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➗ Mathematics — Number, Numeration and Algebra

JAMB UTME Mathematics — Number, Numeration and Algebra: Revision Notes

This block of the UTME Mathematics syllabus (Section I: Number and Numeration, and Section II: Algebra) tests fast, accurate work with numbers and standard algebraic processes. On number bases, the syllabus covers operations in bases 2 to 10, including conversion from one base to another and conversion of fractional parts. Under fractions, decimals, approximations and percentages, two formulas are essential: simple interest I = PRT/100 (P = principal, R = rate per cent per annum, T = time in years) and percentage error = (error ÷ actual measurement) × 100%. Ratio, proportion and rate questions apply these same skills to sharing, scale and speed-type problems.

Indices, logarithms and surds are heavily examined. By the first law of indices, am × an = am+n; also a0 = 1 (a ≠ 0) and a-n = 1/an. For logarithms, loga(MN) = logaM + logaN and loga(M/N) = logaM − logaN, while the change-of-base formula is logbN = logaN ÷ logab. To rationalise a fraction such as 1/(√a + √b), multiply numerator and denominator by the conjugate (√a − √b). In Sets, remember that a set with n elements has exactly 2n subsets, and practise two- and three-circle Venn diagram problems.

In the Algebra section, memorise these core results:

Inequalities and linear programming require you to solve linear and quadratic inequalities, represent solutions on the number line or graph, and identify feasible regions. Exam tip: check the discriminant before classifying roots, and confirm |r| < 1 before applying the sum to infinity.

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Sample questions (35)

1. According to the JAMB UTME Mathematics syllabus, candidates are expected to perform operations on number bases and convert from one base to another within which range of bases?

  1. Base 2 to base 16
  2. Base 8 to base 60
  3. Base 2 to base 10
  4. Base 10 to base 16

The syllabus specifies number base work covering bases 2 to 10, including conversion from one base to another and conversion of fractional parts. (JAMB UTME Mathematics Syllabus, Section I: Number and Numeration — Number bases)

2. Convert 101101 in base 2 to a number in base 10.

  1. 43
  2. 45
  3. 46
  4. 41

101101 in base 2 = (1x32)+(0x16)+(1x8)+(1x4)+(0x2)+(1x1) = 45. (JAMB UTME Mathematics Syllabus, Section I — Number bases (conversion to base 10))

3. Convert 39 (base 10) to base 2.

  1. 100101
  2. 101001
  3. 100111
  4. 110011

39 = 32 + 4 + 2 + 1, so in base 2 this is written as 100111. (JAMB UTME Mathematics Syllabus, Section I — Number bases (conversion from base 10))

4. Convert 347 in base 8 to base 10.

  1. 229
  2. 231
  3. 233
  4. 235

347 in base 8 = (3x64)+(4x8)+(7x1) = 192+32+7 = 231. (JAMB UTME Mathematics Syllabus, Section I — Number bases (conversion of octal numbers))

5. Convert 152 (base 10) to base 8.

  1. 231
  2. 220
  3. 230
  4. 232

152 = (2x64)+(3x8)+(0x1), so 152 in base 8 is 230. (JAMB UTME Mathematics Syllabus, Section I — Number bases (conversion to base 8))

6. Which digit can never appear in a number written in base 5?

  1. 2
  2. 5
  3. 3
  4. 4

A base 5 (quinary) system uses only the digits 0 to 4, so 5 cannot appear as a digit. (JAMB UTME Mathematics Syllabus, Section I — Number bases)

7. Add 1011 and 1101 in base 2, giving your answer in base 2.

  1. 11010
  2. 10111
  3. 11000
  4. 11001

1011 in base 2 (11) + 1101 in base 2 (13) = 24, which is written as 11000 in base 2. (JAMB UTME Mathematics Syllabus, Section I — Number bases (operations in base 2))

8. Subtract 101 (base 2) from 1100 (base 2).

  1. 101
  2. 110
  3. 1000
  4. 111

1100 in base 2 (12) minus 101 in base 2 (5) equals 7, which is written as 111 in base 2. (JAMB UTME Mathematics Syllabus, Section I — Number bases (operations in base 2))

9. Multiply 101 (base 2) by 11 (base 2), giving your answer in base 2.

  1. 1110
  2. 1011
  3. 1101
  4. 1111

101 in base 2 (5) multiplied by 11 in base 2 (3) equals 15, which is written as 1111 in base 2. (JAMB UTME Mathematics Syllabus, Section I — Number bases (operations in base 2))

10. Convert the decimal fraction 0.625 to base 2.

  1. 0.110
  2. 0.011
  3. 0.101
  4. 0.100

0.625 = 1/2 + 1/8, so repeated multiplication by 2 gives 0.101 in base 2. (JAMB UTME Mathematics Syllabus, Section I — Number bases (conversion of fractional parts))

11. If 24 in base n is equivalent to 20 in base 10, find the value of n.

  1. 6
  2. 9
  3. 8
  4. 7

24 in base n = 2n + 4 = 20, so 2n = 16 and n = 8. (JAMB UTME Mathematics Syllabus, Section I — Number bases (conversion involving an unknown base))

12. Which base uses only the digits 0 to 7?

  1. Base 10
  2. Base 5
  3. Base 8
  4. Base 2

The octal (base 8) system uses only the digits 0 to 7. (JAMB UTME Mathematics Syllabus, Section I — Number bases)

13. Convert 88 (base 10) to base 5.

  1. 332
  2. 313
  3. 322
  4. 323

88 = (3x25)+(2x5)+(3x1), so 88 in base 5 is 323. (JAMB UTME Mathematics Syllabus, Section I — Number bases (conversion to base 5))

14. By the Remainder Theorem, when a polynomial f(x) is divided by (x - a), the remainder is equal to?

  1. f(1)
  2. f(0)
  3. f(-a)
  4. f(a)

The Remainder Theorem states that dividing f(x) by (x - a) leaves a remainder equal to f(a). (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (factor and remainder theorems))

15. Which of the following is a factor of f(x) = x^2 - 5x + 6, according to the Factor Theorem?

  1. (x - 1)
  2. (x + 2)
  3. (x - 4)
  4. (x - 2)

f(2) = 4 - 10 + 6 = 0, so by the Factor Theorem (x - 2) is a factor of f(x). (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (factor and remainder theorems))

16. For the quadratic equation ax^2 + bx + c = 0 (a not equal to 0), which formula correctly gives the roots?

  1. x = [-b +/- sqrt(b^2 - 4ac)] / (2a)
  2. x = [b +/- sqrt(b^2 - 4ac)] / (2a)
  3. x = [-b +/- sqrt(b^2 + 4ac)] / (2a)
  4. x = [-b +/- sqrt(b^2 - 4ac)] / a

The general quadratic formula for ax^2 + bx + c = 0 is x = [-b +/- sqrt(b^2 - 4ac)] / (2a). (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (solution of quadratic equations))

17. The roots of the quadratic equation ax^2 + bx + c = 0 are real and equal when the discriminant satisfies which condition?

  1. b^2 - 4ac < 0
  2. b^2 - 4ac > 0
  3. b^2 + 4ac = 0
  4. b^2 - 4ac = 0

Real and equal roots occur when the discriminant b^2 - 4ac equals zero. (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (symmetric properties of roots))

18. If a and b are the roots of 2x^2 - 7x + 3 = 0, find the value of a + b.

  1. -7/2
  2. 3/2
  3. -3/2
  4. 7/2

The sum of the roots equals -b/a, so a + b = -(-7)/2 = 7/2. (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (sum and product of roots))

19. If a and b are the roots of 3x^2 + 5x - 2 = 0, find the value of ab.

  1. 5/3
  2. -5/3
  3. 2/3
  4. -2/3

The product of the roots equals c/a, so ab = -2/3. (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (sum and product of roots))

20. A quadratic equation has roots 3 and -2. Which of the following equations has these roots?

  1. x^2 + x - 6 = 0
  2. x^2 - 5x + 6 = 0
  3. x^2 - x + 6 = 0
  4. x^2 - x - 6 = 0

Sum of roots = 1 and product = -6, so the equation is x^2 - (sum)x + (product) = x^2 - x - 6 = 0. (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (forming equations from given roots))

21. Factorise x^2 - 9 completely.

  1. (x - 9)(x + 1)
  2. (x - 3)^2
  3. (x - 3)(x + 3)
  4. (x + 3)^2

x^2 - 9 is a difference of two squares, which factorises as (x - 3)(x + 3). (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (factorisation))

22. Factorise x^2 + 7x + 12 completely.

  1. (x + 2)(x + 6)
  2. (x + 1)(x + 12)
  3. (x + 3)(x + 4)
  4. (x - 3)(x - 4)

Two numbers that multiply to 12 and add to 7 are 3 and 4, so x^2 + 7x + 12 = (x + 3)(x + 4). (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (factorisation))

23. Solve the quadratic equation x^2 - 5x + 6 = 0.

  1. x = -2 or x = -3
  2. x = 2 or x = -3
  3. x = -2 or x = 3
  4. x = 2 or x = 3

Factorising gives (x - 2)(x - 3) = 0, so x = 2 or x = 3. (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (solution of quadratic equations))

24. Using the quadratic formula, solve 2x^2 + 3x - 2 = 0.

  1. x = -1/2 or x = 2
  2. x = 1/2 or x = -2
  3. x = 2 or x = -1/2
  4. x = -1/2 or x = -2

The discriminant is 3^2 - 4(2)(-2) = 25, so x = (-3 +/- 5)/4, giving x = 1/2 or x = -2. (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (solution of quadratic equations))

25. Expand (x + 5)(x - 5).

  1. x^2 + 10x - 25
  2. x^2 - 10x + 25
  3. x^2 + 25
  4. x^2 - 25

(x + 5)(x - 5) is a difference of two squares, which expands to x^2 - 25. (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (expansion and factorisation))

26. Find the remainder when f(x) = x^3 - 2x^2 + x - 5 is divided by (x - 2).

  1. 5
  2. -5
  3. 3
  4. -3

By the Remainder Theorem, the remainder equals f(2) = 8 - 8 + 2 - 5 = -3. (JAMB UTME Mathematics Syllabus, Section II: Algebra — Polynomials (factor and remainder theorems))

27. Convert 3/8 to its decimal equivalent.

  1. 0.375
  2. 0.385
  3. 0.325
  4. 0.415

Dividing 3 by 8 gives 0.375, obtained by expressing the fraction as a decimal through division. (JAMB UTME Mathematics Syllabus, Section I: Number and Numeration — Fractions, Decimals, Approximations and Percentages)

28. Express 0.45 as a fraction in its lowest terms.

  1. 9/25
  2. 4/9
  3. 9/20
  4. 3/8

0.45 = 45/100, which simplifies to 9/20 after dividing numerator and denominator by 5. (JAMB UTME Mathematics Syllabus, Section I: Number and Numeration — Fractions, Decimals, Approximations and Percentages)

29. What is 15% of 240?

  1. 30
  2. 36
  3. 40
  4. 24

15% of 240 = 0.15 x 240 = 36. (JAMB UTME Mathematics Syllabus, Section I: Number and Numeration — Fractions, Decimals, Approximations and Percentages)

30. A trader buys an item for ₦2,500 and sells it for ₦2,875. What is the percentage profit?

  1. 12%
  2. 20%
  3. 15%
  4. 25%

Profit = ₦375, so percentage profit = (375/2500) x 100% = 15%. (JAMB UTME Mathematics Syllabus, Section I: Number and Numeration — Fractions, Decimals, Approximations and Percentages)

31. Using simple interest I = PRT/100, find the interest on ₦50,000 for 3 years at 8% per annum.

  1. ₦10,000
  2. ₦15,000
  3. ₦8,000
  4. ₦12,000

I = (50000 x 8 x 3)/100 = ₦12,000, applying the simple interest formula directly. (JAMB UTME Mathematics Syllabus, Section I — Fractions, Decimals, Approximations and Percentages (simple interest, I = PRT/100))

32. Round off 47.3861 to 2 decimal places.

  1. 47.38
  2. 47.40
  3. 47.39
  4. 47.37

The digit after the second decimal place is 6, so the second decimal digit 8 is rounded up to 9, giving 47.39. (JAMB UTME Mathematics Syllabus, Section I: Number and Numeration — Fractions, Decimals, Approximations and Percentages)

33. The length of a board was measured as 24.0 cm but its actual length is 24.3 cm. Find the percentage error, correct to 2 decimal places, using percentage error = (error/actual value) x 100%.

  1. 1.30%
  2. 1.20%
  3. 1.25%
  4. 1.23%

Error = 0.3 cm, so percentage error = (0.3/24.3) x 100% = 1.23% to 2 decimal places. (JAMB UTME Mathematics Syllabus, Section I — Fractions, Decimals, Approximations and Percentages (percentage error))

34. Approximate 0.0048531 to 2 significant figures.

  1. 0.0048
  2. 0.0049
  3. 0.0050
  4. 0.00485

The first two significant figures are 4 and 8; the next digit, 5, rounds the 8 up to 9, giving 0.0049. (JAMB UTME Mathematics Syllabus, Section I: Number and Numeration — Fractions, Decimals, Approximations and Percentages)

35. Simplify 2/3 + 1/6.

  1. 1/2
  2. 5/9
  3. 5/6
  4. 7/6

Using a common denominator of 6: 4/6 + 1/6 = 5/6. (JAMB UTME Mathematics Syllabus, Section I: Number and Numeration — Fractions, Decimals, Approximations and Percentages)

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