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Frage 1 Bericht
Solve for (x, y) in the equation 2x + y = 4: x^2 + xy = -12
Antwortdetails
2x + y = 4......(i)
x^2 + xy = -12........(ii)
from eqn (i), y = 4 - 2x
= x2 + x(4 - 2x)
= -12
x2 + 4x - 2x2 = -12
4x - x2 = -12
x2 - 4x - 12 = 0
(x - 6)(x + 2) = 0
sub. for x = 6, in eqn (i) y = -8, 8
=(6,-8); (-2, 8)
Frage 2 Bericht
Find the area of a regular hexagon inscribed in a circle of radius 8cm
Antwortdetails
Area of a regular hexagon = 8 x 8 x sin 60o
= 32 x √32
=
Area = 16√3
x 6 = 96 √3
cm2
Frage 3 Bericht
If a{x+1x−2−x−1x+2 } = 6x. Find a in its simplest form
Antwortdetails
a{x+1x−2−x−1x+2
} = a{(x+1)(x+2)−(x−1)(x−2)(x−2)(x+2)
}
= 6
6xx2−4
= 6x
a = x2 - 4
Frage 4 Bericht
In a class of 120 students, 18 of them scored an A grade in mathematics. If the section representing the A grade students on a pie chart has angle Zo at the centre of the circle, what is Z?
Antwortdetails
Total students = 120
grade = 18
Zo = 18120
x 3601
= 54o
Frage 5 Bericht
Write h in terms of a, b, c, d if a = b(1?ch)a?dh
Antwortdetails
a = b(1?ch)a?dh
a = b?bch1?dh
= a - adh
= b - bch
a - b = bch + adn
a - b = adh
a - b = h(ad - bc)
h = a?bad?bc
Frage 6 Bericht
If f(x - 2) = 4x2 + x + 7, find f(1)
Antwortdetails
f(x - 2) = 4x2 + x + 7
x - 2 = 1, x = 3
f(x - 2) = f(1)
= 4(3)2 + 3 + 7
= 36 + 10
= 46
Frage 7 Bericht
PRSQ is a trapezium of area 14cm2 in which PQ||RS. If PQ = 4cm and SR = 3cm, Find the area of SQR in cm2
Antwortdetails
Area of trapezium = 14cm2
Area of trapezium = 12
(a + b)h
14 = 12
(4 + 3)h
14 = 72
h
h = 14×27
= 4
Area of SQR = 12
(3 x 4)
= 122
= 6.0
Frage 8 Bericht
If the hypotenuse of right angled isosceles triangle is 2, what is the length of each of the other sides?
Antwortdetails
45o = x2
, Since 45o = 1√2
x = 2 x 1√2
= 2√22
= √2
Frage 9 Bericht
Antwortdetails
Simple Space: (1, 2, 3, 4, 5, 6, 7, 8, 9, 10 = 10)
Prime: (2, 3, 5, 7)
multiples of 3: (3, 6, 9)
Prime or multiples of 3: (2, 3, 5, 6, 7, 9 = 6)
Probability = 610
= 35
Frage 10 Bericht
In the figure, GHIJKLMN is a cube of side a. Find the length of HN.
Antwortdetails
HJ2 = a2 + a2 = 2a2
HJ = √2a2=a√2
HN2 = a2 + (a√2 )2 = a2 + 2a2 = 3a2
HN = √3a2
= a√3 cm
Frage 11 Bericht
Two points X and Y both on latitude 60oS have longitude 147oE and 153oW respectively. Find to the nearest kilometer the distance between X and Y measured along the parallel of latitude (Take 2π R = 4 x 104km, where R is the radius of the earth)
Antwortdetails
Length of an area = θ360
× 2π
r
Longitude difference = 147 + 153 = 300oN
distance between xy = θ360
× 2π
R cos60o
= 300360
× 4 × 104 × 12
= 1.667 × 104km (1667 km)
Frage 12 Bericht
Find the missing value in the table below
x |
-4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
y=4?3x?x^3
|
80 | 18 | 8 | 4 | 0 | -10 | -32 |
Antwortdetails
When x = -3, y = 4 - 3(3) - (-3)3
= 4 + 9 + 27
= 13 + 27
= 40
Frage 13 Bericht
If cos θ = √32 and θ is less than 90o. Calculate cos90−θsin2θ
Antwortdetails
cos90−θsin2θ
= tanθsin2θ
Sin2θ
=14
cos(90−θ)sin2θ
= 1√3
= 4√3
Frage 14 Bericht
Without using table, calculate the value of 1 + sec2 30o
Antwortdetails
1 + sec2 30o = sec 30o
= 2√3
(2)23
= 43
1 + sec2 30o = sec 30o
= 1 + 4√3
= 213
Frage 15 Bericht
In the figure, MNQP is a cyclic quadrilateral. MN and Pq are produced to meet at X and NQ and MP are produced to meet at Y. If MNQ = 86∘
and NQP = 122∘
find (x∘
, y∘
)
Antwortdetails
y∘ = 180∘ - (86∘ + 58∘ )
180 - 144 = 36∘
x∘ = 180 - (94 + 58)
180 -152 = 28
(x∘ , y∘ ) = (28∘ , 36∘ )
Frage 16 Bericht
List all integers satisfying the inequality -2 ≤
2 x -6 < 4
Antwortdetails
-2 ≤
2x - 6 < 4 = 2x - 6 < 4
= 2x < 10
= x < 5
2x ≥
-2 + 6 ≥
= x ≥
2
∴ 2 ≤
x < 5 [2, 3, 4]
Frage 17 Bericht
In the figure, PQ is the tangent from P to the circle QRS with SR as its diameter. If QRS = θ
∘
and RQP = ϕ
∘
, which of the following relationships between θ
∘
and ϕ
∘
is correct
Antwortdetails
180 - ϕ ∘ = θ ∘ + ϕ ∘ (Sum of opposite interior angle equal to its exterior angle)
180 = 2ϕ + θ ∘
Frage 18 Bericht
At what real value of x do the curves whose equations are y = x3 + x and y = x2 + 1 intersect?
Antwortdetails
y = x3 + x and y = x2 + 1
x−2−1012Y=x3+x−10−20210y=x2+152125
The curves intersect at x = 1
Frage 19 Bericht
(√2+4√2 ) (√3+√2 ) (√3+√2 ) is equal to
Antwortdetails
(√2+4√2
) (√3+√2
) (√3+√2
)
(√3+√2
+ (√3+√2
) = 3 + (√6−√6
) - 2
= 1
Frage 20 Bericht
Solve the simultaneous equations 2x - 3y + 10 = 10x - 6y = 5
Antwortdetails
2x - 3y = -10; 10x - 6y = -5
2x - 3y = -10 x 2
10x - 6y = 5
4x - 6y = -20 .......(i)
10x - 6y = 5.......(ii)
eqn(ii) - eqn(1)
6x = 15
x = 156
= 52
x = 212
Sub. for x in equ.(ii) 10(52
) - 6y = 5
y = 312
Frage 21 Bericht
A bag contains 4 white balls and 6 red balls. Two balls are taken from the bag without replacement. What is the probability that they are both red?
Antwortdetails
P(R1) = 610
= 23
P(R1 n R11) = P(both red)
35
x 59
= 13
Frage 22 Bericht
If three numbers P, Q, R are in ratio 6 : 4 : 5, find the value of 3p−q4q+r
Antwortdetails
P : Q : r = 6 : 4 : 5
5 = 6 + 4 + 5
= 15
P = 615
, q = 415
, r = 515
= 13
To find 3p−q4q+r
3p - q = 3 x 615
- 415
1815
- 415
= 1415
∴ 4q + r = 4 x 415
+ 515
1615
= 1615
+ 515
= 2115
1415
x 1521
= 1421
= 23
Frage 23 Bericht
If the quadratic function 3x2 - 7x + R is a perfect square, find R
Antwortdetails
3x2 - 7x + R. Computing the square, we have
x2 - 73
= -R3
(x1?76
)2 = -R3
+ 4936
?R3
+ 4936
= 0
R = 4936
x 31
= 4912
Frage 24 Bericht
If ex = 1 + x2 + x21.2 + x31.2.3 + .....Find 1ex
Antwortdetails
ex = 1 + x2 + x21.2 + x31.2.3 + x41.23.4
Frage 25 Bericht
In △
XYZ, XKZ = 90O, XK = 15cm, XY = 35cm and YK = 8cm. Find the area of △
XYZ
Antwortdetails
By Pythagoras, KZ2 = 252 - 52
KZ2 = (25 + 15)(25 - 15) = 400
KZ = √400 = 20
area of XYZ = 12×28×15
= 210 sq. cm
Frage 26 Bericht
In a restaurant, the cost of providing a particular type of food is partly constant and partially inversely proportional to the number of people. If cost per head for 100 people is 30k and the cost for 40 people is 60k, Find the cost for 50 people?
Antwortdetails
C = a + k
1N
= c
= aN+kN
CN = aN + K
30(100) = a(100) + k
3000 = 100a + k.......(i)
60(40) = a(40) + k
2400 = 40a + k.......(ii)
eqn (i) - eqn (ii)
600 = 60a
a = 10
subt. for a in eqn (i) 3000 = 100(10) + K
3000 - 1000 = k
k = 2000
CN = 10N + 2000. when N = 50,
50C = 10(50) + 2000
50C = 500 + 2000
C = 250050
= 50k
Frage 27 Bericht
a sum of money was invested at 8% per annum simple interest. If after 4 years the money amounts to N330.00. Find the amount originally invested
Antwortdetails
S.I = PTR100
T = 4yrs, R = 8%, a = N330.00
330 - P = PTR100
, A = P + I
i.e. A = P + PTR100
330 = P + P(4)(8)100
33000 = 32P + 100p
132P = 33000
P = N250.00
Frage 28 Bericht
In the figure, the area of the shaded segment is
Antwortdetails
Area of sector = 120360×π×(3)2=3π
Area of triangle = 12×3×3×sin120o
= 92×√32=9√34
Area of shaded portion = 3π−9√34
= 3 π−3√34
Frage 29 Bericht
Factorize abx2 + 8y - 4bx - 2axy
Antwortdetails
abx2 + 8y - 4bx - 2axy = (abx2 - 4bx) + (8y - 2axy)
= bx(ax - 4) 2y(ax - 4) 2y(ax - 4)
= (bx - 2y)(ax - 4)
Frage 31 Bericht
Find correct to two decimals places 100 + 1100 + 31000 + 2710000
Antwortdetails
100 + 1100
+ 31000
+ 2710000
1000,000+100+30+2710000
= 1,000.15710000
= 100.02
Frage 32 Bericht
In △
XYZ, XY = 13cm, YZ = 9cm, XZ = 11cm and XYZ = θ
. Find cosθ
o
Antwortdetails
cosθ
= 132+92−1122(13)(9)
= 169+81−2126×9
cosθ
= 12926×9
= 4378
Frage 33 Bericht
Find the values of p for which the equation x2 - (p - 2)x + 2p + 1 = 0
Antwortdetails
Equal roots implies b2 - 4ac = 0
a = 1b = - (p - 2), c = 2p + 1
[-(p - 2)]2 - 4 x 1 x (2p + 1) = 0
p2 - 4p + 4 - 4(2p + 1) = 0
p2 - 4p = 4 - 8p - 4 = 0
p2 - 12p = 0
p(p - 12) = 0
p = 0 or 12
Frage 34 Bericht
John gives one-third of his money to Janet who has ₦105.00. He then finds that his money is reduced to one-fourth of what Janet now has. Find how much money john has at first
Antwortdetails
Let x be John's money, Janet already had ₦105, 13
of x was given to Janet
Janet now has 132
x + 105 = x+3153
John's money left = 23
x
= 14(x+315)3
= 23
24x = 3x + 945
∴ x = 45
Frage 35 Bericht
The factors of 9 - (x2 - 3x - 1)2 are
Antwortdetails
9 - (x2 - 3x - 1)2 = [3 - (x2 - 3x - 1)] [3 + (x2 - 3x - 1)]
= (3 - x2 + 3x + 1)(3 + x2 - 3x - 1)
= (4 + 3x - x2)(x2 - 3x + 2)
= (4 - x)(1 + x)(x - 1)(x - 2)
= -(x - 4)(x + 1) (x - 1)(x - 2)
Frage 36 Bericht
Solve the following equation 22r−1 - 53 = 1r+2
Antwortdetails
22r−1
- 53
= 1r+2
22r−1
- 1r+2
= 53
2r+4−2r+12r−1(r+2)
= 53
5(2r+1)(r+2)
= 53
5(2r - 1)(r + 2) = 15
(10r - 5)(r + 2) = 15
10r2 + 20r - 5r - 10 = 15
10r2 + 15r = 25
10r2 + 15r - 25 = 0
2r2 + 3r - 5 = 0
(2r2 + 5r)(2r + 5) = r(2r + 5) - 1(2r + 5)
(r - 1)(2r + 5) = 0
r = 1 or −52
Frage 37 Bericht
The bearing of a bird on a tree from a hunter on the ground is ₦72oE. What is the bearing of the hunter from the birds?
Antwortdetails
Bearing of Hunter from Bird is
180 + 27 = 207o
Note that bearing is always taken from the north
207o = S 27o W
Frage 38 Bericht
Which of these angles can be constructed using ruler and a pair of compasses only?
Frage 39 Bericht
Arrange the following numbers in ascending order of magnitude 67
, 1315
, 0.8650
Antwortdetails
67
, 1315
, 0.8650
In ascending order, we have 0.8571, 0.8650, 0.8666
i.e. 67
< 0.8650 < 1315
Frage 40 Bericht
If 32y + 6(3y) = 27. Find y
Antwortdetails
32y + 6(3y) = 27
This can be rewritten as (3y)2 + 6(3y) = 27
Let 3y = x
x2 + 6x - 27 = 0
(x + 9)(x - 3) = 0
when x - 3 = 0, x = 3
sub. for x in 3y = x
3y = 3
log33 = y
y = 1
Frage 41 Bericht
The ratio of the length of two similar rectangular blocks is 2 : 3. If the volume of the larger block is 351cm3, then the volume of the other block is
Antwortdetails
Let x represent total vol. 2 : 3 = 2 + 3 = 5
35
x = 351
x = 351×53
= 585
Volume of smaller block = 23
x 585
= 234.00
Frage 42 Bericht
A solid sphere of radius 4cm has a mass of 64kg. What will be the mass of a shell of the same metal whose internal and external radii are 2cm and 3cm respectively?
Antwortdetails
1√3(12)2
= 4√3
= √3√3
= 4√3√3
m = 64kg, V = 4πr33
= 4π(4)33
= 256π3
x 10-6m3
density(P) = MassVolume
= 64256π3×10−6
= 64×3×10−6256
= 34×10−6
m = PV = 34π×10−6
x 43
π
[32 - 22] x 10-6
34×10−6
x 43
x 5 x 10-6
= 5kg
Frage 43 Bericht
The number of goals scored by a football team in 20 matches is shown below:
No. of goals012345No. of matches357310
What are the values of the mean and the mode respectively?
Antwortdetails
xffx03015527143412414500
∑f
= 20
∑fx
= 35
Mean = ∑fx∑f
= 3520
= 74
= 1.75
Mode = 2
= 1.75, 2
Frage 44 Bericht
2212 % of the Nigerian Naira is equal to 17110 % of a foreign currency M. What is the conversion rate of the M to the Naira?
Antwortdetails
N = 2212
%, M = 17110
%
M = 17110
%, N = 452
452
x 10171
= 225171
= 1 54171
= 1 1857
Frage 45 Bericht
In the equation below, Solve for x if all the numbers are in base 2: 11x = 1000x+101
Antwortdetails
11x
= 1000x+101
= 11(x + 101)
1000x = 11x + 1111
1000x - 11x = 1111
101x = 1111
x = 1111101
x = 11
Frage 46 Bericht
If two fair coins are tossed, what is the probability of getting at least one head?
Antwortdetails
Prob. of getting at least one head
Prob. of getting one head + prob. of getting 2 heads
= 14
+ 24
= 34
Frage 47 Bericht
Find x if log9x = 1.5
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