1.
Bala sold an article for N6,900.00 and made a profit of 15%. calculate his percentage profit if he had sold it for N6,600.00.
Answer: C
selling price = N6,900.00
Profit percent = 15%
percentage profit if he had sold if for N6,600.00
let C.P represent cost price
S.P represnt selling Price
% represent Profit percent
C.P = S.P - P
but put P = %P X C.P
put P = 15% x C.P
P = 0.15 C.P
Gives = C.P = S.P - ( 0.15 - C.P )
but S.P = 6,900.00
C.P = 6,900 - (0.15 C.P )
Group like terms
C.P + 0.15 C.P = 6,900.00
1.5 C.P⁄1.5 = 6,900.00⁄1.5
therefore cost price of the article is 6,000
If the owner of the article had sold it N6,600.000
Profit = Selling price - Cost Price
= 6,600 - 6000
N600.00
since profit is know
percent profit = P⁄C.P x 100
= 600⁄6000 x 100
10%
2.
1.639 X 10-2
1.639 X 10-1
1.639 X 101
1.639 X 102
Answer: C
3.
42
45
54
64
Answer: C
Change the mixed fractions to improper fractions
Rule
Rules
Thus, reciprocate/switch the numerator and denominator at the right and change the division to multiplication
Thus, the numerators multiply each other and the denominators multiply each other
Multiply both sides by 96
96 x 9/16 = x
16 goes into 96, 6 times
6 x 9 = x
x = 54
4.
The total surface area of a solid cylinder is 165cm2.
If the base diameter is 7cm, calculate its height [Take ϖ = 22⁄7]Answer: B
diameter = 7cm
Area of cylinder = 165cm2 ϖ = 22⁄7
since radius = 35
165 = 8(22⁄7)(3.5)(h) + 2(22⁄7)(3.5)2
165 = 22h + 44⁄7 (12.25)
165 =22h + 77
f=group like terms
165 - 77 = 22h
88 =22h
divide through by 22
h=4
5.
solve 4x2 - 16x + 15 = 0
A. x = -1 1⁄2 or -2 1⁄2
B. x = 1 1⁄2 or -2 1⁄2
C. x = 1 1⁄2 or 2 1⁄2
D. x = 1 1⁄2 or -2 1⁄2
Answer: C
look for two numbers If you add it will give you the co-efficient of x and multipy will give 60
The two numbers are -10,-6
substitute -10,-6 in place of the co-efficient of x which is -16
Gives 4x2 - 10x - 6x + 15 = 0
Group like terms
(4x2 - 6x ) - (10x + 15)
2x(2x - 3) - 5(2x - 3) = 0
(2x -5)(2x -3) = 0
2x -5 = 0
2x⁄2 = 5⁄2
x = 5⁄2 = 2 1⁄2
2x - 3 = 0
2x⁄2 = 3⁄2
x = 3⁄2 = 1 1⁄2
x = 1 1⁄2 or 2 1⁄2
6.
3p = 4q and 9p = 8q - 12, find the value of pq
Answer: D
3p = 4q ...... 1eqn
9p = 8p - 12 .......2eqn
multiply equation 1 by 3
3 x 3p = 3 x 4q
9p = 12q ..... 3eqn
put 3eqn into 2eqn
12q = 8q - 12
12q - 8q = -12
4q⁄4 =-12⁄4
q = 3
put q = -3 into 1
3p = 4(-3)
3P⁄3 = -12⁄3
Pq = (-4)(-3)
12
7.
Two friends, Dede and Kofi decided, to buy the same type of car. They found out that the car cost ₦3,000,000.00. The amount of money (f) which Dede had was not enough to buy the car but Kofi had enough money (k) to buy the car. Which of the of the following inequalities is true?
f≤₦3,000,000.00 ≤ k
f> ₦3,000,000.00 > k
f< ₦3,000,000.00 ≤ k
f≥ ₦3,000,000.00 ≥ k
Answer: C
The amount of money(f) which Dede has was not enough to buy the car
⇒ f(Dede's amount) < (less than) the cost of car (₦3,000,000.00)
but Kofi had enough money(k) to buy the car
⇒ Kofi can have the exact amount of the car or even more than the cost of the car
⇒k≥₦3,000,000.00
⇒ f< ₦3,000,000.00 ≤ k
Read as Dede's amount(f) is less than ₦3,000,000.00 and
₦3,000,000.00 is less than or equal to Kofi's amount(k)
8.
If log102 = 0.3010 and log102y = 1.8060, find, correct to the nearest whole number, the value of y.
7
6
5
4
Answer: B
Applying logabc = c x logab = clogab
log102y = y x log102
But log102y = 1.8060
⇒ y x log102 = 1.8060
But log102 = 0.3010
Hence y x 0.3010 = 1.8060
Dividing both sides by 0.3010
⇒ y = 1.8060/0.3010 = 6
9.
Answer: B
Answer is B
Cost of car = ₦5,000,000.00
Deposite = ₦3,000.000.00
Rate = 8% compound intesrest per annum = 8⁄100 = 0.08
Balance = ₦5,000,000.00 - ₦3,000,000.00 = ₦2,000,000.00
Hence the interest will only be charged on the ₦2,000,000.00 balance after the first year
Fisrt year
Interest = Principal x Rate x Time
Time = 1 (Compound interest is calculated year by year)
Interest = 0.08 x ₦2,000,000 = ₦160,000
Total amount to be paid by the end of the first year
⇒ ₦2,000,0000 + ₦160,000 = ₦2,160,000
Since he pays ₦1000,000 at each year
Amount left to pay the subsequent year
⇒ ₦2,160,000 - ₦1,000,000 = ₦1,160,000
Second Year
Interest = 0.08 x ₦1,160,000 = ₦92,800
Amount to pay
⇒ ₦92,800 + ₦1,160,000 = ₦1,250,8000
He pays another ₦1,000,000 by the end of the second year.
Hence amount to pay after the second year
⇒ ₦1,252,800 - ₦1,000,000.00 = ₦252,800
NB: For compound interest, the new principal amount is the interest + the previous amount to be paid
10.
2x2 -4x + 6 = 0
4x2 -4x - 3 = 0
2x2 +3x + 4 = 0
4x2 -4x + 3 = 0
Answer: B
Change the mixed fraction to improper fraction
Rule
If α and β are the two roots of a quadratic equation, then the formula to construct the quadratic equation is:
x2 - (α + β)x + αβ = 0
Thus, x2 - (sum of roots)x + product of roots = 0
⇒ x2 - (-1/2 + 3/2)x + (-1/2 x 3/2) = 0
-1/2 + 3/2 = (-1+3)/2 = 2/2 = 1
Rule
-1/2 x 3/2 = (-1x3)/(2x2) = -3/4
⇒ x2 - (1)x + (-3/4) = 0
x2 - x - 3/4 = 0
NB: + - = - ⇒ + - 3/4 = -3/4
Multiply both sides by 4
4x - 4x - 3 = 0
11.
find the equation of a straight line passing through the point (1,-5) and having gradient of 3⁄4
3x-4y-23 = 0
3x-4y+23 =0
3x+4+23 =0
3x+4y-23= 0
Answer: A
Equation of a line y = mx + c
If point (1,-5) is given and gradient 3⁄4
y = -5
x = 1
y = mx + c
Gives : -5 = 3⁄4 (1) + c
-5 = 3⁄4 + c
Group like terms
-5-3⁄4 = c
-5⁄1 -3⁄4 = c
-20-3⁄4 = c
-23⁄4
y = 3⁄4x - 23⁄4
multipy through by 4
4 x y = 4 X 3⁄4x - 23⁄4 x 4
4y =3x - 23
0 =3x - 23 - 4y = 0
3x - 4y - 23 = 0
12.
Not drawn to scale
In ∆XYZ, ∣YZ∣ = 32 cm, ∠YXZ = 52 ° and ∠XZY = 90 °.
Find, the correct to nearest centimetre. ∣XZ∣.
Answer: C
∅ = 52
opposite ∣YZ∣ = 32
Adjacement ∣XZ∣ = ?
since opposite is known and we are ask to find adjacent (∣XZ∣)
tan∅ formular will be appiead.
which says tan∅ = opposite⁄Adjacement ∣YZ∣⁄∣XZ∣
tan(32) = 32⁄∣XZ∣
cross multiply
tan32⁄1 = 32⁄∣XZ∣
∣XZ⁄ tan(52) =32
On your scientific calculator tan(52) = 0.52
∣XZ (∅.28) = 32
making ∣XZ∣ the subject divide through by 0.53
∣XZ⁄0.53 = 32⁄1.28
∣XZ = 32⁄1.28
∣XZ∣ = 25cm
13.
Answer: C
let represent initial term d represent difference
a = 6
d = common diffence
To find common diffence is the second term minus the first time that's current minus intial term
d = p - 6 = 14 -p
p - 6 = 14 - p
Group like terms
p + p = 14 + 6
2p⁄2 = 20⁄2
p = 10
14.
Answer: B
1st expand the equation by multipyling
number y from zero, alway number it from your right to left do same to the other side of the equatiion
(2xy1) + (3 x y0) = (1x23)+ (1x22) + (1x21) + (1x20)
2y + 3 = 8 + 4 + 2 + 1
Grouping like terms
2y = 8 + 4 + 2 + 1
2y⁄2= 12⁄2
y = 6
15.
y varies inversely as the square of x. When x = 3, y = 100. Find the value of x when y = 25.
x = 30
x = 12
x = 6
x = 5
Answer: C
y ∝ 1⁄x2
NB: ∝ is the symbol for proportional
inversely proportional is 1 over the expression
directly proportional is just the expression (It would had been y ∝ x2)
a constant(=k) is introduced in place of the ∝
y = kx1⁄x2
y = k⁄x2
Use the known values to calculate the constant(k) to get the general expression
y = 100 and x = 3
⇒ 100 = k⁄32
100 = k⁄9
100x9 = k
⇒ k = 900
∴ y = 900⁄x2
when y = 25
⇒ 25 = 900⁄x2
25 x x2 = 900
x2 = 900⁄25
x2 = 36
√x2 = √36
x = 6
NB: The square root of a squared number or variable is equal to the number or variable
16.
An empty rectangular tank is 250 cm long and 120 cm wide. If 180 litres of water is poured into the tank, calculate the height of the water.
4.5 cm
5.0 cm
5.5 cm
6.0 cm
Answer: D
Let h = height of the water
Volume of a cuboid = Length x Breadth x Height
Since the length and breadth are in cm, convert the litres to cm3
1 litre = 1000 cm3
⇒ 180 litres = 180 x 1000 cm3
Length = 250 cm, breadth = 120 cm
Volume = 180 litres = 180 x 1000 cm3
⇒ 250 cm x 120 cm x h cm = 180 x 1000 cm3
Divide both sides by 250 cm x 120 cm
h = (180 x 1000) / (250 x 120) = 6
17.
Arrange the following in ascending order of magnitude: 0.45,¾ and 25%
¾, 0.45, 25%
¾, 25%, 0.45
0.45, 25%, ¾
25%, 0.45, ¾
Answer: D
Change the fraction and decimal into percentage so that they will all be in percentages.
Percentage is simply multiplying by 100.
0.45 ⇒ 0.45 x 100 = 45%
¾ ⇒ ¾ x 100 = 75%
25% is already in percentage
Now arrange in ascending order
25%, 45%, 75%
45% = 0.45
¾ = 75%
Hence the ascending order is 25%,0.45,¾
NB: Ascending is from lowest to highest
18.
If (0.25)y find the value of y.
Answer: A
(25⁄100)y = 32
(1⁄4)y 32
log(1⁄4)y = log32
ylog 0.25 = log32
y = log32⁄log0.25
y = -2.5
y = -25⁄10
y = -5⁄2
19.
There are 8 boys and 4 girls in a lift.what is the probability that the first person who steps out of the lift will be a boy
1⁄4
2⁄3
1⁄3
3⁄4
Answer: B
Number of Boys = 5
Number of Girls = 4
Total pupils = 8 + 4
To get the probability of boys out of lift is the number of boys over the total number of pupils
P(a boys steps out of the lift) = 8⁄12
2⁄3
20.
In the diagram,POS and ROT are straight lines.QPQP is a parallelogram,
|QS| = |QT| and ∆OST = 50°; Calculate tyhe value of ∆OPQ
Answer: D
∆OST = OTS (Isocicoles)
OTS = 50°
considering ∆OST
50 + 50 +∆SOT = 180 100 + ∆SOT =180 Group like terms ∆SOT = 180 ∆SOT = 80° ∆SOT = ∆ROP = 80° (vertical opposite angles are equal 80 + OPQ = 180 OPQ =180 - 80 OPQ = 100
21.
Answer: A
Answer is A
3 x 9 1 + x = 27-x
Concept
Ab x Ac = Ab+c
Thus if the bases are the same, we can add the indexes (superscripts)
3 can go into the other numbers(9,27) hence we change then to have base 3
9 = 31x 31 = 31 + 1=32
27 = 31x 31 x 31 =31+1+1 = 33
⇒ 3 x 32(1 + x) = 33(-x)
But 3 = 31, thus every number is raised to the power 1
⇒ 31 x 32(1 + x) = 33(-x)
Applying Ab x Ac = Ab + c
3 1 x 32(1+ x) = 31 + 2(1 + x )
31+2(1+ x) = 33(-x)
Concept
If Ab = Ac ⇒ b = c
Thus if the base at the left side of the equation, is the same as the right side of the equation, then the powers are the same
∴ 1 + 2(1 + x ) = 3(-x)
Concept
a(b + c ) = a x b + a x c
a(b - c ) = a x b - a x c
Thus the number outside the bracket multiplies each of the number in the bracket and the sign ( + or - ) is(are) maintained
-x = -1 x x
3(-x) = 3 x (-1) x x = -3x
1 + 2(1 + x ) = 1 + 2 x 1 + 2 x x
= 1 + 2 + 2x
= 3 + 2x
1 + 2(1+x) = 3(-x)
⇒ 3 + 2x = -3x
Grouping the x at the left and the rest at the right
Concept
If the sign is positive(+) and crosses to the other side of the equation the sign changes to negative ( - ) and the vice versa
2x + 3x = -3
5x = -3
Concept
Divide both sides by the number multiplying the number or variable (x)
5x⁄5 = -3⁄5
x = -3⁄5
22.
Answer: D
Answer is D
Use a venn diagram to illustrate
In a class of 39 student ⇒ n(u)=39
let F represent students who offer Fante ⇒ n(F)= 25
Let T represent students who offer Twi ⇒ n(T)= 19
Five student do not offer any of the two languages ⇒ 5 are outside the sets F and T
Let n represent number of student who offer both languages
NB: You need to substract the number of the students who offer both languaes from each of the languages to get the students who offer only that languages.
When we add all the parts in the venn diagram we are to get the total number of students n(u) = 39
(25 - n )+ n + (19 - n) + 5 =39
25 - n + n + 19 -n + 5 = 39
-n + 49 = 39
-n = 39 - 49
-n = -10
-n⁄-1 = -10⁄ -1
n = 10
Only Twi = 19-n
⇒ Only Twi = 19 - 10 = 9
23.
In the diagram, RT is a tangent to the circle at R,∠PQR =70° , ∠QRT =52°re; ∠QSR = y and ∠PRQ = x. Use the diagram to answer question 40 and 41
Answer: B
∠TRQ = ∠RSQ = 52°
y = ∠RPQ = 52
24.
Answer: B
The answer is B
Starting zeros are not significant and so the number of significant digits starts from 3597. The 3 digits lands on 9 but the digit next to it is more than 5 (7) and so 1 is added to the 9 to make it 10 and so the 0 is written down and the 1 is added to the 5 next to it which makes it 6.
NOTE
Zeros between other digits are significant. Only starting zeros are insignificant. For instance 3.052306, all the zeros are significant because they are not starting zeros after a decimal point.
25.
If T = {prime numbers} and M = {odd number} are subset of U = {x : 0 < x ≤}, and x is an interger,
find (T1 n M 1)
Answer: D
T ={ 2,3,5,7}
M ={1,3,5,7,9}
U {1,2,3,4,5,6,7,8,9,10}
T1 is called T compliment, which means numbers found in universal set but can't be found in set T
M1 is called M compliment, which means numbers found in universal set but can't be found in set M
T1 = {1,4,6,8,9,10}
M 1 = { 2,4,6,8,10}
T 1 M 1 = { 4,6,8,10}
26.
Find the angle which arc of the length 22 cm subtends at the centre of a circle of radius 15cm. [Take π = 22⁄7 ]
Answer: C
Length of an arc = 22 cm
raduis =
π = 22⁄7
angle (∅)= ?
formular for length of an arc = ∅⁄360 x 2πr
since length of an arc =22 cm
r = 15 and π = 22⁄7
22 = π⁄360 X 2(22)⁄7 X 15
22 = π⁄360 X 44⁄7
22 = π⁄360 X 94.286
22 = 94.2860⁄360
22⁄0.262 = 0.2620⁄0.262
π =22⁄0.262
π = 84
The angle is 84 °
27.
(27.63)2 - (12.37)2
Answer: A
diffence of two squards
(27.63 - 12.37)(27.63 + 12.37)
610.4
Run it to three dicimal place
since 4 is less than 5 you maintian it
28.
Answer: C
Commision on selling price let selling price of house = x
commision on selling price = x x 8⁄100
$0.08x
Eric share of the house = 17,760.00
Total selling price of the house = Agent commision + Eric share x = $117,760 + $0.80x
group like terms
x - 0.08x = 117,760.00
0.92x⁄0.92 = 117760⁄0.92
x = $178,000.000
The total selling price of the house is 128,000
29.
Answer: B
30.
If 2a = √64 and b⁄a = 3, evaluate a2 + b2
Answer: B
Considering 2a = √64
2a = √64
what raise the power 2 will give you 64
26 = √64
2a = 261⁄2
since √ = 1⁄2
2
divide 2 by 6
2a = 23(1)
2a = 22
since the base arethe same we equate the exponent considering b⁄a = 3
b⁄a = 3
cross multiple
b⁄a = 3⁄l
b(1) = 3(a)
b =3a
since a = 3
b = 3(3)
b = 9
therefore a2 + b2 = 92 + 32
81 + 9
90
31.
Answer: C
C
NB: Change the mixed fractions to improper fractions
Explanation:
Multiply the whole number(A) by the denominator(c) and add the numerator(b) to the result(Axc). Divide everything by the denominator(c).
⇒ 21⁄2 = (2x2 + 1)⁄2 = 5⁄2
⇒ 11⁄2 = (1x2 + 1)⁄2 = 3⁄2
Concepts
a = a/1
b = b/1
Explanation:
Change the ÷ to x (multiplication). To do so:
Reciprocate the fraction at the right side of the ÷
Reciprocate means the top goes to the down and the down comes to the top
Explanation:
The numerators multiply each other over the denominators multiplying each other.
Thus the products of the numerators over the products of the denominators
Product means multiplication
Applying all the above concepts
32.
If the sequence x,4,16,y is Geometric Progression (GP), find the ratio of x:y.
64:1
8:1
1:3
1:64
Answer: D
Geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, none-one number called the common ratio
Rules
For the sequence a1, a2, a3, a4 ...
Where a1 = first term
Where a2 = second term
Where a3 = third term
Where a4 = forth term
r = a2 / a1 = a3 / a2 = a4 / a3
Where r = common ratio
Thus the common ratio (r) is the same for all calculations
Hence 4/x = 16/4
4/x = 4
Multiplying both sides by x
x x 4/ x = 4 x x
4x = 4
Dividing both sides by 4
x = 1
16/4 = y/16
4 = y/16
Multiplying both sides by 16
16 x 4 = y/16 x 16
y = 64
⇒ x:y = 1:64
33.
A ladder, 10 m long, touches a side of a building at a height of 8 m. At what height would a ladder with length 12 m touch the building, if it makes the same angle with the ground?
(Assume that the ladder and building are on the same horizontal ground)
10.6 m
10.4 m
10.0 m
9.6 m
Answer: D
Represent them by diagrams
First Ladder
Second Ladder
The angle θ can be calculated using any of SOH, CAH , TOA
SOH = sin θ = Opposite (O) / Hypotenuse (H)
CAH = cos θ = Adjacent (A) / Hypotenuse (H)
TOA = tan θ = Opposite (O) / Adjacent (A)
You have to know which of the sides are known (Adjacent, Opposite or Hypotenuse)
The hypotenuse is the side facing the right angle triangle (90o), the ladder
The adjacent is the side adjacent to the angle θ, ground
The opposite is the side opposite (facing) the angle θ, the building
From both diagrams, the Opposite and Hypotenuse sides are known
Hence SOH can be used to calculate the angle
SOH = sin θ = Opposite / Hypotenuse
For the first ladder
Opposite = 8 m and Hypotenuse = 10 m
Hence sin θ = 8 m / 10 m = 8 / 10
For the second ladder
Opposite = h, Hypotenuse = 12 m
Hence sin θ = h m/ 12 m
Since the angle (θ) is the same as indicated in the question, it means the value of sin θ is the same for both the first and second ladders
Hence 8/10 = h/12
Multiply both sides by 12
12 x 8/10 = h
∴ h = 9.6
34.
Calculate the value x.
Answer: C
Considering ‣RPQ
Sum of interior angle is 180°
∠PRT + ∠RQP + ∠RPQ = 180
x+70+52 = 180
x+122 = 180
x =180 - 122
x=58°
35.
Find the truth set of the equation (x -2)2 + 3 = (x + 1)2 - 6.
{-2}
{-1}
{1}
{2}
Answer: D
Concept
(a+b)2 = a2+2xaxb + b2 = a2+2ab + b2
(a-b)2 = a2-2xaxb + b2 = a2-2ab + b2
Alternatively
(a+b)2 = (a+b)x(a+b) [Expand and simplify]
(a-b)2 = (a-b)x(a-b) [Expand and simplify]
⇒ (x-2)2 = x2-2xxx2+22 = x2-4x+4
(x+1)2 = x2+2xxx1+12 = x2+2x+1
(x-2)2+3 = (x+1)2 - 6
x2-4x+4+3 = x2+2x+1 - 6
x2-4x+7 = x2+2x-5
x2-x2-4x-2x = -5 - 7
-6x = -12
-6x⁄-6 = -12⁄-6
x = 2
36.
Answer: C
37.
Answer: A
3 can go into 48, hence express 48 as a multiplication of 3 and 16
√48 = √(16x3)
Applying √(axb) = √a x √b
⇒ √(16x3) = √16 x √3
√16 = 4
Hence √16 x √3 = 4 x √3 = 4√3
NB: 4√3 - √3 is similar to 4x - 3x, thus you can simplify like terms.
Express 72 too as a multiplication of 24 and 3 so we can cancel the √3 at the numerator and denominator
√72 = √(24 x 3) = √24 x √3
24 = 4 x 6
√24 = √(4 x 6) = √4 x √6
√4 = 2
√24 = 2 x √6 = 2√6
⇒ √72 = √(24 x 3) = √24 x √3 = 2√6 x √3
⇒ √72 /3√3 = 2√6 x √3 / 3√3
√3 cancels each other
Hence 2√6/3
38.
Answer: B
39.
If y varies inversely as x and y = 6 when x = 3, find y when x = 9
4
3
2
1
Answer: C
y ∝ 1 / x
NB: Inversely = 1 / the relation
Varies directly = Relation 1 ∝ Relation 2
Find the constant of the relation
Let k = the constant
y = k x 1/x
When y = 6, x is 3
⇒ 6 = k/3
Multiplying both sides by 3
⇒ k = 6 x 3 = 18
Hence y = 18/x
When x = 9
⇒ y = 18 / 9 = 2
40.
The graph is for the relation: y = d - x.
Find the value of d.
2
1
0
-1
Answer: B
Trace any point on the graph where the line passes and use that x and y coordinates to substitute and find y
The line passes through the points (0,1), hence x = 0 and y = 1
y = d - x ⇒ 1 = d - 0 ⇒ 1 = d ⇒ d = 1
NB: The line also passes through the points (1,0), (4, -3 ). Any point on the line you use should arrive at the same answer
41.
The interior angle of a polygon are 3x°;2x°;4x°; 3x°; and 6x°. Find the size of the smallest angle of the polygon.
Answer: C
Since the angle of the polygon are five
It means the polygon is pentagon
interior angles of polygon = n-2x180⁄n
number of side (n) = 5
Interior angles of pentagon = (5-2)X180⁄5
=3X180⁄5
=540⁄5
108
since 180 is the interior
3x which is the smalllest angle
3x⁄3 = 108⁄3
x =60°
the smallest angle is 60°
42.
Solve the equation: t - 9⁄5 = -11⁄15 .
t = 3⁄5
t = 11⁄15
t = 4⁄5
t = 13⁄15
Answer: B
Change the mixed fraction to an improper fraction.
Explanation:
Multiply the whole number(A) by the denominator(c) and add the numerator(b). Divide everything by the denominator(c).
⇒ 11⁄15 = (1x15+1)⁄15 = 16⁄15
⇒ t - 9⁄5 = -16⁄15
Get ride of the denominators by multiplying both sides by the least common multiple(LCM)
The denominators are 5 and 15,5 can go into 15, hence the LCM is 15
tx15 -15x9⁄5 = 15x-16⁄15
15t - 27 = -16
15t = -16+27
15t = 11
15t⁄15 = 11⁄15
t = 11⁄15
43.
Answer: A
44.
Answer: B
45.
x+2⁄x-2
x-2⁄x+4
x+7⁄x-7
x-7⁄x-7
Answer: A
46.
A rectangular board has length 15 cm and width xcm.If the side are doubled,find its new area
15x cm2
30x cm2
45x cm2
60x cm2
Answer: D
Length = 15cm
width = x cm
If the side are double
New length = 15 X 2
30cm
New width = 2 X x
2x
2x
Area of new rectangular box = Length(L) X (W)
30cm X 2xcm
60x cm2
47.
Make x the subject of the relation:
Answer: C
Move the z to the left side, + becomes - and - becomes +
E - z = kx2/2y
Multiply both sides by 2y to get rid of the 2y at the denominator
2y x (E-z) = kx2
2y(E-z) = kx2
Divide both sides by k
2y(E-z)/k = x2
To get rid of a square on an alphabet, take the square root of both sides
48.
Answer: D
Change the decimal number to a standard form with a complete whole number
∴ 0.064 = 64 x 10-3
⇒ (0.064)-1/3 = (64 x 10-3)-1/3
Applying, (a x b)n = an x bn ⇒ (64 x 10-3)-1/3 = 64-1/3 x 10(-3 x -1/3)
64-1/3 x 10(-3 x -1/3) = 64-1/3 x 101
Note:-3 x -1/3, negatives cancels each other likewise the 3s
Change the 64 to an indices form in order to cancel the powers
64 = 2 x 2 x 2 x 2 x 2 x 2 = 26
⇒ 64-1/3 x 101 = (26)-1/3 x 10
3 goes into 6, 2 times ⇒ 6 x -1/3 = -2
∴ (26)-1/3 x 10 = 2-2 x 10
Applying, a-n = 1 / an
⇒ 2-2 = 1 /22
22 = 4
∴ 2-2 x 10 = 1/4 x 10
Using 2 to cancel each other, 2 goes into 4, 2 times and 10, 5 times
⇒ 1/4 x 10 = 1/2 x 5 = 5/2
Note 1/2 x 5 = 1/2 x 5/1 , every number is divided by 1
Applying a/b x c/d = axc / bxd
⇒ 1/2 x 5/1 = 1 x 5 / 2 x 1 = 5/2
∴ (0.064)-1/3 = 5/2
49.
Factorize completely
(2x2y)(x-y) + (2x -2y)(x + Y).
2(x - y)
2(x - y)(x + Y)
4(x-y
4(x - y)(x + y)
Answer: D
Expand the bracket
2x(x) +2y(x-y) +2x(x+y) 2y(x+y)
2x2 - 2xy + 2xy + - 2y2 + 2x2 + 2xy - 2xy - 2y2
2x2 - 2y2 + 2x2 + 2y2
Group like terms
2x2 - 2y2 - 2y2
factorize 4 out
4(x - y2)
(x - y2) = (x - y)(x + y)
4[(x - y)(x + y)]
50.
In the diagram O is the centre of the circle with raduis 18 cm. If the angle ∠ZXY = 70° , calculate the length of arc ZY. [Take π = 22⁄7
Answer: C
∠ZXY = ∠ZOY = 70°
Length of an arc = Δ⁄2πr
since angle (Δ)π = 22⁄7 and r = 18cm
length of an arc =70⁄360 x 2 x 22⁄7 x 18
cancelation will take place here
70⁄360 x 2 x 22⁄7 x 18
1⁄18 x 22 x 18
22cm
Cross multiple:
5(5y – x) = 1(8y + 3X)
Expand the bracket:
25Y – 5X = 8Y + 3X
Group like terms and add:
25Y- 8Y = 3X + 5X
17Y = 8X
Divide through by 8.
Divide through by Y to make X the numerator
The angle of a sector of a circle with radius 22cm is 60o. If the sector is folded such that the straight edges coincide, forming a cone,calculate correct to one decimal place, the:
(a)
radius;
(b)
height;
(c)
volume;
From the above diagram, the length of the arc becomes the circumference of the base of the cone.
radius of the circle = 22cm
θ = 60o
(a)
⇒ The radius of the cone = 3.67cm
(b)
The height (h) can be calculated using pythagoras theorem
The c (hypotanus is always the side facing the right angle (⊾))
⇒ 222 = 3.672 + h2
222 - 3.672 = h2
470.53 = h2
Take the square root (√) of both sides to get ride of the square
√470.53 = h
21.69 = h
⇒ The height of the cone = 21.69cm
(c)
Volume = ⅓ x 22⁄7x3.672 x 21.69
Volume = 306.05cm3
NB: Unit of area = unit square (e.g cm2,m2) and unit of volume = unit cubic (e.g cm3, m3)
Run the decimal point to the right side to get whole number for easy calculation:
Cancellation will take place here, 5 will go into 15 (3times) and 189 will go into 567 (3times). Which will give us:
Group like terms:
Multiply like terms
9 x 10-6– (-5)
9 x 10-1
NB: Always leave your answer in standard form.
4b2– ab + (a + 9b)2 – a2.
4b2 – ab + (a + 9b)2 – a2
step 1: Expand (a + 9b)2 to give (a + 9b) (a + 9b)
4b2 – ab + (a + 9b) (a + 9b) – a2
step 2: Expand the bracket:
=4b2 – ab + a2 + 18ab + 81b2 – a2
step 3: Group like terms:
= 4b2 + 81b2 – ab + 18ab + a2 - a2
= 4b2 + 81b2 – ab + 18ab
= 85b2 + 17ab
step 4: Factorize 17b out:
= 17b(5b + a)
The first three terms of an Arithmetic Progression (A.P:) are (x + 1),(4x - 2) and (6x - 3)
respectively.If the last term is 18, find the:(a)
vale of x;
(b)
Sum of the terms of the progression.
Notes
Arithmetic Progression
Is a sequence of numbers such that the difference between the consecutive terms is constant.
E.g The sequence 1,3,5,7,9.. is an arithmetic progression with a common difference of 2
an = a1 + (n - 1)d
an = The value of the sequence number of on the nth position.
a1 = the first number in the sequence
d = the common difference
d = an + 1 - an (The difference between two consecutive sequence numbers)
d is the same for all consecutive numbers used for the calculator.
Thus an + 2 - an + 1 = an + 1 - an
For instance, the common difference for the sequence a1, a2, a3, a4, a5, a6 .... will be calculated as:
a2 - a1 = a3 - a2 = a5 - a4 = a6 - a5
NB: The common difference (d) is the higher sequence number value - lower previos sequence number.
For the sequence 1,3,5,7,9
⇒ d = 3 - 1 = 5 - 3 = 7 - 5 = 9 - 7 = 2
Summation of Arithmetic Progression
Sn = n/2 (a1 + an)
Sn = n/2 [2a1 + (n - 1)d]
If the last number is know the summation can also be writen as
Sn = n/2 [a1 + l]
where l = last term
(a)
A. P = (x + 1), (4x - 2) and (6x - 3)
⇒ From the knowledge of common difference calculation
(4x - 2) - (x + 1) = (6x - 3) - (4x - 2)
NB Common difference is the same for all consecutive terms
⇒ 4x - 2 - x - 1 = 6x - 3 - 4x - -2
4x - 2 - x - 1 = 6x - 3 - 4x + 2
3x - 3 = 2x - 1
3x - 2x = -1 + 3
x = 2
(b)
Since we are given the last term, we are to use the summation formula with the last term in it.
Sn = n⁄2 [a1 + l]
We need to calculate the (n) position for the last number (l)
an = a1 + (n - 1)d
x = 2
⇒ first term = x + 1 = 2 + 1 = 3
Second term = 4x - 2 = 4 x 2 - 2 = 6
Third term = 6 x 2 - 3 = 9
The sequence = 3,6,9...18
⇒ a1 = 3
d = 6 - 3 = 9 - 6 = 3
But an = a1+(n-1)d
an = 18
a1 = 3
d = 3
⇒ 18 = 3+(n-1)x3
18 = 3+3n-3
18 = 3n
3n⁄3 = 18⁄3
n = 6
⇒ Sn = n⁄2[a1+l]
Sn = 6⁄2[3+18]
Sn = 63
Since Z = 8, x = 4, y = 1/8, substitute them in place of the variables.