and z = 4 when x = 7. Find the value of x (other than x = 0) for which y = z, giving your answer correct to 3 significant figures.(4)
(Total for Question 18 is 7 marks)
Mark scheme · A8 Direct and Inverse Proportion
Question 1
B1 B cao
Answer: B) y = 3x
Question 2
B1 B cao
Answer: B) y = k/x2
Question 3
M1 finds the constant of proportionality, k = 3.5 (from 21 = 6k)
A1 y = 35 cao
Answer: y = 35
Question 4
M1 finds the constant of proportionality, k = 90 (from k = 5 x 18)
A1 y = 10 cao
Answer: y = 10
Question 5
M1 finds k = 2 (from 32 = k(4)2, i.e. 32 = 16k)
M1 substitutes q = 7 into p = 2q2
A1 p = 98 cao
Answer: p = 98
Question 6
M1 finds k = 72 (from k = 8 x 32)
M1 substitutes n = 6 into m = 72/n2
A1 m = 2 cao
Answer: m = 2
Question 7
(a) M1 finds k = 5 (from 15 = k √9, i.e. 15 = 3k)
(a) A1 y = 5 √x cao
(a) Answer: y = 5 √x
(b) M1 substitutes x = 25 into their formula from part (a)
(b) A1 y = 25 cao
(b) Answer: y = 25
Question 8
(a) M1 finds k = 20 (from k = 5 x 4)
(a) A1 T = 20/n cao
(a) Answer: T = 20/n
(b) B1 T = 2.5 hours cao
(b) Answer: T = 2.5 hours
(c) B1 valid assumption, e.g. all pumps work at the same constant rate and do not interfere with each other oe
(c) Answer: All pumps work at the same constant rate
Question 9
(a) M1 finds k = 5 (from 40 = k(2)3, i.e. 40 = 8k)
(a) M1 substitutes x = 5 into y = 5x3
(a) A1 y = 625 cao
(a) Answer: y = 625
(b) M1 sets up 1080 = 5x3 and rearranges to x3 = 216
(b) A1 x = 6 cao
(b) Answer: x = 6
Question 10
(a) M1 finds k = 3 (from 27 = k(3)2, i.e. 27 = 9k)
(a) A1 E = 3x2 cao
(a) Answer: E = 3x2
(b) B1 E = 75 cao
(b) Answer: E = 75
(c) M1 sets up 108 = 3x2 and solves x2 = 36
(c) A1 x = 6 cm cao (rejects x = -6 since extension is positive)
(c) Answer: x = 6 cm
Question 11
(a) M1 finds k = 180 (from k = 45 x 22)
(a) A1 F = 180/d2 cao
(a) Answer: F = 180/d2
(b) B1 F = 5 cao
(b) Answer: F = 5
(c) M1 sets up 20 = 180/d2 and solves d2 = 9
(c) A1 d = 3 cm cao
(c) Answer: d = 3 cm
(d) B1 F decreases (towards zero) oe, since F is inversely proportional to d2
(d) Answer: F decreases towards zero as d increases
Question 12
(a) M1 finds k = 6 (from 12 = k(8)/22, i.e. 12 = 2k)
(a) A1 z = 6x/y2 cao
(a) Answer: z = 6x/y2
(b) M1 substitutes x = 10, y = 5 into their formula
(b) A1 z = 2.4 cao
(b) Answer: z = 2.4
(c) M1 sets up 2 = 6(27)/y2 and solves y2 = 81
(c) A1 y = 9 cao
(c) Answer: y = 9
Question 13
M1 lets the original y = kx2
M1 finds the new y when x is doubled: k(2x)2 = 4kx2 = 4y
A1 correct conclusion: the increase is 4y - y = 3y, which is 300% of the original value, as required
Answer: y increases by 300% (shown)
Question 14
M1 lets the original p = k/q3
M1 finds the new p when q is trebled: k/(3q)3 = k/(27q3) = p/27
A1 correct conclusion: p is divided by 27, as required
Answer: p is divided by 27 (shown)
Question 15
M1 finds k = 4 (from 12 = 3k), so y = 4x
M1 sets up 4x = x2 - 7 and rearranges to x2 - 4x - 7 = 0
M1 applies the quadratic formula with a = 1, b = -4, c = -7, giving x = (4 ± √16 + 28)/2
A1 x = 2 + √11 or x = 2 - √11 cao (both values, exact surd form)
Answer: x = 2 + √11 or x = 2 - √11
Question 16
(a) M1 finds k = 5 (from 20 = k(2)2, i.e. 20 = 4k)
(a) A1 y = 5x2 cao
(a) Answer: y = 5x2
(b) M1 finds the second constant of proportionality, k2 = 500 (from k2 = 20 x 25)
(b) A1 y = 500/w cao
(b) Answer: y = 500/w
(c) M1 finds y = 125 using their formula from part (a) (y = 5(5)2)
(c) M1 substitutes their y = 125 into their formula from part (b) to form 125 = 500/w
(c) A1 w = 4 cao
(c) Answer: w = 4
Question 17
(a) M1 finds k = 2 (from 320 = k(4)2(10), i.e. 320 = 160k)
(a) A1 V = 2w2 h cao
(a) Answer: V = 2w2 h
(b) M1 substitutes w = 5, h = 6 into their formula
(b) A1 V = 300 cm3 cao
(b) Answer: V = 300 cm3
(c) M1 sets up 144 = 2w2(8) and solves w2 = 9
(c) A1 w = 3 cm cao
(c) Answer: w = 3 cm
Question 18
(a) B1 gradient = 7 cao
(a) Answer: Gradient = 7
(b) M1 substitutes x = 1/2 into y = 7x3
(b) A1 y = 7/8 (or 0.875) cao
(b) Answer: y = 7/8
(c) M1 finds the constant for z, k2 = 28 (from k2 = 4 x 7), so z = 28/x
(c) M1 sets up 7x3 = 28/x
(c) M1 multiplies both sides by x and divides by 7 to obtain x4 = 4