Unit 2 Practice NAB

Marking

1. Show that (x + 1) is a factor of f(x) = x3 + 2x2 – 5x – 6, and express f(x) in fully factorised form.

Outcome 1

1 2 -5 -6-1

1

-1

1

-1

-6

6

0

(x + 1) is a factor

→ f(x) = (x + 1)

(x2 + x – 6)

= (x + 1)(x – 2)(x + 3)

2. Use the discriminant to determine the nature of the roots of the equation 2x2 – 3x + 2 = 0.

using b2 – 4ac a = 2, b = -3, c = 2

= (-3)2 – 4(2)(2)

= 9 – 16

= -7

→ no real roots

Threshold 4 out of 6

3.Find ∫ 2 dxx2

= ∫ 2x-2 dx

= 2x-1 + c-1

= -2x-1 + c

∫x2(4 – x) dx0

4

∫(4x2 – x3) dx0

4

= 4x3 – x4 3 4[ ]

0

4

= 4(4)3 – 44 3 4

( ) – 4(0)3 – 04 3 4

( )

= 64/3 units2

Decimal Acceptable for answer

Finding Limits using y = y

x2 – 2x + 2 = x + 2

x2 – 3x = 0

x(x – 3) = 0

x = 0 or x = 3

∫ (x + 2) – (x2 – 2x + 2) dx0

3

Threshold 8 out of 11

2cos2x=1 for 0≤x<π6. Solve algebraically the equation

2cosA = 1 where A = 2x

cosA = 0.5

cos-1(0.5) = 600

A = 600 or 3000

2x = 600 or 3000

x = 300 or 1500

= π/6 or 5π/6 radians

513

a) sin x = 4/5 cos x = 3/5 sin y = 5/13 cos y = 12/13

b) cos(x + y) = cosx cosy – sinx siny

= 3/5 X 12/13 - 4/5 X 5/13

= 16/65

sinx°cos10°+ cosx°sin10°= 2/3 for 0 ≤x< 180

8. (a) Express sinx ° cos 10° + cos x° sin 10° in the form sin(A + B) °. (b) Use the result of (a) to solve the equation

a) sin(x + 10)0

b) sin(x + 10)0 = 2/3

sin A = 2/3 where A = x + 10

sin-1(2/3) = 41.80

A = 41.80 or 138.20

x + 10 = 41.80 or 138.20

x = 31.80 or 128.20

Threshold 9 out of 12

using (x – a)2 + (y – b)2 = r2

→ (x + 3)2 + (y – 2)2 = 16

9. (a) A circle has radius 4 units and centre (–3, 2). Write down the equation of the circle.

(b) A circle has equation x2 + y2 + 6x – 8y – 11 = 0. Write down its radius and the coordinates of its centre.

using x2 + y2 + 2gx+ 2fy + c = 0

g = 3, f = -4, c = -11

centre (-3 , 4) radius √(32 + (-4)2 + 11)

= √36 = 6

substitute → x2 + (2x – 3)2 + 2x – 4 = 0

x2 + 4x2 – 12x + 9 + 2x – 4 = 0

5x2 – 10x + 5 = 0

5(x2 – 2x + 1) = 0

5(x – 1)(x – 1) = 0

x = 1

only 1 solution → Tangent

10. Show that the straight line with equation y = 2x – 3 is a tangent to the circle with equation x2 + y2 + 2x – 4 = 0.

Either Or

using discriminant

(-10)2 – 4(5)(5)

= 0

equal roots → tangent

mPQ = -1 + 5

3 – 1

= 2

mtangent = - ½ (m1m2 = -1)

Equation y + 5 = -½(x – 1)

etc.

Threshold 10 out of 14