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Mariam's Pay: a) Determine the number of extra hours she taught. b) Calculate her total pay for the month. c) Find the total deduction made before tax.

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here's the solution for question 1 from Section B.

1 (i) Mariam's Pay

a) Determine the number of extra hours she taught.

  1. Identify total hours and normal hours.

    Total hours worked =50= 50 hours.
    Normal contracted hours =36= 36 hours.

  2. Calculate extra hours.

    Extra hours=Total hours−Normal hours\text{Extra hours} = \text{Total hours} - \text{Normal hours} Extra hours=50−36=14 hours\text{Extra hours} = 50 - 36 = 14 \text{ hours}

    The number of extra hours she taught is 14 hours\boxed{14 \text{ hours}}.

    b) Calculate her total pay for the month.

  3. Calculate pay for normal hours.

    Normal rate =4,500= 4,500 FCFA/hour.

    Pay for normal hours=36 hours×4,500 FCFA/hour=162,000 FCFA\text{Pay for normal hours} = 36 \text{ hours} \times 4,500 \text{ FCFA/hour} = 162,000 \text{ FCFA}

  4. Calculate the rate for extra hours.

    Extra hour rate =150% of normal rate=1.50×4,500 FCFA/hour=6,750 FCFA/hour= 150\% \text{ of normal rate} = 1.50 \times 4,500 \text{ FCFA/hour} = 6,750 \text{ FCFA/hour} .

  5. Calculate pay for extra hours.

    Pay for extra hours=14 hours×6,750 FCFA/hour=94,500 FCFA\text{Pay for extra hours} = 14 \text{ hours} \times 6,750 \text{ FCFA/hour} = 94,500 \text{ FCFA}

  6. Calculate total pay.

    Total pay=Pay for normal hours+Pay for extra hours\text{Total pay} = \text{Pay for normal hours} + \text{Pay for extra hours} Total pay=162,000 FCFA+94,500 FCFA=256,500 FCFA\text{Total pay} = 162,000 \text{ FCFA} + 94,500 \text{ FCFA} = 256,500 \text{ FCFA}

    Her total pay for the month is 256,500 FCFA\boxed{256,500 \text{ FCFA}}.

    c) Find the total deduction made before tax.

  7. Calculate pension deduction.

    Pension deduction =5% of total pay=0.05×256,500 FCFA=12,825 FCFA= 5\% \text{ of total pay} = 0.05 \times 256,500 \text{ FCFA} = 12,825 \text{ FCFA} .

  8. Identify Social Insurance deduction.

    Social Insurance deduction =15,000= 15,000 FCFA.

  9. Calculate total deduction.

    Total deduction=Pension deduction+Social Insurance deduction\text{Total deduction} = \text{Pension deduction} + \text{Social Insurance deduction} Total deduction=12,825 FCFA+15,000 FCFA=27,825 FCFA\text{Total deduction} = 12,825 \text{ FCFA} + 15,000 \text{ FCFA} = 27,825 \text{ FCFA}

    The total deduction made before tax is 27,825 FCFA\boxed{27,825 \text{ FCFA}}.

    d) Find the income that was taxed.

  10. Calculate income after deductions.

    Income after deductions=Total pay−Total deduction\text{Income after deductions} = \text{Total pay} - \text{Total deduction} Income after deductions=256,500 FCFA−27,825 FCFA=228,675 FCFA\text{Income after deductions} = 256,500 \text{ FCFA} - 27,825 \text{ FCFA} = 228,675 \text{ FCFA}

  11. Subtract the tax-free amount.

    Tax-free amount =50,000= 50,000 FCFA.

    Taxable income=Income after deductions−Tax-free amount\text{Taxable income} = \text{Income after deductions} - \text{Tax-free amount} Taxable income=228,675 FCFA−50,000 FCFA=178,675 FCFA\text{Taxable income} = 228,675 \text{ FCFA} - 50,000 \text{ FCFA} = 178,675 \text{ FCFA}

    The income that was taxed is 178,675 FCFA\boxed{178,675 \text{ FCFA}}.

    e) Find the tax paid.

  12. Calculate tax paid.

    Tax rate =10%= 10\%.

    Tax paid=10% of Taxable income\text{Tax paid} = 10\% \text{ of Taxable income} Tax paid=0.10×178,675 FCFA=17,867.50 FCFA\text{Tax paid} = 0.10 \times 178,675 \text{ FCFA} = 17,867.50 \text{ FCFA}

    The tax paid is 17,867.50 FCFA\boxed{17,867.50 \text{ FCFA}}.

    1 (ii) Polynomial f(x)=2x3+x2−18x+9f(x) = 2x^3 + x^2 - 18x + 9

    a) Show that (x+3)(x+3) is a factor of f(x)f(x).

  13. Apply the Factor Theorem.

    According to the Factor Theorem, if (x+3)(x+3) is a factor of f(x)f(x), then f(−3)f(-3) must be equal to 0.
    Substitute x=−3x=-3 into f(x)f(x):

    f(−3)=2(−3)3+(−3)2−18(−3)+9f(-3) = 2(-3)^3 + (-3)^2 - 18(-3) + 9 f(−3)=2(−27)+9+54+9f(-3) = 2(-27) + 9 + 54 + 9 f(−3)=−54+9+54+9f(-3) = -54 + 9 + 54 + 9 f(−3)=0f(-3) = 0

    Since f(−3)=0f(-3)=0, (x+3)(x+3) is a factor of f(x)f(x).

    b) Factorise f(x)f(x) completely.

  14. Perform polynomial division to divide f(x)f(x) by (x+3)(x+3).

    Using synthetic division:

    -3 | 2   1   -18   9
   |    -6    15  -9
   -----------------
     2  -5    -3   0

    The quotient is 2x2−5x−32x^2 - 5x - 3.
    So, f(x)=(x+3)(2x2−5x−3)f(x) = (x+3)(2x^2 - 5x - 3).

  15. Factorise the quadratic factor 2x2−5x−32x^2 - 5x - 3.

    We look for two numbers that multiply to 2×(−3)=−62 \times (-3) = -6 and add to −5-5. These numbers are −6-6 and 1.
    Rewrite the middle term:

    2x2−6x+x−32x^2 - 6x + x - 3

    Factor by grouping:

    2x(x−3)+1(x−3)2x(x-3) + 1(x-3) (2x+1)(x−3)(2x+1)(x-3)

  16. Write the complete factorisation of f(x)f(x).

    f(x)=(x+3)(2x+1)(x−3)f(x) = (x+3)(2x+1)(x-3)

    The complete factorisation of f(x)f(x) is (x+3)(2x+1)(x−3)\boxed{(x+3)(2x+1)(x-3)}.

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