2020 FURTHER MATH ANSWERS

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FURTHER MATH ANSWERS

(1)

(2i)

F(x) = x³ - 6x² + 9x

FD/dx (fx) = 3x² - 12x + 9

Using standard deviation

(2ii) Gradient of f(x) at point A (2,2)

d/dx f(x) = 3x² - 12x + 9

At point A , x=2

= 3(2)² - 12(2) + 9

= 3(4) - 12(2) + 9

= 12 -24 + 9

= -3

(2iii)

Equation of Tangent at point A

y-y¹= m ( x-x¹)

but m= -3

at point A, y¹= 2¹ x¹= 2

y-2=-3(x-2)

y-2 =-3x+6

y=-3x +6 + 2=> y= 8-3x

(4)

(1-√3)²(x+y√3) = 2√3 -2

(1-√3)(1-√3)(x+y√3) = 2√3 -2

(1- √3 - √3 + √9)(x+y√3) = 2√3 - 2

(1-2√3 + 3)(x+y√3) = 2√3 - 2

(1-2√3 + 3)(x+y√3) = 2√3 - 2

(1+3 - 2√3)(x+y√3)= 2√3 - 2

(4-2√3)(x+y√3) = 2√3 - 2

x+y√3 = 2√3 - 2/4 - 2√3

x+y√3 = 2(√3 - 1)/2(2- √3)

x+y√3 = √3 -1/2- √3

x+y√3 = √3 - 1/2 - √3 * 2+√3/2+√3

x+y√3 = (√3 -1)(2+√3)/(2- √3)(2+ √3)

x+y√3 = 2√3 + √9 - 2- √3/4+2√3 - 2√3 - √9

x+y√3 = 2√3 + 3 - 2 - √3/4+2√3 - 2√3 - 3

x+y√3 = 1+ 2√3 - √3/4-3+2√3 - 2√3

x+y√3 = 1+ √3/1

x+y√3 = 1+ √3

x=1 and y=1

(5)

Mass,m=150g, g=9.8m/s²

When the lift moves with a constant velocity acceleration

a=o

(i) Reaction,R=w=mg

R=mg

=150×9.8

=1470N

(ii) When the lift moves up word with acceleration 4.5m/s²

F=ma=R-mg

: . R=ma+mg

R=m(a+g)

R=150(4.5+9.8)

=150×14.3

=2145N

======================================

(9b)

Set the RHS of both equations equal since the LHS are equal

:. 294/r² = 56-2r/r

:. 294/r = 56-2r

294 = 56r - 2r²

= 2r² - 56r + 294 = 0

= r² - 28r + 147 = 0

= (r-7)(r-21) = 0

Either:

r-7 = 0

r=7

OR

r-21 = 0

r=21

And,

When r=21

θ=294/21² = 0.7rad

From equ(1) and when r=7

θ=294/7² =6rad

======================================

10a)

2x² — 5x — 3 = 0

Following general quadratic formula 

@ + ẞ = — b/a 

@ẞ= c/a 

x² — 5/2x — 3/2 = 0

— 5/2 = —(@ + ẞ) : (@ + ẞ) = 5/2 

— 3/2 = @ẞ

Find 1/@ + 1/ẞ

: (@ + ẞ)/@ẞ = 5/2 ÷ (—3/2)

5/2 × (— 2/3) = *— 5/3....*

*Thus , 1/@ + 1/ẞ = — 5/3..*

@² + ẞ² = (@ + ẞ)² — 2@ẞ

= (5/2)² — 2(—3/2) 

= 25/4 + 3

= *(25 + 12)/4 : 37/4..*

*Hence @² + ẞ² = 37/4...*

*NUMBER 10B*

*solution...* 

Since they have equal roots 

D = 0

b² = 4ac

(q + 2)² = 4(q)²

q² + 4q + 4 = 4q²

3q² — 4q — 4 = 0

3q² — 6q + 2q — 4 = 0

3q(q — 2) + 2(q — 2) = 0

*thus : q = — 2/3 and 2...*

(13ai)

Given: mass ,m =10kg

Force,F = 40N

Time, t = 0.5secs

Impulse, I = Ft = 40×0.5 = 20Ns

(13aii)

 Ft = m(v-u) where u= 0 (at rest)

20 = 10(v-0)

20 = 10v

V = 20/10 = 2m/s

Final speed = 2m/s

(13aiii)

Given: u=0 ; v=2m/s ; t=0.5secs

S= 1/2(u+v)t

S= 1/2(0+2)×0.5

S= 0.5 metres

Distance = 1/2 metre or 50cm

(13b)

Range R , = Time of flight × Horizontal component of speed

75 = T×35×cos38°

T = 75/35cos38° = 2.719secs

Vertical displacement= vertical component × Time of flight of speed

= Usinθ × T

= 35sin38 × 75/35cos38

= 75Tan38°

= 58.596 metres

~ 58.6 metres

============

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