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1. Notion: Number bases

   1. Convert 101101₂ to base ten.

   A43B47C45D41
   Correction

   C. 45

   • 101101₂ = 1×2⁵ + 0×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰.
   • = 32 + 0 + 8 + 4 + 0 + 1 = 45.

2. Notion: Indices

   2. Evaluate 25^(−3/2).

   A1/25B1/125C125D−125
   Correction

   B. 1/125

   • A negative power means a reciprocal: 25^(−3/2) = 1 ÷ 25^(3/2).
   • 25^(1/2) = 5, so 25^(3/2) = 5³ = 125.
   • The answer is 1/125.

3. Notion: Logarithms

   3. Simplify log₂8 + log₂4 − log₂2.

   A4B3C5D6
   Correction

   A. 4

   • log₂8 = 3, log₂4 = 2 and log₂2 = 1.
   • 3 + 2 − 1 = 4.

4. Notion: Indices

   4. Solve 2^(x + 1) = 32.

   Ax = 5Bx = 3Cx = 16Dx = 4
   Correction

   D. x = 4

   • Write 32 as a power of 2: 32 = 2⁵.
   • So x + 1 = 5, which gives x = 4.

5. Notion: Surds

   5. Simplify √12 + √27.

   A√39B6√3C5√3D13√3
   Correction

   C. 5√3

   • √12 = √(4 × 3) = 2√3.
   • √27 = √(9 × 3) = 3√3.
   • 2√3 + 3√3 = 5√3.

6. Notion: Modular arithmetic

   6. Evaluate 5 × 4 (mod 7).

   A20B6C2D3
   Correction

   B. 6

   • 5 × 4 = 20.
   • 20 ÷ 7 = 2 remainder 6, so 20 ≡ 6 (mod 7).

7. Notion: Simple interest

   7. Find the simple interest on ₦20,000 at 5% per annum for 3 years.

   A₦3,000B₦1,000C₦23,000D₦3,150
   Correction

   A. ₦3,000

   • Simple interest = PRT ÷ 100.
   • = 20,000 × 5 × 3 ÷ 100 = ₦3,000.

8. Notion: Quadratic equations

   8. If α and β are the roots of 2x² − 7x + 3 = 0, find α + β.

   A−7/2B3/2C7D7/2
   Correction

   D. 7/2

   • For ax² + bx + c = 0, the sum of the roots is −b/a.
   • Here a = 2 and b = −7, so α + β = 7/2.

9. Notion: Simultaneous equations

   9. Solve x + y = 7 and x − y = 1.

   Ax = 3, y = 4Bx = 5, y = 2Cx = 4, y = 3Dx = 4, y = −3
   Correction

   C. x = 4, y = 3

   • Add the equations: 2x = 8, so x = 4.
   • Substitute: 4 + y = 7, so y = 3.

10. Notion: Variation

    10. y varies inversely as x, and y = 6 when x = 4. Find y when x = 8.

    A12B3C2D48
    Correction

    B. 3

    • Inverse variation: y = k/x.
    • 6 = k/4, so k = 24.
    • When x = 8, y = 24/8 = 3.

11. Notion: Sequences

    11. Find the 5th term of the arithmetic progression 3, 7, 11, ...

    A19B15C23D20
    Correction

    A. 19

    • The first term a = 3 and the common difference d = 4.
    • Tₙ = a + (n − 1)d, so T₅ = 3 + 4 × 4 = 19.

12. Notion: Sequences

    12. Find the sum of the first 4 terms of the geometric progression 2, 6, 18, ...

    A26B54C162D80
    Correction

    D. 80

    • a = 2 and r = 3.
    • Sₙ = a(rⁿ − 1)/(r − 1) = 2(3⁴ − 1)/(3 − 1).
    • = 2 × 80 ÷ 2 = 80.

13. Notion: Sets

    13. n(A) = 12, n(B) = 9 and n(A ∩ B) = 4. Find n(A ∪ B).

    A21B25C17D13
    Correction

    C. 17

    • n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
    • = 12 + 9 − 4 = 17.

14. Notion: Matrices

    14. Find the determinant of the matrix with rows (3, 2) and (1, 4).

    A14B10C11D−10
    Correction

    B. 10

    • For rows (a, b) and (c, d), the determinant is ad − bc.
    • = 3 × 4 − 2 × 1 = 12 − 2 = 10.

15. Notion: Differentiation

    15. Find the gradient of the curve y = 3x² − 4x + 1 at x = 2.

    A8B5C12D6
    Correction

    A. 8

    • dy/dx = 6x − 4.
    • At x = 2: 6 × 2 − 4 = 8.

16. Notion: Integration

    16. Find ∫(4x³ + 2x) dx.

    A12x² + 2 + cB4x⁴ + 2x² + cCx⁴ + 2x² + cDx⁴ + x² + c
    Correction

    D. x⁴ + x² + c

    • Raise each power by 1 and divide by the new power.
    • ∫4x³ dx = x⁴ and ∫2x dx = x².
    • Add the constant: x⁴ + x² + c.

17. Notion: Statistics

    17. Find the median of 3, 8, 5, 9, 4, 7.

    A5B7C6D6.5
    Correction

    C. 6

    • Arrange in order: 3, 4, 5, 7, 8, 9.
    • There are 6 numbers, so the median is the mean of the 3rd and 4th: (5 + 7) ÷ 2 = 6.

18. Notion: Probability

    18. A bag holds 5 red and 3 blue balls. Two balls are picked one after the other without replacement. Find the probability that both are red.

    A25/64B5/14C5/8D5/16
    Correction

    B. 5/14

    • P(first red) = 5/8.
    • One red ball is gone, so P(second red) = 4/7.
    • 5/8 × 4/7 = 20/56 = 5/14.

19. Notion: Trigonometry

    19. If sin θ = 3/5 and θ is acute, find tan θ.

    A3/4B4/3C4/5D5/4
    Correction

    A. 3/4

    • Draw a right-angled triangle with opposite = 3 and hypotenuse = 5.
    • Adjacent = √(5² − 3²) = 4.
    • tan θ = opposite ÷ adjacent = 3/4.

20. Notion: Mensuration

    20. Find the volume of a cylinder of radius 7 cm and height 10 cm. (Take π = 22/7.)

    A440 cm³B3080 cm³C220 cm³D1540 cm³
    Correction

    D. 1540 cm³

    • V = πr²h.
    • = 22/7 × 7 × 7 × 10 = 22 × 7 × 10 = 1540 cm³.