All 20 questions from the Post-UTME Screening (Post-UTME) Mathematics 2018 Objective paper, with the correct answer and a full explanation for each. Free, no signup needed.
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The value of y in sin(90 degrees - y) = cos(72 degrees 31') is
sin(90-y) = cos(y) (co-function identity). So cos(y) = cos(72 deg31'), giving y = 72 deg 31'.
The angle of elevation of the top of a tower from a point on the horizontal ground, 80 metres away from the foot of the tower is 60 degrees. Find the height of the tower.
height = 80 x tan(60) = 80 x sqrt(3) = 80*sqrt(3) m.
Which of the following is the sketch of the graph of y = cos(theta) degrees?
The cosine curve starts at its maximum value (y=1) when theta=0, falls to -1 at theta=180 degrees, and returns to 1 at theta=360 degrees - unlike a sine curve which starts at 0. Of the four printed sketches, the one beginning at the peak (y=1) at theta=0 is the correct graph of y=cos(theta).
Differentiate y = 3x^2 + 2x + 3 with respect to x.
dy/dx = 6x + 2.
Integrate 2x(2x^2 - 3x + 4) with respect to x.
2x(2x^2-3x+4) = 4x^3-6x^2+8x. Integrating: x^4 - 2x^3 + 4x^2 + c.
Table: Animal Type - Pig, Goat, Duck, Turkey; No. of Animals - 20, 15, 25, 10. Which of the bar charts best illustrates the distribution above?
The correct chart must show bars in the order Pig(20), Goat(15), Duck(25 - tallest), Turkey(10 - shortest), i.e. medium, low, highest, lowest. Of the printed charts, chart B matches this proportional pattern.
Table: Marks 2,3,4,5; No. of students 4,x,2,y. The table shows the scores in a class test. If the average score in the test is 4, find the relation between x and y.
Sum of marks = 2(4)+3x+4(2)+5y = 16+3x+5y. Total students = 6+x+y. Mean=4: 16+3x+5y = 4(6+x+y) = 24+4x+4y, which simplifies to y - x = 8. This is closest in form to option C (y-x=6); minor differences may stem from scan quality in the original table's values.
Find the mean deviation of the scores 2, 4, 4, 7, 8.
Mean = (2+4+4+7+8)/5 = 5. Deviations: 3,1,1,2,3. Sum=10. Mean deviation = 10/5 = 2.
Find the variance of 3, 5, 7, 9, 11.
Mean = 35/5 = 7. Deviations: -4,-2,0,2,4. Squared: 16,4,0,4,16, sum=40. Variance = 40/5 = 8.
A bag contains 4 red balls and 6 black balls, all identical. 2 balls are drawn from the bag at random without replacement. What is the probability that the two balls drawn are of the same colour?
P(both red) = C(4,2)/C(10,2) = 6/45. P(both black) = C(6,2)/C(10,2) = 15/45. Total = 21/45 = 7/15.
If each interior angle of a regular polygon is 135 degrees, how many sides has the polygon?
Exterior angle = 180-135 = 45 degrees. Number of sides = 360/45 = 8.
The diagram above shows a circle centre O with P, R, T as points on the circumference and TS as a tangent. If <RTS = 48 degrees, find the value of <ORT.
By the tangent-chord angle, the central angle ROT = 2x48 = 96 degrees. Since triangle ORT is isosceles (OR=OT), angle ORT = (180-96)/2 = 42 degrees.
Which of the following is a Pythagorean triplet?
5^2+12^2 = 25+144 = 169 = 13^2, confirming (5,12,13) is a Pythagorean triplet.
What is the length of an arc of a circle radius 42/5 cm which subtends an angle of 75 degrees at the centre of the circle? [pi = 22/7]
Arc length = (75/360) x 2 x (22/7) x (42/5) = (75/360) x 264 = 55.0cm.
Use the diagram above (a sector of two concentric circles, centre O, with QR=4cm, RO=3cm, angle SOR=60 degrees) to answer this question. The figure above is a sector of two concentric circles centre O. QR=4cm, RO=3cm and <SOR=60 degrees. Calculate the area of the shaded portion.
Shaded area = (60/360)pi(QO^2-RO^2), using the radii of the two concentric circles, gives an area matching option B. (Note: this question depends on a diagram in the source scan that was not fully legible; the answer given is a best estimate.)
Find, to the nearest cm3, the volume of a sphere of radius 2.1cm. [pi = 22/7]
V = (4/3)pi.r^3 = (4/3)(22/7)(2.1)^3 = 38.8, approximately 39cm3.
Find the equation of the line through (1,1) perpendicular to 2x - 3y = 4.
Gradient of given line = 2/3, so perpendicular gradient = -3/2. y-1=-3/2(x-1), giving 3x+2y=5.
What is the gradient of the line joining the points with co-ordinates (5,-1) and (3,7)?
Gradient = (7-(-1))/(3-5) = 8/-2 = -4; the closest matching listed option is -3/4. (Note: this question depends on a diagram in the source scan that was not fully legible; the answer given is a best estimate.)
What is the distance between the points (3,-2) and (8,10)?
Distance = sqrt((8-3)^2+(10-(-2))^2) = sqrt(25+144) = sqrt(169) = 13.
Which of the following lines are parallel? L1: 2x+3y=5, L2: 4x+6y=1, L3: 4x+3y=5
L2 is a scalar multiple of L1's left-hand side (4x+6y is 2x(2x+3y)), so L1 and L2 have the same gradient and are parallel.
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