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VCE 11 Methods 2023

1.03 Substitution

Worksheet
Substitution into expressions
1

Evaluate:

a
m + 5 for m = 7
b
9 + m for m = 3
c
x - 2 for x = 8
d
8 - c for c = 1
e
6 x + 4 for x = 9
f
3 x^{2} - 7 x - 4 for x = 3
g
- 2 x^{3} + 4 x^{2} + 8 x + 4 for x = - 1
2

Evaluate:

a
\dfrac{x + 6}{x + 5} for x = 8
b
\dfrac{4 y - 1}{y - 1} for y = 6
c
\dfrac{z^{3}}{z^{2} - 6} for z = 4
d
\dfrac{y}{y^{3} - 43} for y = 4
e
\dfrac{x^{2} + 3 x + 2}{x^{2} - 4 x - 12} for x = 7
f
\dfrac{3 x + 2}{x^{2} - 4 x - 12} for x = 7
3

Evaluate:

a
7 x + 5 for x = \dfrac{1}{7}
b
4 x^{2} + 7 x + 3 for x = \dfrac{1}{4}
c
4 x^{3} + 10 x^{2} for x = \dfrac{1}{2}
d
\dfrac{2 x - 9}{5} for x = \dfrac{29}{2}
e
\dfrac{11 x + 5}{7 x + 9} for x = - \dfrac{13}{3}
4

Evaluate y + 5 for:

a
y = 3
b
y = 6
5

Evaluate u + v for:

a
u = 7 and v = 3
b
u = 3 and v = 16
6

Evaluate a + b for a = 5 and b = 1.1

7

Evaluate a - b for a = 9 and b = 5.

8

Evaluate u + v + 25 for u = 32 and v = 48.

9

Evaluate a + b + c for a = 10, b = 15, and c = 11.

10

If a = 2, b = 7 and c = - 8, evaluate the following expressions:

a

\dfrac{a b c}{a}

b

a^{ 2 b} + c

c

\dfrac{a}{b} + \dfrac{c}{a}

11

If a = 0.4, b = - 2.5 and c = 5, evaluate the following expressions:

a

\dfrac{a b}{c}

b

\left( a b\right)^{2} - c

c

a + b c - a

Substitution into formulas
12

The perimeter of a triangle is defined by the formula P = p + q + r.

Find the value of P if the length of each of its three sides are:

a
p = 17 cm, q = 16 cm and r = 14 cm
b
p = 6 cm, q = 3 cm and r = 7 cm
13

The perimeter of a square with side lengths of a is given by the formula P = 4a.

Find the value of P if the length of each side is 7 cm.

14

The area of a rectangle is given by the formula A=\text{ Length }\times \text{ Width}. If the length of a rectangle is 2 cm and its width is 3 cm, find its area.

15

The perimeter of a rectangle is given by the formula P = 2 \times \left(l + w\right) , where l is the length and w is the width. If the width of a rectangle is 10 cm and its length is 5 cm, find its perimeter.

16

The area of a triangle is given by the formula A = \dfrac{1}{2} \times \text{base} \times\text{height}. If the base of a triangle is 5 cm and its height is 8 cm, find its area.

17

The area of a rhombus is given by the formula A = \dfrac{1}{2} x y, where x and y are the lengths of the diagonals. If the diagonals of a rhombus have lengths of2 cm and 4 cm, find the area of the rhombus.

18

The area of a square with side lengths of s is given by the formula A = s^{2}. Find the value of A if the length of each side is 6 cm.

19

The equation of a straight line is given by the formula y = m x + c. Given that m = 6, x = - 4 and c = 9, find the value of y.

20

For many 3 dimensional shapes, we can find the number of edges, E, on the shape by using the formula: E = V + F - 2 where V is the number of vertices and F is the number of faces.

Find the number of edges of a 3 dimensional shape which has:

a

7 vertices and 7 faces

b

8 vertices and 6 faces

21

The surface area of a rectangular prism is given by formula S = 2 \left( l w + w h + l h\right), where l, w and h are the dimensions of the prism. Given that a rectangular prism has a length of \\ 8 cm, a width of 7 cm and a height of 9 cm, find its surface area.

22

The volume of a rectangular prism is given by the formula V = l \times w \times h, where l , w and h are the dimensions of the prism. Given that a rectangular prism has a length of 4 cm, a width of 8 cm and a height of 5 cm, find its volume.

23

The simple interest generated by an investment is given by the formula I = \dfrac{P \times R \times T}{100}.

Given that P = 1200, R = 5 and T = 9, find the interest generated.

24

Converting a measure of temperature from Celsius to Fahrenheit is given by the formula: F = 32 + \dfrac{9 C}{5}

Given that C = 15, evaluate F.

25

Converting a measure of temperature from Fahrenheit to Celsius is given by the formula:C = \dfrac{5}{9} \left(F - 32\right)

a

Given that F = 50, evaluate C.

b

The temperature inside a freezer is 86 \degree\text{F}. What is this temperature in Celsius?

26

In physics, Newton's second law states that F = m a, where F is the force of on object (measured in Newtons), m is the mass of the object (in kilograms) and a is the acceleration of that object (measured in metres/second^{2}).

Calculate the force, F, created from an object with:

a
A mass of 6 kilograms and an acceleration of 13\text{ m}/\text{s}^{2}.
b
A mass of 1540 grams and an acceleration of 19\text{ m}/\text{s}^{2}.
27

Newton's second law of motion states that a = \dfrac{F}{m}, where F is the force acting on an object (in Newtons), m is the mass of the object (in kilograms) and a is the acceleration of that object (in \text{m}/\text{s}^{2}). Calculate the acceleration of an object with a mass of 25 kilograms and a force of 775 Newtons acting on it.

28

The volume of a sphere can be calculated using the formula V = \dfrac{4}{3} \pi r^{3}, where r is the radius of the sphere. Given that a sphere has a radius of 2 cm, calculate its volume correct to two decimal places.

29

The value of a variable T is defined by the formula T = a + \left(n - 1\right) d. Given that a = 6,\\ n = 5 and d = 8, find the value of T.

30

Sally bought a television series online. The television series was 7.2 gigabytes large and took 2 hours to download. The formula b = 8 B converts gigabytes, B, to gigabits b. Find the size of the television series in gigabits.

31

The sum of n terms in an arithmetic sequence is defined by the formula S = \dfrac{n}{2} \left( 2 a + \left(n - 1\right) d\right)Given that n = 10, a = 3 and d = 9, find the value of S.

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Outcomes

U1.AoS2.12

substitute integer, simple rational and irrational numbers in exact form into expressions, including rules of functions and relations, and evaluate these by hand

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