1. Practice NAB Unit 3

2. ( ) ( ) ( ) ( ) ( ) ( ) Outcome 1 ii) BC = c – b i) AC = c – a – – = =
1. (a) A, B and C have coordinates (1, 2, 3), (4, –4, 12) and (5, –6, 15).
(i) Write down the components of AC
(ii) Hence show that the points A, B and C are collinear.
ii) BC = c – b
i) AC = c – a
( )
( )
( )
( ) – –
( )
( ) = =
Thus AC = 4BC with common point C, therefore collinear
(could also have used AB)

3. ( ) ( ) ( ) ( ) KL = 2LM l – k = 2(m – l) l – k = 2m – 2l 3l = 2m + k = 2 +
( )
3l =
coordinates L (3,-2,5)
( )
l =

4. a) 2(3) + -1(0) + 2(4) = 14
b) cosϴ = PQ . PR
PQ = √22 + (-1)2 + 22
PQ PR
= √9
= 3
cosϴ = 14
PQ = √
3(5)
= √25
= 5
ϴ = cos-1(14/15)
ϴ = 21.00
Threshold 9 out of 12

5. → -3cosx dy/dx = -½sinx Outcome 2
3. (a) Differentiate –3sin x with respect to x.
→ -3cosx
dy/dx = -½sinx
(b) Given y = ½cosx, find dy/dx

6. 4. Find f/(x) when, f(x) = (x + 3)-5
let y = U-5 where U = x + 3
dy/dU = -5U dU/dx = 1
dy/dx = -5U-6.1
= -5(x + 3)-6

7. b) Integrate –¾sinx with respect to x → ¾cosx + c
5) a) Find ∫4cosx dx
→ 4sinx + c
b) Integrate –¾sinx with respect to x
→ ¾cosx + c

8. [ ] ∫ = (x – 1)5 1(5) = (3 – 1)5 – (2 – 1)5 5 5 = 31/5 units2
c) Evaluate (x – 1)4 dx ∫ 2
[ ] 3
= (x – 1)5
1(5) 2
= (3 – 1)5
– (2 – 1)5 5 5
= 31/5 units2
Threshold 8 out of 11

9. Outcome 3 loga24 = log825 – log84 = log832 – log84 = log88 = 1
6. (a) Simplify loga12 + loga2.
loga24
(b) Simplify 5 log82 – log8 4.
= log825 – log84
= log832 – log84
= log88
= 1

10. (c) If y = 102.9, find an approximation for y.
7) a) If x = loge7 find an approximation for x
x = 1.40 (using calculator)
(b) Given that log10y = 3.4, write down an expression for the exact value of y.
y = 103.4
(c) If y = 102.9, find an approximation for y.
794.3 (using calculator)
loge4
Threshold 5 out of 8

11. kcos(x – α) = kcosxcosα + ksinx sinα
Outcome 4
8) Express 2cosx + 3sinx in the form kcos(x-α) where k>0 and 0≤α<3600
2cosx + 3sinx
kcos(x – α) = kcosxcosα + ksinx sinα
equating coefficients kcosα = 2, ksinα = 3
k = √( ) tanα = 3/2
tan-1(3/2) = 56.30
Quad i → α = 56.30
→ √13cos(x – 56.30)
= √13
Threshold 3 out of 5