MATHEMATICS (POST JAMB)
MATHEMATICS (POST JAMB)
Make q the subject of the formula in the equation mna2−pqb2=1mna2−pqb2=1
Explanation to the correct answer
mna2−pqb2=1mna2−pqb2=1
mna2−1=pqb2mna2−1=pqb2
mn−a2a2=pqb2mn−a2a2=pqb2
pq=b2(mn−a2)a2pq=b2(mn−a2)a2
q=b2(mn−a2)a2pq=b2(mn−a2)a2p
The angle of elevation of the top of a tree from a point on the ground 60m away from the foot of the tree is 78°. Find the height of the tree correct to the nearest whole number.
Explanation to the correct answer
tan78=h60tan78=h60
h=60tan78h=60tan78
h=60×4.705=282.27mh=60×4.705=282.27m
≊≊ 282m to the nearest whole number.
A binary operation ⊗⊗ is defined by m⊗n=mn+m−nm⊗n=mn+m−n on the set of real numbers, for all m, n ∈∈ R. Find the value of 3 ⊗⊗ (2 ⊗⊗ 4).
Explanation to the correct answer
m⊗n=mn+m−nm⊗n=mn+m−n
3 ⊗⊗ (2 ⊗⊗ 4)
2 ⊗⊗ 4 = 2(4) + 2 - 4 = 6
3 otimesotimes 6 = 3(6) + 3 - 6 = 15
The table below shows the number of pupils in a class with respect to their ages. If a pie chart is constructed to represent the age, the angle corresponding to 8 years old is
Age in years 7 8 9 10 11
No of pupils 4 13 30 44 9
Explanation to the correct answer
Total number of pupils : 4 + 13 + 30 + 44 + 9 = 100
The number of 8 - year olds = 13
The angle represented by the 8-year olds on the pie chart = 13100×360°13100×360°
= 46.8°
In a class of 50 students, 40 students offered Physics and 30 offered Biology. How many offered both Physics and Biology?
Explanation to the correct answer
n(Total) = 50
n(Physics) = 40
n(Biology) = 30
Let n(Physics and Biology) = x
n(Physics only) = 40 -x
n(Biology only) = 30 - x
40 - x + 30 - x + x = 50
70 - x = 50
x = 20
Rationalize √2+√3√2−√32+32−3
Find the length of the chord |AB| in the diagram shown above.
Explanation to the correct answer
Length of chord = 2rsin(θ2)2rsin(θ2)
= 2(3)sin(682)2(3)sin(682)
= 6sin346sin34
= 6×0.5596×0.559
= 3.354 cm ≊≊ 3.4 cm
Given sin58°=cosp°sin58°=cosp°, find p.
Explanation to the correct answer
sinθ=cos(90−θ)sinθ=cos(90−θ)
sinθ=cos(90−58)sinθ=cos(90−58)
= cos32cos32
23÷4514+35−1323÷4514+35−13
Explanation to the correct answer
23÷4514+35−1323÷4514+35−13
23÷45=23×5423÷45=23×54
= 5656
14+35−13=15+36−206014+35−13=15+36−2060
= 31603160
∴23÷4514+35−13=56÷3160∴23÷4514+35−13=56÷3160
= 56×603156×6031
= 50315031
If 6x3+2x2−5x+16x3+2x2−5x+1 divides x2−x−1x2−x−1, find the remainder.
If a fair coin is tossed 3 times, what is the probability of getting at least two heads?
Explanation to the correct answer
The outcomes are {HHH, HHT, HTT, HTH, THH, THT, TTH, TTT}
P(at least two heads) = 4848
= 1212
In how many ways can the word MATHEMATICIAN be arranged?
Explanation to the correct answer
MATHEMATICIAN = 13 letters with 2M, 3A, 2T, 2I.
Hence, the word MATHEMATICIAN can be arranged in 13!2!3!2!2!13!2!3!2!2!
= 129729600 ways
find MT+2MMT+2M
Given matrix M = ∣∣
∣∣−2040−16563∣∣
∣∣|−2040−16563|
Explanation to the correct answer
M = ∣∣
∣∣−2040−16563∣∣
∣∣|−2040−16563|
MTT = ∣∣
∣∣−2050−16463∣∣
∣∣|−2050−16463|
2M = ∣∣
∣∣−4080−21210126∣∣
∣∣|−4080−21210126|
MTT + 2M = ∣∣
∣∣−60130−31814189∣∣
∣∣|−60130−31814189|
Find the mean of the data.
Score (x) 0 1 2 3 4 5 6
Freq (f) 5 7 3 7 11 6 7
Explanation to the correct answer
Mean = ∑fx∑f∑fx∑f
= 1504615046
= 3.26
Find the variance
Score (x) 0 1 2 3 4 5 6
Freq (f) 5 7 3 7 11 6 7
Explanation to the correct answer
Variance = ∑f(x−¯x)∑f∑f(x−x¯)∑f
= 170.88846170.88846
= 3.72
The locus of a point which moves so that it is equidistant from two intersecting straight lines is the
Explanation to the correct answer
The locus of a points equidistant from two intersecting straight lines is a pair of bisectors that bisect the angles formed by the two intersecting lines.
From the cyclic quadrilateral MNOP above, find the value of x
Explanation to the correct answer
The sum of two opposite angles of a cyclic quadrilateral = 180°
∴∴ (2x + 18)° + 84° = 180°
2x + 102° = 180° ⟹⟹ 2x = 78°
x = 39°
If 4sin2x−3=04sin2x−3=0, find the value of x, when 0° ≤≤ x ≤≤ 90°
Explanation to the correct answer
4sin2x−3=04sin2x−3=0
4sin2x=3⟹sin2x=344sin2x=3⟹sin2x=34
sinx=√32sinx=32
∴x=sin−1(√32)∴x=sin−1(32)
x = 60°
In the figure above, |CD| is the base of the triangle CDE. Find the area of the figure to the nearest whole number.
Area of rectangle ABCD = length x breadth
= 7 x 4
= 28 cm22
Area of triangle CDE = 1212 base x height
= 12×3×412×3×4
= 6 cm22
Area of the figure = 28 cm22 + 6 cm22
= 34 cm22
Explanation to the correct answer
The marks scored by 30 students in a Mathematics test are recorded in the table below:
Scores (Mark) 0 1 2 3 4 5
No of students 4 3 7 8 6 2
What is the total number of marks scored by the children?
Explanation to the correct answer
Scores (Mark)012345
No of students437862
fx 0 3 14 24 24 10 75
If given two points A(3, 12) and B(5, 22) on a x-y plane. Find the equation of the straight line with intercept at 2.
Explanation to the correct answer
The equation of a straight line is given as y=mx+by=mx+b
where m = the slope of the line
b = intercept
Given points A(3, 12) and B(5, 22), the slope = 22−125−322−125−3
= 102102 = 5
Hence, the equation of the line is y=5x+2y=5x+2.
If P(2, m) is the midpoint of the line joining Q(m, n) and R(n, -4), find the values of m and n.
Explanation to the correct answer
Q(m, n) and R(n, -4)
Midpoint : P(2, m)
⟹(m+n2,n−42)=(2,m)⟹(m+n2,n−42)=(2,m)
m+n=2×2⟹m+n=4...(i)m+n=2×2⟹m+n=4...(i)
n−4=2×m⟹n−4=2m...(ii)n−4=2×m⟹n−4=2m...(ii)
Solving (i) and (ii) simultaneously,
m = 0 and n = 4.
If ∣∣∣2−4x9∣∣∣=58|2−4x9|=58, find the value of x.
Explanation to the correct answer
∣∣∣2−4x9∣∣∣=58|2−4x9|=58
⟹(2×9)−(−4×x)=58⟹(2×9)−(−4×x)=58
18+4x=58⟹4x=58−18=4018+4x=58⟹4x=58−18=40
x=10x=10
If y=6x3+2x−2−x−3y=6x3+2x−2−x−3, find dydxdydx.
Explanation to the correct answer
y=6x3+2x−2−x−3y=6x3+2x−2−x−3
dydx=18x2−4x−3+3x−4dydx=18x2−4x−3+3x−4
ddx[log(4x3−2x)]ddx[log(4x3−2x)] is equal to
Explanation to the correct answer
ddx[log(4x3−2x)]ddx[log(4x3−2x)] ... (1)
Let u = 4x33 - 2x.
ddx(log(4x3−2x))=(ddu)(dudx)ddx(log(4x3−2x))=(ddu)(dudx)
ddu(logu)ddu(logu) = 1u1u
dudx=12x2−2dudx=12x2−2
∴ddx[log(4x3−2x)]=12x2−2u∴ddx[log(4x3−2x)]=12x2−2u
= 12x2−24x3−2x12x2−24x3−2x
If f(x)=3x3+4x2+x−8f(x)=3x3+4x2+x−8, what is the value of f(-2)?
Explanation to the correct answer
f(x)=3x3+4x2+x−8f(x)=3x3+4x2+x−8
f(−2)=3(−2)3+4(−2)2+(−2)−8f(−2)=3(−2)3+4(−2)2+(−2)−8
= −24+16−2−8−24+16−2−8
= -18
Solve for x in 4x−63≤3+2x24x−63≤3+2x2
Explanation to the correct answer
4x−63≤3+2x24x−63≤3+2x2
2(4x - 6) ≤≤ 3(3 + 2x)
8x - 12 ≤≤ 9 + 6x
8x - 6x ≤≤ 9 + 12
2x ≤≤ 21
x≤212x≤212
Solve the inequality: -7 ≤≤ 9 - 8x < 16 - x
Explanation to the correct answer
-7 ≤≤ 9 - 8x < 16 - x
-7 ≤≤ 9 - 8x and 9 - 8x < 16 - x
-7 - 9 ≤≤ -8x and -8x + x < 16 - 9
-16 ≤≤ -8x and -7x < 7
∴∴ x ≤≤ 2 and -1 < x
-1 < x ≤≤ 2.
The nth term of a sequence is given by 22n−12n−1. Find the sum of the first four terms.
Explanation to the correct answer
Tn=22n−1Tn=22n−1
T1=22(1)−1T1=22(1)−1
= 2
T2=22(2)−1T2=22(2)−1
= 8
T3=22(3)−1T3=22(3)−1
= 32
T4=22(4)−1T4=22(4)−1
= 128
T1+T2+T3+T4=2+8+32+128T1+T2+T3+T4=2+8+32+128
= 170
Integrate ∫2−1(2x2+x)dx∫−12(2x2+x)dx
Explanation to the correct answer
∫2−1(2x2+x)dx∫−12(2x2+x)dx
= [2x2+13+x1+12]2−1[2x2+13+x1+12]−12
= [2x33+x22]2−1[2x33+x22]−12
= (2(2)33+222)−(2(−1)33+(−1)22)(2(2)33+222)−(2(−1)33+(−1)22)
= (163+2)−(−23+12)(163+2)−(−23+12)
= 223−(−16)223−(−16)
= 223+16223+16
= 152152
= 712
If P varies inversely as the square root of q, where p = 3 and q = 16, find the value of q when p = 4.
Explanation to the correct answer
p∝1√qp∝1q
⟹p=k√q⟹p=kq
when p = 3, q = 16.
3=k√163=k16
k=3×4=12k=3×4=12
∴p=12√q∴p=12q
when p = 4,
4=12√q⟹√q=1244=12q⟹q=124
√q=3⟹q=32q=3⟹q=32
q=9q=9
Tade bought 200 mangoes at 4 for ₦2.50. 30 out of the mangoes got spoilt and the remaining were sold at 2 for ₦2.40. Find the percentage profit or loss.
Explanation to the correct answer
200 mangoes at 4 for N2.50
⟹⟹ Total cost price = 2004×N2.502004×N2.50
= N 125.00
Since 30 mangoes got spoilt ⟹⟹ Left over = 200 - 30
= 170 mangoes
170 mangoes at 2 for N 2.40
⟹⟹ Total selling point = 1702×N2.401702×N2.40
= N 204.00
Profit : N (204.00 - 125.00) = N 79.00
% profit = 79125×10079125×100
= 63.2% profit.
The simple interest on ₦8550 for 3 years at x% per annum is ₦4890. Calculate the value of x to the nearest whole number.
Explanation to the correct answer
S.I = PRT100PRT100
⟹⟹ N 4890 = 8550×3×x1008550×3×x100
x=4890×1008550×3x=4890×1008550×3
x=19.06x=19.06
x≊19x≊19
Simplify 81−34−34 x 251212 x 2432525
Explanation to the correct answer
81−34−34 x 251212 x 2432525
= (4√81)−3×√25×(5√243)2(814)−3×25×(2435)2
= 5×323−35×323−3
= 5353
Find the value of (0.5436)30.017×0.219(0.5436)30.017×0.219 to 3 significant figures.
Explanation to the correct answer
(0.5436)30.017×0.219(0.5436)30.017×0.219
= 0.160630.017×0.2190.160630.017×0.219
= 43.1 (to 3 s.f)
If S = (4t + 3)(t - 2), find ds/dt when t = 5 secs.
Explanation to the correct answer
s=(4t+3)(t−2)s=(4t+3)(t−2)
dsdt=(4t+3)(1)+(t−2)(4)dsdt=(4t+3)(1)+(t−2)(4)
= 4t+3+4t−84t+3+4t−8
= 8t - 5
dsdt(t=5secs)=8(5)−5dsdt(t=5secs)=8(5)−5
= 40 - 5
= 35 units per sec
The angles of a polygon are given by 2x, 5x, x and 4x respectively. The value of x is
Explanation to the correct answer
Since there are 4 angles given, the polygon is a quadrilateral.
Sum of angle in a quadrilateral = 360°
∴∴ 2x + 5x + x + 4x = 360°
12x = 360°
x = 30°
The weight of a day-old chick was measured to be 0.21g. If the actual weight of the chick is 0.18g, what was the percentage error in the measurement?
Explanation to the correct answer
Actual weight = 0.18g
Error = 0.21g - 0.18g
= 0.03g
% error = 0.030.18×1000.030.18×100
= 16.7%
Evaluate (60.32÷20.084)−1(60.32÷20.084)−1 correct to 1 decimal place.
Explanation to the correct answer
(60.32÷20.084)−1(60.32÷20.084)−1
= (60032÷200084)−1(60032÷200084)−1
= (60032×842000)−1(60032×842000)−1
= (6380)−1(6380)−1
= 80638063
= 1.3 (to 1 decimal place)
If 2x+yx+y = 16 and 4x−y=132x−y=132, find the values of x and y.
Explanation to the correct answer
2x+yx+y = 16 ; 4x−yx−y = 132132.
⟹2x+y=24⟹2x+y=24
x+y=4...(1)x+y=4...(1)
22(x−y)=2−522(x−y)=2−5
22x−2y=2−522x−2y=2−5
⟹2x−2y=−5...(2)⟹2x−2y=−5...(2)
Solving the equations (1) and (2) simultaneously, we have
x = 3434 and y = 134
Simplify 0.0839×6.3815.44 to 2 significant figures.
Find the value of x and y in the simultaneous equation: 3x + y = 21; xy = 30
Explanation to the correct answer
3x + y = 21 ... (i);
xy = 30 ... (ii)
From (ii), y=30x. Putting the value of y in (i), we have
3x + 30x = 21
⟹ 3x2 + 30 = 21x
3x2 - 21x + 30 = 0
3x2 - 15x - 6x + 30 = 0
3x(x - 5) - 6(x - 5) = 0
(3x - 6)(x - 5) = 0
3x - 6 = 0 ⟹ x = 2.
x - 5 = 0 ⟹ x = 5.
If x = 2, y = 302 = 15;
If x = 5, y = 305 = 6.
Points X and Y are 20km North and 9km East of point O, respectively. What is the bearing of Y from X? Correct to the nearest degree.
Explanation to the correct answer
tanθ=920=0.45
θ=tan−1(0.45)
= 24.23°
∴ The bearing of Y from X = 180° - 24.23°
= 155.77°
= 156° (to the nearest degree)
If P=(Q(R−T)15)13, make T the subject of the formula.
Explanation to the correct answer
P=(Q(R−T)15)13
P3=Q(R−T)15
Q(R−T)=15P3
R−T=15P3Q
T=R−15P3Q
In the diagram above, O is the centre of the circle ABC, < ABO = 26° and < BOC = 130°. Calculate < AOC.
Explanation to the correct answer
< BAC = 1302 (angle subtended at the centre)
< BAC = 65°
Also, x = 26° (theorem)
y = 65° - 26° = 39°
< AOC = 180° - (39° + 39°)
= 102°
Each of the interior angles of a regular polygon is 140°. Calculate the sum of all the interior angles of the polygon.
Explanation to the correct answer
Since each interior angle = 140°;
Each exterior angle = 180° - 140° = 40°
Number of sides of the polygon = 360°40°
= 9
Sum of angles in the polygon = 140° x 9
= 1260°
A man bought a car newly for ₦1,250,000. He had a crash with the car and later sold it at the rate of ₦1,085,000. What is the percentage gain or loss of the man?
Explanation to the correct answer
Cost price of the car = N 1,250.00
Selling price = N 1,085.00
Loss = N (1250 - 1085)
= N 165.00
% loss = 1651250×100
= 13.2% loss
If the volume of a frustrum is given as V=πh3(R2+Rr+r2), find dVdR.
Explanation to the correct answer
V=πh3(R2+Rr+r2)
V=πR2h3+πRrh3+πr2h3
dVdR=2πRh3+πrh3
= π3(2R+r)
Express (0.0439÷3.62) as a fraction.
Explanation to the correct answer
(0.0439÷3.62)
= 0.01213
≊ 0.012
= 121000
If 251−x×5x+2÷(1125)x=625−1, find the value of x.
Explanation to the correct answer
251−x×5x+2÷(1125)x=625−1
(52)(1−x)×5(x+2)÷(5−3)x=(54)−1
52−2x×5x+2÷5−3x=5−4
5(2−2x)+(x+2)−(−3x)=5−4
Equating bases, we have
2−2x+x+2+3x=−4
4+2x=−4⟹2x=−4−4
2x=−8
x=−4