💯 FURTHER MATHS FROM MIratech
1
a) 8m + 2³m
(2³)m + 2³m = 1/4
2³m + 2³m = 1/4
2(2³m) = 1/4
2³m = 1/8
2³m = 8 - ¹
2³m = 2-³
Since bases are the same, powers are same
3m = - 3
m = - 3/3
m = -1
1b) log15 base 4 = x
x = log15 base/log4 base 10 = 1.1761/0.6020
= 1.9536
therefore x approximately 1.954
2a)
(-2) = m(-2)² + n(-2)+2=0
4m-2n = - - - - - - - - (1)
F(1) = m(1)²+n(1)+2=3
m+n+2=3
m+n=3-2
m+n=1 - - - - - - - - - - - -(2)
X Equate (2) by 4
4m+4n=4 - - - - - - - - - - (3)
4m-2n= -2
-4m+4n=4/-6n = - 6
n=1
4)
Given y= 5/x² + 3
Y = 5(x² +3)-1
dy/dx = anxn-1
y = 5(x² + 3)-1
dy = 5(-1) (x2 +3)-1^1
dy/dx = 10x (x² + 3)-²
dy/dx = - 10/(x² +3)²
5a)
^nP5 ÷ ^nC4 = 24
n!/(n-5)! ÷ n!/(n-4)!4! =24
n!/(n-5)! * (n-4)!4!/n! =24
(n-4)!4!/(n-5)!=24
(n-4)(n-5)!4!/(n-5)! = 24
n-4=1
n=4+1
n=5
b)PR= 5C3 (1/6)³ (5/6)^5-3
5!/(5-3)!3! (1/6)³ (5/6)²
5!/2!3! (1/6)³ (5/6)²
10 (1/216) (25/36)
=0.03215
Pr = 1-0.03215
=0.9678
6) make a table containing
MARKS,TALLY,F & FX
UNDER F - 2,9,4,2,2,1
UNDER FX - 2,18,12,8,10,6
€F= 20
€FX = 56
6b) Mean = €fx / €f
= 56 / 20
= 2.8
7
(a)
Given
h=15.4t - 4.9t²
Velocity v = dh/dt = 15.4 - 9.8t
At maximum height, V =0
15.4 - 9.8t = 0
9.8t = 15.4
t = 15.4/9.8
t = 1.6secs
Time to reach maximum height = 1.6secs
b) maximum height = 15.4 t - 4.9t²
= 15.4(1.6) - 4.9(1.6)²
=24.64 - 12.544
= 12.096
Maximum height = 12.1m
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8 a)
60, 56,70 ,63 ,50, 72, 65,60
mean= £ x/ n =60 + 56+ 70 +63 + 50+ 72+ 65 + 60/ 8
mean= 62
8 b)
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 = sqr 45. 25= 6 .73
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9a)X+6/(x+1)^2 = A/x+1 + B/(x+1)² + C/(x+1)³
X+6/ ~(x+1)³~ =A(x+1)² + B(x+1) + C / ~(x+1)³~
X+6 =A (x+1)² + B(x+1) + C
Let X+1=0 , X=-1
-1+6= A (-1+1)² + B(-1+1) +C
5= C
Therefore:- C=5
X+6= A(X2+2x+1) + Bx + B + C
X+6= Ax²+2Ax+A+Bx+B+C
comparing the coefficient of X²
A=0
Comparing the coefficient of X
1=2A+B
1=2(0)+B
B=1
x+6/(x+3)² = o/x+1 + 1/(x+1)² + 5(x+1)³
1/(x+1)² + 5/(x+1)³
9b)
S²1 1/(x+1)² + 5/(x+1)³ DX
1n (x+1)³ + 5 1n (X+1)⁴ |²1
1n 3(3) + 5 1n(3)⁴ - 1n +2³ - 5 1n +2⁴
3.2958+21.9722-2.0794-13.8629
=9.3297
10a)
3x^2+x-2 <= 0
3x^2+3x-2x-2 <= 0
3x(x+1) -2 (x+1) <= 0
(3x-2)(x+1) <= 0
3x-2 <= 0 or 8+1 <= 0
3x <= 2 or x <= -1
X<= 2/3
Therefore range of value is
-1 >= x <= 2/3
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