Question

1. The width w of a rectangular swimming pool is x+4. The area A of the pool is 2x^3-29+12. what is an expression for the length of the pool?

a. 2x^2+8x+3
b. 2x^2-8x-3
c. 2x^2-8x+3
d. 2x^2+8-3

2. simplify x/6x-x^2
a. 1/6-x; where x = 0,6
b. 1/6-x; where x=6
c. 1/6; where x=0
d. 1/6

3. simplify -12x4/x4+8x^5
a. -12/1+8x; where x= -1/8
b. -12/1+8x; where x= -1/8,0

4. simplify x+5/ x^2+6x+5
a. 1/x+1; where x= -1
b. 1/x+1; where x=-1, -5

5. simplify x^2-3x-18/x+3
a. x-3
b. x-6; where x= -3
c. x-6; where x= 6
d. 1/x+3; where x= -3

6. simplify 2/3a . 2/a^2
a. 4/3a^2; where a=0
b. x-6; where a=0
c. 4/3a^3; where a=0
d. 4/3a^2

7. multiply x-5/4x+8 times (12x^2+32x+8)
a. (3x+2)/ 4(x-5)
b. (x-5) (3x+2)/ 4
c. (x-5) (3x+2)
d. (x-5) (12x+8)

8. divide. (x^2-16/x-1) / x+4
a. x-4/x-1
b. x+4/ x-1
c. (x+4) (x-3)/ x-1
d. x-4/x+1

9. divide. x^2+2x+1/x-2 / x^2-1/ x^2-4
a. (x+1) (x+2)/ x-1
b. (x-1) (x-2)/ x+1
c. (x+1) (x-2)/ x+1
d. (x-1) (x+2)/ x-1

10. divide (24w^10 + 8w^12) divide by (4x^4)
a. 6w^6+2x^8
b. 6w^6 + 8w^12
c. 24x^10+2w^8
d. 6w^10 + 2w^12

11. divide (-6m^9-6m^8-16m^6) divided by (2m^3)
a. -3m^9-3m^8-8m^6
b. -3m^6-6m^8-16m^3
c. -3m^6-3m^5-8m^3
d. -3m^6-3m^5-16m^3

12. simplify into one fraction -4x/x+7 - 8/x-7
a. -4x+8/x+7
b. x-8/ x
c. -4x-8/x+7
d. x+7/-4x

13. simplify into one fraction 3/x-3 - 5/x-2
a. -2x+9/ (x-3) (x-2)
b. -2x/ (x-3) (x-2)
c. 2x+9/ (x-3) (x-2)
d. 2x+9 / ( x-3) (x-2)

14. simplify into one fraction 9/x-1 - 5/ x+4
a. 4x+5/ (x-1) (x+4)
b. 4x+41/ (x-1) (x+4)
c. 4/ (x-1) (x+4)
d. 14/ (x-1) (x+4)

15. simplify into one fraction -3/x+2 - -5/x+3
a. -8x-19/ (x+2) (x+3)
b. -8/ (x+2) (x+3)
c. 2/ (x+2) (x+3)
d. 2x+1/ (x+2) (x+3)

16. solve 4/x + 5/x = -3
a. x=27
b. x=3
c. x=-3
d. x=-27

17. solve 1/3x-6 - 5/x-2 = 12
a. x= 34/9
b. x= -29/18
c. x= -34/9
d. x=29/18

18. what is the solution of the equation ? 1/x - 6/x^2 = -12
a. x=3/4 or x= -2/3
b. x= 3/4 or x=2/3
c. x= -3/4 or x= 2/3
d. x= -3/4 or x= -2/3

19. Dorothy and Rosanne are baking cookies for party, working alone Rosanne can finish the cookies in 6 hours, Dorothy can finished them in eight hours working alone. How long would it take for them to bake the cookies if they were working together ?
a. 7.00 hours
b. 3.43 hours
c. 0.29 hours
d. 14.00 hours

20 . The pressure, p , for gas varies inversely with it's volume, v ,. pressure is measured in units of pa. Suppose that a particular amount of gas as initially at a pressure of 104 pa at a volume of 108 L. If the volume is expanded to 432 L, what will the new pressure be ?
a. 26 pa
b. 27 pa
c. 416 pa
d. 1728 pa

21. do the data in that table represent a direct variation or an inverse variation?
x: 1,3,5,10
y: 4,12,20,40

a. direct variation y=4x
b. direct variation xy=1/4
c. inverse variation xy= 4
d. inverse variation xy=1/4

22. what are the excluded values of the function? y= 3/4x+64
a. x=0
b. x=-64
c. x=-16
d. x=-8

Answer 1

Question 1

To find the width of the rectangle, we divide the area by the length

`2x^(3)-29x+12`÷
`x+4`
We use the method of long division to get the answer. The method is shown in the first diagram below

Answer:
`2x^(2)-8x+3`

Question 2:

`(x)/(6x-x^(2) ) = (x)/(x(6-x)) = (1)/(6-x)`

Question 3:

`(-12 x^(4) )/(x^(4)+8 x^(5) )= (-12 x^(4) )/( x^(4)(1+8x))= (-12)/(1+8x)`

Question 4:

`(x+5)/(x^(2)+6x+5)= (x+5)/((x+1)(x+5))= (1)/(x+1)`


Question 5:

`\frac{x^(2)-3x-18} {x+3}= ((x-6)(x+3))/(x+3)= (x-6)/(1)=x-6`

Question 6:

`(2)/(3a)`×
`(2)/(a^(2))`=
`(4)/(3a^(3) )` where
`a \\eq 0`

Question 7: (Question is not written well)

`(x-5)/(4x+8)`×
`(12x^(2)+32x+8)`

`(12 x^(3)-28 x^(2) -152x-40 )/(4x+8)`
By performing long division we get an answer
`3 x^(2) -x-36` with remainder of 248

Question 8:

`( \frac{x^(2)-16} {x-1})`÷
`(x+4)`

`( ( x^(2)-16 )/(x-1))`×
`(1)/(x+4)`

`((x+4)(x-1))/(x-1)`×
`(1)/(x+4)`
Cancelling out
`x+4` we obtain
`(x+1)/(x-1)`

Question 9:

`\frac{x^(2)+2x+1} {x-2}`÷
`(x^(2-1) )/(x^(2)-4 )`

`( x^(2)+2x+1 )/(x-2)`×
`(x^(2)-4 )/(x^(2)-1)`
Factorise all the quadratic expression gives

`((x+1)(x+1))/(x-2)`×
`((x-2)(x+2))/((x+1)(x-1))`
Cancelling out
`(x+1)` and
`(x-2)` gives a simplest form

`((x+1)(x+2))/(x-1)`

Question 10:


`(24 w^(10)+8w^(12) )/(4 x^(4) )= (24w^(10) )/(4 x^(4) ) + (8 w^(12) )/(4 x^(4) )`
Cancelling out the constants of each fraction

`(6w^(10) )/(x^(4) )+ (2w^(12) )/(x^(4))= (6w^(10)+2w^(12) )/( x^(4))`

Question 11:


`(-6m^(9)-6m^(8)-16m^(6) )/(2m^(3) ) = (-2m^(6)(3m^(3)-3m^(2)-8))/(2m^(3) )`
Cancelling
`2m^(3)` gives us the simplified form

`-m^(3)(3m^(3)-3m^(2)-8) = -3m^(6)+3m^(5)+8m^(3)`

Question 12:


`(-4x)/(x+7) - (8)/(x-7) = (-4x(x-7)-8(x+7))/((x+7)(x-7))`

`(-4 x^(2) +28x-8x-56)/((x+7)(X-7))= (-4 x^(2) +20x-56)/((x+7)(x-7))`
Factorising the numerator expression

`((-4x+28)(x-2))/((x+7)(x-7)) = (-4(x-7)(x-2))/((x+7)(x-7))`
Cancelling out
`x-7` gives the simplified form

`(-4x+8)/(x-7)`

Question 13:


`(3)/(x-3) - (5)/(x-2)= (x3(x-2)-5(x-2))/(y(x-3)(x-2))`

`(3x-6-5x+15)/((x-3)(x-2))= (-2x+9)/((x-3)(x-2))`

Question 14:


`(9)/(x-1)- (5)/(x+4)= (9(x+4)-5(x-1))/((x-1)(x+4))`
`(9x+36-5x+5)/((x-1)(x+4))= (4x+41)/((x-1)(x+4))`

Question 15:


`(-3)/(x+2)- ((-5))/(x+3)= (-3(x+3)-(-5)(x+2))/((x+2)(x+3))`

`(-3x-9+5x+10)/((x+2)(x+3))= (2x+1)/((x+2)(x+3))`

Question 16:


`(4)/(x)+ (5)/(x)=-3`

`(9)/(x)=-3`

`x=-3`

Question 17:


`(1)/(3x-6)- (5)/(x-2)=12`

`((x-2)-5(3x-6))/((3x-6)(x-2)) = (x-2-15x+30)/((3x-6)(x-2))= (-14x+28)/((3x-6)(x-2))`

Question 18

Answer 2

** CORRECTIONS: Q1: It's 2x^3-29x+12; Q2,3,4,5,6: All conditions have ≠ symbol; Q7: it's (12x^2+32x+16); Q10: Option D should be divided by x^4; **

(1) Given:

Width = W = x+4

Area = A =
`2x^3-29x+12`

Length = L = ?

Since the pool is rectangular in shape:

area = width * length

A = W * L

Substitute:

`2x^3-29x+12 = (x+4) * L \\ L =(2x^3-29x+12)/(x+4)`

The long division is attached with the answer (below in the picture). Hence the correct answer is
`2x^2-8x+3` (Option C)

(2) Given expression:

`(x)/(6x-x^2) \\ (x)/(x(6-x)) \\ (1)/(6-x)`

Where x ≠ 6. (Option B)

(3) Given :

`(-12 x^(4) )/(x^(4)+8 x^(5) )`

Now simplify:

`(-12 x^(4) )/(x^(4)+8 x^(5) )= (-12 x^(4) )/( x^(4)(1+8x))= (-12)/(1+8x)`

Where x ≠ -1/8 (Option A)

(4) Given:

`(x+5)/(x^(2)+6x+5)`

Simplify:

`(x+5)/(x^(2)+6x+5)= (x+5)/((x+1)(x+5))= (1)/(x+1)`

Where x ≠ -1 (Option A)

(5) Given:

`\frac{x^(2)-3x-18} {x+3}`

Simplify:

`\frac{x^(2)-3x-18} {x+3}= ((x-6)(x+3))/(x+3)= (x-6)/(1)=x-6` where x≠6 (Option C)

(6) Given:

`(2)/(3a) .(2)/(a^2)`

Simplify:

`(4)/(3a^(1+2)) = (4)/(3a^(3))`

Where a ≠ 0 (Option C)

(7) Mathematically:

`(x-5)/(4x + 8) * (12x^2+32x+16)`

Simplify:

`(x-5)/(4x + 8) * (12x^2+32x+16) \\ (x-5)/(4(x + 2)) * 12x^2 + (x-5)/(4(x + 2)) * 32x + (x-5)/(4(x + 2)) * 16 \\ (x-5)/((x + 2)) * 3x^2 + (x-5)/((x + 2)) * 8x + (x-5)/((x + 2)) * 4 \\ ((x-5)(3x^2 + 8x + 4x) )/((x+2)) \\ ((x-5)(3x^2 -6x - 2x + 4x) )/((x+2)) \\ ((x-5)(3x+2)(x+2) )/((x+2)) \\ =(x-5)(3x+2)`

(Option C)

(8) Simplify:

`\frac{( \frac{x^(2)-16} {x-1}) }{(x+4)} \\ \frac{( \frac{(x+4)(x-4)} {x-1}) }{(x+4)} \\ = \frac{(x-4)} {(x-1)}`

(Option A)

(9) Simplify:

`( (x^2+2x+1)/(x-2))/((x^2-1)/(x^2-4 )) \\ ( ((x+1)(x+1))/(x-2))/(((x+1)(x-1))/((x-2)(x+2) )) \\ ( ((x+1))/(1))/(((x-1))/((x+2) )) \\ = ((x+1)(x+2))/((x-1))`

(Option A)

(10) Given:

`(24 w^(10)+8w^(12) )/(4 x^(4) )`

Simplify:

`(24 w^(10)+8w^(12) )/(4 x^(4) )= (24w^(10) )/(4 x^(4) ) + (8 w^(12) )/(4 x^(4) ) = (6w^(10) )/(x^(4) )+ (2w^(12) )/(x^(4))= (6w^(10)+2w^(12) )/( x^(4))`

(Option D)

(11) Given:

`(-6m^(9)-6m^(8)-16m^(6) )/(2m^(3) )`

Simplify:

`(-6m^(9)-6m^(8)-16m^(6) )/(2m^(3) ) = (-2m^(6)(3m^(3)+3m^(2)+8))/(2m^(3) ) = -m^(3)(3m^(3)+3m^(2)+8)\\ = -3m^(6)-3m^(5)-8m^(3)`

(Option C)

(12) Simplify:

`(-4x)/(x+7) - (8)/(x-7) = (-4x(x-7)-8(x+7))/((x+7)(x-7)) \\ (-4 x^(2) +28x-8x-56)/((x+7)(X-7))= (-4 x^(2) +20x-56)/((x+7)(x-7)) \\ ((-4x+28)(x-2))/((x+7)(x-7)) = (-4(x-7)(x-2))/((x+7)(x-7)) = (-4x+8)/(x+7)`

(Option A)

(13) Simplify:

`(3)/(x-3) - (5)/(x-2) \\ = (x3(x-2)-5(x-2))/(y(x-3)(x-2)) \\ (3x-6-5x+15)/((x-3)(x-2)) \\= (-2x+9)/((x-3)(x-2))`

(Option A)

(14) Simplify:

`(9)/(x-1)- (5)/(x+4)= (9(x+4)-5(x-1))/((x-1)(x+4)) \\ (9x+36-5x+5)/((x-1)(x+4))= (4x+41)/((x-1)(x+4))`

(Option B)

(15) Simplify:

`(-3)/(x+2)- ((-5))/(x+3)\\= (-3(x+3)-(-5)(x+2))/((x+2)(x+3)) \\ = (-3x-9+5x+10)/((x+2)(x+3))\\= (2x+1)/((x+2)(x+3))`

(Option D)

(16) Given:

4/x + 5/x = -3

Simplify:

(4+5)/x = -3

-3x = 9

x = -3 (Option C)

(17) Simplify:

`(1)/(3x-6) - (5)/(x-2) = 12 \\ ((x-2)-5(3x-6))/((3x-6)(x-2)) = 12 \\ ((x-2)-5*3(x-2))/((3x-6)(x-2)) = 12 \\ (-14(x-2))/((3x-6)(x-2)) = 12 \\ (-14)/((3x-6)) = 12\\ -14 = 12(3x-6) \\ -14 = 36x - 72 \\ 36x = 58 \\ x=(29)/(18)`

(Option D)

(18) Simplify:

`(1)/(x) - (6)/(x^2) = -12 \\ (x - 6)/(x^2) = -12 \\ x-6 = -12x^2 \\ 12x^2 + x - 6 = 0 \\ 12x^2 + 9x - 8x - 6 = 0 \\ 3x(4x + 3) -2(4x + 3) =0 \\ (3x-2)(4x+3) =0 \\ => x =(2)/(3) , x =(-3)/(4)`

(Option C)

(19) Dorothy's rate (alone) will be:

`R_D =(1)/(6)`

Rosanne's rate (alone) will be:

`R_R =(1)/(8)`

If both work together, add both the rates:

`R_T = R_D + R_R = (1)/(6) + (1)/(8) = (7)/(24)` (in 1/hours)

To find the hours, flip the rate:

`(24)/(7) = 3.43` hours (Option B)

(20) As pressure (p) is inversely proportional with volume (v):

p = k/v (where k is constant of proportionality)

k = pv

Find constant using initial values:

k = (104)(108)

k = 11232

Now new pressure is:

p = k/v = 11232/432 = 26 Pa (Option A)

(21)

x: 1,3,5,10

y: 4,12,20,40

Direct variation is the value of y increases with x. So,

y = 4x

If x = 1,y=4(1)=4

If x = 3,y=4(3)=12

If x = 5,y=20

If x = 10,y=40 (Option A)

(22)
`(3)/(4x+64)`

If x=-16,4(-16) + 64 = 0;denominator will become zero,which means that there will be discontinuity at x = -16. Hence, x=-16 (Option C) should be excluded.