ID 27

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***SECTION A***

***_(Answer ALL QUESTIONS From This Section)_***

**(1a)**

**6^n+1*9^n*4^2n/18^n*2^n*12^2n**

**=(2*3)^n+1*(3*3)^n*(2*2)^2n/(2*3*3)^n*2^n*(2*2*3)^2n**

**=2^n+1*2^n+1*3^n*3^n*2^2n*2^2n/2^n*3^n*3^n*2^n*2^n*2^n*3^2n**

**= 2^n*2^1*2^n*2^1*3^n*3^n*2^n*2^n*2^n*2^n/2^n*3^n*3^n*2^n*2^n*2^n*2^n*2^n*3^n*3^n**

**= 2^6n*3^2n*2*2/2^6n*3^4n**

**= (2^6n / 2^6n) *(3^2n / 3^4n)*2*2**

**=2^6n - 6n * 3^2n - 4n * 4**

**= 2^0 * 3^-2n * 4**

**= 4*3^-2n**

**= 4 * 1/3^2n**

**= 4/3^2n**

**(1b)**

**Log 243/log 27**

**= log 3^5/log 3^3**

**= 5log3/3log3 = 5/3 = 1 whole no 2/3**

**============================================**

**(2a) **

**3t - 2p = 8Ã—2 ...(I) **

**2t - 3p = 14Ã—2...(II) **

**6t - 4p = 16 ...(III) **

**6t - 9p = 42 ...(IV) **

** 9P - 4P = -42 + 16**

** 5p = -26**

**P = -26/5 = -5 1/5**

**Put p = -26/5 into eqn (I) **

**3t - 2(-26/5) = 8**

**3t + 52/5 = 8**

**3t = 8 - 52/5**

**3t = 40 - 52/5 = -12/5**

**Divide both sides by 3**

**36/3 = -12/5 / 3**

**t = -12/5 Ã— 1/3**

**= -12/15 = -4/5**

**(2b)**

**3/âˆš3(2/âˆš3 - âˆš12/6)**

**2/âˆš3 Ã— 3/âˆš3 - 3/âˆš3 Ã— âˆš12/6**

**6/âˆš9 - 3âˆš12/6/âˆš3**

**6/3 - 3âˆš12/6âˆš3**

**2 - âˆš12/2âˆš3**

**2 Ã— 2âˆš3 - âˆš12/2âˆš3**

**4âˆš3 - (âˆš4 Ã— 3)/2âˆš3**

**= 4âˆš3 - 2âˆš3/2âˆš3**

**= 2âˆš3/2âˆš3 = 1**

**============================================**

**(3)**

**P(k) = 2/3**

**P(Y) = 5/8**

**P(I) = 3/4**

**P(k fail) = 1- 2/3 = 1/3**

**P(Y fail) 1 - 5/8 = 3/8**

**P(I fail) = 1 - 3/4 = 1/4**

**(a) 2/3*5/8*3/4 = 5/16**

**(b) 1/3*3/8*1/4 = 1/32**

**(c) 2/3*5/8*1*4 = 5/48**

**==============================================**

**(4)**

**Draw the triangle **

**<GFN = 180 - ( 68+90)**

**Sum of angles in a Î”**

**= 180 - 158 = 22°**

**Sin 22/|GN| = Sin 68/8**

**|GN|= 8sin22/sin68**

**= 8Ã—0.3746/0.9272**

**= 2.9969/0.9272**

**= 3.2322cm**

**|GT| = |TN| = 3.2322**

**= 1.616â‰ˆ 1.62cm**

**Let tita = <FTG**

**(i) tan tita = 8/1.6161 = 4.9502**

**tita = tan^-¹(4.9502)**

**tita = 78.6°**

**FTN = 180 - tita**

**= 180 - 78.6**

**(sum of angles on a straight line) = 101.4**

**(ii) >TFN **

**= 180 - (68+101.4)**

**Sum of angles in a Î”**

**180 - 169.4**

**=10.6°**

**==============================================**

**(5a)**

**4 1/2 - 3(y - 2) = 2y + 1/3**

**= 9/2 - 3y + 6 = 2y + 1/3**

**= -3y + 6 + 9/2 = 2y + 1/3**

**6 + 9/2 = 2y + 1/3 + 3y**

**12 + 9/2 = 2y + 1/3 + 3y**

**21/2 = 2y + 1 + 9y/3**

**21 Ã— 3 = 2(2y + 1 + 9y)**

**63 = 4y + 2 + 18y**

**63 = 22y + 2**

**22y = 63 - 2**

**22y = 61**

**y = 61/22 = 2 17/22**

**(5b) **

**Let the number be y **

**7 - 2y >_ 16**

**-2y >_16 - 7**

**-2y >_ 9**

**y _< _9/2 = 4 1/2**

**The greatest possible values of y are; -4, -3, -2, -1, 0**

**===============================================**

***SECTION B***

**_(All Candidates should Answer FOUR while SECRETARIAL & BUSINESS CANDIDATES Should Answers Only TWO from this Section)_**

**(6a)**

**Let the water melon be x**

**Let the mango be y**

**12x + 24y = 432 ...(i) Ã— 24**

**24x + 12y = 360...(ii) Ã— 12**

**288x + 576y = 10368 ...(iii) **

**288x + 144y = 4320 ... (iv) **

**-144y + 576y = -4320+10368**

**432y = 6048**

**y = 6018/432 = 14**

**Put y = 14 into eqn (i) **

**12x + 24(14) = 432**

**12x + 336 = 432**

**12x = 432 - 336**

**12x = 96**

**X = 96/12 = 8**

**(i) Water melon per kg = 8**

**(ii) Mango per kg = 14**

**(iii) (3 Ã— 8) + (2 Ã— 14)**

**= 24 + 28**

**= 52**

**(6b) **

**123x = 83ten**

**1*X² + 2*X¹ + 3*X^0**

**= 83**

**X² + 2x + 3 = 83**

**X² + 2x + 3 - 83 = 0**

**X² + 2x - 80 = 0**

**X² + 10x -8x - 80 = 0**

**X(x+10) - 8(x+10) = 0**

**Then either **

**X - 8 = 0 or X + 10 = 0**

**X = 8 or X = -10**

**================================================**

**(8a) **

**Distance /xy/ = tita/360 Ã— 2Ï€r**

**tita = 45° - 15° = 30°**

**r = RcosÎ±**

**r = 6400 Ã— cos40°**

**r = 6400 Ã— 0.7600**

**r = 4902.4km**

**Distance /xy/ = 30/360 Ã— 2 Ã— 22/7 Ã— 4902.4**

**= 30 Ã— 2 Ã— 22 Ã— 4902.4/2520**

**=6471168/2520 = 2567.9238**

**= 2600km**

**(8bi) **

**Time taken to fly from X to y**

**Speed = Total distance/Total time taken **

**850 = 2600/t**

**t = 2600/850 = 3.0588hrs**

**t = 3hours**

**Time taken to fly to Y from X**

**Time = Distance/Speed**

**= 6700/850 = 7.8824hrs = 8hrs**

**Total time = 3hrs + 8hrs = 11hrs**

**(8bii) **

**Latitude of Z**

**Distance /yz/ = tita/360 Ã— 2Ï€r**

**6700 = Z-40/360Ã—2Ã—22/7Ã—6400**

**6700Ã—360Ã—7 =(Z - 40)281600**

**16884000 = 281600Z - 11264000**

**281600Z = 16884000 + 11264000**

**281600Z = 28148000**

**Z = 28148000/281600**

**Z = 99.9574 = 100°**

**=================================================**

**(9a)**

**DRAW THE TRIANGLE DAIGRAM**

**|YX| = x + 620**

**From âˆ† YFP**

**Tan56 = R/620**

**R = 620 x tan56**

**= 620 x 1.4826**

**h = 919.19m**

**To find x**

**Tan 20 = h/x**

**X = h/tan20 = 919.19/0.3639**

**X = | XF | = 2525.45m**

**|YX| = |XF| + |FY|**

**|YX| = 2525.45 + 620**

**= 3145.5.45m**

**Hence the value of |YX| correct to the four significant figure is 3145m**

**|XY| = 3145m**

**(9b)**

**DRAW THE DAIGRAM**

**|YX|² = |OY|² + |OX|² - 2|OY| |OX| cosÃ˜**

**(6)² = (5)² + (5)² - 2 (5) (5) cosÃ˜**

**36 = 25 +25 â€“ 50cosÃ˜**

**36 = 50 â€“ 50cos Ã˜**

**50cosÃ˜ = 50 â€“ 36**

**50cos = 14**

**cosÃ˜ = 14/50 = 0.28**

**Ã˜ = cos-¹ (0.028) = 73.34º**

**Therefore <XZY = ½ (73.74) (Angle at the center twice the angle at the circumference)**

**<XZY = ½ (73.74) = 36.87º**

**<XZY = 37º (nearest degree)**

**(9bii)**

**Let |XZ| = |YZ| = y**

**|XY|² = |YZ|² + |XZ|² - 2 |YZ| |XZ| Cosã€†**

**(6)² = y² + y² - 2y² cos 36.87**

**36 = 0.4y²**

**y² = 36/0.4 = 60**

**y = âˆš60 = 7.75m**

**hence, |XZ| = 7.8m (1dp)**

**=================================================**

**(11a)**

**In a tabular form **

**Under scores(x)**

**2, 3, 4, 5, 6, 7**

**Under frequency(f) **

**2, 4, 5, 3, 4, 2 Ef, = 20**

**Under fx **

**4, 12, 20, 15, 24, 14 Efx = 89**

**(i) Modal score = 4**

**(ii) Median = n1+n2 = 4+4/2 = 8/2 = 4**

**(iii) Mean X = Efx/Ef = 89/20 = 4.45**

**(11b) **

**A = P(1+r/100)n **

**A = amount in compound interest.**

**P = principal, r = rates, n = no of years the compound interest is charged. **

**A = 53000(1+7/100)^5**

**= 53000(1+0.07)^5 = 53000(1.07)^5**

**In a tabular form **

** No | Log**

**53000| 4.7243 = 4.7243**

** | 0.0294Ã—5 = 0.1469**

** 4.8712**

**Antilog of 4.8712 = â‚¦74436.14**

**Compound interest = 74436.14 - 53000 = â‚¦21336.14**

**=================================================**

***SECTION C***

**_(Only For SECRETARIAL & BUSINESS Candidates. They shoud answers only TWO from this section)_**

**(12a) **

**If passengers paid â‚¦8,400.00 per trip **

**Commission on each trip = 5/100 Ã— 8400 = â‚¦420.00**

**For 52 trips, his commission would be 52 Ã— â‚¦420 = â‚¦21,840.00**

**His take home for the month = â‚¦6000.00 + â‚¦21,840.00 = â‚¦27,840.00**

**(12bi) **

**Gross pay = â‚¦125018.35 per month. **

**5% income tax = 5/100 Ã— 125018.35 = 6250.9175**

**2% Union dues = 2/100 Ã— 125018.35 = 2500.3670**

**1% housing fund = 1/100 Ã— 125018.35 = 1250.1835**

**7.5% pension scheme = 7.5/100 Ã— 125018.35 = 9376.3763**

**2.5% health insurance scheme = 2.5/100 Ã— 125018.35 = 3125.4588**

**10% cooperative contribution = 10/100 Ã— 125018.35 = 12501.8350**

**Total deductions = 6250.9175+2500.3670+1250 1835+9376.3763+3125.4588+12501.8350 = 35005.14**

**(12bii)**

**His net pay = â‚¦125018.35 - 35005.14 = â‚¦90013.21**

**=====================================================**

**(13a)**

**4 - point moving average. **

**16, 18, 20, 27, 28, 31, 32**

**16+11+20+27/4 = 81/4 = 20.25**

**18+20+27+28/4 = 93/4 = 23.25**

**20+27+28+31/4 = 106/4 = 23.25**

**20+27+28+31/4 = 106/4 = 26.5**

**27+28+31+32/4 = 118/4 =**

** 29.5**

**(13b)**

**First 20weeks, she earns â‚¦400Ã—20 = â‚¦8000**

**The next 20weeks she earns â‚¦720Ã—20 = â‚¦14400**

**Total earns for 40weeks = â‚¦8000 + â‚¦14400 = â‚¦22400**

**Total earns for the whole**

** year = â‚¦52Ã—600 = â‚¦31200**

**Amount earns for the remaining 12months = â‚¦31200 - â‚¦22400 = â‚¦8800**

**Average weekly earns for the last 12 months = 8800/12 = â‚¦733.33**

**======================================================**

**(15a)**

**Rent = 15/100 * â‚¦365,500 = â‚¦54,825**

**Personal allowance = 12/100 Ã—â‚¦365,500 = â‚¦43,860**

**Annual tax free allowance **

**= â‚¦54825 + â‚¦43860 = â‚¦98685**

**(15b) **

**Taxable income = â‚¦(365500 - 98685) = â‚¦266,815**

**Tax calculation **

**The first â‚¦65,000 = 0% = â‚¦0**

**Next â‚¦100,000 = 10/100 Ã— 100,000 = â‚¦10,000**

**Next 100,000 = 15/100 Ã— 100,000 = â‚¦15,000**

**The remaining â‚¦1815 = 20/100 Ã— 1815 = â‚¦363**

**Annual income tax = â‚¦(0 + 10,000 + 15,000 + 363)**

**=â‚¦25,363**

**(15c) **

**Percentage of his salary paid as tax = 25,363/365,500 Ã— 100 = 6.94%**

**=======================================================**

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