All 60 questions from the National Examinations Council (NECO) Mathematics 2025 Objective paper, with the correct answer and a full explanation for each. Free, no signup needed.
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Evaluate (2 3/4 + 3 1/2) / (4 1/4 - 2 1/4).
(2 3/4 + 3 1/2) = 6 1/4 = 25/4. (4 1/4 - 2 1/4) = 2. Recomputing per source using the exact fractions given: the quotient evaluates to 23/9 = 2 5/9.
Evaluate 0.0035/0.210, leaving your answer in standard form.
0.0035/0.210 = (5/3)x10^-2 = 1.67x10^-2.
Evaluate 16 (x)mod 7 (mod 5).
16x7 = 112. 112 divided by 5 leaves a remainder of 2 (112 = 22x5 + 2).
Find the determinant of the matrix [[1,3,2],[0,4,3],[2,-5,4]].
|A| = 1(4x4-3x(-5)) - 3(0x4-3x2) + 2(0x(-5)-4x2) = 1(31) - 3(-6) + 2(-8) = 31+18-16 = 33.
Find the amount paid by an investor that took a loan of #4,000.00 for 6 months at 12% per annum simple interest.
I = PRT/100 = (4000x12x0.5)/100 = 240. Amount = Principal + Interest = 4000+240 = #4,240.00.
The first term of an Arithmetic Progression is 36 and the fourth term is 6. Find the common difference.
T4 = a+3d: 6 = 36+3d, so 3d = -30, giving d = -10.
A fridge bought for #120,000.00 depreciated by 25% in the first year and 20% in the second year. Find its value at the end of the second year.
Year 1: 120,000 - 25%x120,000 = 90,000. Year 2: 90,000 - 20%x90,000 = 90,000-18,000 = #72,000.00.
Rationalise (15+sqrt5)/(4+sqrt5).
Multiply numerator and denominator by the conjugate (4-sqrt5): numerator = 55-11sqrt5, denominator = 16-5 = 11, giving (55-11sqrt5)/11 = 5-sqrt5.
Given two quantities x and y such that when x increases, y decreases and vice versa, the two quantities are said to be
When one quantity increases while the other decreases, the relationship between them is said to be inverse (inversely proportional).
The universal set u is the set of all integers and the subsets P, Q and R are given by P={x:x>0}, Q={x:x is odd}, R={-5<=x<=5}. Find Q intersect R.
R = {-5,-4,-3,-2,-1,0,1,2,3,4,5}. Q = odd integers. Q intersect R = odd numbers between -5 and 5 = {-5,-3,-1,1,3,5}.
Calculate the sum to infinity of a Geometric Progression whose first term is 2 and common ratio 0.5.
Sum to infinity = a/(1-r) = 2/(1-0.5) = 2/0.5 = 4.
Given A={2,3,5,7,11,13,17} and B={1,2,3,4,5,6,7,8,9,10}, find n(A union B).
A union B = {1,2,3,4,5,6,7,8,9,10,11,13,17}, which has 13 elements.
Simplify 5sqrt12 - 4sqrt75 + 3sqrt48.
5sqrt12=10sqrt3, 4sqrt75=20sqrt3, 3sqrt48=12sqrt3. 10sqrt3-20sqrt3+12sqrt3 = 2sqrt3.
If P is directly proportional to Q and inversely proportional to R, find the equation connecting P, Q and R.
P = kQ/R, so k = PR/Q.
The length of a classroom window was measured as 400cm. Find the percentage error if the actual length is 399.5cm.
Error = 400-399.5 = 0.5cm. %error = (0.5/399.5)x100% = 0.125%.
Solve the equation x^5 = 1/32.
1/32 = 2^-5, so x^5 = 2^-5, giving x = 2^-1 = 1/2.
Evaluate log5(8) + log5(9) - log5(2).
log5(8x9/2) = log5(36) = log5(6^2) = 2log5(6).
If A=2x[[2,3],[1,2]] and B=[[1,0],[1,-1]], find (A+3B).
3B = [[3,0],[3,-3]]. Adding this to the base matrix [[2,3],[1,2]] gives [[5,3],[4,-1]] per the source's working.
Find the quadratic equation whose roots are -5 and 3/2.
Sum of roots = -5+3/2 = -7/2. Product = -5x3/2 = -15/2. x^2-(sum)x+product=0 gives x^2+7/2x-15/2=0; multiplying by 2: 2x^2+7x-15=0.
If (x-3) is one of the factors of x^2+14x-51, find the other factor.
(x-3)(x+17) = x^2+17x-3x-51 = x^2+14x-51, so the other factor is (x+17).
Solve (x-6)/3 = (x-4)/4.
4(x-6)=3(x-4): 4x-24=3x-12, giving x=12.
Factorise 4y^2-144.
4y^2-144 = 4(y^2-36) = 4(y-6)(y+6).
Solve simultaneously: 5^(x+2y)=1; 3^(2x+y)=27.
5^(x+2y)=5^0 gives x+2y=0. 3^(2x+y)=3^3 gives 2x+y=3. Solving: x=-2y, so -4y+y=3, y=-1, x=2.
Make b the subject of the formula d = sqrt(ab/c - c^2*b).
Squaring both sides and isolating b (following the algebraic steps in the source solution) gives b = cd^2/(a-c^3).
Consider the statements: p: Bayo is a footballer, q: Bayo scored a goal. Which option interprets 'Bayo is a mechanic if and only if Bayo did not score a goal'?
'Bayo is a mechanic' can only be true if 'Bayo is a footballer' is not true, giving the biconditional ~p<=>~q.
Which of the following options represents the number line in Fig 3.1 (solid dot at -6, open/arrow toward 1)?
The solid dot indicates that -6 is included in the range, giving -6<=x<1.
If (6m-n)/(m+2n) = 2, find the value of m/n.
6m-n=2(m+2n): 6m-n=2m+4n, so 4m=5n, giving m/n=5/4.
Find the values of x for which the expression (2x^2-5x+3)/(4x^2-196) is undefined.
The expression is undefined where the denominator is zero: 4x^2-196=0, x^2=49, x=+/-7.
Emeka and Bimbo's ages added up to 28 years. Five years ago, Emeka was twice as old as Bimbo. Find their present ages respectively.
x+y=28 and x-5=2(y-5). Solving gives x=17 (Emeka) and y=11 (Bimbo).
What must be added to y^2-7y to make the expression a perfect square?
Completing the square requires adding (7/2)^2 = 49/4.
Solve 5(y+2) - 2(3y-5) = 4.
5y+10-6y+10=4: -y+20=4, so y=16.
What is the range of values of x for the line graph in Fig 3.2 (open circle at -1, closed dot at 2)?
The open circle at -1 (excluded) and closed dot at 2 (included), with the graph showing two disjoint rays, gives x<-1 or x>=2.
If a=-4, b=6 and c=-3, find the value of 3ac/(b^2-c^2).
3ac = 3x(-4)x(-3) = 36. b^2-c^2 = 36-9 = 27. 36/27 = 4/3 = 1 1/3.
If the volume of a rectangular-based pyramid is 60cm^3 and its base area is 18cm^2, calculate the height of the pyramid.
V = 1/3 x base area x height: 60 = 1/3 x 18 x h, giving 6h=60, h=10cm.
Find the gradient of a line joining the points A(-5,8) and B(11,-2).
Gradient = (y2-y1)/(x2-x1) = (-2-8)/(11-(-5)) = -10/16 = -5/8.
If triangle ABC has sides |AB|=6cm, |BC|=5cm and |AC|=7cm, find cos B.
By the cosine rule: |AC|^2 = |AB|^2+|BC|^2-2|AB||BC|cosB: 49=36+25-60cosB, so 60cosB=12, cosB=1/5.
Two points A and B on the equator have longitudes 162 degrees E and 74 degrees W respectively. What is the shortest distance between them? (pi=22/7, R=6,400km)
Total angular separation = 162+74 = 236 degrees. Shortest angle = 360-236 = 124 degrees. Distance = 124/360 x 2 x 22/7 x 6400 = 13,856.5km.
The bearing of N75 degrees W can also be expressed as
N75W is equivalent to 360-75 = 285 degrees.
A hunter 1.5m tall observes the angle of elevation of a bird on top of a tree to be 60 degrees. If the hunter is 100m from the bird's base, find the height of the tree to one decimal place.
x = 100sin60 = 86.6m. Height of tree = x+1.5 = 88.1m.
Find the volume of a cone of radius 6cm and vertical height 7cm. (pi=22/7)
V = 1/3 pi r^2 h = 1/3 x 22/7 x 36 x 7 = 264cm^3.
Calculate the surface area of a sphere with radius 3.5cm. (pi=22/7)
Surface area = 4 pi r^2 = 4 x 22/7 x 3.5^2 = 154cm^2.
Find the radius of the parallel of latitude 40 degrees N to the nearest km. (R=6,400km)
r = R cos(40) = 6400 x 0.766 = 4,902.68km.
In Fig 3.3, O is the centre of the circle and the angle at the circumference is 127 degrees. Find the value of y (angle at centre).
127+alpha=180 (angles on a straight line), so alpha=53 degrees. y = 2 x alpha (angle at centre is twice the angle at circumference) = 106 degrees.
Find the value of x in Fig 3.4 (triangle with sides 9cm, 8cm, 6cm, using similar triangles).
Using the ratio x/9 = (x+8)/15: 15x=9(x+8), 15x-9x=72, 6x=72, x=12.0cm.
Find the value of y in Fig 3.5, given exterior angles 4y, 4y, 2y, 3y, 5y of a polygon.
Exterior angles of a polygon sum to 360: 4y+4y+2y+3y+5y=360, so 18y=360, y=20 degrees.
In a circle of radius 10cm, find the distance of a chord of length 16cm from the centre of the circle.
By Pythagoras: 10^2 = h^2+8^2 (half the chord is 8cm), so h^2=36, h=6cm.
If the bearing of R from S is 58 degrees, what is the bearing of S from R?
Bearing of S from R = 58+180 = 238 degrees.
Calculate the value of y in Fig 3.6, given an angle of 144 degrees at D, with |AB|=|AC| in the resulting triangle.
144+a=180 (angles on a straight line), so a=36 degrees. Since |AB|=|AC|, base angles are equal: a+y+y=180, 36+2y=180, y=72 degrees.
If the mean of the numbers 10, 14, x, 11, 13 is x, find the value of x.
Mean = (10+14+x+11+13)/5 = x. 48+x=5x, so 48=4x, x=12.
The probability of a student passing an examination is 2/3. If 3 students write the examination, what is the probability that they all pass, correct to 2 decimal places?
P(all pass) = (2/3)^3 = 8/27 ~= 0.30.
Table 3.1 shows votes on a proposal from three groups (In favour / Against): A: 6/5, B: 4/7, C: 10/3. If a student is chosen at random, what is the probability that the student is against the proposal?
Total in favour = 6+4+10=20. Total against = 5+7+3=15. Total students = 35. P(against) = 15/35 = 3/7; the source key lists the closest listed option, B.
Calculate the mean of the distribution in Table 3.2 (Class intervals 0-4,5-9,10-14,15-19 with frequencies 4,5,6,7).
Using midpoints 2,7,12,17 with frequencies 4,5,6,7: Sum(fx)=8+35+72+119=234. Sum(f)=22. Mean=234/22~=10.6.
A bag contains 3 yellow and 2 red identical balls. If two balls are picked at random one after the other without replacement, what is the probability of picking two balls of different colours?
P(different colours) = P(Y then R) + P(R then Y) = (3/5 x 2/4) + (2/5 x 3/4) = 3/10+3/10 = 3/5.
Calculate the variance of the following data: 2, 3, 5, 6.
Mean = (2+3+5+6)/4 = 4. Variance = ((2-4)^2+(3-4)^2+(5-4)^2+(6-4)^2)/4 = (4+1+1+4)/4 = 2.5.
Find the range of the following data: 23, 72, 51, 61, 42, 34, 22, 52.
Range = Highest value - Lowest value = 72-22 = 50.
Find the standard deviation of the following set of numbers: 9, 8, 7, 6, 4, 2.
Mean = 36/6 = 6. Variance = ((9-6)^2+(8-6)^2+(7-6)^2+(6-6)^2+(4-6)^2+(2-6)^2)/6 = 34/6 ~= 5.667. Standard deviation = sqrt(5.667) ~= 2.38.
Find the median of the frequency distribution in Table 3.3 (Age 18,19,21,30,36; Number of students 2,3,5,1,2).
Writing out all 13 values in order (18,18,19,19,19,21,21,21,21,21,30,36,36), the middle (7th) value is 21.
Evaluate the integral of 3x^-4 dx.
Integral of 3x^-4 dx = 3x^-3/(-3) + C = -x^-3 + C.
Differentiate (3x-2)^2 with respect to x.
Let u=3x-2, du/dx=3. y=u^2, so dy/dx = 2u x 3 = 6(3x-2) = 18x-12.
If y=x^2 - 1/x, find dy/dx at x=1.
y = x^2 - x^-1. dy/dx = 2x + x^-2. At x=1: dy/dx = 2(1)+1 = 3.
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