num-title-ct-pathc

Cartesian Plane, SMB-020

Prove the points `(1,-1), (-1,1)` and `(-sqrt3,-sqrt3)` are the vertices of a equilateral triangle. (4 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{Proof (See worked solutions)}`

Show Worked Solution

`text{Let points be:}\ A(1,-1), B(-1,1) and C(-sqrt3,-sqrt3)`

`text(Using the distance formula):`

`d_(AB)`	`=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}`
	`=sqrt{(1-(-1))^2+(-1-1)^2}`
	`=sqrt8`

`d_(BC)`	`=sqrt{(-1-(-sqrt3))^2+(1-(-sqrt3))^2}`
	`=sqrt{(-1+sqrt3)^2+(1+sqrt3)^2}`
	`=sqrt(1-2sqrt3+3 +1+2sqrt3+3)`
	`=sqrt8`

`d_(AC)`	`=sqrt{(1-(-sqrt3))^2+(-1-(-sqrt3))^2}`
	`=sqrt{(1+sqrt3)^2+(-1+sqrt3)^2}`
	`=sqrt(1+2sqrt3+3 +1-2sqrt3+3)`
	`=sqrt8`

`text{Since}\ AB=BC=AC`

`:. ΔABC\ text{is equilateral.}`

Cartesian Plane, SMB-019

A straight line passes through points `Q(3,-2)` and `R(-1,4)` .

Find the equation of `QR` and express in general form. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`2y+3x-5=0`

Show Worked Solution

`text{Line goes through}\ (3,-2) and (-1,4).`

`text(Using the gradient formula):`

`m`	`=(y_2-y_1)/(x_2-x_1)`
	`=(-2-4)/(3-(-1))`
	`=-3/2`

`text{Find equation through}\ (3,-2), m=-3/2:`

`y-y_1`	`=m(x-x_1)`
`y-(-2)`	`=-3/2(x-3)`
`2(y+2)`	`=-3(x-3)`
`2y+4`	`=-3x+9`
`2y+3x-5`	`=0`

Cartesian Plane, SMB-018

A straight line passes through points `A(-2,-2)` and `B(1,5)` .

Find the equation of `AB` and express in form `y=mx+c`. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`y=7/3x+8/3`

Show Worked Solution

`text{Line goes through}\ (-2,-2) and (1,5).`

`text(Using the gradient formula):`

`m`	`=(y_2-y_1)/(x_2-x_1)`
	`=(5-(-2))/(1-(-2))`
	`=7/3`

`text{Find equation through}\ (1,5), m=7/3:`

`y-y_1`	`=m(x-x_1)`
`y-5`	`=7/3(x-1)`
`y-5`	`=7/3x-7/3`
`y`	`=7/3x+8/3`

Cartesian Plane, SMB-017

Albert drew a straight line through points `P` and `Q` as shown on the graph below.

Find the equation of Albert's line and express in general form. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`3y-5x+2=0`

Show Worked Solution

`text{Line goes through}\ (-2,-4) and (1,1).`

`text(Using the gradient formula):`

`m`	`=(y_2-y_1)/(x_2-x_1)`
	`=(1-(-4))/(1-(-2))`
	`=5/3`

`text{Find equation through}\ (1,1), m=5/3:`

`y-y_1`	`=m(x-x_1)`
`y-1`	`=5/3(x-1)`
`3y-3`	`=5x-5`
`3y-5x+2`	`=0`

Cartesian Plane, SMB-016

Calculate the value(s) of `p` given that the points `(p,3)` and `(1,p)` are exactly 10 units apart. (3 marks)

--- 7 WORK AREA LINES (style=lined) ---

Show Answers Only

`p=9\ text{or}\ -5`

Show Worked Solution

`(p,3),\ \ (1,p)`

`text{Using the distance formula:}`

`d`	`=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}`
`10`	`=sqrt{(p-1)^2+(3-p)^2}`
`10`	`=sqrt{p^2-2p+1+9-6p+p^2}`
`10`	`=sqrt{2p^2-8p+10}`
`100`	`=2p^2-8p+10`
`0`	`=2p^2-8p-90`
`0`	`=p^2-4p-45`
`0`	`=(p-9)(p+5)`

`:.p=9\ text{or}\ -5`

Cartesian Plane, SMB-015

Calculate the distance between the points `(2,-3)` and `(-5,4)`.

Round your answer to the nearest tenth. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`9.9\ text{units}`

Show Worked Solution

`(2,-3),\ \ (-5,4)`

`text{Using the distance formula:}`

`d`	`=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}`
	`=sqrt{(2-(-5))^2+(-3-4)^2}`
	`=sqrt{49+49}`
	`=sqrt{98}`
	`=9.899…`
	`=9.9\ text{units (nearest tenth)}`

Cartesian Plane, SMB-014

Calculate the distance between the points `(6,-5)` and `(0,3)`. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`10\ text{units}`

Show Worked Solution

`(6,-5),\ \ (0,3)`

`text{Using the distance formula:}`

`d`	`=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}`
	`=sqrt{(6-0)^2+(-5-3)^2}`
	`=sqrt{36+64}`
	`=sqrt{100}`
	`=10\ text{units}`

Cartesian Plane, SMB-013

Calculate the distance between the point `(-6,2)` and the origin.

Give your answer in exact form. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`2sqrt{10}\ \ text{units}`

Show Worked Solution

`(-6,2),\ \ (0,0)`

`text{Using the distance formula:}`

`d`	`=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}`
	`=sqrt{(-6-0)^2+(2-0)^2}`
	`=sqrt{36+4}`
	`=sqrt{40}`
	`=2sqrt{10}\ \ text{units}`

Cartesian Plane, SMB-012

What is the equation of the line `l`? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`y = -2x + 2`

Show Worked Solution

`l\ text{passes through (0, 2) and (1, 0)}`

`text(Gradient)`	`= (y_2-y_1)/(x_2-x_1)`
	`= (0-2)/(1-0)`
	`= -2`

`y\ text(intercept = 2)`

`:.y = -2x + 2`

Cartesian Plane, SMB-011

Rambo drew a line as shown on the grid below.

What is the gradient of Rambo's line? (3 marks)

Show Answers Only

`-7/5`

Show Worked Solution

`text{Line goes through}\ (-3,4) and (2,-3).`

`text(Using the gradient formula):`

`m`	`=(y_2-y_1)/(x_2-x_1)`
	`=(-3-4)/(2-(-3))`
	`=-7/5`

Cartesian Plane, SMB-010

The point `C(-2,3)` is the midpoint of the interval `AB`, where `B` has coordinates `(-1,0).`

What are the coordinates of `A`? (3 marks)

Show Answers Only

`(-3,6)`

Show Worked Solution

`text(Using the midpoint formula):`

`(x_A + x_B)/2`	`= x_C`	`(y_A + y_B)/2`	`= y_C`
`(x_A-1)/2`	`= -2`	`(y_A + 0)/2`	`= 3`
`x_A`	`= -3`	`y_A`	`= 6`

`:. A\ text(has coordinates)\ (-3,6).`

Cartesian Plane, SMB-009

Given `C(-3,-5)` and `D(-5,1)`, find the midpoint of `CD`. (2 marks)

Show Answers Only

`(-4, -2)`

Show Worked Solution

`C(-3,-5),\ \ \ D(-5,1)`

`M`	`= ( (x_1 + x_2)/2, (y_1 + y_2)/2)`
	`= ( (-3-5)/2, (-5+1)/2)`
	`= (-4, -2)`

Cartesian Plane, SMB-008

Find `M`, the midpoint of `PQ`, given `P(2, -1)` and `Q(5, 7)`. (2 marks)

Show Answers Only

`M(7/2, 3)`

Show Worked Solution

`P(2,-1)\ \ \ Q(5,7)`

`M`	`= ( (x_1 + x_2)/2, (y_1 + y_2)/2)`
	`= ( (2+5)/2, (-1+7)/2)`
	`= (7/2, 3)`

Cartesian Plane, SMB-004 MC

The graph of `y = 2x-3` will be drawn on this grid.

Which two points will the straight line pass through?

`C and D`
`D and A`
`B and D`
`A and C`

Show Answers Only

`D`

Show Worked Solution

`text(Solution 1)`

`y = 2x-3\ text(passes through)\ (0, -3)`

`text(with a gradient of 2.)`

`:. A and C`

`text(Solution 2)`

`text{Substitute the coordinates of each point into the equation:}`

`A(-1, -5), \ B(1, -5), \ C(3, 3), \ D(-2, 1)`

`text(Only)\ A and C\ text(satisfy the equation) \ \ y = 2x-3.`

`=>D`

Transformations, SMB-003 MC

Jamie is drawing a square on this grid.

He has drawn two corner points as shown.

Jamie draws the third corner at (5, 6).

Where will the other corner of the square be?

`(1,2)`
`(2,1)`
`(1,6)`
`(2,2)`

Show Answers Only

`A`

Show Worked Solution

`text{Corner at point}\ (1, 2).`

`=>A`

Transformations, SMB-002 MC

Bobby places a disc at (4, −2).

He then moves the disc up 4 units.

What are the coordinates of the new position of the disc?

`(8,-2)`
`(4,2)`
`(4,-6)`
`(0,-2)`

Show Answers Only

`B`

Show Worked Solution

`=>B`

Transformations, SMB-001 MC

Point `Q` is translated down 3 units.

What are the new coordinates of `Q`?

`(4,5)`
`(1,2)`
`(7,2)`
`(4,-1)`

Show Answers Only

`D`

Show Worked Solution

`Q\ text(has coordinates)\ (4,2)`

`text(When translated 3 down, the)\ ytext(-value decreases by 3.)`

`:.\ text{New coordinates are (4, −1)}.`

`=>D`

Linear Relationships, SMB-001 MC

Leo drew a straight line through the points (0, 5) and (3, -2) as shown in the diagram below.

What is the gradient of the line that Leo drew?

`7/3`
`3/7`
`-3/7`
`-7/3`

Show Answers Only

`-7/3`

Show Worked Solution

`text{Line passes through (0, 5) and (3, – 2)}`

`text(Gradient)`	`= (y_2-y_1)/(x_2-x_1)`
	`= (5-(-2))/(0-3)`
	`= -7/3`

Algebraic Fractions, SMB-050

Solve the equation `(2p+2)/3+1 = (p-5)/5`, leaving your answer as a fraction. (3 marks)

Show Answers Only

`p=-40/7`

Show Worked Solution

`underbrace{(2p+2)/3+1}_text(multiply × 15)`	`=underbrace{(p-5)/5}_text(multiply × 15)`
`5(2p+2)+15`	`= 3(p-5)`
`10p+10+15`	`=3p-15`
`7p`	`= -40`
`p`	`=-40/7`

Algebraic Fractions, SMB-049

Solve the equation `(x-1)/2+(2x+3)/3 = 2`, leaving your answer as a fraction. (3 marks)

Show Answers Only

`x=9/7`

Show Worked Solution

`underbrace{(x-1)/2+(2x+3)/3}_text(multiply × 6)`	`=2xx6`
`3(x-1)+2(2x+3)`	`= 12`
`3x-3+4x+6`	`=12`
`7x`	`= 9`
`x`	`=9/7`

Algebraic Fractions, SMB-048

Solve `(2x+1)/3-(x+1)/8=1` for `x`. (3 marks)

Show Answers Only

`x=19/13`

Show Worked Solution

`(2x+1)/3-(x+1)/8`	`=1`
`(24(2x+1))/3-(24(x+1))/8`	`=24`
`8(2x+1)-3(x+1)`	`=24`
`16x+8-3x-3`	`=24`
`13x`	`=19`
`y`	`=19/13`

Algebraic Fractions, SMB-047

Solve the equation `(3a)/7 = (2a + 1)/2-3`, leaving your answer as a fraction. (3 marks)

Show Answers Only

`a=35/8`

Show Worked Solution

`underbrace{(3a)/7}_text(multiply × 14)`	`=underbrace{(2a + 1)/2-3}_text(multiply × 14)`
`2xx3a`	`= 7xx(2a+1)-14xx3`
`6a`	`=14a+7-42`
`8a`	`= 35`
`a`	`=35/8`

Algebraic Fractions, SMB-046

Solve `(3y-1)/4-(y+1)/3=2` for `y`. (3 marks)

Show Answers Only

`y=31/5`

Show Worked Solution

`(3y-1)/4-(y+1)/3`	`=2`
`(12(3y-1))/4-(12(y+1))/3`	`=24`
`3(3y-1)-4(y+1)`	`=24`
`9y-3-4y-4`	`=24`
`5y`	`=31`
`y`	`=31/5`

Algebraic Fractions, SMB-045

Solve `(2a-5)/3-(a+7)/5=3` for `a`. (3 marks)

Show Answers Only

`a=13`

Show Worked Solution

`(2a-5)/3-(a+7)/5`	`=3`
`(5(2a-5))/15-(3(a+7))/15`	`=3`
`5(2a-5)-3(a+7)`	`=45`
`10a-25-3a-21`	`=45`
`7a`	`=91`
`a`	`=13`

Quadratics and Cubics, SMB-041

By completing the square, solve for `b` given

`b^2-10b-125=0.` (3 marks)

Show Answers Only

`5+-5sqrt(6)`

Show Worked Solution

`b^2-10b-125`	`=0`
`b^2-10b+25-150`	`=0`
`(b-5)^2`	`=150`
`b-5`	`= +-sqrt(150)`
`b`	`=5+-sqrt(25xx6)`
`b`	`=5+-5sqrt(6)`

Quadratics and Cubics, SMB-040

By completing the square, solve for `x` given

`x^2-12x=-4.` (3 marks)

Show Answers Only

`6+-4sqrt(2)`

Show Worked Solution

`x^2-12x`	`=-4`
`x^2-12x+36`	`=-4+36`
`(x-6)^2`	`=32`
`x-6`	`= +-sqrt(32)`
`x`	`=6+-4sqrt(2)`

Quadratics and Cubics, SMB-039

By completing the square, solve for `y` given

`y^2-14y+37=0.` (3 marks)

Show Answers Only

`7+-2sqrt(3)`

Show Worked Solution

`y^2-14y+37`	`=0`
`y^2-14y+49-12`	`=0`
`(y-7)^2`	`=12`
`y-7`	`= +-sqrt(12)`
`y`	`=7+-2sqrt(3)`

Quadratics and Cubics, SMB-038

By completing the square, solve for `x` given

`x^2+4x-1 = 0.` (3 marks)

Show Answers Only

`-2+-sqrt(5)`

Show Worked Solution

`x^2+4x-1`	`=0`
`x^2+4x+4-5`	`=0`
`(x+2)^2`	`=5`
`x+2`	`= +-sqrt(5)`
`x`	`=-2+-sqrt(5)`

Quadratics and Cubics, SMB-037

Using the quadratic formula, find `p` given

`p^2+2p-4 = 0.` (3 marks)

Show Answers Only

`-1 +- sqrt(5)`

Show Worked Solution

`p^2+2p-4 = 0`

`text(Using)\ x = (-b +- sqrt( b^2-4ac) )/(2a)`

`p`	`= (-2 +- sqrt{(2)^2-4 xx 1 xx(-4) })/ (2 xx 1)`
	`= (-2 +- sqrt(20) )/2`
	`=(-2 +- 2sqrt(5) )/2`
	`= -1 +- sqrt(5)`

Quadratics and Cubics, SMB-036

Using the quadratic formula, find `a` given

`5a^2+7a-1 = 0.` (3 marks)

Show Answers Only

`(-7 +- sqrt(69) )/10`

Show Worked Solution

`5a^2+7a-1 = 0`

`text(Using)\ a = (-b +- sqrt( b^2-4ac) )/(2a)`

`a`	`= (-7 +- sqrt{(7)^2-4 xx 5 xx(-1) })/ (2 xx 5)`
	`= (-7 +- sqrt(49+20) )/10`
	`= (-7 +- sqrt(69) )/10`

Quadratics and Cubics, SMB-035

Using the quadratic formula, solve

`3x^2-4x-2 = 0`. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`(2 +- sqrt(10) )/3`

Show Worked Solution

`3x^2-4x-2 = 0`

`text(Using)\ x = (-b +- sqrt( b^2-4ac) )/(2a)`

`x`	`= (4 +- sqrt{(-4)^2-4 xx 3 xx(-2) })/ (2 xx 3)`
	`= (4 +- sqrt(16+24) )/6`
	`= (4 +- sqrt(40) )/6`
	`= (4 +- 2sqrt(10) )/6`
	`= (2 +- sqrt(10) )/3`

Quadratics and Cubics, SMB-034

Solve the equation `21-4b^2=5b` for `b.` (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`b=7/4 \ text{or}\ -3`

Show Worked Solution

`21-4b^2`	`=5b`
`4b^2+5b-21`	`=0`
`(4b-7)(b+3)`	`=0`

`4b-7`	`=0`	`text{or}\ \ \ \ b=-3`
`b`	`=7/4`

Quadratics and Cubics, SMB-033

Solve the equation `12a^2+8a-15=0` for `a.` (2 marks)

Show Answers Only

`x=5/6 \ text{or}\ -3/2`

Show Worked Solution

`12a^2+8a-15`	`=0`
`(6a-5)(2a+3)`	`=0`

`6a-5`	`=0`	`text{or}\ \ \ \ a=-3/2`
`a`	`=5/6`

Quadratics and Cubics, SMB-033

Solve the equation `6p^2-p-7=0` for `p`. (2 marks)

Show Answers Only

`x=7/6 \ text{or}\ -1`

Show Worked Solution

`6p^2-p-7`	`=0`
`(6p-7)(p+1)`	`=0`

`6p-7`	`=0`	`text{or}\ \ \ \ p=-1`
`p`	`=7/6`

Quadratics and Cubics, SMB-032

Solve the equation `6x^2-3x-9=0` for `x`. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`x=3/2 \ text{or}\ -1`

Show Worked Solution

`6x^2-3x-9`	`=0`
`3(2x^2-x-3)`	`=0`
`3(2x-3)(x+1)`	`=0`

`2x-3`	`=0`	`text{or}\ \ \ \ x=-1`
`x`	`=3/2`

Quadratics and Cubics, SMB-031

Solve the equation `3q^2-10q-8=0` for `q.` (2 marks)

Show Answers Only

`q=-2/3 \ text{or}\ 4`

Show Worked Solution

`3q^2-10q-8`	`=0`
`(3q+2)(q-4)`	`=0`

`3q+2`	`=0`	`text{or}\ \ q=4`
`q`	`=-2/3`

Algebraic Techniques, SMB-066

Expand and simplify `(2x+3)(4x^2-6x+9).` (2 marks)

Show Answers Only

` 8x^3+27`

Show Worked Solution

`(2x+3)(4x^2-6x+9)`	`=8x^3-12x^2+18x+12x^2-18x+27`
	`= 8x^3+27`

Non-Linear, SMB-024

The volume of a sphere is given by `V = 4/3 pi r^3` where `r` is the radius of the sphere.

If the volume of a sphere is `220\ text(cm)^3`, find the radius, to 1 decimal place. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`3.7\ \ text{cm (to 1 d.p.)}`

Show Worked Solution

`V`	`= 4/3 pi r^3`
`3V`	`= 4 pi r^3`
`r^3`	`= (3V)/(4 pi)`

`text(When)\ \ V = 220`

`r^3`	`= (3 xx 220)/(4 pi)`
	`= 52.521…`
`:. r`	`=root3 (52.521…)`
	`= 3.744…\ \ \ text{(by calc)}`
	`= 3.7\ \ text{cm (to 1 d.p.)}`

Non-Linear, SMB-023

Make `r` the subject of the equation `V = 4/3 pir^3`. (3 marks)

Show Answers Only

`r = root(3)((3V)/(4pi))`

Show Worked Solution

`V`	`= 4/3 pir^3`
`3V`	`=4pir^3`
`(3V)/4`	`= pir^3`
`r^3`	`= (3V)/(4pi)`
`r`	`= root(3)((3V)/(4pi))`

Equations, SMB-001

Make `p` the subject of the equation `c = 5/3p + 15`. (2 marks)

Show Answers Only

`p = 3/5 c-9`

Show Worked Solution

`c`	`= 5/3p + 15`
`5/3p`	`= c-15`
`p`	`= 3/5 (c-15)`
	`= 3/5 c-9`

Indices, SMB-029

Expand and simplify `(4sqrt(3)+sqrt(2))(4sqrt(8)-sqrt(12))`. (2 marks)

Show Answers Only

`30sqrt(6)-8`

Show Worked Solution

`(4sqrt(3)+sqrt(2))(4sqrt(8)-sqrt(12))`

`=16sqrt(24)-4sqrt(36)+4sqrt(16)-sqrt(24)`

`=15sqrt(4xx6)-24+16`

`=30sqrt(6)-8`

Indices, SMB-028

Expand and simplify `(sqrt(20)+2sqrt(10))(3sqrt(6)-sqrt(3))`. (2 marks)

Show Answers Only

`4sqrt(30)+10sqrt(15)`

Show Worked Solution

`(sqrt(20)+2sqrt(10))(3sqrt(6)-sqrt(3))`

`=3sqrt(120)-sqrt(60)+6sqrt(60)-2sqrt(30)`

`=3sqrt(4xx30)+5sqrt(4xx15)-2sqrt(30)`

`=6sqrt(30)+10sqrt(15)-2sqrt(30)`

`=4sqrt(30)+10sqrt(15)`

Algebraic Techniques, SMB-073

Simplify `(4p-12p^2)/3 xx (6p)/(3p^2-p)`. (3 marks)

Show Answers Only

`-8p`

Show Worked Solution

`(4p-12p^2)/3 xx (6p)/(3p^2-p)`	`= (4p(1-3p))/3 xx (6p)/(p(3p-1))`
	`= (8p(1-3p))/(3p-1)`
	`= (-8p(3p-1))/(3p-1)`
	`=-8p`

Algebraic Techniques, SMB-072

Simplify `(9x^2)/(x+3) -: (3x)/(x^2-9)`. (3 marks)

Show Answers Only

`3x(x-3)`

Show Worked Solution

`(9x^2)/(x+3) -: (3x)/(x^2-9)`	`=(9x^2)/(x+3) xx (x^2-9)/(3x)`
	`=(9x^2)/(x+3) xx ((x-3)(x+3))/(3x)`
	`=3x(x-3)`

Indices, SMB-026

Find `a` and `b` such that `a,b` are real numbers and

`(6sqrt3-sqrt5)/(2sqrt5)= a + b sqrt15`. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`a= -1/2, \ b=3/5`

Show Worked Solution

`(6sqrt3-sqrt5)/(2sqrt5)`	`=(6sqrt3-sqrt5)/(2sqrt5) xx (sqrt5)/(sqrt5)`
	`=(sqrt5(6sqrt3-sqrt5))/(2 xx5)`
	`=(6sqrt15-5)/10`
	`=-1/2 + 3/5 sqrt15`

`:. a= -1/2, \ b=3/5`

Indices, SMB-025

Show working to find `a` and `b` such that `a,b` are real numbers and

`(sqrt32-6)/(3sqrt2) = a + bsqrt2`. (2 marks)

Show Answers Only

`:. a = 4/3, \ b = -1`

Show Worked Solution

`(sqrt32-6)/(3sqrt2) xx (sqrt2)/(sqrt2)`	`= (sqrt2(4sqrt2-6))/6`
	`= (8-6sqrt2)/6`
	`= 4/3-sqrt2`

`:. a = 4/3, \ b = -1`

Indices, SMB-024

Show working to simplify `a` and `b` such that `a, b` are real numbers and

`(8-sqrt27)/(2sqrt3) = a + bsqrt3`. (2 marks)

Show Answers Only

`:. a =-3/2, \ b = 4/3`

Show Worked Solution

`(8-sqrt27)/(2sqrt3) xx (sqrt3)/(sqrt3)`	`=(sqrt3(8-3sqrt3))/(2xx3)`
	`= (8sqrt3-9)/6`
	`= -3/2 + 4/3sqrt3`

`:. a = -3/2, \ b = 4/3`

Indices, SMB-023

Rationalise the denominator of `1/(4sqrt 3)`. (2 marks)

Show Answers Only

`sqrt 3/12`

Show Worked Solution

`1/(4sqrt 3) xx (sqrt 3)/(sqrt 3)`	`= (sqrt 3)/(4xx3)`
	`= sqrt 3/12`

Algebraic Techniques, SMB-071

Simplify the expression `4/(y^2-16)+1/(3y-12)` (3 marks)

Show Answers Only

`(y+16)/(3(y^2-16))`

Show Worked Solution

`4/(y^2-16)+1/(3y-12)`	`=4/((y-4)(y+4))+1/(3(y-4))`
	`=12/(3(y-4)(y+4))+(y+4)/(3(y-4)(y+4))`
	`=(y+16)/(3(y-4)(y+4))`

Algebraic Techniques, SMB-070

Simplify the expression `(5a-15)/3xx(6a)/(a^2-9)` (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`(10a)/(a+3)`

Show Worked Solution

`(5a-15)/3xx(6a)/(a^2-9)`	`=(5(a-3))/3xx(2xx3xxa)/((a-3)(a+3))`
	`=(10a)/(a+3)`

Algebraic Techniques, SMB-069

Factorise the expression `5/(x^2-1)-3/(x-1)` (3 marks)

Show Answers Only

`(2-3x)/((x-1)(x+1))`

Show Worked Solution

`5/(x^2-1)-3/(x-1)`	`=5/((x+1)(x-1))-3/(x-1)`
	`=5/((x+1)(x-1))-(3(x+1))/((x+1)(x-1))`
	`=(5-3(x+1))/((x-1)(x+1))`
	`=(2-3x)/((x-1)(x+1))`

Algebraic Techniques, SMB-068

Factorise the expression `5/(b^2-b)-2/(b^2-1)` (3 marks)

Show Answers Only

`(3b+5)/(b(b-1)(b+1))`

Show Worked Solution

`5/(b^2-b)-2/(b^2-1)`	`=5/(b(b-1))-2/((b+1)(b-1))`
	`=(5(b+1))/(b(b-1)(b+1))-(2b)/(b(b-1)(b+1))`
	`=(5b+5-2b)/(b(b-1)(b+1))`
	`=(3b+5)/(b(b-1)(b+1))`

Algebraic Techniques, SMB-067

Factorise the expression `(x^2-7x+12)/(x-3)` (2 marks)

Show Answers Only

`x-4`

Show Worked Solution

`(x^2-7x+12)/(x-3)`	`=((x-3)(x-4))/(x-3)`
	`=x-4`

Algebraic Techniques, SMB-066

Factorise the expression `-12y^2+18y-6` (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`6(2y-1)(1-y)\ \ text(or)\ \ 6(1-2y)(y-1)`

Show Worked Solution

`-12y^2+18y-6=-6(2y^2-3y+1)`

`-6(2y^2-3y+1)=>\ \ `	`text{P}`	`=2xx1=2`
	`text{S}`	`=-3`
	`text{F}`	`=-2,-1`

`-6(2y^2-3y+1)`	`=(-6(2y-1)(2y-2))/2`
	`=-6(2y-1)(y-1)`
	`=6(2y-1)(1-y)\ \ text(or)\ \ 6(1-2y)(y-1)`

Algebraic Techniques, SMB-065

Fully factorise the expression `6y^2+10y-4` (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`2(3y-1)(y+2)`

Show Worked Solution

`6y^2+10y-4=2(3y^2+5y-2)`

`2(3y^2+5y-2)=>\ \ `	`text{P}`	`=3xx-2=-6`
	`text{S}`	`=5`
	`text{F}`	`=-1,6`

`2(3y^2+5y-2)`	`=(2(3y-1)(3y+6))/3`
	`=2(3y-1)(y+2)`

Algebraic Techniques, SMB-064

Factorise the expression `2x^2+x-6` (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`(2x-3)(x+2)`

Show Worked Solution

`2x^2+x-6=>\ \ `	`text{P}`	`=2xx-6=-12`
	`text{S}`	`=1`
	`text{F}`	`=-3,4`

`2x^2+x-6`	`=((2x-3)(2x+4))/2`
	`=(2x-3)(x+2)`

Algebraic Techniques, SMB-063

Fully factorise the expression `3a^2+12ab-2a-8b` (2 marks)

Show Answers Only

`(3a-2)(a+4b)`

Show Worked Solution

`3a^2+12ab-2a-8b`	`=3a(a+4b)-2(a+4b)`
	`=(3a-2)(a+4b)`

Algebraic Techniques, SMB-062

Fully factorise the expression `2x^2-6xy-3x+9y` (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`(2x-3)(x-3y)`

Show Worked Solution

`2x^2-6xy-3x+9y`	`=2x(x-3y)-3(x-3y)`
	`=(2x-3)(x-3y)`

Algebraic Techniques, SMB-061

Factorise the expression `5b^3-5b` (2 marks)

Show Answers Only

`5b(b+1)(b-1)`

Show Worked Solution

`5b^3-5b`	`=5b(b^2-1)`
	`=5b(b+1)(b-1)`

Algebraic Techniques, SMB-060

Expand and simplify the expression `(a+b)^2-(a-b)^2` (2 marks)

Show Answers Only

`4ab`

Show Worked Solution

`(a+b)^2-(a-b)^2`	`=a^2+2ab+b^2-(a^2-2ab+b^2)`
	`=a^2+2ab+b^2-a^2+2ab-b^2`
	`=4ab`