Mathematics OBJ:

1-10: CABDDADCBC

11-20: BCBCBBAACD

21-30: DCCBCCABBA

31-40: AABDDCDDCC

41-50: BBCDCCBCCD

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

**(1a)**

**110ₓ = 40₅**

**change all to base ten.**

**110ₓ = 40₅**

**1x²+1x ¹+0x⁰ = 45¹+0*5⁰**

**x² + x + 0 = 20**

**x² + x – 20 = 0**

**Solve quadratically**

**x² + 5x – 4x – 20 = 0**

**x(x+5) – 4(x+5) = 0**

**(x-4)(x+5) = 0**

**Either x-4 = 0 or x + 5 = 0**

**x = -5 or 4**

**therefore, x= 4**

**(1b)**

**15/√75 + √108 + √432**

**= 15/√253 + √363 + √144*3**

**= 15/5√3 + 6√3 + 12√3**

**= 3/√3 + 18√3**

**= 3√3/3 = 18√3**

**= √3 + 18√3**

**=19√3**

**CLICK HERE FOR THE IMAGE**

**(2a)**

**y2-y1/x2-x1 = y-y1/x-x1**

**Given : A(-2,7) and B(2,-3)**

**-3-7/2+2 = y-7/x+2**

**-10/4=y-7/x+2**

**4(y-7)= -10(x+2)**

**4y-28= -10x-20**

**4y= -10x+28-20**

**4y= -10x+8**

**2y= -5x+4**

**OR**

**5x+2y=4**

**(2b)**

**5b-a/8b+3a=1/5**

**(5b-a)/b/(8b+3a)/b =1/5**

**5-(a/b)/8+3(a/b) =1/5**

**8+3(a/b)= 5 [ 5-(a/b)]**

**8+3(a/b)=25-5(a/b)**

**3(a/b) + 5(a/b)=25-8**

**8(a/b)=17**

**a/b=17/8**

**CLICK HERE FOR THE IMAGE**

**(3a)**

**Total ratio =3+5+8=16**

**Ali’s share =3/16 * 420,000/1**

**=210,000**

**Son’s of Ali and Yusuf’s share = N78,750+ N210,000**

**= N288,750**

**(3b)**

**2(1/8)^x=32^x-1**

**2^1(2^-3)^x=2^5(x-1)**

**2^1 * 2^-3x=2^5(x-1)**

**2^-3x+1=2^5(x-1)**

**-3x+1=5(x-1)**

**-3x+1=5x-5**

**-5x-3x= -5-1**

**-8x/-8= -6/-8**

**x=3/4**

**CLICK HERE FOR THE IMAGE**

**(4)**

**Since <PQR = <PRS = 90°**

**Using Pythagoras theorem**

**|PR|² = |PQ|² + |QR|²**

**|PR|² = 3² + 4²**

**|PR|² = 9 + 16**

**|PR|² = 25 PR = √25**

**|PR| = 5cm**

**Considering PRS**

**|PS|² = |PR|²+|SR|²**

**13² = 5² + |SR|²**

**169 = 25 + |SR|²**

**|SR|² = 169 – 25**

**|SR|² = 144**

**|SR| = √144 = 12cm**

**Hence the area of the quadrilateral = Area of triangle PQR + area of PRS**

**= 1/2bh + 1/2bh**

**= 1/2×4×3 + 1/2×12×5**

**= 6+30 = 36cm**

**CLICK HERE FOR THE IMAGE**

**(5a)**

**No of red balls = 3**

**No of green balls = 5**

**No of blue balls = x**

**Prob.(red ball) = no of total outcome/no of possible outcome**

**Pr(red) = 3/3+5+x = 1/6**

**3/8+x = 1/6**

**6(3) = 1(8+x)**

**18 = 8 + x**

**X = 18 – 8 = 10**

**Therefore the no of blue ball = 10**

**(5b)**

**Probability of picking a green ball**

**P(g) = no of green balls/no of possible outcome**

**P(g) = 5/3+5+10 = 5/18**

**=5/18**

**CLICK HERE FOR THE IMAGE**

**(6ai)**

**F α M1M2/d²**

**F = KM1M2/d²**

**Given F = 20N, M1= 25kg, M2 = 10kg and d = 5m**

**20 = k(25)(10)/5²**

**250k = 500**

**k = 500/250 = 2**

**Expression is**

**F = 2M1M2/d²**

**(6aii)**

**Making d subject**

**d = √2M1M2/F**

**d = √2 ×7.5×4/30**

**d = √60/30 = √2**

**d = √2m or 1.41m**

**(6b)**

**Draw the diagram**

**X+X+60+X+80+X+40+X+20 = 540(sum of angles in a Pentagon)**

**5x + 200 = 540**

**5x = 540 – 200**

**5x = 340**

**X = 340/5**

**X = 68**

**CLICK HERE FOR THE IMAGE**

**(8a)**

**1/3x – 1/4(x+2)>_ 3x -1⅓**

**1/3x – 1/4(x+2)> 3x – 4/3**

**Multiply through by the L. C. M(12), we have****4x – 3(x + 2)>_36x – 16****4x – 3x – 6 >****36x – 16**

**-6+16 >**

*36x + 3x – 4x*

**10 >****35x**

**35x _< 10**

**X = 10/35**

**X = 2/7**

**(8bi)**

**Draw the triangle**

**|AB|/66 = sin35**

**|AB| = 66sin35 = 66×0.5736 = 37.8576**

**Draw the right angled triangle**

**|AD|/|AB| = Tan52**

**|AD| = 37.8576 × Tan52° = 37.8576 × 1.2799 = 48.45m**

**Height of tower = 48.45m**

**(8bii)**

**|AC|/66 = Cos35°**

**|AC| = 66 x cos35°**

**= 66 x 0.8192**

**= 54.0672**

**Tan = 41.86°**

**Angle of elevation of top of tower from c = 41.85°**

**CLICK HERE FOR THE IMAGE**

**(10)**

**130kg of tomatoes for #52,000**

**Half of the tomatoes**

**130/2 = 65kg sold at 30%**

**Profit = #52,000/2 = 26,000**

**#26,000 = 100%**

**X = 130%**

**X = 26000 × 130/100**

**= #33,800**

**Then 65kg was then sold at reduction of 12% per kg**

**Recall that the initial cost price = 52000/130**

**=400kg**

**65kg sold at = 33,000/65**

**=#520/kg**

**Then for 12% reduction**

**520 × 88/100 = 457.6/kg**

**(a)**

**The new selling price per kg = #457.6/kg**

**(b) 65kg – 5kg = 60kg**

**(60kg×457.6kg)+33,800**

**= #61,256.00**

**#profit = selling price /cost price × 1000/1**

**=61256/52000×100/1= 117.8**

**= 17.8%**

**CLICK HERE FOR THE IMAGE**

**(11ai)**

**ar² = 1/4 ……(1)**

**ar^5= 1/32 …..(2)**

**Divide eqn (2) by eqn(1)**

**ar^5/ar² = 1/32÷1/4**

**r³ = 1/32 × 4/1**

**r³= 1/8**

**r³ = 2-³**

**r = 2-¹**

**r = 1/2**

**Common ratio = 1/2**

**Put this into eqn (1)**

**a(1/2)² = 1/4**

**a(1/4) = 1/4**

**a = (1/4)/(1/4) = 1**

**First term, a = 1**

**(11aii)**

**Seventh term, T7 = ar^6**

**=(1)(1/2)^6**

**=1/64**

**(11b)**

**Given : X = 2 and X = -3**

**(X – 2)(X + 3) = 0**

**X² + 3x – 2x – 6 , 0**

**X² + x – 6 = 0**

**Comparing with ax²+bx+c = 0**

**a = 1**

**b = 1**

**C = -6**

**CLICK HERE FOR THE IMAGE**

**(12a)**

**Given : siny = 8/17**

**Draw the right angle**

**From Pythagorean triple, third side is 15**

**Draw the right angle triangle**

**tan y = 8/15**

**tan y/1+2tany = 8/15/1+2(8/15) = 8/15/1+16/15**

**tany /1+2tan y = 8/31**

**(12b)**

**Amount shared = #300,000**

**Otobo’s share = #60,000**

**Ade’s share = 5/12 × #(300,000-60,000)**

**= 5/12 × #240,000**

**=#100,000**

**Adeobi’s share = #300,000 – (#60,000 + #100,000)**

**= 300,000 – 160,000**

**=#140,000**

**Ratio : Otobo : Ade : Adeola**

**60,000 : 100,000 : 140,000**

**60 : 100 : 140**

**6 : 10 : 14**

**3 : 5 : 7**

**CLICK HERE FOR THE IMAGE**

**(13a)**

**RT^0=RS^0=90° (radius o targets)**

**T^0S=2TU^0 (angle at centre = 2 time angle at circumstance)**

**TOs=2*68°=136°**

**Now RT^0+RS^0+T^0S+SRT^0=360°(sum of angle in a quadrilateral)**

**90° + 90° + 136° + x = 360°**

**X+316°=360°**

**X=360°-316**

**X=44°**

**(13b)**

**Let tank B hold x litres**

**; Tank A hold (x+600)literally**

**3(x-100)=(x+600-100)**

**3x-300=x+500**

**3x-x=500+300**

**2x=800**

**X=800/2=400**

**Tank B holds 400 litres**

**Tank A holds (400+600)=1000litres**

**CLICK HERE FOR THE IMAGE**

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