**15a)**

DRAW THE DIAGRAM

15b)

(2.2) + 80(0.6) = 300(1.2) + 100 (3)

2.2B + 48 = 360 + 300

2.B + 48 = 660

2.2B = 660 – 48

2.2B = 612

DRAW THE DIAGRAM

15b)

(2.2) + 80(0.6) = 300(1.2) + 100 (3)

2.2B + 48 = 360 + 300

2.B + 48 = 660

2.2B = 660 – 48

2.2B = 612

**Reation at B = 278.18N
Taking moment about B
80(2.1) + 300(1) = 2.2A + 100 (0.8)
224 + 300 = 2.3A + 80
524 – 80 = 2.2A
A = 444/2.2
Raction a+ A = 201.82N
**

2)

Log 3x – 3logx³+ 2 = 0

Log 3x – 3log3³/log3x + 2 = 0

Log3x – 3/log3x + 2 = 0

P – 3/p + 2 = 0 where P = log3x

P² – 3 + 2p = 0

P² + 2p – 3 = 0

(p + 3) ( p – 1) = 0

P = -3 OR p = 1

But log3x = p

when p = -3

Log3x = -3

X = 3-³ = 1/27

And when P = 1

Log3x = 1

X = 3¹ = 3

And X = 1/27 OR 3

2)

Log 3x – 3logx³+ 2 = 0

Log 3x – 3log3³/log3x + 2 = 0

Log3x – 3/log3x + 2 = 0

P – 3/p + 2 = 0 where P = log3x

P² – 3 + 2p = 0

P² + 2p – 3 = 0

(p + 3) ( p – 1) = 0

P = -3 OR p = 1

But log3x = p

when p = -3

Log3x = -3

X = 3-³ = 1/27

And when P = 1

Log3x = 1

X = 3¹ = 3

And X = 1/27 OR 3

3a)

U=x-2 hence x=u+2

therefore

x^2+5/(x-2)^4

=(u+2)^3+5/(u+2-2)^2

=(u+2)^3+5/u^4

3a)

U=x-2 hence x=u+2

therefore

x^2+5/(x-2)^4

=(u+2)^3+5/(u+2-2)^2

=(u+2)^3+5/u^4

**3b)
(u+2)^3+5/u^4
=(u+2)^3/u^4+5/u^4
**

4)

sn=n/2[2a+(n-1)d]

s12=12/2[2a+11d]

s12=6(2a+11d]=168

2a+11d=28 eq—–(1)

multipply 1by2 to obtain

2a+4d=14 eq——(3)

subtract (3)from(2)

7d=14

d=2

sub for d in (1)

a+4=7

a+7-4-3

a=3

common difference d=2

first term a=3

4)

sn=n/2[2a+(n-1)d]

s12=12/2[2a+11d]

s12=6(2a+11d]=168

2a+11d=28 eq—–(1)

multipply 1by2 to obtain

2a+4d=14 eq——(3)

subtract (3)from(2)

7d=14

d=2

sub for d in (1)

a+4=7

a+7-4-3

a=3

common difference d=2

first term a=3

5)

tabulate

X,y,d,d^2

A 8,6,2,4

B 5,3,2,4

C 1,4,-3,9

D 7,8,-1,1

E 2,5,-3,9

F 6,7,-1,1

G 3,1,2,4

H 4,2,2,4

Ed^2 =4+4+9+1+9+1+4+4

=36

r=1-6£d^2/n(n^2-1)

=1-6*36/8(8^2-1)=1-216/504

=1-0.43

=0.57

5)

tabulate

X,y,d,d^2

A 8,6,2,4

B 5,3,2,4

C 1,4,-3,9

D 7,8,-1,1

E 2,5,-3,9

F 6,7,-1,1

G 3,1,2,4

H 4,2,2,4

Ed^2 =4+4+9+1+9+1+4+4

=36

r=1-6£d^2/n(n^2-1)

=1-6*36/8(8^2-1)=1-216/504

=1-0.43

=0.57

1)

X-3(12-2x+8)+4(30-5x-4)+3(-20-4)=24

X-3(20-2x)+4(26-5x)+3(-24)= -24

20x-2x^2+60+6z+104-20x-72=24

20x^2+6x-60+104-72+24=0

-2x^2+6x-4=0

1)

X-3(12-2x+8)+4(30-5x-4)+3(-20-4)=24

X-3(20-2x)+4(26-5x)+3(-24)= -24

20x-2x^2+60+6z+104-20x-72=24

20x^2+6x-60+104-72+24=0

-2x^2+6x-4=0

**Divide 2x by -2
x2^2-3x+2=0
x2^2-2x-x+2=0
x(x-2)-1(x-2)-0
(x-1)(x-2)=0
x-1 or x=2
**

6)

It follow that:

P(n) = 1/3 and p(k)¹ = 1 – 1/3 = 2/3

P(T) = 1/5 and P(T)¹ = 1- 1/5 = 4/5

6)

It follow that:

P(n) = 1/3 and p(k)¹ = 1 – 1/3 = 2/3

P(T) = 1/5 and P(T)¹ = 1- 1/5 = 4/5

**Hence, probability that only one if the them will be solve the questions will be:
= (1/3 × 4/5) + (1/5 × 2/3)
= 4/15 + 2/15
= 6/15
= 2/5
**

**
**8a)

m1 = 20kg

u1 = 8ms-1

m2 = 30kg

u2 = 50ms-1

(a) in same direction

m1u1 + m2u2 = (m1+m2)v

20 × 80 + 30 × 50 = (20+30)v

1600+1500 = 50v

3100/50 = 50/50

V = 62ms-1

8a)

m1 = 20kg

u1 = 8ms-1

m2 = 30kg

u2 = 50ms-1

(a) in same direction

m1u1 + m2u2 = (m1+m2)v

20 × 80 + 30 × 50 = (20+30)v

1600+1500 = 50v

3100/50 = 50/50

V = 62ms-1

**8b)
M1=20kg
u1=80mls
m2=30kg
u2=50mls
v=?
M1u1+m2u2=v(m1+m2)
(20*80)+(30*(-50))=v(20+30)
1600-1500=50v
100=50v
v=100/50
v=2mls
**

7)

m = 3i – 2j ; n = 2i + 3j ; p = i + 6j

Therefore:

4(3i – 2j) +2(2i +3j) -3(-i + 6j)

12i – 8j + 4i + 6j + 3i – 18j

12i + 4i + 3i – 8j + 6j – 18j

19i = 20j

7)

m = 3i – 2j ; n = 2i + 3j ; p = i + 6j

Therefore:

4(3i – 2j) +2(2i +3j) -3(-i + 6j)

12i – 8j + 4i + 6j + 3i – 18j

12i + 4i + 3i – 8j + 6j – 18j

19i = 20j

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