# Q3 PICK OUT THE SOLUTION FROM THE VALUES GIVEN IN THE BRACKET NEXT TO EACH EQUATION SHOW THAT THE OT

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**1. In a group of cows & chickens, the number of legs was 14 more than twice the number of heads. The number of cows was:(a) 5, (b) 7, (c) 10, (d) 12, (e) 14Solution:****Let the number of cows be x và their legs be 4x. Let the number of chicken be y và their legs be 2x.Total number of legs = 4x + 2y.Total number of heads = x + y.The number of legs was 14 more than twice the number of heads.Therefore, 2 × (x + y) + 14 = 4x + 2y. Or, 2x + 2y + 14 = 4x + 2y.or, 2x + 14 = 4x .or, 14 = 4x – 2x .or, 14 = 2x.or, x = 7 **

**2. The roots of the equation ax2 + bx + c = 0 will be reciprocal if:(a) a = b, (b) a = bc, (c) c = a, (d) c = b, (e) c = ab.Solution:**

**Let k be one of the root of the given equation.According to the problem,1/k will be the other root of the given equation.We know that, product of the roots of the equation = c/a.Therefore, k × 1/k = c/a.or, 1 = c/a.or, a = c**.The roots of the equation ax2 + bx + c = 0 will be reciprocal if a = c.Therefore, a = c or c = a.Answer: (c)

**3. If 8 ∙ 2x = 5(y+8), then when y = -8, x = (a) -4, (b) -3, (c) 0, (d) 4, (e) 8Solution:**

**8 ∙ 2x = 5(y+8).or, 8 ∙ 2x = 5(-8+8).or, 2x = 50.or, 2x = 1**.Taking log on both sides, log 2x = log 1.or, x log 2 = 0 .or, x = 0/log 2 .Therefore, x = 0.Answer: (c)

**4. A circle of radius 10 inches has its center at the vertex C of an equilateral triangle ABC và passes through the other two vertices. The side AC extended through C intersects the circle at D. The number of degrees of angle ADB is:(a) 15, (b) 30, (c) 60, (d) 90, (e) 120**

**Answer: (b)**

**5. The expression 1 – 1/(1 + √3) + 1/(1 - √3) equals:(a) 1 - √3, (b) 1, (c) - √3, (d) √3, (e) 1 + √3.Solution:**

6. If x-1 – 1 is divided by x - 1 the quotient is:(a) 1, (b) 1/(x-1), (c) - 1/(x-1), (d) 1/x, (e) – 1/x Solution:

6. If x-1 – 1 is divided by x - 1 the quotient is:(a) 1, (b) 1/(x-1), (c) - 1/(x-1), (d) 1/x, (e) – 1/x Solution:

**x-1 – 1 is divided by (x – 1)= x-1 – 1 × 1/(x-1)= 1/x – 1 × 1/(x-1)= (1 – x/x) × 1/(x-1)= - (1 – x)/x × 1/(x-1)= – 1/x**.Answer: (e)

**7. The root(s) of 15/(x2 - 4) – 2/(x - 2) = 1 is (are):(a) -5 and 3, (b) ± 2, (c) 2 only, (d) -3 & 5, (e) 3 only.Solution:**

**15/(x2 - 4) – 2/(x - 2) = 1.15/(x2 - 22) – 2/(x - 2) = 1.15/(x + 2)(x - 2) – 2/(x - 2) = 1.<15 – 2(x + 2)>/<(x + 2)(x - 2)> = 1.<15 – 2x - 4> /<(x + 2)(x - 2)> = 1.<11 - 2x>/<(x + 2)(x - 2)> = 1.11 – 2x = (x + 2)(x - 2).11 – 2x = x2 - 4.x2 - 4 = 11 – 2x.x2 + 2x – 4 – 11 = 0.x2 + 2x – 15 = 0.x + 5) (x - 3) = 0.x + 5 = 0 or, x – 3 = 0x = -5 or, x = 3.Therefore, x = -5 & 3.Answer: (a)**

**8. A 7-in. Pizza costs $8 và a 14-in. Pizza costs $20. Tommy says the smaller pizza is a better buy because the larger pizza is twice as big và more than twice as expensive. Bởi vì you agree with his reasoning? If not, explain why not.Solution:****No, a pizza looks lượt thích a circle.And we know that area of a circle depends on the radius.**

**Area of a circle = πr2, here pi (π) is constant.**

So, we can clearly say that area of a circle is proportional lớn the square of the diameter.Similarly, the cost a pizza depends on its area not its diameter.

**9. Suppose that Jody drove 80 miles in 2 hours. Dividing 80 by 2 tells us how many miles Jody drove in each hour. The units for this rate are miles per hour (mi/hr). If we divide 2 by 80 what information would this give us? Give an interpretation of the rate. What units would be used for this rate?Solution:****We know that, speed = distance/time.When total distance is divided by total time we get the speed.Here, speed = distance covered in 1 hour.According to lớn the question,Total distance 80 miles divided by total time 2 hours = 80/2. Miles/hour = 40 mile/hour.Therefore, Jody drove 40 miles in 1 hour.But if we divide total time by total distance then, we get time taken to lớn cover 1 mile.Similarly, if we divide 2 by 80 then, we get time taken to lớn cover 1 mile.Therefore, unit used for this case = hours/mile.Xem thêm: Nhà Trường Chỉ Cho Chúng Ta Chiếc Chìa Khóa Tri Thức Học Trong Cuộc Sống Là Công Việc Cả Đời**

**10. 19/6 = 4/27. Which method did you use to lớn determine whether this proportion is true or false?Solution:****19/6 = 4/27.Cross multiplication:19 × 27 = 513.6 × 4 = 24.We see that 513 are not equal khổng lồ 24.Therefore, we determine 19/6 = 4/27 is not proportion so, the answer is false**.

**11. On Thursday Mabel handled 90 transactions. Anthony handled 10% more transactions than Mabel, Cal handled 2/3rds of the transactions that Anthony handled, and Jade handled 16 more transactions than Cal. How much transactions did Jade handled?Solution:****Mabel handled 90 transactionsAnthony handled 10% more transactions than MabelAnthony = 90 + 90 × 10%= 90 + 90 × 0.10= 90 + 9= 99Cal handled 2/3rds of the transactions than Anthony handledCal = 2/3 × 99 = 66Jade handled 16 more transactions than Cal.Jade = 66 + 16= 82Jade handled = 82 transactions.Answer: 82**

**12. If a man buys trăng tròn lollipops for $90 and sold them for $2 dollars determine his loss.Solution:**** Cost of 20 lollipops = $90 Sold each lollipop for $2 So he sold đôi mươi lollipops for $(20 × 2) = $40 Loss = $(90 – 40) = $50 Therefore, loss = $ 50.**

**13 . If C is a whole number, C+1 is a whole number after that. If A is a whole number, what is a whole number before that?**

**Solution:**

We know that the number ‘0’ together with the natural numbers gives us the numbers 0, 1, 2, 3, 4, 5, …………… which are called **whole numbers**.**If A is a whole number và it is greater than 0 then the whole number before that is A – 1, which will always be a whole number.Again, when A is a whole number and it is equal khổng lồ 0 then the whole number before that is A – 1 which means 0 – 1 = -1, which is not a whole number.Therefore, if A is a whole number then the number before that will not always be a whole number.**

**14. Solve và leave answer in fraction- (2-5/4)/9Solution:**

Answer: 3/2**16. The annual cost of owning and operating a car, C dollars, is a linear function of the distance, d kilometers, it is driven.**c = md + b**The cost is $4600 for 10 000 km và $9100 for 25 000 km. (a) Determine the values of m và b.(b) Write c as a function of d.**

**Solution:**

**(a)**

c = md + b**The cost is $ 4600 for 10 000 km4600 = m (10 000) + b10 000 m + b = 4600 b = -10 000 m + 4600 ------- (i)The cost is $ 9100 for 25 000 km9100 = m (25 000) + b9100 = 25 000 m + b25 000 m + b = 9100 b = -25 000 m + 9100 --------- (ii)From equation (i) and (ii) we get;-10 000 m + 4600 = -25 000 m + 9100-10 000 m + 25 000 m = 9100 – 460015 000 m = 4500m = 4500/15 000m = 3/10 m = 0.3****Now, put the value of m = 0.3 in equation (i) we get;b = -10 000 (0.3) + 4600b = -3000 + 4600b = 1600****Answer: m = 0.3 & b = 1600****(b)** c = md + b**m = 0.3b = 1600c = 0.3d + 1600Answer: c = 0.3d + 1600**

**17. Solve using synthetic division or long division(x2 + 13x + 40) ÷ (x + 8)Solution:**

**We solved the division in both the ways;(x2 + 13x + 40) ÷ (x + 8) by using synthetic division**

Quotient = x + 5Remainder = 0Answer: x + 5

Quotient = x + 5Remainder = 0Answer: x + 5

**18. Solve using on synthetic division or long division (2x2 - 23x + 63) ÷ (x - 7)Solution:**

**We solved the division in both the ways;(2x2 - 23x + 63) ÷ (x - 7) by using synthetic division**,

Quotient = 2x - 9

Quotient = 2x - 9

**Remainder = 0Answer: 2x - 9****19. Let f(x) = 4x3 + 7x2 – 13x – 3 and g(x) = x + 3. Find f(x)/g(x)Solution:****f(x) = 4x3 + 7x2 – 13x – 3g(x) = x + 3f(x)/g(x) = (4x3 + 7x2 – 13x – 3)/(x + 3)We solved the division in both the ways;(4x3 + 7x2 – 13x – 3) ÷ (x + 3) by using synthetic division**

Answer:Quotient = 4x2 – 5x + 2Remainder = -9Answer:Quotient = 4x2 – 5x + 2Remainder = -9

**OR****(4x3 + 7x2 – 13x – 3) ÷ (x + 3) by using long division**

Answer:Quotient = 4x2 – 5x + 2Remainder = -9Answer:Quotient = 4x2 – 5x + 2Remainder = -9

**20. Let f(x) = 3x3 - 4x – 1 và g(x) = x + 1. Find f(x)/g(x)Solution:****f(x) = 3x3 - 4x – 1g(x) = x + 1f(x)/g(x) = (3x3 - 4x – 1)/(x + 1)We first need lớn rearrange all the terms of the dividend according to lớn descending powers of x. The dividend then becomes 3x3 - 4x – 1, with 3 understood as the coefficient of the first term. No x2 term is there in the polynomial, but we take a zero as a place holder in the x2 position, so the dividend is written as 3x3 + 0x2 - 4x – 1.We solved the division in both the ways;(3x3 + 0x2 - 4x – 1) ÷ (x + 1) by using synthetic division**

Quotient: 3x2 – 3x – 1Remainder: 0ORQuotient: 3x2 – 3x – 1Remainder: 0OR

**No x2 term is there in the polynomial, but we take a zero as a place holder in the x2 position, so the dividend is written as 3x3 + 0x2 - 4x – 1.(3x3 + 0x2 - 4x – 1) ÷ (x + 1) by using long division**

Quotient: 3x2 – 3x – 1Remainder: 0Answer: 3x2 – 3x – 1Quotient: 3x2 – 3x – 1Remainder: 0Answer: 3x2 – 3x – 1

**21. Let f(x) = x4 – 8x3 + 16x2 – 19 and g(x) = x - 5. Find f(x)/g(x)Solution:****f(x) = x4 – 8x3 + 16x2 – 19g(x) = x – 5f(x)/g(x) = (x4 – 8x3 + 16x2 – 19)/(x - 5)We first need khổng lồ rearrange all the terms of the dividend according to descending powers of x. The dividend then becomes x4 – 8x3 + 16x2 – 19, with 1 understood as the coefficient of the first term. No x term is there in the polynomial, but we take a zero as a place holder in the x position, so the dividend is written as x4 – 8x3 + 16x2 + 0x – 19.We solved the division in both the ways;(x4 – 8x3 + 16x2 + 0x – 19) ÷ (x - 5) by using synthetic division**

Quotient = x3 - 3x2 + x + 5Remainder = 6ORQuotient = x3 - 3x2 + x + 5Remainder = 6OR

**No x term is there in the polynomial, but we take a zero as a place holder in the x position, so the dividend is written as x4 – 8x3 + 16x2 + 0x – 19.(x4 – 8x3 + 16x2 + 0x – 19) ÷ (x - 5) by using long division**

Quotient = x3 - 3x2 + x + 5Remainder = 6Answer:Quotient = x3 - 3x2 + x + 5Remainder = 6Quotient = x3 - 3x2 + x + 5Remainder = 6Answer:Quotient = x3 - 3x2 + x + 5Remainder = 6

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