31 reviewed questions with answers and explanations.
These adapted questions come from a publisher-labelled 2022 collection. The original UTME sitting and question numbers are unconfirmed. Practice selection: full-paper coverage has not been confirmed.
Answer 10 distinct reviewed Mathematics questions from this 2022 collection in 30 minutes, then see your raw score and explanations. This is a practice selection, not a confirmed complete UTME paper.
Start 10-question 2022 Mathematics quick practiceMyschool 2022 collection · Original sitting and question number unconfirmed.
For real k, (1/8)^(k+2)=1. Find k.
B: −2
A positive base other than 1 gives a result of 1 only when its exponent is zero. Set k+2=0, so k=−2.
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Cube the binary number 101₂ and express the result in base two.
C: 1111101₂
The binary value 101₂ is 5 in base ten. Cubing gives 125, and 125=64+32+16+8+4+1, so its binary digits are 1111101₂.
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A fair coin is tossed twice. What is the probability of at least one head?
C: 3/4
The equally likely outcomes are HH, HT, TH and TT. Three contain at least one head, so the probability is 3/4; equivalently subtract P(TT)=1/4 from 1.
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Six numbers are 2, 5, x+1, x+2, 7 and 9. Their mean is 6. Find their median.
C: 6.5
A mean of 6 requires a total of 36. The six expressions sum to 26+2x, giving x=5. In order the values are 2, 5, 6, 7, 7, 9; the middle pair averages to 6.5.
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If sin θ=−3/5 and θ lies in the third quadrant, find cos θ.
D: −4/5
Using sin²θ+cos²θ=1 gives cos²θ=1−9/25=16/25. Both sine and cosine are negative in quadrant III, so cosθ=−4/5.
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Simplify 1/(3−√2) in the form p+q√2.
D: 3/7+√2/7
Multiply numerator and denominator by 3+√2. The denominator becomes 9−2=7, giving 3/7+√2/7.
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Find the length of a chord 3 cm from the centre of a circle of radius 5 cm.
A: 8 cm
The perpendicular from the centre bisects the chord. Half the chord is √(5²−3²)=4 cm, so the full chord is 8 cm.
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Mr Adu spends his annual salary on food (28%), rent (15%), car maintenance (20%), gifts (14%), savings (10%) and miscellaneous expenses (13%). If these amounts are represented on a pie chart, what angle represents food?
B: 100.8°
Food uses 28% of the total. Its sector angle is 28/100×360°=100.8°. The six source percentages sum to 100%.
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The tenth term of an arithmetic progression is 32 and the first term is 3/2. Find its fourth term.
A: 35/3
Use a₁₀=a+9d: 32=3/2+9d, so d=61/18. Then a₄=3/2+3(61/18)=35/3. Check that a+9d returns 32.
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L, N and Z lie on a straight line. Rays NM and NY lie on the same side of that line, in the order NL, NM, NY, NZ. If angle LNM is g degrees and angle YNZ is h degrees, find angle MNY.
A: 180°−g°−h°
Adjacent angles on a straight line add to 180°. Here g + angle MNY + h = 180°, so angle MNY = 180°−g°−h°. Both given angles must be subtracted.
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How many different arrangements use all seven letters of ZOOLOGY?
B: 840
There are seven positions, but the three O letters are indistinguishable. Divide the 7! possible orderings by 3!, giving 5040/6=840 distinct arrangements.
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Which expression is an antiderivative of (2x+1)^3?
A: (2x+1)^4/8+C
Set u=2x+1, so du=2 dx. Then the integral is one half of u^4/4, or (2x+1)^4/8+C. Differentiating this result returns (2x+1)^3.
AI-reviewed · calculation checkedMyschool 2022 collection · Original sitting and question number unconfirmed.
Matrix A has rows (2,1), (2,3), (1,2), and matrix B has rows (3,2), (4,2). Find AB.
D: Rows (10,6), (18,10), (11,6)
Multiply each row of A by each column of B: (2×3+1×4,2×2+1×2)=(10,6); (2×3+3×4,2×2+3×2)=(18,10); (1×3+2×4,1×2+2×2)=(11,6). Thus AB has rows (10,6), (18,10), (11,6).
AI-reviewed · calculation checkedMyschool 2022 collection · Original sitting and question number unconfirmed.
For h>0, r=√(3V/(πh)). Make V the subject of the formula.
B: πhr²/3
Square both sides to get r²=3V/(πh). Multiply by πh and divide by 3, giving V=πhr²/3.
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Find the determinant of the matrix with rows (2, 3) and (1, 3).
C: 3
For a 2×2 matrix, multiply the main diagonal and subtract the other diagonal: 2×3−3×1=3.
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What is the value of log₂(8√2)?
C: 3.5
Write 8 as 2³ and √2 as 2^(1/2). Their product is 2^(7/2), so its base-two logarithm is 7/2=3.5.
AI-reviewed · calculation checkedMyschool 2022 collection · Original sitting and question number unconfirmed.
What is the new value of 96/50 when its numerator is decreased by 37.5% and its denominator by 20%?
A: 1.5
A 37.5% decrease leaves 62.5% of 96, namely 60. A 20% decrease leaves 80% of 50, namely 40. The new fraction is 60/40=1.5.
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Find the limit of (x³+6x−7)/(x−1) as x approaches 1.
C: 9
Factor the numerator as (x−1)(x²+x+7). Cancel x−1 for x≠1, then substitute x=1 into x²+x+7 to obtain 9.
AI-reviewed · calculation checkedMyschool 2022 collection · Original sitting and question number unconfirmed.
Find the median of 14, 17, 10, 13, 18 and 10.
C: 13.5
Sort the six values as 10, 10, 13, 14, 17, 18. The median is the mean of the third and fourth values: (13+14)/2=13.5.
AI-reviewed · calculation checkedMyschool 2022 collection · Original sitting and question number unconfirmed.
Convert 10110.011₂ to a decimal number.
B: 22.375
The whole-number bits contribute 16+4+2=22. The fractional bits contribute 1/4+1/8=0.375, so the number is 22.375.
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A school's running cost varies directly with student count. If 20 students cost ₦10,000, how many students correspond to ₦1,000,000 at the same rate?
B: 2000
At the stated rate each student accounts for ₦10,000/20=₦500. Divide ₦1,000,000 by ₦500 to get 2,000 students.
AI-reviewed · calculation checkedMyschool 2022 collection · Original sitting and question number unconfirmed.
What is the midpoint of the points (−3, 5) and (2, 10)?
B: (−1/2, 15/2)
Average corresponding coordinates: (−3+2)/2=−1/2 and (5+10)/2=15/2. The midpoint is (−1/2, 15/2).
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Evaluate C(n²+1, n+5) when n=3.
C: 45
With n=3, the expression is C(10,8), equal to C(10,2). Choose two from ten in 10×9/2=45 ways.
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A particle has displacement s=3t²+5t−10 metres. What is its velocity at t=0?
B: 5 m/s
Velocity is the derivative ds/dt=6t+5. At t=0 this equals 5 m/s; the constant displacement term has no effect on velocity.
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An operation is defined by a*b=a−b+(ab)². Find 3*5.
A: 223
Apply the definition to a=3,b=5: 3−5+(3×5)²=−2+225=223. Square the entire product ab.
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If y varies inversely as x, and x=3 when y=4, find x when y=12.
B: 1
Inverse variation means xy=k. The initial values give k=3×4=12. When y=12, x=12/12=1.
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The numbers 8,p,q,26 form an arithmetic progression. Find p and q.
D: p=14,q=20
There are three equal steps from 8 to 26, so the common difference is (26−8)/3=6. The terms are 8,14,20,26; hence p=14,q=20.
AI-reviewed · calculation checkedMyschool 2022 collection · Original sitting and question number unconfirmed.
Factorize the expression 4a²−9b².
A: (2a−3b)(2a+3b)
Recognize a difference of squares: 4a²−9b²=(2a)²−(3b)²=(2a−3b)(2a+3b). Expanding the factors cancels the cross terms.
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If n(P)=20, n(Q)=30 and n(P∪Q)=40, find n(P∩Q).
A: 10
Use n(P∪Q)=n(P)+n(Q)−n(P∩Q). Thus 40=20+30−n(P∩Q), so the intersection contains 10 elements.
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Three times a number x, minus 2, is less than that number minus 6. Find the possible values of x.
A: x<−2
Translate the statement into 3x−2<x−6. Subtract x and add 2 to obtain 2x<−4. Dividing by positive 2 gives x<−2.
AI-reviewed · calculation checkedMyschool 2022 collection · Original sitting and question number unconfirmed.
A dress sells for ₦3,800 after a 20% discount. What was its original price?
D: ₦4,750
After a 20% discount the selling price is 80% of the original. Thus 0.8P=3800 and P=3800/0.8=₦4,750. Check: 20% of ₦4,750 is ₦950; subtracting gives ₦3,800.
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