2026/2027 NECO Further Mathematics Questions and Answers (Objective and Essay)

2026/2027 NECO Further Mathematics Questions and Answers (Objective and Essay)

PAPER I – OBJECTIVE TEST (PART 1)

Time Allowed: 1 Hour 30 Minutes

INSTRUCTIONS TO CANDIDATES

1. Answer all thirty (30) questions in this section.
2. Each question is followed by four options labeled A–D.
3. Choose the option that best answers the question.
4. Show all rough work in the space provided.
5. Each question carries equal marks.

1. Solve the equation 2x−5=92x – 5 = 92x−5=9.

A. 5
B. 6
C. 7
D. 8

Answer: C

2. Simplify (x+4)(x−2)(x + 4)(x – 2)(x+4)(x−2).

A. x2+2x−8x^2 + 2x – 8×2+2x−8
B. x2−2x−8x^2 – 2x – 8×2−2x−8
C. x2+6x+8x^2 + 6x + 8×2+6x+8
D. x2−6x−8x^2 – 6x – 8×2−6x−8

Answer: A

3. Solve x2−16=0x^2 – 16 = 0x2−16=0.

A. ±2
B. ±4
C. ±8
D. 4 only

Answer: B

4. If log⁡10100=x\log_{10}100 = xlog10​100=x, then x=x =x=

A. 1
B. 2
C. 10
D. 100

Answer: B

5. The roots of x2−7x+12=0x^2 – 7x + 12 = 0x2−7x+12=0 are

A. 2 and 6
B. 3 and 4
C. 1 and 12
D. 5 and 2

Answer: B

6. The value of sin⁡30∘\sin 30^\circsin30∘ is

A. 1
B. 12\frac{1}{2}21​
C. 32\frac{\sqrt3}{2}23​​
D. 0

Answer: B

7. The value of cos⁡60∘\cos 60^\circcos60∘ is

A. 12\frac1221​
B. 1
C. 0
D. 32\frac{\sqrt3}{2}23​​

Answer: A

8. Convert 180∘180^\circ180∘ to radians.

A. π\piπ

B. π2\frac{\pi}{2}2π​

C. 2π2\pi2π

D. π4\frac{\pi}{4}4π​

Answer: A

9. Differentiate 5x45x^45×4.

A. 20x320x^320×3

B. 5x35x^35×3

C. 4x54x^54×5

D. 25x225x^225×2

Answer: A

10. Evaluate

∫6x dx\int 6x\,dx∫6xdx

A. 6x+C6x+C6x+C

B. 3×2+C3x^2+C3x2+C

C. 6×2+C6x^2+C6x2+C

D. 2x+C2x+C2x+C

Answer: B

11. If

y=x3+2x,y=x^3+2x,y=x3+2x,

find

dydx\frac{dy}{dx}dxdy​

A. 3×2+23x^2+23×2+2

B. x2+2x^2+2×2+2

C. 3x+23x+23x+2

D. x3x^3×3

Answer: A

12. The derivative of a constant is

A. 1

B. Constant

C. 0

D. x

Answer: C

13. The sum of interior angles of a pentagon is

A. 360°

B. 540°

C. 720°

D. 900°

Answer: B

14. The equation of a straight line is

A. y=mx+cy=mx+cy=mx+c

B. x2+y2=r2x^2+y^2=r^2×2+y2=r2

C. xy=cxy=cxy=c

D. x+y=0x+y=0x+y=0

Answer: A

15. The gradient of the line joining (2,3) and (6,11) is

A. 1

B. 2

C. 3

D. 4

Answer: B

16. The determinant of

∣3003∣\begin{vmatrix} 3&0\\ 0&3 \end{vmatrix}​30​03​​

is

A. 3

B. 6

C. 9

D. 0

Answer: C

17. Two matrices can be added only when they have

A. Equal determinants

B. The same order

C. Equal values

D. Equal diagonals

Answer: B

18. If

A=(1234)A= \begin{pmatrix} 1&2\\ 3&4 \end{pmatrix}A=(13​24​)

the order of A is

A. 1×2

B. 2×1

C. 2×2

D. 4×4

Answer: C

19. The magnitude of vector (6,8) is

A. 8

B. 10

C. 12

D. 14

Answer: B

20. The unit vector has magnitude

A. 0

B. 1

C. 2

D. 10

Answer: B

21. The mean of 4, 6, 8, 10 is

A. 6

B. 7

C. 8

D. 9

Answer: B

22. The median of 5, 8, 10, 12, 15 is

A. 8

B. 10

C. 12

D. 15

Answer: B

23. The mode is

A. Highest value

B. Most frequent value

C. Average value

D. Middle value

Answer: B

24. The probability of an impossible event is

A. 1

B. 2

C. 0

D. ½

Answer: C

25. The SI unit of force is

A. Joule

B. Newton

C. Watt

D. Pascal

Answer: B

26. Momentum equals

A. Force × Time

B. Mass × Velocity

C. Mass × Weight

D. Weight × Velocity

Answer: B

2026 NECO Further Mathematics Objective and Essay Questions and Answers

27. The acceleration due to gravity is approximately

A. 5 m/s²

B. 9.8 m/s²

C. 12 m/s²

D. 15 m/s²

Answer: B

28. The work done when a force of 20 N moves an object through 5 m is

A. 4 J

B. 25 J

C. 100 J

D. 200 J

Answer: C

29. The SI unit of power is

A. Joule

B. Newton

C. Watt

D. Volt

Answer: C

30. Mechanical energy is the sum of

A. Heat and sound energy

B. Potential and kinetic energies

C. Electrical and heat energies

D. Chemical and electrical energies

Answer: B

31. Differentiate y=4×3−2x+7y = 4x^3 – 2x + 7y=4×3−2x+7.

A. 12×2−212x^2 – 212×2−2

B. 12×3−212x^3 – 212×3−2

C. 4×2−24x^2 – 24×2−2

D. 8x−28x – 28x−2

Answer: A

32. Evaluate

∫(3×2) dx\int (3x^2)\,dx∫(3×2)dx

A. x3+Cx^3 + Cx3+C

B. 3×3+C3x^3 + C3x3+C

C. 6x+C6x + C6x+C

D. 9x+C9x + C9x+C

Answer: A

33. If y=sin⁡xy=\sin xy=sinx, then

dydx\frac{dy}{dx}dxdy​

is

A. cos⁡x\cos xcosx

B. −cos⁡x-\cos x−cosx

C. tan⁡x\tan xtanx

D. sec⁡x\sec xsecx

Answer: A

34. The derivative of exe^xex is

A. exe^xex

B. xexexe

C. 1/ex1/e^x1/ex

D. xex^exe

Answer: A

35. The integral of a constant kkk is

A. kkk

B. kx+Ckx+Ckx+C

C. k2k^2k2

D. k/xk/xk/x

Answer: B

36. The distance between the points (2,3)(2,3)(2,3) and (6,6)(6,6)(6,6) is

A. 3

B. 4

C. 5

D. 6

Answer: C

37. The midpoint of the line joining (2,4)(2,4)(2,4) and (8,10)(8,10)(8,10) is

A. (5,7)

B. (6,7)

C. (4,6)

D. (3,8)

Answer: A

38. The equation of a circle with centre (0,0)(0,0)(0,0) and radius 4 is

A. x+y=4x+y=4x+y=4

B. x2+y2=16x^2+y^2=16×2+y2=16

C. x2+y2=8x^2+y^2=8×2+y2=8

D. 4x+4y=164x+4y=164x+4y=16

Answer: B

39. The slope of a vertical line is

A. Zero

B. One

C. Undefined

D. Two

Answer: C

40. If z=3+4iz=3+4iz=3+4i, then ∣z∣|z|∣z∣ equals

A. 3

B. 4

C. 5

D. 7

Answer: C

41. The conjugate of 5−2i5-2i5−2i is

A. 5+2i5+2i5+2i

B. −5+2i-5+2i−5+2i

C. 2+5i2+5i2+5i

D. 5−2i5-2i5−2i

Answer: A

42. In the expansion of (a+b)n(a+b)^n(a+b)n, the coefficients are obtained from

A. Logarithm table

B. Pascal’s Triangle

C. Probability table

D. Matrix table

Answer: B

43. The coefficient of x2x^2×2 in (x+1)2(x+1)^2(x+1)2 is

A. 0

B. 1

C. 2

D. 3

Answer: B

44. Evaluate 5!5!5!

A. 25

B. 60

C. 120

D. 240

Answer: C

45. The number of ways of arranging 4 different books is

A. 16

B. 12

C. 24

D. 48

Answer: C

2026 NECO Further Mathematics Objective and Essay Questions and Answers

46. The number of combinations of selecting 2 objects from 5 is

A. 5

B. 10

C. 15

D. 20

Answer: B

47. If a body travels with uniform velocity, its acceleration is

A. Positive

B. Negative

C. Zero

D. Infinite

Answer: C

48. The kinetic energy of a body is given by

A. mghmghmgh

B. 12mv2\frac12mv^221​mv2

C. FdFdFd

D. PtPtPt

Answer: B

49. Potential energy depends mainly on

A. Height

B. Speed

C. Pressure

D. Density

Answer: A

50. The impulse experienced by a body equals

A. Force × Time

B. Mass × Height

C. Weight × Distance

D. Force × Distance

Answer: A

51. The identity sin⁡2θ+cos⁡2θ\sin^2\theta+\cos^2\thetasin2θ+cos2θ equals

A. 0

B. 1

C. 2

D. sin⁡θ\sin\thetasinθ

Answer: B

52. tan⁡45∘\tan45^\circtan45∘ equals

A. 0

B. 1

C. 2

D. 3\sqrt33​

Answer: B

53. The standard deviation measures

A. Central location

B. Spread of data

C. Frequency only

D. Probability

Answer: B

54. Variance is

A. Square root of standard deviation

B. Square of standard deviation

C. Mean divided by frequency

D. Median multiplied by mode

Answer: B

55. A histogram is used to represent

A. Continuous data

B. Pie chart

C. Line graph

D. Pictogram

Answer: A

56. The feasible region in linear programming is obtained from

A. The graph of constraints

B. Logarithm tables

C. Quadratic equations

D. Binomial expansion

Answer: A

57. The optimum solution in linear programming occurs at

A. Any point

B. Origin only

C. Corner point of the feasible region

D. Midpoint

Answer: C

58. The probability of a certain event is

A. 0

B. 0.5

C. 1

D. 2

Answer: C

59. If two events are mutually exclusive, then

A. They occur together

B. They cannot occur simultaneously

C. They are identical

D. They have equal probabilities

Answer: B

60. A singular matrix is one whose determinant is

A. 1

B. -1

C. 0

D. 2

Answer: C

2026 NECO Further Mathematics Objective and Essay Questions and Answers

FURTHER MATHEMATICS

PAPER II – ESSAY

Time Allowed: 2 Hours 30 Minutes
Maximum Marks: 100

INSTRUCTIONS TO CANDIDATES

1. Answer Five (5) questions only.
2. Question 1 is compulsory.
3. All questions carry 20 marks.
4. Show all necessary working. Credit will be given for correct methods even where the final answer is incorrect.
5. Use approved mathematical tables where necessary.

QUESTION 1 (COMPULSORY)

(a) Solve the quadratic equation:

x2−7x+12=0x^2-7x+12=0x2−7x+12=0

(b) Solve simultaneously:

2x+y=112x+y=112x+y=11 x−y=1x-y=1x−y=1

MODEL ANSWER

(a)

Factorize:

x2−7x+12=0x^2-7x+12=0x2−7x+12=0 (x−3)(x−4)=0(x-3)(x-4)=0(x−3)(x−4)=0

Therefore,

x=3 or x=4\boxed{x=3 \text{ or } x=4}x=3 or x=4​

(b)

Given

2x+y=112x+y=112x+y=11 x−y=1x-y=1x−y=1

Add both equations:

3x=123x=123x=12 x=4x=4x=4

Substitute into

x−y=1x-y=1x−y=1 4−y=14-y=14−y=1 y=3y=3y=3

Answer

x=4,  y=3\boxed{x=4,\;y=3}x=4,y=3​

QUESTION 2

(a) Differentiate

y=5×4−3×2+7y=5x^4-3x^2+7y=5×4−3×2+7

(b) Find the gradient when

x=2x=2x=2

MODEL ANSWER

Differentiate:

dydx=20×3−6x\frac{dy}{dx}=20x^3-6xdxdy​=20×3−6x

At

x=2x=2x=2 20(8)−1220(8)-1220(8)−12 160−12160-12160−12 148\boxed{148}148​

QUESTION 3

(a) Evaluate

∫(6×2+4) dx\int(6x^2+4)\,dx∫(6×2+4)dx

(b) Find

∫132x dx\int_{1}^{3}2x\,dx∫13​2xdx

MODEL ANSWER

(a)

∫6x2dx=2×3\int6x^2dx=2x^3∫6x2dx=2×3 ∫4dx=4x\int4dx=4x∫4dx=4x

Therefore,

2×3+4x+C\boxed{2x^3+4x+C}2×3+4x+C​

(b)

∫132x dx\int_1^32x\,dx∫13​2xdx =x2∣13=x^2\Big|_1^3=x2​13​ =9−1=9-1=9−1 8\boxed{8}8​

QUESTION 4

Given

A=(2153)A= \begin{pmatrix} 2&1\\ 5&3 \end{pmatrix}A=(25​13​)

(a) Find the determinant.

(b) Find the inverse of A.

MODEL ANSWER

Determinant

=(2)(3)−(1)(5)=(2)(3)-(1)(5)=(2)(3)−(1)(5) =6−5=6-5=6−5 1\boxed{1}1​

Inverse

Since

∣A∣=1|A|=1∣A∣=1 A−1=(3−1−52)A^{-1} = \begin{pmatrix} 3&-1\\ -5&2 \end{pmatrix}A−1=(3−5​−12​)

QUESTION 5

The scores of ten students are:

12, 15, 18, 18, 20, 22, 24, 24, 25, 32

Calculate:

(a) Mean

(b) Median

(c) Mode

MODEL ANSWER

Sum

=210

Mean

=21010=\frac{210}{10}=10210​ =21=\boxed{21}=21​

Median

=20+222=\frac{20+22}{2}=220+22​ =21=\boxed{21}=21​

Mode

18 and 24\boxed{18 \text{ and }24}18 and 24​

(Bimodal distribution)

2026 NECO Further Mathematics Objective and Essay Questions and Answers

QUESTION 6

(a) Expand

(2x+3)4(2x+3)^4(2x+3)4

using the Binomial Theorem.

(b) Find the coefficient of

x2x^2×2

MODEL ANSWER

Using

(a+b)4=a4+4a3b+6a2b2+4ab3+b4(a+b)^4 =a^4+4a^3b+6a^2b^2+4ab^3+b^4(a+b)4=a4+4a3b+6a2b2+4ab3+b4

Expansion:

16×4+96×3+216×2+216x+8116x^4 +96x^3 +216x^2 +216x +8116×4+96×3+216×2+216x+81

Coefficient of

x2x^2×2

is

216\boxed{216}216​

QUESTION 7

A manufacturer produces two products, A and B.

The constraints are

x+y≤10x+y\le10x+y≤10 2x+y≤162x+y\le162x+y≤16 x≥0,  y≥0x\ge0,\;y\ge0x≥0,y≥0

The objective function is

P=5x+4yP=5x+4yP=5x+4y

(a) Draw the feasible region.

(b) Determine the maximum value of P.

MODEL ANSWER

Corner points:

• (0,0)
• (0,10)
• (6,4)
• (8,0)

Evaluate:

Point	P = 5x + 4y
(0,0)	0
(0,10)	40
(6,4)	46
(8,0)	40

Maximum occurs at

(6,4)\boxed{(6,4)}(6,4)​

Maximum value

P=46\boxed{P=46}P=46​