Maths General Notes PDF

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General Math Notes – HSC 2013

Algebraic Skills and Techniques 1) Substitution  A pronumeral takes the place of a number.  A collection of pronumerals and numbers connected by operational symbols is called an expression.  An algebraic expression can be evaluated by substituting values for pronumerals.  Replacing the pronumeral in an expression with a value

e.g solve if a=6 and b=2.

√ a−2b=√ 6−2 ( 2) √ 6− 4 √ 2=1.41 2) Simplifying algebraic expressions  Algebraic expressions can be simplified by collecting like terms.  Like terms are the only terms that can be added or subtracted.  (Example) Just as ‘5 x 10’ is the same as ‘10 x 5’, ‘pq’ is the same as ‘qp’. ‘x’ and ‘x 2’ are not like terms.  When expanding brackets, multiply by every term inside the brackets.  Pronumeral x pronumeral = pronumeral2, negative x negative = positive.

e.g a) Simplify 4x – x2+ 7x =

4 x −x2 +7 x =11 x − x 2

b) E.g expand and simplify: 3+2(2-5x)

3+2 ( 2−5 x )

¿ 3+4−10 x

¿ 7−10 x

c) Expand and simplify: x(2-x)-4(x-3)

x ( 2−x ) −4 ( x−3 )

¿ 2 x− x2−4 x +12

¿− 2 x − x 2 +12

3) Multiplying and dividing algebraic expressions  

Numbers first then pronumerals The algebraic terms do not have to be like terms when multiplying or dividing.



If there are algebraic fractions then they may be cancelled in the same way as numerical fractions. In division cancel by dividing top and bottom.

 e.g

a) simplify: 7x x (-x3) =

4

−7

b) E.g expand and simplify: 3+2(2-5x) =

¿ 3 + 4 −10 x

3+2 ( 2−5 x )

¿ 7−10 x

c) Expand and simplify: x(2-x)-4(x-3) =

x ( 2−x ) −4 ( x−3 )

2

¿ 2 x− x −4 x +12

¿−2 x−x 2 +12

4) Multiplying and dividing algebraic expressions  Numbers first then pronumerals  The algebraic terms do not have to be like terms when multiplying or dividing.  If there are algebraic fractions then they may be cancelled in the same way as numerical fractions.  In division cancel by dividing top and bottom.  If no number is in front of the pronumeral, put a 1. If no number is to the power of a pronumeral it is a 1.  Add the powers for multiplication and subtract for division.  Pronumeral x pronumeral = pronumeral2, negative x negative = positive.  Change decimals into fractions.  When multiplying/dividing fractions, multiply/divide across e.g Multiplication

7 x1 ×( −1 x 3) =7 4 b) Simplify: (-7x)2= −7 x ×−7 x=14 x a) Simplify: 7x x (-x3) =

Division

5 15 p q 13 p 3 3−15 a = 2 b) (-15a)/20a3 = 2 2 4 20 a a 4 a a b ab x = c) x x x2 a) 15pq/3p=

d)

1 14 b 7 = 2b 2 b

5) Equations reviews    

Addition and subtraction Multiplication and division Substitution Simplification

6) Practical equations  A formula links two or more pronumerals according to a rule.  If one pronumeral is expressed in terms of the others, it is called the subject of the formula.  If the value to be found id the subject of the formula then it is found directly from substitution.  If the value to be found is not the subject, then following substitution, the resulting equation is solved to find the value.  When the square root of both sides is taken ± is used e.g find the value of t if v = 117, u = 5, a=8 and v= u+at

v =u+at

117=5+8 t

117−5=8t

112=8 t

112 =t 8

t=14

7) Changing the subject  To rearrange formulas  The subject of the formula is the letter on the left hand side of the equation followed by the equals sign. The formula p=rx+d has p as the subject because it is expressed in terms of the variables r, x and d. e.g a) Make u the subject of b) make x the subject of

2 ap=

h( x+ y ) ×2 2

v 2=u 2+2as 1 p= h( x + y ) 2 2 p h ( x+ y ) = h h

v 2−2as=u 2 u=± √ v 2 −2as 1 h ( x+ y ) ¿ × × 1 2 1 2p − y= (x + y ) − y h

8) Scientific notation  Multiplying by 10n shifts the decimal place n places to the left.  Mm to m to km, no cm.  Add the powers when multiplying.  A x 10n, A = no. between 1-10, n = decimal place shift.

x10-3 mm

X103

x10-3 m

X103

km

x=

2p −y h

When moving from a smaller measurement to a larger measurement (TOP) you ‘×’ (or add 3) When moving from a larger measurement to a smaller measurement (BOTTOM) you ‘÷’ (or subtract 3)

Distance:

Weight: x10-3

mg

g

X103

x10-3

x10-3

t

kg

X103

X103

e.g convert: a) 6.42 x 104km to mm 6.42 x 104 x 103m =5.42 x 107m (powers added, can write x4+3 since 1000 = 103) 6.42 x 107+3 = 6.42 x 1010mm b) 8.3 x 104mg to kg 8.3 x 104-3g = 8.3 x 101g 8.3 x 101-3mg = 8.3 x 10-2mg Further Applications of Area and Volume

1. Area of Circles  Area of a circle: A = πr2, where r = radius of the circle.  Half the diameter to get the radius. Always use radius, NOT diameter.  Circle: full circle shape.  Sector: part of a circle (looks like a pie piece).  Annulus: the shaded region, often the outer section of a circle with a piece cut out of the middle. Circle Sector Annulus

∅

r d

A = πr2

A=

A = πR2 – πr2 (R = big) (r = small)

∅ ×π r2 360

Or A = π(R2-r2) (HSC Formula)

e.g a) What fraction of the circle is drawn and the area to the nearest cm2?

1 × π r2 4

90 1 = 360 4

6.8 ¿ 2 6.8cm

1 × π r2× ¿ 4 ¿ 36.3=36 cm 2

b) 2 2 A=π ( R −r )

(

A=π (

)

11 2 5 2 ) −( ) 2 2 2

A=¿ 75 4cm

If the annulus is a half semicircle shape, simply divide the end result by 2.

5cm 2. Area of Ellipses 11cm  A plane with a curved boundart is the ellipse. The ellipse does not have a radius, as there are many different distances from the centre to the boundary.  We use two different measurements:  The length of the semi- major axis.  The length of the semi-minor axis.  The semi major axis is the longest distance from the centre of the ellipse to the boundary.  The semi-minor is the shortest distance from the centre of the ellipse to the boundary.  If diameter is given, divide by 2. α is the length of the semi-major axis. b is the length of the semi-minor axis. b The area of an ellipse: a A=πab

3. Simpson’s Rule  If a property being surveyed has an irregular boundary, like a river, then Simpsons rule is used to find the area.  If there are 6 vertical line measurements two calculations must be done and added together.  If no measurement is given, the measurement is 0.

Df +Dm + Dl h A≑ ¿ 3 Df: first measurement Dm: middle measurement

dM dF

dL h

h

Dl: last measurement H: equal distance between successive measurements.

4. Surface Area of a Cylinder

r r h

Closed cylinder:

2 πrh+2 π r

h

A = πr2

Open cylinder: 2 πr x h

2

One end open: 2

5. Surface Area of Spheres

Sphere:

4 π r2

Open hemisphere:

4 π r2 2

Closed hemisphere:

4πr

2

6. Volume of composite shapes  Prism: V=Ah  Rectangular prism: V=lbh    

1 2 Cylinder: V =π r 2 h 1 Pyramid: V = Ah 3 4 2 Sphere: V = π r 3 2 Cone: V = π r h

7. Accuracy of Measurement  Limit of reading is the smallest unit. E.g 0.1 from 21.2  A given measurement is accurate to ±0.5 of the smallest division on the scale.  Absolute error: limit of reading x 0.5  Upper limit: original measurement + absolute error  Lower limit: original measurement – absolute error

Trigonometry 1. Review of Right-angled Triangles  Right angled triangles use sine, tan and cosine.   

Opposite Hypotenuse Adjacent Cosine = cos=CAH = Hypotenuse Opposite Tan = tan =TOA= Adjacent Sine =

sin=SOH =

Opposite

Hypotenuse

Adjacent 2. Bearings  Always 3 digits e.g 042°T.  Compass bearing: direction.  True bearing: degrees from North.  Bearing: Clockwise

3. Area of a Triangle  

1 A= × b ×h 2 1 A= × a ×b × SinC 2



Need 2 sides and the included angle

4. The Sine rule  Use brackets.  Angles all add to 180°  Non-right angled  The sine rule works in pairs, you must match a side with the angle opposite. The sine rule is used to find:  A side given two angles and one side, or  An angle given 2 sides and one angle (opposite a side), to find the second angle.     Side

b a = SinA SinB

. Put the unknown on the left side.

The letters on top indicate sides and the bottom indicates angles When finding an angle, flip the formula so the angles are on top. If you can draw an X it’s sine rule

Angle

Use brackets here.

5. Cosine rule  If you can’t draw an X it’s cosine rule.  The cosine rule is used to find the 3rd side given 2 sides and the included angle or an angle given 3 sides. 

Side: a2=b2 +c 2−2 ab ×CosA



Angle: CosA=

(b2 +c 2−a2) 2 bc

6. Radial surveys  Two types of radial survey:  Plain table radial survey  Compass radial survey  The cosine rule is used to find the length of each boundary and hence the perimeter. 

Credit and

Borrowing

1. Flat

Rate Loans Simple

 interest loans

Prn 100



I=

    

p - principal R - rate N – time Convert months to years dividing by 12. Non compounding

2. Reducing balance loans  Reducing balance loans.  Monthly reducible loans.  Tables of values.  Graphs

3. Comparing loans  Comparing loans with interest rates that are applied in different ways

  

Effective interest rate. By changing different rates to the effective interest rate, the rate is then expressed as an annual rate of interest, as if the loan was compounded annually. The formula used to convert compound (nominal) interest rate to the effective n

interest rate is 1+r ¿ −1

E=¿

   

E = effective rate of interest per annum as a decimal. R = stated interest rate per compounding period as a decimal. N = number of time periods. This expresses rates as an annual rate of interest as if the loan was compounded annually.

4. Credit cards  Annual fee charged.  Minimum payment that must be listed.  Many banks follow an interest free period and providing the balance is paid by the due date there are no credit charges on purchases imposed by the bank.  If the interest free period is exceeded two charges are imposed.  An initial charge equal to one month’s interest on the amount outstanding.  A daily compound interest charge from the end of the interest free period.  Compound interest formula is used.

Annuities and Loan repayments 1) Future value  An annuity is a f...