FURTHER MATHS OBJ:
11-20: DBEDDCBBAA
21-30: CEABCCCBDC
31-40: BBBEAEACAB

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FURTHER MATHS THEORY

(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

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(10a)
2x² — 5x — 3 = 0
@ + 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

(10b)
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

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(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 