Grade-9 Revision Sheet
1. If 𝑎 = −1, 𝑏 = 3 𝑎𝑛𝑑 𝑐 = −3, evaluate:
𝑎2 +𝑎𝑐+𝑏3
(i)
𝑎+𝑏
𝑐 3 + 𝑏𝑐 + 𝑎2
(ii)
2. Solve for x:
𝒙𝟐 +𝟏
𝟐
−
𝒙𝟐 −𝟏
𝟒
=
𝒙+𝟏
𝟖
3. Simplify:
7𝑝7 𝑞 5
𝑎. (𝑝2
e.
b.
𝑞)3
(
3𝑥2 𝑦 −3 4
)
2𝑥 −2
(3𝑥 3 )2 ×4𝑥 5
6𝑥 4 ×𝑥
3
c. 18𝑛−1 ÷ 9𝑛−2
d.
16 4
(𝑥 4 )
4. Expand and simplify:
a.
b.
(1 − 𝑥)(2 − 3𝑥)(𝑥 + 2)
3𝑥 + 1 − (𝑥 + 3)2
5. Fully Factorize:
(a). 81𝑥 2 − 16𝑦 2
(b). 𝑎𝑏 2 − 9𝑎2 𝑏
(𝑐). 6𝑥 − 𝑥 2 − 9
6. Express the following ratio in the simplest form
1 2 1
1 : :
2 3 6
7. A shop applies a 45% mark up on cost price of fresh fish and sells it for $25.50 per kg. Find the
cost price of the fish.
8. A student claims that 83 × 8−5 is greater than 1. Is the students’ claim correct? Justify the
statement with complete working.
(L-4)
9. The ratio of the number of apples to orange in stall A and stall B are 2:5 and 5:9 respectively. if
the total number of fruits in stall A is twice than in stall B,
a. Find the ratio of the number of apples in stall A to the number of apples in stall B
b. Due to the increase in land rates the rent for fruit stalls have increased in the ratio 12:8, what
is the new rent of Stall A , if the old rent was $ 240 ?
10. By using only suitable identity , evaluate the following :
(1003)2
11. A boy named Richard has $1000 to invest in one of two long term plans. He intends withdraw
only the interest at the end of each year, but leave his $1000 invested. (Cr C & D)
The two plans are as follows:
Plan 1: This plan pays $50 interest each year as long as the original investment is untouched.
This $50 is called 5% simple interest.
Plan 2: This plan pays $500 interest at the end of the first year. (This $500 interest is 50% of the
original investment.) At the end of each of the following years, the $1000 investment earns half as
much interest as the year before. (So in the second year it would only earn $250 in interest.)
a)
After 3 years, what is the total interest that Richard would make from each plan?
b)
Which option should he choose to maximize the total interest if he invests for 7 years?
c)
If Richard left his $1000 invested for a very long time, which plan do you think would
provide the greatest total interest? Justify by complete working.
12. Mr. Griffin’s class is studying the solar system. He explains the students how difficult it is to
write these large distances and teaches them to write these numbers in a simpler method using
scientific notation and standard form. Help the students solve the given questions on the solar
system.
a) The circumference of the Earth at the equator is about 24,900 miles. Express this number in
scientific notation.
b) The speed of light is approximately 6.71×10 miles per hour. Express this number in standard
form.
c) If light travels 6.71×10 miles in one hour. How many miles will it travel in 1 minute?
d) If it takes light 8.3 minutes to reach the Sun from the Earth, what is the distance of the Sun
from the Earth?
13. A Software Company is planning to review its growth. The CEO presents the statistics of the
expenditure incurred by the company in the previous financial year. The graph given below
gives the expenditure done by a company annually read the graph and helps them review the
situation for planning;
a.
How much fraction of percentage of does the company spend on Taxes? Can they reduce the
amount given in taxes? If yes how? Suggest and justify your statement.
b. Find the ratio of the percentage spend on infrastructure and transport ?How will finding the
ratio help them in future. Use appropriate mathematical language to explain the situation.
c. If the company plans to increase the salary by 10 %, how will it affect the overall expenditure of
the company? Is it necessary to increase the salary of the employees for the growth of the
company in the coming years?
14.
a.
A shopkeeper has a DVD which he bought for £12. He decided that he wants to have a
profit of 10%. What price will he have to sell it for so that his profit will be 10% of £12?
b. The DVD doesn’t sell, so he reluctantly decides to slash its selling price by 10%. What is the
new selling price?
c. His teen-age assistant points out that he didn’t make very much money on the transactions and
suggests that he first takes a 10% discount on the buying price and then adds 10% of this
amount to get a final price. The shopkeeper is doubtful, but whips out his calculator.
d. What does the shopkeeper discover?
e. Would the same thing happen if the DVD were some other price?
15. The distance a ball falling from rest varies directly as the square of the time it falls (ignoring air
resistance). If a ball falls 144 feet in three seconds.
How far will the ball fall in seven seconds?
a.
What will be the distance after 5 seconds.
b.
Will the distance increases with the time duration? Justify your
answer.
16. The frequency of a vibrating guitar string varies inversely as its
length. Suppose a guitar string 0.65 meters long vibrates 4.3 times per second.
a. What frequency would a string 0.5 meters long have?
b. If the frequency is increased how will it affect the length
appropriate mathematical strategies?
of the string? Explain using
17. The volume of a cylinder of fixed height varies in direct proportion to the square of the base
radius. Find the change in volume when the base radius is increased by 18%.
18. Calculate the value of xy in the following table. Hence, determine whether x and y are
inversely proportional. If an inverse proportion exists, determine the law connecting the
variables and draw the graph of y against x.
x
7
6
12
1
y
12
14
7
84
19. Change the subject:
i. ‘x’ the subject of 𝑦 = 4𝑥 2 − 7
𝑎
ii. ‘b’ the subject of 𝐴 = 4√𝑏
20. The first term of an arithmetic sequence is 7 and the sixth term is 22. Find
(a) the common difference;
(b) the twelfth term;
(c) the sum of the first 100 terms.
21.
22.
INVESTIGATION
MAXIMISING THE PERIMETER
Identical shapes can be joined to make larger shapes
1.
Equilateral triangle of side 1 cm may be joined edge to edge, for example
but not like this.
i.
The diagram below shows a shape made of 4 equilateral triangles, with a perimeter of
cm and a shape made of 5 equilateral triangles with the perimeter 7cm
ii.
The diagram below shows a shape, made of 6 equilateral triangles, with a perimeter greater
than 6 cm.
iii.
The diagram below shows a shape, made of 7 equilateral triangles, with a perimeter greater
than 7 cm.
iv.
This table shows the greatest possible perimeters of shapes made of an equilateral triangles.
Complete the table
Number of equilateral triangles
2
3
Greatest perimeter (cm)
4
5
4
5
6
7
8
10
6
v.
Write down an expression, in terms of x, for the greatest perimeter for a shape made of x
equilateral triangles.
vi.
Write down the greatest perimeter for a shape made of 20 equilateral triangles.
2. Squares of side 1 cm may be joined edge to edge, for example
But not like this
i.
Complete the table
Number of Squares
2
3
Greatest perimeter (cm)
6
8
ii.
4
5
12
6
7
8
9
10
22
Write down an expression, in terms of x, for the greatest perimeter for a shape made of x
squares.
iii.
Write down the greatest perimeter for a shape made of 17 squares.
3. The table shows the greatest perimeters for the shape made of regular hexagons (6 sides) of side 1 cm
complete the table:
i.
Number of Hexagons
2
Greatest perimeter (cm)
8
3
4
5
6
22
26
7
ii. Write down an expression, in terms of x, for the greatest perimeter for a shape made of x hexagons.
4. Find an expressions, in terms of x for the greatest perimeter for the shape made of x, regular octagons.
(8 sides)
5. i. Observe the rule in Q.1(v), Q.2 (ii) , Q.3 (i) and Q.4---- to state an expression in terms of x
and y, for the greatest perimeter for the shape made of x, regular polygons each with y sides .
ii. Find the greatest perimeter for a shape made of 26 sides.