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NIGERIAN ANSWERS

1a)

g(x)=y

y=x+6

x=y-6

g^- f(x-6)

=4-5(x-6)/2=4-5x+30/2

=34-5x/2

g(x)=y

y=x+6

x=y-6

g^- f(x-6)

=4-5(x-6)/2=4-5x+30/2

=34-5x/2

1b)

coodinate=(x1+x2/2 ,y1+y2/2)

=(7-2/2,7-5/2)=(5/2,2/2)

=(5/2,1)

coodinate=(x1+x2/2 ,y1+y2/2)

=(7-2/2,7-5/2)=(5/2,2/2)

=(5/2,1)

(14ai)

SKETCH THE DIAGRAM

SKETCH THE DIAGRAM

14ii)

Using lami’s theory

T1/sin60=T2/sin30

48N/sin60=T2/sin30

48N/0.8660=T2/0.5

0.5(48)/0.8660=T2(0.8660)/0.8660

T2=24/0.8660

T2=27.7N

T1/sin60=T2/sin30

48N/sin60=T2/sin30

48N/0.8660=T2/0.5

0.5(48)/0.8660=T2(0.8660)/0.8660

T2=24/0.8660

T2=27.7N

(14b)

Using the equation of motion

H=U^2/2g

H=(20)^2/2*10

=20*20/20

H=20m

Timetaken to reach the maximum height

S=Ut+1/2at^2

20=0+1/2(100)t^2

20/5=5t^2/5

t^2=4

t=sqroot4

t=2S

Using the equation of motion

H=U^2/2g

H=(20)^2/2*10

=20*20/20

H=20m

Timetaken to reach the maximum height

S=Ut+1/2at^2

20=0+1/2(100)t^2

20/5=5t^2/5

t^2=4

t=sqroot4

t=2S

( 5a)

pr(age)=4/5

pr(fully)=3/4

pr(must)=2/3

pr(age not admitted)=1-4/5

=1/5

pr(fully not admitted)=1-3/4

=1/4

pr(must not admitted)=1-2/3

=1/3

Therefore pr(none admitted)=1/5*1/4*1/3

=1/60

pr(fully)=3/4

pr(must)=2/3

pr(age not admitted)=1-4/5

=1/5

pr(fully not admitted)=1-3/4

=1/4

pr(must not admitted)=1-2/3

=1/3

Therefore pr(none admitted)=1/5*1/4*1/3

=1/60

5b)

pr(only age and fully gained

admission)=4/5*3/4*1/3

=1/5

pr(only age and fully gained

admission)=4/5*3/4*1/3

=1/5

4)

(x^2+5x+1)sqroot(2x^3+mx^2+nx+11)=(2x-5)

remainder:30x+16

(x^2+5x+1)(2x-5)

=2x^3+10x^2+2x-5x^2-25x-5

=2x^3+10x^2-5x^2-25x-5

=2x^3+5x^2-23x+30x+16-5

=2x^3+5x^2+7x+11

Therefore m=5, n=7

(x^2+5x+1)sqroot(2x^3+mx^2+nx+11)=(2x-5)

remainder:30x+16

(x^2+5x+1)(2x-5)

=2x^3+10x^2+2x-5x^2-25x-5

=2x^3+10x^2-5x^2-25x-5

=2x^3+5x^2-23x+30x+16-5

=2x^3+5x^2+7x+11

Therefore m=5, n=7

12a)

tabulate

Marks| 1-10, 11-20, 21-30, 31-40, 41-50,

51-60, 61-70, 71-80, 81-90, 91-100

F| 3, 17, 41, 85, 97, 115, 101, 64, 21, 6

C.B| 0.5-105, 10.5-205, 20.5-305, 30.5-405,

40.5-505, 50.5-605, 60.5-705, 70.5-805,

80.5-905, 90.5-1005

C.F| 0+3=3, 3+17=20, 20+41=61, 61+85=146,

146+77=243, 243+115=358, 358+101=459,

459+64=523, 523+21=544, 544+6=550

Marks| 1-10, 11-20, 21-30, 31-40, 41-50,

51-60, 61-70, 71-80, 81-90, 91-100

F| 3, 17, 41, 85, 97, 115, 101, 64, 21, 6

C.B| 0.5-105, 10.5-205, 20.5-305, 30.5-405,

40.5-505, 50.5-605, 60.5-705, 70.5-805,

80.5-905, 90.5-1005

C.F| 0+3=3, 3+17=20, 20+41=61, 61+85=146,

146+77=243, 243+115=358, 358+101=459,

459+64=523, 523+21=544, 544+6=550

13ai)

M=2

P=5

C=3

total=10

If the books of the same subject are to stand

together

No of arrangements=2!*5!*3!*3!

=2*120*6*6

=8640arrangements

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P=5

C=3

total=10

If the books of the same subject are to stand

together

No of arrangements=2!*5!*3!*3!

=2*120*6*6

=8640arrangements

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(13aii)

Only the physics textbook must stand together

No of arrangements=5!*6!

=120*720

=86400arrangements

No of arrangements=5!*6!

=120*720

=86400arrangements

(13b)

P=13/20

q=1-13/20=7/20

pr(atleast 3 speak E)=1-Pr(2 speak E)

=(1-8C1p^1q^7+8C2p^2q^6)

=1-(8*(13/20)*(7/20)^7+28(13/20)^2*(7/20)^6

=1-(0.003346+0.0217467)

=1-0.0251

=0.9749

=0.975(3s.f)

P=13/20

q=1-13/20=7/20

pr(atleast 3 speak E)=1-Pr(2 speak E)

=(1-8C1p^1q^7+8C2p^2q^6)

=1-(8*(13/20)*(7/20)^7+28(13/20)^2*(7/20)^6

=1-(0.003346+0.0217467)

=1-0.0251

=0.9749

=0.975(3s.f)

( 4)

(x^2+5x+1)sqroot(2x^3+mx^2+nx+11)=(2x-5)

remainder:30x+16

(x^2+5x+1)(2x-5)

=2x^3+10x^2+2x-5x^2-25x-5

=2x^3+10x^2-5x^2-25x-5

=2x^3+5x^2-23x+30x+16-5

=2x^3+5x^2+7x+11

Therefore m=5, n=7

remainder:30x+16

(x^2+5x+1)(2x-5)

=2x^3+10x^2+2x-5x^2-25x-5

=2x^3+10x^2-5x^2-25x-5

=2x^3+5x^2-23x+30x+16-5

=2x^3+5x^2+7x+11

Therefore m=5, n=7

(12a)

tabulate

Marks| 1-10, 11-20, 21-30, 31-40, 41-50,

51-60, 61-70, 71-80, 81-90, 91-100

F| 3, 17, 41, 85, 97, 115, 101, 64, 21, 6

C.B| 0.5-105, 10.5-205, 20.5-305, 30.5-405,

40.5-505, 50.5-605, 60.5-705, 70.5-805,

80.5-905, 90.5-1005

C.F| 0+3=3, 3+17=20, 20+41=61, 61+85=146,

146+77=243, 243+115=358, 358+101=459,

459+64=523, 523+21=544, 544+6=550

tabulate

Marks| 1-10, 11-20, 21-30, 31-40, 41-50,

51-60, 61-70, 71-80, 81-90, 91-100

F| 3, 17, 41, 85, 97, 115, 101, 64, 21, 6

C.B| 0.5-105, 10.5-205, 20.5-305, 30.5-405,

40.5-505, 50.5-605, 60.5-705, 70.5-805,

80.5-905, 90.5-1005

C.F| 0+3=3, 3+17=20, 20+41=61, 61+85=146,

146+77=243, 243+115=358, 358+101=459,

459+64=523, 523+21=544, 544+6=550

FURTHERMATHEMATICS NIGERIAN ANSWERS

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1a)

g(x)=y

y=x+6

x=y-6

g^- f(x-6)

=4-5(x-6)/2=4-5x+30/2

=34-5x/2

g(x)=y

y=x+6

x=y-6

g^- f(x-6)

=4-5(x-6)/2=4-5x+30/2

=34-5x/2

1b)

coodinate=(x1+x2/2 ,y1+y2/2)

=(7-2/2,7-5/2)=(5/2,2/2)

=(5/2,1)

coodinate=(x1+x2/2 ,y1+y2/2)

=(7-2/2,7-5/2)=(5/2,2/2)

=(5/2,1)

9a)1/1-cos tita + 1/1+cos tita

=1+ cos tita+ 1- cos 0/1-cos^2 tita

= 2/1- cos^2 tita

= 2/sin^2 tita

=1+ cos tita+ 1- cos 0/1-cos^2 tita

= 2/1- cos^2 tita

= 2/sin^2 tita

11a)Given:

f(x)={(4x-x^2)dx

f(x)=2x^2 – x^3/3 + K

f(3)=2(3)^2 – (3)^2/3 + K =21

18 – 9 + K=11

9+K=21

K=21-9

K=12

Therefore

f(x)= -x^3 + 2x^2 + 12

f(x)={(4x-x^2)dx

f(x)=2x^2 – x^3/3 + K

f(3)=2(3)^2 – (3)^2/3 + K =21

18 – 9 + K=11

9+K=21

K=21-9

K=12

Therefore

f(x)= -x^3 + 2x^2 + 12

11b)Given

AP: T2=a+d

T4=a+3d

T8=a+7d

Since they form the consecutive numbers of a G.P.

a+3d/a+d=a+7d/a+3d

a^2+6ad+9d^2=a^2+8ad+7d^2

6ad+9d^2=8ad+7d^2

6a+9d=8a+7d

9d-7d=8a-6a

2d=2a

a=d——Eqn(1)

Also:

S3+S5=20

3/2(2a+2d)+5/2(2a+4d)=20

Sub eqn(1) into eqn(2)

3/2(2a+2a)+5/2(2a+4a)=20

6a+15a=20

21a=20

a=20/21

i)T1=a=20/21

T2=a+d=40/21

T3=a+2d=20/7

T4=a+3d=80/21

AP: T2=a+d

T4=a+3d

T8=a+7d

Since they form the consecutive numbers of a G.P.

a+3d/a+d=a+7d/a+3d

a^2+6ad+9d^2=a^2+8ad+7d^2

6ad+9d^2=8ad+7d^2

6a+9d=8a+7d

9d-7d=8a-6a

2d=2a

a=d——Eqn(1)

Also:

S3+S5=20

3/2(2a+2d)+5/2(2a+4d)=20

Sub eqn(1) into eqn(2)

3/2(2a+2a)+5/2(2a+4a)=20

6a+15a=20

21a=20

a=20/21

i)T1=a=20/21

T2=a+d=40/21

T3=a+2d=20/7

T4=a+3d=80/21

ii)S10=10/2[2(20/21)+9(20/21)]

=(10/2)(20/21)(11)

=1100/21

=(10/2)(20/21)(11)

=1100/21

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(14ai)

SKETCH THE DIAGRAM

SKETCH THE DIAGRAM

14ii)

Using lami’s theory

T1/sin60=T2/sin30

48N/sin60=T2/sin30

48N/0.8660=T2/0.5

0.5(48)/0.8660=T2(0.8660)/0.8660

T2=24/0.8660

T2=27.7N

Using lami’s theory

T1/sin60=T2/sin30

48N/sin60=T2/sin30

48N/0.8660=T2/0.5

0.5(48)/0.8660=T2(0.8660)/0.8660

T2=24/0.8660

T2=27.7N

(14b)

Using the equation of motion

H=U^2/2g

H=(20)^2/2*10

=20*20/20

H=20m

Timetaken to reach the maximum height

S=Ut+1/2at^2

20=0+1/2(100)t^2

20/5=5t^2/5

t^2=4

t=sqroot4

t=2S

Using the equation of motion

H=U^2/2g

H=(20)^2/2*10

=20*20/20

H=20m

Timetaken to reach the maximum height

S=Ut+1/2at^2

20=0+1/2(100)t^2

20/5=5t^2/5

t^2=4

t=sqroot4

t=2S

(9a)

1/1-cos tita + 1/1+cos tita

=1+cos tita + 1-cos tita//(1-cos tita) (1+cos tita)

= 2/1+cos tita – cos tita – cos^2 tita

= 2/1-cos^2 tita

1/1-cos tita + 1/1+cos tita

=1+cos tita + 1-cos tita//(1-cos tita) (1+cos tita)

= 2/1+cos tita – cos tita – cos^2 tita

= 2/1-cos^2 tita

Recall that :

Cos^2 tita + sin^2 tita = 1

Cos^2 tita + sin^2 tita = 1

.:. Cos^2 tita = 1-sin^2 tita

.:. 1/1-cos^2 tita + 1/1+cos tita

= 2/1-(1-sin^2 tita)

.:. 1/1-cos^2 tita + 1/1+cos tita

= 2/1-(1-sin^2 tita)

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(9b)

At stationary points,

dy/dx=0.

y=x^0(x-3)

At stationary points,

dy/dx=0.

y=x^0(x-3)

Let u=x^2,v=x-3.

du/dx=2x dv/dx=1.

dy/dx= Udv/dx + Vdu/dx

dy/dx=x^2(1)+(x-3)(2x)

du/dx=2x dv/dx=1.

dy/dx= Udv/dx + Vdu/dx

dy/dx=x^2(1)+(x-3)(2x)

.:. dy/dx=x^2+2x^2-6x

dy/dx=3x^2-6x

At stationary point,

dy/dx=0..

dy/dx=3x^2-6x

At stationary point,

dy/dx=0..

.:.3x^2-6x=0

Equation of line=> 3x^2-6x=0

Equation of line=> 3x^2-6x=0

10b)

X1 Y2

(3, 1)

r=sqr(x2-x1)^2+(y2-y1)^2

r=sqr(3+3)^2+(1-1)^2

r=sqr6^2+0=sqr36=6

the equatuon of a circle

(x-a)^2+(y-b)^2=r^2

(x-(-3))^2+(y-1)^2=6^2

(x+3)^2+(y-1)^2=36

x^2+6x*9+y^2-2y+1=36

x^2+y^2+6x-2y+9+1-36=0

x^2+y^2+6x-2y-26=0

X1 Y2

(3, 1)

r=sqr(x2-x1)^2+(y2-y1)^2

r=sqr(3+3)^2+(1-1)^2

r=sqr6^2+0=sqr36=6

the equatuon of a circle

(x-a)^2+(y-b)^2=r^2

(x-(-3))^2+(y-1)^2=6^2

(x+3)^2+(y-1)^2=36

x^2+6x*9+y^2-2y+1=36

x^2+y^2+6x-2y+9+1-36=0

x^2+y^2+6x-2y-26=0

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GHANA ANSWERS

2)

2)

x+1/3x^2-x-2

3x^2-x-2/-6(-3 2)

3x^2-3x+2x-2

3x(x-1)+2(x-1)

(3x+2)(x-1)

x+1/(3x+2)(x-1)=A/3x+2+B/x-1

x+1/(3x+2)(x-1)=A(x-1)+B(3x+2)/(3x+2)(x-1)

x+1=A(x-1)+B(3x+2)

3x+2=0

x=-2/3

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8a)

60,56,70,63,50,72,65,60

mean=£x/n=60+56+70+63+50+72+65+60/8

mean=62

8b)

variance=£(x-x^-)^2/n

=(62-60)^2+(62-56)^2+(62-70)^2+(62-63)^2+(62-50)^2+(62-72)^2+(62-63)^2+(62-60)^2/8

=362/8=45.25

SD=sqr variane =sqr45.25=6.73

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8a)

60,56,70,63,50,72,65,60

mean=£x/n=60+56+70+63+50+72+65+60/8

mean=62

8b)

variance=£(x-x^-)^2/n

=(62-60)^2+(62-56)^2+(62-70)^2+(62-63)^2+(62-50)^2+(62-72)^2+(62-63)^2+(62-60)^2/8

=362/8=45.25

SD=sqr variane =sqr45.25=6.73

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