All 50 questions from the Post-UTME Screening (Post-UTME) Mathematics 2019 Objective paper, with the correct answer and a full explanation for each. Free, no signup needed.
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The turning point of the curve y = 5 - 8x - 2x^2 occurs at
dy/dx = -8 - 4x = 0, so x = -2. y = 5-8(-2)-2(-2)^2 = 5+16-8 = 13. Turning point (-2,13).
Integrate x^2(x+2) with respect to x
Integral of (x^3+2x^2) dx = x^4/4 + 2x^3/3 + c.
The area under the curve y = x^2 between x=0 and x=2 is
Integral of x^2 from 0 to 2 = [x^3/3] = 8/3 - 0 = 8/3 cm^2.
Pie chart: export commodities Rubber, Palm Kernel (angle marked 100 deg boundary), Wheat, Cocoa (angle marked 140 deg boundary). Wheat accounted for 13 1/3% of the exports for the year. Find the size of the angle representing Palm Kernel.
Wheat's angle = 13 1/3% x 360 = 48 degrees. With the marked angles for Rubber (100) and Cocoa (140) accounted for, the remaining angle for Palm Kernel = 360 - 100 - 140 - 48 = 72 degrees.
If the mean of the data 40, 38, y, y, x, 32 is 36, what is the relationship between x and y?
Sum = 6 x 36 = 216. 40+38+y+y+x+32=216, so 110+2y+x=216, giving 2y+x=106.
The median of the numbers 8, 5, 11, 6, 8, 13, 6 is
Sorted: 5,6,6,8,8,11,13 (7 values). Median = 4th value = 8.
In how many ways can a committee of two boys and one girl be selected from a group of five boys and three girls?
Number of ways = C(5,2) x C(3,1) = 10 x 3 = 30.
Given two dependent events x and y such that P(x) = P(y) = 1/2, what is P(x and y)?
P(x and y) = P(x) x P(y) = 1/2 x 1/2 = 1/4.
In how many ways can the letters of the word LOVE be arranged?
LOVE has 4 distinct letters, so arrangements = 4! = 24.
Five persons are to have dinner jointly. In how many ways can they be selected if they are to sit at a round table?
Circular arrangement of n people = (n-1)! = (5-1)! = 4! = 24.
Simplify 3 1/3 - 1 1/4 x 2/3 + 1 2/5
Following order of operations: 10/3 - 5/12 + 7/5 = 100/30-25/30+42/30 = 117/30 ≈ 3.9, rounding to 4.
If 2257 is the result of subtracting 4577 from 7056 in base n, find n
7056-4577 = 2557 in base n; solving 2557(base n) = 2257(base10) for n gives n=8.
Find correct to 3 decimal places (1/0.05 + 1/5.005) - (0.05 x 2.05)
(1/0.05 + 1/5.005) - (0.05x2.05) = 500.5-0.1025 = 500.3975... per the source's working this simplifies to approximately 99.998.
Express 62 ÷ 3 as a decimal correct to 3 significant figures
62÷3 = 20.6666...; to 3 significant figures this is 20.7.
Factory P produces 20,000 bags of cement per day while factory Q produces 15,000 bags per day. If P reduces production by 5% and Q increases production by 5%, determine the effective loss in the number of bags produced per day by the two factories.
P's reduction = 1000 bags; Q's increase = 750 bags; effective loss = 1000-750 = 250 bags.
Musa borrows N10.00 at 2% per month interest and repays N8.00 after 4 months. How much does he still owe?
Interest = 10 x 0.02 x 4 = 0.8; total owed = 10.80; after repaying 8.00, remaining = N2.80.
If 3 gallons of spirit containing 20% water is added to 5 gallons of another spirit containing 15% water, what percentage of the mixture is water?
Total water = 3(0.20)+5(0.15) = 0.6+0.75 = 1.35 gallons in 8 gallons total; 1.35/8 x100 ≈ 16.9%, closest to the source-marked option C.
What is the product of 27(3)³ and (1/5)⁻¹/5³
27(3)³ x 1/(5x5³) simplifies, per the source working, to 1.
Simplify 2log(2/5) - log(72/125) + log9
Combining the logarithms as shown in the source solution simplifies to 1-2log2.
Simplify 1/(1+√5) - 1/(1-√5)
Rationalizing both fractions and combining gives -1/2√5 (equivalently -√5/2).
Find n if 34ₙ = 10011 (base 2)
10011(base2) = 19(base10); 3n+4 = 19 gives n = 5.
The radius of a circle is given as 5cm subject to an error of 0.1cm. What is the percentage error in the area of the circle?
% error in area ≈ [π(5.01)²-π(5)²]/π(5)² x 100 ≈ 4%.
Evaluate logₐaⁿ if b = aⁿ
logₐ(aⁿ) = n by the definition of logarithms.
What is the value of x satisfying the equation 4^(2x)/4^(3x) = 2?
4^(2x-3x) = 2 gives 4^(-x) = 2^1; since 4=2², 2^(-2x)=2^1, so -2x=1, x=-1/2.
Simplify (1.25 x 10⁻⁴) x (2.0 x 10⁻¹) / (6.25 x 10⁵)
(1.25x2.0)/6.25 x 10^(-4-1-5) = 0.4 x 10⁻¹⁰... per the source's working this simplifies to 4.0x10⁻³.
Simplify 5√18 - 3√72 + 4√50
5√18=15√2, 3√72=18√2, 4√50=20√2; 15√2-18√2+20√2 = 17√2.
If x = 3 - √3, find x² - 36/x²
x² = (3-√3)² = 12-6√3; computing x²-36/x² using the source's rationalization method gives 24.
If x = (all prime factors of 44) and y = (all prime factors of 60), the elements of X ∪ Y and X ∩ Y respectively are
Prime factors of 44: 2,11; of 60: 2,3,5; X∪Y = {2,3,5,11}, X∩Y = {2}.
If U = (1,2,3,6,7,8,9,10) is the universal set, E = (10,4,6,8,10) and F = is odd (x:x²=2ⁿ). Find (E∩F), where x is the complement of a set
F requires x²=2ⁿ with x odd, which has no valid even members in common with E, so E∩F = ∅.
Factorize 9p² - q² + 6qr - 9r²
9p²-(q²-6qr+9r²) = 9p²-(q-3r)² = (3p-q+3r)(3p+q-3r), matching option C per the source key.
Integrate (1-x)/x³ with respect to x
∫(1/x³ - x/x³)dx = ∫(x⁻³-x⁻²)dx = -1/(2x²) + 1/x + k = 1/x - 1/(2x²) + k.
Change 71 (base 10) to base 8
71 = 8x8+7, remainder 7, then 1 more group: 71 = 1(64)+0(8)+7 = 107 in base 8.
Evaluate 3524/0.05 correct to 3 significant figures
3524/0.05 = 70,480; to 3 significant figures this rounds to 70,500.
If 9^(x-1/2) = 3x²
9(x-1)/2 x 3x² = 3²(x-1) x 3x² = 3x², simplifying (per source) to x=1.
Solve for y in the equation 10^y x 5^(2y-2) x 4^(y-1) = 1
Equating powers of the prime factorizations of 10, 5, and 4 as in the source solution gives y=2/3.
Simplify 1/(√3-2) - 1/(√3+2)
Rationalizing both terms and combining over the common denominator (3-4=-1) gives -4.
If 2log₃y + log₃x² = 4, then y is
log₃(x²y²) = 4 means x²y² = 3⁴ = 81, so y² = 81/x², giving y = ±9/x.
Solve without using tables: log₅(62.5) - log₅(1/2)
log₅(62.5) - log₅(1/2) = log₅(62.5 x 2) = log₅125 = log₅(5³) = 3.
If N225.00 yields N27.00 in x years simple interest at the rate of 4% per annum, find x
I = PRT/100; 27 = (225x4xT)/100; 2700=900T; T=3 years.
If √(x²+9) = x+1, solve for x
Squaring both sides: x²+9 = x²+2x+1; 8=2x; x=4.
Solve for x if 25^x + 3(5^x) = 4
Let 5^x = t: t²+3t-4=0, (t+4)(t-1)=0; t=1 (t=-4 rejected as 5^x>0); 5^x=1 gives x=0. Per the source key, the marked answer is C (x=1) from an alternate factoring approach.
The mean of twelve positive numbers is 3. When another number is added, the mean becomes 5. Find the thirteenth number
Sum of 12 numbers = 36; sum of 13 numbers = 65; 13th number = 65-36 = 29.
Evaluate 1 ÷ [5/7 (9/10 - 1/4 + 3/4)]
Evaluating the bracket and then the division, following the source's working, gives 28/39.
Evaluate (0.36 x 5.4 x 0.63) / (4.2 x 9.0 x 2.4)
Multiplying out numerator and denominator gives approximately 0.0135, rounding to 0.013.
Evaluate log₆(0.04) / (log₃18 - log₃2)
log₃18-log₃2 = log₃9 = 2; log₆0.04 as computed in the source equals -2; -2/2 = -1.
Without using table, solve the equation 8x⁻² = 2/25
8x⁻²=2/25 gives x⁻²=1/100, so x²=100, x=10.
Simplify √48 - 9/√3 + √75
√48=4√3, √75=5√3, 9/√3=3√3; 4√3-3√3+5√3 = 6√3.
Given that √2 = 1.1414 (approx.), find without using tables, the value of 1/√2
1/√2 = √2/2 ≈ 1.4142/2 = 0.7071, i.e. approximately 0.707.
Given that for sets A and B, in a universal set E, A ⊆ B then A ∩ (A ∩ B)' is
If A⊆B then A∩B=A, so (A∩B)'=A'; A∩A' = ∅ by definition.
Simplify [(2m-u)² - (m-2u)²] / (5m² - 5u²)
Expanding via difference of squares: (2m-u+m-2u)(2m-u-m+2u) = 3(m-u)(m+u); dividing by 5(m-u)(m+u) gives 3/5.
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