JAMB UTME - Mathematics - 1985
Question 2 Rapport
Détails de la réponse
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
Question 3 Rapport
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?
Détails de la réponse
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
Question 4 Rapport
If \(e^x = 1 + \frac{x}{2} + \frac{x^2}{1.2} + \frac{x^3}{1.2.3} + .....\) Find \(\frac{1}{e^x}\)
Détails de la réponse
Question 5 Rapport
Make c the subject of formula \(v = 1 - \frac{a}{5}\left(b + \frac{3c}{7}\right)\)
Détails de la réponse
Question 6 Rapport
Write h in terms of a, b, c, d if \(a = \frac{b(1-ch)}{a-dh}\)
Détails de la réponse
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
Question 7 Rapport
\((\sqrt{2} + 4\sqrt{2})(\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})\) is equal to
Détails de la réponse
(√3+√2 + (√3+√2 ) = 3 + (√6−√6 ) - 2
= 1
Question 8 Rapport
In the figure, PQ is the tangent from P to the circle QRS with SR as its diameter. If QRS = \( \theta^\circ \) and RQP = \( \phi^\circ \), which of the following relationships between \( \theta^\circ \) and \( \phi^\circ \) is correct
Détails de la réponse
180 = 2ϕ + θ ∘
Question 9 Rapport
In \( \triangle \) XYZ, XY = 13cm, YZ = 9cm, XZ = 11cm and XYZ = \( \theta \). Find cos\( \theta \)o
Détails de la réponse
= 169+81−2126×9
cosθ = 12926×9
= 4378
Question 10 Rapport
Find correct to two decimals places \(100 + \frac{1}{100} + \frac{3}{1000} + \frac{27}{10000}\)
Détails de la réponse
1000,000+100+30+2710000 = 1,000.15710000
= 100.02
Question 11 Rapport
Détails de la réponse
180 + 27 = 207o
Note that bearing is always taken from the north
207o = S 27o W
Question 12 Rapport
Détails de la réponse
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
Question 13 Rapport
In \( \triangle \) XYZ, XKZ = 90O, XK = 15cm, XY = 35cm and YK = 8cm. Find the area of \( \triangle \) XYZ
Détails de la réponse
KZ2 = (25 + 15)(25 - 15) = 400
KZ = √400 = 20
area of XYZ = 12×28×15
= 210 sq. cm
Question 14 Rapport
Détails de la réponse
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
Question 15 Rapport
Détails de la réponse
x2 - 73 = -R3
(x1?76 )2 = -R3 + 4936
?R3 + 4936 = 0
R = 4936 x 31
= 4912
Question 16 Rapport
Détails de la réponse
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
Question 17 Rapport
In the figure, GHIJKLMN is a cube of side a. Find the length of HN.
Détails de la réponse
HJ = √2a2=a√2
HN2 = a2 + (a√2 )2 = a2 + 2a2 = 3a2
HN = √3a2
= a√3 cm
Question 18 Rapport
In the figure, the area of the shaded segment is
Détails de la réponse
Area of triangle = 12×3×3×sin120o
= 92×√32=9√34
Area of shaded portion = 3π−9√34
= 3 π−3√34
Question 19 Rapport
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 = \(86^\circ\) and NQP = \(122^\circ\) find (\(x^\circ\), \(y^\circ\))
Détails de la réponse
180 - 144 = 36∘
x∘ = 180 - (94 + 58)
180 -152 = 28
(x∘ , y∘ ) = (28∘ , 36∘ )
Question 20 Rapport
Détails de la réponse
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
Question 21 Rapport
List all integers satisfying the inequality \( -2 \le 2x - 6 < 4 \)
Détails de la réponse
-2 ≤
2x - 6 < 4 = 2x - 6 < 4
= 2x < 10
= x < 5
2x ≥
-2 + 6 ≥
= x ≥
2
∴ 2 ≤
x < 5 [2, 3, 4]
Question 22 Rapport
Détails de la réponse
= 2√3
(2)23
= 43
1 + sec2 30o = sec 30o
= 1 + 4√3
= 213
Question 23 Rapport
If \(a\left\{\frac{x+1}{x-2}-\frac{x-1}{x+2}\right\}=6x\). Find a in its simplest form
Détails de la réponse
= 6
6xx2−4 = 6x
a = x2 - 4
Question 24 Rapport
22\( \frac{1}{2} \)% of the Nigerian Naira is equal to 17\( \frac{1}{10} \)% of a foreign currency M. What is the conversion rate of the M to the Naira?
Détails de la réponse
M = 17110 %, N = 452
452 x 10171
= 225171
= 1 54171
= 1 1857
Question 25 Rapport
Factorize abx2 + 8y - 4bx - 2axy
Détails de la réponse
abx2 + 8y - 4bx - 2axy = (abx2 - 4bx) + (8y - 2axy)
= bx(ax - 4) 2y(ax - 4) 2y(ax - 4)
= (bx - 2y)(ax - 4)
Question 26 Rapport
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 |
Détails de la réponse
= 4 + 9 + 27
= 13 + 27
= 40
Question 27 Rapport
Solve for (x, y) in the equation 2x + y = 4: x^2 + xy = -12
Détails de la réponse
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)
Question 28 Rapport
Détails de la réponse
x - 2 = 1, x = 3
f(x - 2) = f(1)
= 4(3)2 + 3 + 7
= 36 + 10
= 46
Question 29 Rapport
Détails de la réponse
x = 2 x 1√2
= 2√22
= √2
Question 30 Rapport
Solve the following equation \( \frac{2}{2r-1} - \frac{5}{3} = \frac{1}{r+2} \)
Détails de la réponse
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
Question 31 Rapport
If three numbers P, Q, R are in ratio 6 : 4 : 5, find the value of \( \frac{3p-q}{4q+r} \)
Détails de la réponse
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
Question 32 Rapport
Détails de la réponse
Question 33 Rapport
Arrange the following numbers in ascending order of magnitude \( \frac{6}{7} \), \( \frac{13}{15} \), 0.8650
Détails de la réponse
67
, 1315
, 0.8650
In ascending order, we have 0.8571, 0.8650, 0.8666
i.e. 67
< 0.8650 < 1315
Question 34 Rapport
Détails de la réponse
= 23
P(R1 n R11) = P(both red)
35 x 59
= 13
Question 35 Rapport
Détails de la réponse
35 x = 351
x = 351×53
= 585
Volume of smaller block = 23 x 585
= 234.00
Question 36 Rapport
Détails de la réponse
= 32 x √32 =
Area = 16√3 x 6 = 96 √3 cm2
Question 37 Rapport
Détails de la réponse
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
Question 38 Rapport
In the equation below, Solve for x if all the numbers are in base 2: \(\frac{11}{x} = \frac{1000}{x+101}\)
Détails de la réponse
1000x = 11x + 1111
1000x - 11x = 1111
101x = 1111
x = 1111101
x = 11
Question 39 Rapport
Détails de la réponse
Janet now has 132 x + 105 = x+3153
John's money left = 23 x
= 14(x+315)3
= 23
24x = 3x + 945
∴ x = 45
Question 40 Rapport
The number of goals scored by a football team in 20 matches is shown below:
| No. of goals | 0 | 1 | 2 | 3 | 4 | 5 |
| No. of matches | 3 | 5 | 7 | 3 | 1 | 0 |
What are the values of the mean and the mode respectively?
Détails de la réponse
∑f = 20
∑fx = 35
Mean = ∑fx∑f
= 3520
= 74
= 1.75
Mode = 2
= 1.75, 2
Question 41 Rapport
The factors of 9 - (x2 - 3x - 1)2 are
Détails de la réponse
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)
Question 42 Rapport
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?
Détails de la réponse
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
Question 43 Rapport
If \( \cos \theta = \frac{\sqrt{3}}{2} \) and \( \theta \) is less than 90o. Calculate \( \cos \frac{90-\theta}{\sin^2\theta} \)
Détails de la réponse
= tanθsin2θ
Sin2θ =14
cos(90−θ)sin2θ
= 1√3
= 4√3
Question 44 Rapport
Two points X and Y both on latitude \(60^\circ\text{S}\) have longitude \(147^\circ\text{E}\) and \(153^\circ\text{W}\) respectively. Find to the nearest kilometer the distance between X and Y measured along the parallel of latitude (Take \(2\pi R = 4 \times 10^4\text{ km}\), where R is the radius of the earth)
Détails de la réponse
Longitude difference = 147 + 153 = 300oN
distance between xy = θ360 × 2π R cos60o
= 300360 × 4 × 104 × 12
= 1.667 × 104km (1667 km)
Question 45 Rapport
Détails de la réponse
x−2−1012Y=x3+x−10−20210y=x2+152125
The curves intersect at x = 1
Question 46 Rapport
Détails de la réponse
Prob. of getting one head + prob. of getting 2 heads
= 14 + 24
= 34
Question 47 Rapport
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?
Détails de la réponse
Total students = 120
grade = 18
Zo = 18120
x 3601
= 54o
