Question 1
y is directly proportional to x.
When y = 36, x = 4
1(a) Find a formula for y in terms of x
ANSWER: Simple text answer
Answer: y = 9x
Workings:
y = kx
36 = 4k so k = 9
y = 9x
Marks = 3
1(b) Find the value of y when x = 3.
ANSWER: Simple text answer
Answer: 27
Workings:
y = 9 × 3 = 27
Marks = 1
Question 2
y is directly proportional to x^2.
When x=2, y = 36.
2(a) Find a formula for y in terms of x.
ANSWER: Multiple choice (Type 1)
A: y = 3x^2
B: y = \dfrac{x}{4}x^2
C: y = 9x^2
D: y = 3x
Answer: C
Workings:
y = kx^2
36 = 4k, k = 9
y = 9x^2
Marks = 3
2(b) Find the value of x when y = 49
Give your answer as a fraction.
ANSWER: Fraction
Answer: x=\dfrac{7}{3}
Workings:
49 = 9x^2 so x^2 = \dfrac{49}{9}
x = \sqrt{\dfrac{49}{9}} = \dfrac{7}{3}
Marks = 1
Question 3
In the table below, d is directly proportional to c
3(a) Find a formula for d in terms of c
ANSWER: Simple text answer
Answer: d = 4c
Workings:
d= kc
12 = 3c so k = 4
d = 4c
Marks = 3
3(b) Hence, or otherwise, find the values of X. Y and Z to complete the table above.
ANSWER: Multiple answers (type 1)
Answer: X = 7, Y = 20, Z = 48
Workings:
Y = 4\times 5 = 20
X = \dfrac{28}{4} = 7
Z = 12\times 4 = 48
Marks = 3
Question 4
x is inversely proportional to y
when x = 7, y = 4.
4(a) Find a formula for y in terms of x
ANSWER: Multiple choice (Type 1)
A: y = \dfrac{28}{x}
B: y = \dfrac{x}{28}
C: y = \dfrac{4k}{7}
D: y = \dfrac{7k}{4}
Answer: A
Workings:
x = \dfrac{k}{y} so 7=\dfrac{k}{4}
So x = \dfrac{28}{y} so y = \dfrac{28}{x}
Marks = 3
4(b) Find the value of x when y =2
ANSWER: Simple text answer
Answer: 14
Workings:
x = \dfrac{28}{2} = 14
Marks = 1
Question 5
y is inversely proportional to the square of x.
y=3 when x = 4.
5(a) Find a formula for y in terms of x
ANSWER: multiple choice (Type 1)
A: y = \dfrac{12}{x^2}
B: y = \dfrac{48}{x}
C: y = \dfrac{48}{x^2}
D: y = \dfrac{3k}{16}
Answer: C
Workings:
y = \dfrac{k}{x^2}
3 = \dfrac{k}{4^2} so k = 48
y = \dfrac{48}{x^2}
Marks = 3
5(b) Find the value of y when x = 5
Give your answer as a fraction.
ANSWER: fraction
Answer: \dfrac{48}{25}
Workings:
y = \dfrac{48}{5^2} = \dfrac{48}{25}
Marks = 1
Question 6
If r is inversely proportional to b^2 and r = 4 when b = 4.
6(a) Find the formula for r in terms of b.
Give your answer as a fraction.
ANSWER: fraction
A: r = \dfrac{64}{b^2}
B: r = \dfrac{16}{b}
C: r = \dfrac{64}{b}
D: r = \dfrac{16}{b^2}
Answer: A
Workings:
r \alpha \dfrac{1}{b^2}
r = \dfrac{k}{b^2}
4 = \dfrac{k}{(4)^2} so k = 64
r = \dfrac{64}{b^2}
Marks = 3
6(b) Find the value of r when b = 2
ANSWER: Simple text answer
Answer: 16
Workings:
r = \dfrac{64}{(2)^2} = \dfrac{64}{4} = 16
Marks = 1
6(c) Find the value of b when r = 2.
Give your answer to 3 significant figures.
ANSWER: Simple text answer
Answer: 5.66
Workings:
b^2 = \dfrac{64}{r} = \dfrac{64}{2} = 32
b = \sqrt{32} = 4\sqrt{2} = 5.66 (3 s.f.)
Marks = 2
Question 7
a, b and c are three variables.
a is proportional to b^2
a is also proportional to \sqrt{c}
b = 4.5 when c = 2.25
Find b when c = 8
Give your answer correct to 3 significant figures.
ANSWER: Simple text answer
Answer: 6.18
Workings:
a = kb^2 and a = m\sqrt{c}
So kb^2 = m\sqrt{c}
So b^2 = \dfrac{m}{k}\sqrt{c}
So b^2 \alpha \sqrt{c}, so b^2 = g\sqrt{c}
g=\dfrac{b^2}{\sqrt{c}}= \dfrac{4.5^2}{\sqrt{2.25}} = 13.5
b^2 = 13.5 \sqrt{8}; b = 6.18
Marks = 6