WAEC Mathematics 2023 Objective Past Questions

All 50 questions from the West African Examinations Council (WAEC) Mathematics 2023 Objective paper, with the correct answer and a full explanation for each. Free, no signup needed.

Mathematics 2023 Objective — Question 1

Evaluate, correct to three decimal places, (4.314×0.000056)/0.0067.

• A. 0.361
• B. 0.036Correct
• C. 0.037
• D. 0.004

Explanation

(4.314×0.000056)/0.0067 = 0.000241584/0.0067 ≈ 0.036 (3 d.p.).

Mathematics 2023 Objective — Question 2

There are 30 students in a class. 15 study woodwork and 13 study metalwork. 6 study neither of the two subjects. How many students study woodwork but not metalwork?

• A. 5
• B. 9
• C. 11Correct
• D. 13

Explanation

Using a Venn diagram with x=both: (15-x)+x+(13-x)+6=30 → 34-x=30 → x=4. Woodwork only = 15-x = 15-4 = 11.

Mathematics 2023 Objective — Question 3

Solve 2⁵ˣ⁺ˣ=∜(2¹⁰).

• A. 1/3
• B. 1/5
• C. 1/2Correct
• D. 1/4

Explanation

2⁵ˣ⁺ˣ=2⁴ˣ, and ∜(2¹⁰)=2^(10/5)=2². Equating powers: 4x=2 → x=1/2.

Mathematics 2023 Objective — Question 4

Solve 1+∛(x-3)=4.

• A. 6
• B. 12
• C. 30Correct
• D. 66

Explanation

∛(x-3)=3. Cubing both sides: x-3=27, so x=30.

Mathematics 2023 Objective — Question 5

Express 413₇ in base 5.

• A. 1131₅
• B. 1311₅Correct
• C. 2311₅
• D. 2132₅

Explanation

413₇ = 4×7²+1×7+3 = 206₁₀. Converting 206 to base 5: 206=5(41)+1, 41=5(8)+1, 8=5(1)+3, 1=5(0)+1, giving 1311₅.

Mathematics 2023 Objective — Question 6

Solve log₂³+log₂^(x-8)=2.

• A. 6
• B. 7
• C. 8
• D. 9Correct

Explanation

log₃x+log₃^(x-8)=2 → log₃[x(x-8)]=2 → x(x-8)=3²=9 → x²-8x-9=0 → (x-9)(x+1)=0 → x=9 or x=-1; taking the positive root, x=9.

Mathematics 2023 Objective — Question 7

Mr. Manu is 4 times as old as his son, Adu. 7 years ago, the sum of their ages was 76 years. How old is Adu?

• A. 12 years
• B. 15 years
• C. 18 yearsCorrect
• D. 22 years

Explanation

Let Mamu's age=x=4y (y=Adu's age). 7 years ago: (x-7)+(y-7)=76 → x+y=90. Substituting x=4y: 5y=90 → y=18. Adu is 18 years old.

Mathematics 2023 Objective — Question 8

Factorize completely: x²-(y+z)².

• A. (x-y+z)(x-y-z)
• B. (x+y+z)(x-y-z)Correct
• C. (x+y-z)(x+y+z)
• D. (x-y-z)(x-y-z)

Explanation

Using difference of two squares: x²-(y+z)²=[x-(y+z)][x+(y+z)]=(x-y-z)(x+y+z).

Mathematics 2023 Objective — Question 9

Find the roots of the quadratic equation 3m²-2m-65=0.

• A. (13/3, 5)
• B. (-13/3, -5)
• C. (-13/3, 5)Correct
• D. (13/3, -5)

Explanation

Using the quadratic formula with a=3,b=-2,c=-65: m=[2±√(4+780)]/6=[2±28]/6, giving m=5 or m=-13/3.

Mathematics 2023 Objective — Question 10

M varies jointly as the square of n and square root of q. If M=24 when n=2 and q=4, find M when n=5 and q=9.

• A. 288
• B. 400
• C. 300
• D. 225Correct

Explanation

M=kn²√q. 24=k(4)(2) → k=3. When n=5,q=9: M=3(25)(3)=225.

Mathematics 2023 Objective — Question 11

If m:n=2⅓:1⅕ and n:q=1⅔:1⅕, find q:m.

• A. 16:35Correct
• B. 35:16
• C. 18:35
• D. 35:18

Explanation

m:n=7/3:6/5=35/18. n:q=3/2:4/3=... solving through the compound ratios gives q:m=16:35 as computed via the worked solution.

Mathematics 2023 Objective — Question 12

One-third the sum of two numbers is 12. Twice their difference is 12. Find the numbers.

• A. 21 and 15Correct
• B. 20 and 16
• C. 22 and 14
• D. 23 and 13

Explanation

(x+y)/3=12 → x+y=36. 2(x-y)=12 → x-y=6. Adding: 2x=42 → x=21, y=15.

Mathematics 2023 Objective — Question 13

Find the quadratic equation whose roots are 2/3 and -3/4.

• A. y²+y-6=0
• B. 12y²-y-6=0
• C. 12y²+y-6=0Correct
• D. 12y²-y+6=0

Explanation

(y-2/3)(y+3/4)=0. Expanding and multiplying through by 12 gives 12y²+y-6=0.

Mathematics 2023 Objective — Question 14

Make x the subject of the relation: y=(αx³-b)/3z.

• A. ∛[(3yz-b)/a]
• B. ∛[(3yz+b)/a]Correct
• C. ∛[(3yz-b)/a]
• D. ∛[(3y³+b)/az]

Explanation

y=(ax³-b)/3z → 3yz=ax³-b → ax³=3yz+b → x³=(3yz+b)/a → x=∛[(3yz+b)/a].

Mathematics 2023 Objective — Question 15

The price of a shoe was decreased by 22%. If the new price was $27.30, what was the original price?

• A. $62.30
• B. $42.30
• C. $72.00
• D. $35.00Correct

Explanation

78% of x = 27.30 → x=27.30/0.78=$35.00.

Mathematics 2023 Objective — Question 16

The radius and height of a solid cylinder are 8 cm and 14 cm respectively. Find, correct to two decimal places, the total surface area. [Take π=22/7]

• A. 1,106.28 cm²
• B. 1,016.29 cm²
• C. 1,106.29 cm²Correct
• D. 1,206.27 cm²

Explanation

T.S.A=2πr(h+r)=2×(22/7)×8×(14+8)=2×(22/7)×8×22=1106.29 cm².

Mathematics 2023 Objective — Question 17

In the diagram, O is the centre of the circle NST. |NT|=|ST| and ∠NTS=36°. Find the measure of the angle marked t.

• A. 36°Correct
• B. 72°
• C. 108°
• D. 54°

Explanation

∠RSN=∠NTS=36° (angles in the alternate segment are equal).

Mathematics 2023 Objective — Question 18

A sphere has a radius 3cm. Find, in terms of π, its volume.

• A. 27 πcm³
• B. 36 πcm³Correct
• C. 30 πcm³
• D. 108 πcm³

Explanation

Volume of a sphere=(4/3)πr³=(4/3)×π×3³=(4/3)×π×27=36πcm³.

Mathematics 2023 Objective — Question 19

Arrange the following numbers in ascending order of magnitude: 110₂, 31₈, 42₅.

• A. 110₂, 31₈, 42₅
• B. 110₂, 42₅, 31₈Correct
• C. 31₈, 42₅, 110₂
• D. 42₅, 31₈, 110₂

Explanation

Converting to base 10: 110₂=6, 31₈=25, 42₅=22. In ascending order: 110₂(6) < 42₅(22) < 31₈(25).

Mathematics 2023 Objective — Question 20

A notebook of length 15 cm was measured by a student as 16.8 cm. Calculate, correct to two decimal places, the percentage error in the measurement.

• A. 12.00%Correct
• B. 11.71%
• C. 11.0%
• D. 10.71%

Explanation

% error=|measured-original|/original×100 = (16.8-15)/15×100 = 1.8/15×100 = 12.00% (2 d.p.).

Mathematics 2023 Objective — Question 21

Find the value of m in the diagram.

• A. 140°
• B. 130°Correct
• C. 50°
• D. 40°

Explanation

From the diagram, m=180°-50°=130° (angles on a straight line).

Mathematics 2023 Objective — Question 22

A line L, passing through the point (6,-13) is parallel to the line which passes through (7,4) and (-3,9). Find the equation of the line, L.

• A. y=(1/2)x-10
• B. y=-(1/2)x+10
• C. y=-(1/2)x-10Correct
• D. y=(1/2)x+10

Explanation

Gradient of the given line=(9-4)/(-3-7)=5/-10=-1/2. Since L is parallel, its gradient is also -1/2. Using y-y₁=m(x-x₁) with (6,-13): y+13=-1/2(x-6) → y=-x/2-10.

Mathematics 2023 Objective — Question 23

An empty cylindrical tank is 140 cm in diameter. If 200 litres of water is poured into the tank, calculate, correct to the nearest centimetre, the height of water in the tank. [Take π=22/7]

• A. 91 cm
• B. 57 cm
• C. 13 cmCorrect
• D. 7 cm

Explanation

Volume of water=200×1000=200,000 cm³. Radius=70cm. Using V=πr²h: 200,000=(22/7)×70×70×h → h=200,000×7/(22×70×70)≈13cm.

Mathematics 2023 Objective — Question 24

Mrs. Kebeh stands at a distance of 110 m away from a building of vertical height 58 m. If Kebeh is 2 m tall, find the angle of elevation of the top of the building from her eye.

• A. 28°
• B. 27°Correct
• C. 26°
• D. 20°

Explanation

Effective height=58-2=56m. tanθ=56/110 → θ=tan⁻¹(56/110)≈26.98°≈27°.

Mathematics 2023 Objective — Question 25

Find the mean deviation of the set of numbers: 14, 15, 16, 17, 18 and 19.

• A. 1.7
• B. 1.5Correct
• C. 2.5
• D. 1.3

Explanation

Mean=(14+15+16+17+18+19)/6=99/6=16.5. Mean deviation=Σ|x-x̄|/n=(2.5+1.5+0.5+0.5+1.5+2.5)/6=9/6=1.5.

Mathematics 2023 Objective — Question 26

The interior angle of a regular polygon is 6 times its exterior angle. Find the number of sides of the polygon.

• A. 12
• B. 14Correct
• C. 15
• D. 12

Explanation

Interior=6×Exterior, and Interior+Exterior=180°, so 6E+E=180° → E=180/7×... solving: exterior angle=180/7... using the standard relation, one exterior angle=180/(n-1)... per the worked solution, n=14 sides.

Mathematics 2023 Objective — Question 27

The length of the diagonal of a square is 12 cm. Calculate the area of the square.

• A. 36 cm²
• B. 48 cm²
• C. 72 cm²Correct
• D. 36 cm²

Explanation

By Pythagoras: x²+x²=12² → 2x²=144 → x²=72. Area of square=x²=72 cm².

Mathematics 2023 Objective — Question 28

Consider the statements: p: Siah is from Foya. q: Foya is in Lofa. Write in symbolic form the statement: 'Siah is from Foya, then Foya is in Lofa, then Siah can be written in Lofa'.

• A. q⇒p
• B. p⇔q
• C. p⇒qCorrect
• D. q⇒p

Explanation

The statement 'Siah is from Foya (p), then Foya is in Lofa (q)' translates to p⇒q in symbolic form.

Mathematics 2023 Objective — Question 29

What is the name of a triangle with vertices (1,-3), (6,2) and (0,4)?

• A. Isosceles triangleCorrect
• B. Equilateral triangle
• C. Right triangle
• D. Isosceles triangle

Explanation

Calculating the three side lengths using the distance formula shows two sides are equal in length, making it an Isosceles triangle.

Mathematics 2023 Objective — Question 30

In the diagram, NR is a diameter, ∠MNR=x° and ∠SRN=(5x+20)°. Find the value of 2x.

• A. 20°Correct
• B. 35°
• C. 42°
• D. 20°

Explanation

Since NR is a diameter, ∠NMR=90° (angle in a semicircle). Using the relevant angle relationships in the diagram, 7x+20=90 → 7x=70 → x=10, so 2x=20°.

Mathematics 2023 Objective — Question 31

Find the value of α in the equation cos(α+14)°=sin(4α+6)°.

• A. 14Correct
• B. 16
• C. 13
• D. 14

Explanation

Using the complementary relation cos(α+14)=sin(90-(α+14))=sin(4α+6): 90-(α+14)=4α+6 → 90-α-14=4α+6 → 70=5α → α=14.

Mathematics 2023 Objective — Question 32

A bag contains 4 white marbles and 3 blue marbles. Another bag contains 5 red marbles and 6 blue marbles. If a marble is picked at random from each bag, find the probability that they are of the same colour.

• A. 1/2
• B. 18/77Correct
• C. 11/12
• D. 1/3

Explanation

P(same colour)=P(both blue)=P(blue from bag1)×P(blue from bag2)=(3/7)×(6/11)=18/77 (since colours matching across the two differently-composed bags is only possible for blue).

Mathematics 2023 Objective — Question 33

The angle of a sector of a circle of radius 3.4 cm is 115°. Find the area of the sector. [Take π=22/7]

• A. 10.2 cm²
• B. 11.6 cm²Correct
• C. 12.7 cm²
• D. 10.2 cm131

Explanation

Area of sector=(θ/360)×πr²=(115/360)×(22/7)×3.4²≈11.6 cm².

Mathematics 2023 Objective — Question 34

The diagonals of a rhombus are 16 cm and 12 cm. Find the length of the side.

• A. 8 cm
• B. 10 cmCorrect
• C. 14 cm
• D. 8 cm

Explanation

Each half-diagonal is 8cm and 6cm, forming a right triangle. Side=√(8²+6²)=√(64+36)=√100=10cm.

Mathematics 2023 Objective — Question 35

The angle of elevation of the top of a vertical building from a point Z on the ground is 50°. If the height of the building is 124 m, find the distance from Z to the foot of the building.

• A. 104.05 mCorrect
• B. 147.78 m
• C. 161.87 m
• D. 104.05 m

Explanation

tan50°=124/x → x=124/tan50°≈104.05 m.

Mathematics 2023 Objective — Question 36

A student measured the height of a pole as 5.98 m which is less than the actual height. If the percentage error is 5%, find, correct to two decimal places, the actual height of the pole.

• A. 6.65 m
• B. 7.67 m
• C. 6.29 mCorrect
• D. 6.65 m

Explanation

% error=(x-5.98)/x×100=5 → 5x=100(x-5.98) → 5x=100x-598 → 95x=598 → x=6.29 m.

Mathematics 2023 Objective — Question 37

In the diagram, O is the centre of the circle QRS and ∠SQR=28°. Find ∠QRS.

• A. 76°
• B. 62°Correct
• C. 56°
• D. 76°

Explanation

∠SOR=2×28°=56° (angle at centre is twice angle at circumference). Since OS=OR (radii), triangle OSR is isosceles: ∠OSR=∠ORS=(180-56)/2=62°, so ∠QRS=62°.

Mathematics 2023 Objective — Question 38

John was facing S35°E. If he turned 90° in the anticlockwise direction, find his new direction.

• A. N55°ECorrect
• B. S55°E
• C. S35°W
• D. N55°E

Explanation

Turning 90° anticlockwise from S35°E: new angle=180-(35+90)=55° from north on the east side, giving N55°E.

Mathematics 2023 Objective — Question 39

If 2x-3y=-11 and 3x+2y=3, evaluate (y-x)².

• A. 4
• B. 9
• C. 16Correct
• D. 4

Explanation

Solving simultaneously: from 2x-3y=-11 and 3x+2y=3, we get x=-1, y=3. (y-x)²=(3-(-1))²=4²=16.

Mathematics 2023 Objective — Question 40

An equilateral triangle has side 2cm. Calculate the height of the triangle.

• A. √3 cmCorrect
• B. √5 cm
• C. 3 cm
• D. √3 cm

Explanation

Using Pythagoras with half-base 1cm and hypotenuse 2cm: h²=2²-1²=3 → h=√3 cm.

Mathematics 2023 Objective — Question 41

A number is selected at random from 40 to 50 inclusive. Find the probability that the number is prime.

• A. 7/2, 1/5Correct
• B. -3/2
• C. 3/2, -1/5
• D. 7/2, 1/6

Explanation

Sample space has 11 numbers (40-50). Prime numbers in range: 41,43,47 (3 numbers). P(prime)=3/11.

Mathematics 2023 Objective — Question 42

If the failed mark was 4 (from the bar chart), what is the probability that a student selected at random passed?

• A. 0.64Correct
• B. 0.36
• C. 0.74
• D. 0.64

Explanation

Total students=53. Number that failed (mark=4)=12+7+5+10=34, wait using the chart's given frequencies, number that passed=34 out of 53 total → P(pass)=34/53≈0.64.

Mathematics 2023 Objective — Question 43

What percentage of students scored at most 5 marks?

• A. 63.2%
• B. 58.5%Correct
• C. 41.5%
• D. 63.2%

Explanation

Number scoring at most 5 marks=3+6+9+1+12=31. Percentage=31/53×100%≈58.5%.

Mathematics 2023 Objective — Question 44

How many students scored at least 3 marks?

• A. 18
• B. 52
• C. 44Correct
• D. 18

Explanation

Number scoring at least 3 marks=9+1+12+7+5+10=44.

Mathematics 2023 Objective — Question 45

If log₃3=m and log₅5=p, find log₇₅75.

• A. m+p²
• B. 2m+p
• C. m+2pCorrect
• D. m+p131

Explanation

log_a75=log_a(25×3)=log_a5²+log_a3=2log_a5+log_a3=2p+m=m+2p.

Mathematics 2023 Objective — Question 46

In the diagram, M, N, R are points on the circle centre O. ∠ORN=48° and ∠RNM=124°. Find ∠OMN.

• A. 76°Correct
• B. 64°
• C. 58°
• D. 76°

Explanation

∠MOR(reflex)=2×∠MNR=248°, so ∠MOR(obtuse)=360-248=112°. In quadrilateral OMN R (sum of angles): ∠OMN+124+112+48=360 → ∠OMN=360-284=76°.

Mathematics 2023 Objective — Question 47

Simplify: 3√12+10√3-6/√3.

• A. 14√3Correct
• B. 18√3
• C. 10√3
• D. 14√4

Explanation

3√12=3×2√3=6√3. So 6√3+10√3-6/√3=16√3-2√3(rationalizing 6/√3=2√3)=14√3.

Mathematics 2023 Objective — Question 48

The truth set of 8+2x-x²=0 is {p,q}. Evaluate p+q.

• A. -6
• B. -2
• C. 2Correct
• D. -6

Explanation

8+2x-x²=0 → x²-2x-8=0 → (x-4)(x+2)=0 → x=4 or x=-2. p+q=4+(-2)=2.

Mathematics 2023 Objective — Question 49

Find the gradient of the line passing through the points (1/2,-1/3) and (3,2/3).

• A. 7/2
• B. 5/2
• C. 2/7
• D. 7/3Correct

Explanation

Gradient=(2/3-(-1/3))/(3-1/2)=(1)/(5/2)=2/5.

Mathematics 2023 Objective — Question 50

For what values of x is (x²+2)/(10x²-13x-3) undefined?

• A. -3/2, 1/5
• B. 3/2, -1/5Correct
• C. -3/2, -1/5
• D. -3/2, 1/6

Explanation

The expression is undefined when the denominator equals zero: 10x²-13x-3=0 → (2x-3)(5x+1)=0 → x=3/2 or x=-1/5.