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Algebraic expressions Questions

 

1.  Simplify Image From EcoleBooks.com  (3mks)

 

2.  Five year ago, a mother’s age was four times that of her daughter. In four years to come,

 she will be 2 ½ times the age of her daughter. Calculate the sum of their present ages  

 

3.  Mutua bought 160 trays of 8 eggs each at shs.150 per tray. On transportation 12 eggs

broke. He later discovered that 20 eggs were rotten. If he sold the rest at shs.180 per

tray, how much profit did he make?

4.  Simplify;

 (a) 6a – 2b + 7a – 4b + 2

Image From EcoleBooks.com  (b) 2x – 2   3x + 2

2x   4x  

 

5.  Simplify   6x2y2 + 13xy-5

ecolebooks.com

   3x2y2 – 13xy + 4

 

6.  Given that x + y = 8 and x2 + y2 = 24

 Find;

 (a) the value of x2 + 2xy + y2

 (b) Find the value of  2xy

 (c) x2 – 2xy + y2  

 (d) x – y

 (e) Value of x and y  

 

7.  Simplify the expression.

 6x2 + 35x – 6

 2x2 – 72

 

8.  Simplify the expression

  2/3 (3x -2) – ¾ (2x -2)  

 

9.  Simplify by factorizing completely:

4y2 – x2

2x2 – yx -6y2

10.  Simplify as far as possible.

  3 – 1


x – y x + y

 

11.  By calculation, find the coordinates of the intersection of the graphs y = x2 + 2x -5

and y = 3x +1

 

12.  Simplify:

(a) y2 + 2y = ¼

y3 – y2 – 6y

 (b) hence solve:- y2 + 2y = ¼

 y3 – y2 – 6y

 

13.  A rectangular field measures 63.9m by 104.6metres find the minimum number of poles to be

  erected for fencing if they are to be at most 2.4meters apart.  

 

14.  Factorize completely the expression

 75x2 – 27y2

 

15.  Every time an insect jumps forward the distance covered is half of the previous jump.

If the insect initially jumped 8.4cm, calculate

 (i) To the nearest two decimal places distance of the sixth jump  

 (ii) The total distance covered after the sixth jump

 

16.  Simplify  P3 –Pq2 + P2q – q3

P2 + 2pq + q2

 

17.  Simplify the expression:- 9x2 – 4y2

12x2 + yx – 6y2  

18.  Given that (x-3) (Ax²+bx+c) = x³-7x-6, find the value of A, B and C

 

19.  a) solve for y in 8x(2²)ʸ=6×2ʸ-1

 b) Simplify completely 2x²-98 ÷ x+7  

3x²-16x-35 3x+5

 

20.  Simplify the expression.:

 

 4x2 – y2

 2x2 – 7xy + 3y2

 

21. Simplify P2 -2Pq + q2

P3 –Pq2+ P2q – q3  

 

22.  The sum of two numbers is 15. The difference between five times the first number and

three times the second number is 19. Find the two numbers

 

23.  Simplify the following expressions by reducing it to a single fraction

2x – 5 – 1 – x – x – 4

  4 3 2

 

24.  Simplify the expression:- 3a2 + 4ab + b2  

4a2 + 3ab – b2

 

Algebraic expressions Answers

 

1.

Image From EcoleBooks.com

 

 

 

M1

 

 

M1

 

A1

 
  

03

 

 

2.  Let the daughter’s age 5yrs ago be x

Mother 4x

come;

Daughter = x + 9

Mother = 4x+ 9

4x + 9 = 5/2 (x +9)

4x + 9 = 2.5x + 22.5

1.5x = 13.5

  x = 9

Mother = 41yrs

14 + 41 =55

 

3.  B.P = 160 x 50 = 24000

S.P = ((160 x 8) – (20 + 12)) x 180

8

= 28080

Profit = 28080 – 24000 = Shs.4080

 

 

4.  a) 6a + 7a – 2b – 4b + 2

= 13a – 6b + 2

 

b) 2x – 2 – 3x + 2 = 2(2x – 2) – (3x + 2)

  2x 4x 4x

= 4x – 3x – 4 – 2

4x

= x – 6

4x

 

5.  6u2y2 + 13uy – 5 = (2uy + 5) (3xy – 1)

3u2y2 – 13uy + X = (uy – 4) (3xy – 1)

(2xy +5) (3xy – 1)

(uy – 4) (3xy -1)

 

= 2xy + 5

Uy – 4

 

6.  a) From x + y and x2 = y2 = 34

X = 8 – y

Substituting for x in x2 – y2 = 34

(8 – y) (8 – y) + y2 = 34

64 – 8y – 8y + y2 + y2 = 34

64 – 16y + 2y2 = 34

 

2y2 – 16y + 64 – 34 = 0

2y2 – 16y + 30 = 0

y2 = 8y + 15 = 0

y (y – 3) – 5 (y-3) = 0 (y-5) (y – 3)

y is either 5 or 3

but x – y = 8

x is either 5 0r 3

 x2 + 2xy + y2 = 32 + 2 x 3 x 5 + 25

= 9 + 30 + 25 = 64

 

b) 2xy = 2 x 3 x 5 = 30

c) x2 – 2xy + y2 = 9 – 2 x 3 x 5 + 25 = 4

 

d) x = y = 8 and x2 + y2 = 34

  x = 8 – y

   (8 – y)2 + y2 = 34

y2 – 8y + 15 = 0

y2 – 3y – 5y + 15 = 0

y(y -3) – 5(y – 3)

(y-3) = o y = 3

  (y-5) = 0 y = 5

 

x + 3 = 8, x = 5 or x + 5 = 8

x = 3

 x is either 3 or 5

  y is either 3 0r 5

 

7.  6x2 + 35x – 6

  2x2 – 72

= 6x(x -+ 6) -1(x + 6)

  2(x2 – 36)

  = (6x – 1) (x + 6)

  2(x – 6) (x + 6)

 = 6x – 1

  2(x – 6)

8.  2/5 (3x -2) – ¾ (2x -2)

= 8(3x-2) -9(2x-2)

Image From EcoleBooks.com 12

= 24 x -16 – 18x + 18

= 6x + 2

  12

= 2(3x +1)

12

= 3x + 1

  6

 

9.  Numerator:

4y2 – x2 = (2y +x) (2y – x)

Denominator :

2x2 + 4yx + 3yx – 6y2

= (2x2 – 4yx) + (3yx – 6y2)

= 2x(x-2y) + 3y(x-2y)

= (2xx+3y) (x-2y)

Combining : (2y + x) (2y-x)

(2x+3y) (x-2y)

2x + 3y or -2x -3y

  2y + x 2y + x

 

10.  3(x + y) – (x –y)

  X 2 – y2

 = 3x + 3y – x + y

x2 – y2

  = 2(x +2y)

  x2 -y2

 

11.   x2 + 2x – 5 = 3x +1

x2 – x – 6 – 6 =0

(x+2) (x-3) = 0

x =-2 or x = 3

When x =-2, y = 3x – 2 + 1 = -5 Point (-2, -5)

When x = 3, y = 3x x 3 +1 = 10 Point (3, 10)

 

12.  (a) y(y + 2)

   y(y2 – y -60

  y(y+2) = y + 2

  y(y2 – y-6) (y+2) (y-3)

 

(b) y + 2 = ¼

(y + 2) (y-3)

4y + 8 = y2 – y – 6

y2 – 5y – 14 = 0

(y-7)(y+2)= 0

y=7

y=-2

 

13.  104.6 = 44 x 2

 2.4

63.9 = 26 x 2

 2.4

= 88 + 54= 142

14.  3 ( 25 x2 – 9y2)

3 (5x + 3y) (5x – 3y)

 

 

15.  i) d = 8.4 r = ½

 6th jump = 8( ½ )6-1


8.4/32

 

= 0.2625 = 0.26cm

 

ii) 56 = 9.4 (1 – ( ½ ) 6

1 – ½

  = 8.4 x 63 x 2

  64

  = 16.54 cm

 

16.  Factorizing the numerator

= p(p2 – q2) + q(p2-q)

= (p+q) (p2-q2)

= (p+q) (p+q) n(p-q)

Factorising the denominator

(p+q) (p+q)

Numerator = p – q

Denominator

 

17.   ( 3x + 2y ) ( 3x – 2y )

 ( 3x + 2y ) ( 3x – 2y )

  3x + 2y

  4x + 3y

 

18.  (x – 3) (AX2 +BX + C) = x3 – 7x – 6

AX3 + BX2 +CX – 3AX2 – 3BX – 3c = x3 – 7x – 6

A = 1

B – 3A = 0

B – 3 x 1 = 0

B = 3

-3c = -6

c = 2

 

19.  a) 8(22)y = 6 x 2y – 1

    let t = 2y

   8t2 = 6t – 1

8t2 – 4t – 2t + 1 = 0

(4t – 1) (2t – 1) = 0

t = ¼ or ½

t = 2y = ¼ = 2-2

y = -2

0r t = 2y = ½ = 2-1

y = -1

y = -2 0r -1

b) Numerator = 2x2 – 98

  = 2(x2 – 49)

 = 2(x+ 7) (x -7)

Denominator = 3x2 – 16x – 35

= 3x2 – 21x + 5x – 35

= 3x(x -7) + 5(x – 7)

= (x – 7) (3x + 5)

2x2 – 98

x + 7 = 2(x + 7) (x – 7) x (3x + 5)

3x2 – 16x – 3 3x + 5 (3x + 5) (x – 7) (x + 7)

 

= 2

20.  (2x- y) (2x + y) √

  (x – 3y) (2x – y) √

  2x + y √

  x – 3y

 

21.  P2 – 2pq + q2 = (p-q)2

P3 – pq2 + p2q – q3

= p(p2-q2) + q (p2 – q2)

= (p+ q) (p2-q2)

(p – q)2
= (p – q)2

Image From EcoleBooks.com(p+ q) (P2 – q2) (p+q)2 (p-q)

  = p- q

  (p + q)2

 

 

22.  Let the numbers be a and b

a + b = 15 – x3

5a – 3b = 19 x 1

 

 3a + 3b = 45

 5a – 3b = 19

  8a = 64

  a = 8

 b = 7  

 

  4 3 2

23.  3(2x-5) – 4(1-x) – 6(x-4)

12

6x – 15 – 4+ 4x – 6x + 24

12

4x – 5

12

 

24.  3a2 + 4ab + b2 = 3a2 + 3ab + ab +b2

4a2 + 3ab – b2
4a2 + 4ab – ab – b2

= 3a(a+b) + b(a+b)

  4a(a+b) – b(a+b)

= (3a+b) (a+b)

  (a+b) (4a-b)

= 3a + b

  4a -b

 


 




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