Using the quadratic formula, find `p` given
`p^2+2p-4 = 0.` (3 marks)
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Using the quadratic formula, find `p` given
`p^2+2p-4 = 0.` (3 marks)
`-1 +- sqrt(5)`
`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)` |
Using the quadratic formula, find `a` given
`5a^2+7a-1 = 0.` (3 marks)
`(-7 +- sqrt(69) )/10`
`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` |
Using the quadratic formula, solve
`3x^2-4x-2 = 0`. (3 marks)
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`(2 +- sqrt(10) )/3`
`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` |
Solve the equation `21-4b^2=5b` for `b.` (2 marks)
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`b=7/4 \ text{or}\ -3`
`21-4b^2` | `=5b` |
`4b^2+5b-21` | `=0` |
`(4b-7)(b+3)` | `=0` |
`4b-7` | `=0` | `text{or}\ \ \ \ b=-3` |
`b` | `=7/4` |
Solve the equation `12a^2+8a-15=0` for `a.` (2 marks)
`x=5/6 \ text{or}\ -3/2`
`12a^2+8a-15` | `=0` |
`(6a-5)(2a+3)` | `=0` |
`6a-5` | `=0` | `text{or}\ \ \ \ a=-3/2` |
`a` | `=5/6` |
Solve the equation `6p^2-p-7=0` for `p`. (2 marks)
`x=7/6 \ text{or}\ -1`
`6p^2-p-7` | `=0` |
`(6p-7)(p+1)` | `=0` |
`6p-7` | `=0` | `text{or}\ \ \ \ p=-1` |
`p` | `=7/6` |
Solve the equation `6x^2-3x-9=0` for `x`. (2 marks)
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`x=3/2 \ text{or}\ -1`
`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` |
Solve the equation `3q^2-10q-8=0` for `q.` (2 marks)
`q=-2/3 \ text{or}\ 4`
`3q^2-10q-8` | `=0` |
`(3q+2)(q-4)` | `=0` |
`3q+2` | `=0` | `text{or}\ \ q=4` |
`q` | `=-2/3` |
Solve the equation `p^2-12p=64` for `p`. (2 marks)
`p=16 \ text{or}\ -4`
`p^2-12p` | `=64` |
`p^2-12p-64` | `=0` |
`(p-16)(p+4)` | `=0` |
`:. p=16 \ text{or}\ -4`
Solve the equation `14x=32-x^2` for `x`. (2 marks)
`x=2 \ text{or}\ -16`
`14x` | `=32-x^2` |
`x^2+14x-32` | `=0` |
`(x-2)(x+16)` | `=0` |
`:. x=2 \ text{or}\ -16`
Solve the equation `c^2-24=5c` for `c`. (2 marks)
`c=8 \ text{or}\ -3`
`c^2-24` | `=5c` |
`c^2-5c-24` | `=0` |
`(c-8)(c+3)` | `=0` |
`:. c=8 \ text{or}\ -3`
Solve the equation `y^2-2y-3=0` for `y`. (2 marks)
`y=3 \ text{or}\ -1`
`y^2-2y-3` | `=0` |
`(y-3)(y+1)` | `=0` |
`:. y=3 \ text{or}\ -1`
Solve the equation `t^2-8t+12=0` for `t`. (2 marks)
`:. t=6 \ text{or}\ 2`
`t^2-8t+12` | `=0` |
`(t-6)(t-2)` | `=0` |
`:. t=6 \ text{or}\ 2`
Expand and simplify `(2x+3)(4x^2-6x+9).` (2 marks)
` 8x^3+27`
`(2x+3)(4x^2-6x+9)` | `=8x^3-12x^2+18x+12x^2-18x+27` |
`= 8x^3+27` |
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)
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`3.7\ \ text{cm (to 1 d.p.)}`
`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.)}` |
Make `r` the subject of the equation `V = 4/3 pir^3`. (3 marks)
`r = root(3)((3V)/(4pi))`
`V` | `= 4/3 pir^3` |
`3V` | `=4pir^3` |
`(3V)/4` | `= pir^3` |
`r^3` | `= (3V)/(4pi)` |
`r` | `= root(3)((3V)/(4pi))` |
Solve `4-x<7`. (2 marks)
`x> -3`
`4-x` | `<7` | |
`-x` | `< 3` | |
`x` | `> -3` |
Solve `3-x/5<8` if `x` is a negative number. (2 marks)
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`-25>x<=0`
`3-x/5` | `<8` | |
`-x/5` | `< 5` | |
`-x` | `<25` | |
`x` | `> -25` |
`text{Given}\ x\ text{is a negative number:}`
`-25>x<=0`
Solve `2x+1>= -3` and graph the solution on a number line. (3 marks)
`x>= -2`
`2x+1` | `>= -3` | |
`2x` | `>= -4` | |
`x` | `>= -2` |
`y = 2x-3`
`y = 4x + 1`
Which value of `x` satisfies both of these equations?
`A`
`text(If)\ x = −2,`
`2(-2)-3` | `= -7` |
`4(-2) + 1` | `= -7` |
`:. x = -2\ \ text(satisfies both.)`
`=>A`
The daily energy requirement, `E` (kilojoules), for a person of mass `m` (kilograms) is calculated using the rule `E = 7m + 7300`.
For Elijah, `E = 7755`.
What is Elijah's mass? (2 marks)
`65\ text{kgs}`
`7755` | `= 7m + 7300` |
`7 m` | `= 455` |
`m` | `= 455/7` |
`= 65\ text(kilograms)` |
In this inequality `n` is a whole number.
`8/n < 5/8`
What is the smallest possible value for `n` to make this inequality true? (2 marks)
`13`
`8/n` | `< 5/8` |
`5n` | `> 64` |
`n` | `> 64/5` |
`> 12.8` |
`:. text(Smallest)\ \ n = 13.`
For the expression `x^2 > 5x`, what is the smallest positive whole number `x` can be that makes the expression correct? (2 marks)
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`6`
`text(If)\ \ x = 6,`
`6^2` | `> 5 xx 6` |
`36` | `> 30\ \ \ =>\ text{correct}` |
`text(All positive whole numbers smaller than 6 result in:)`
`x^2 = 5x\ \ \ text{(for}\ x=5 text{)}`
`x^2 < 5x\ \ \ text{(for}\ x=4,3,2, …text{)}`
`4/45 = 3/40 + 1/x`
Find the value of `x?` (2 marks)
`72`
`1/x` | `= 4/45-3/40` |
`= 1/72\ \ \ text{(by calculator)}` | |
`:. x` | `= 72` |
The weight (`w` kilograms) and age (`a` years) of a turtle are related by the following inequality:
\begin{array} {|l|}
\hline
\rule{0pt}{2.5ex}w < 8a-13\ \ \text{for all values of}\ a\ \text{between 1 and 10}\rule[-1ex]{0pt}{0pt} \\
\hline
\end{array}
Which pair of values satisfy this inequality?
`C`
`text(Test each option by trial and error.)`
`text(Consider)\ \ w = 18,\ a = 4,`
`18 < 8 xx 4-13`
`18 < 19\ \ text{(correct)}`
`:. w = 18,\ a = 4\ text(satisfies the equation).`
`=>C`
`2/13 < 3/x` where `x` is a positive whole number.
What is the highest possible value for `x`? (2 marks)
`19`
`2/13` | `< 3/x` |
`2x` | `< 39` |
`x` | `< 19.5` |
`:. x = 19`
`2(3p-5)-3=17`
Solve for `p`. (2 marks)
`p=5`
`2(3p-5)-3` | `=17` | |
`6p-10-3` | `=17` | |
`6p-13` | `=17` | |
`6p` | `=30` | |
`:.p` | `=5` |
`17y+3(5-3y)-5=26`
What value of `y` makes this equation true? (2 marks)
`y=2`
`17y+3(5-3y)-5` | `=26` | |
`17y+15-9y-5` | `=26` | |
`8y+10` | `=26` | |
`8y` | `=16` | |
`:.y` | `=2` |
`6(3a+4)-12=8a-18`
What value of `a` makes this equation true? (2 marks)
`a=-3`
`6(3a+4)-12` | `=8a-18` | |
`18a+24-12` | `=8a-18` | |
`10a` | `=-12-18` | |
`10a` | `=-30` | |
`:.a` | `=-3` |
`2 (4x-2) + 1 +` |
?
|
`= 9x-3` |
What term makes this equation true for all values of `x`? (2 marks)
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?
|
`= x` |
`2(4x-2) + 1 +` |
?
|
`= 9x-3` |
`8x-4 + 1 +` |
?
|
`= 9x-3` |
`8x-3 +` |
?
|
`= 9x-3` |
?
|
`= x` |
Make `p` the subject of the equation `c = 5/3p + 15`. (2 marks)
`p = 3/5 c-9`
`c` | `= 5/3p + 15` |
`5/3p` | `= c-15` |
`p` | `= 3/5 (c-15)` |
`= 3/5 c-9` |
Rationalise the denominator of the surd fraction `(sqrt(12))/(sqrt(6)-2)`. (3 marks)
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`3sqrt(2)+2sqrt(3)`
`(sqrt(12))/(sqrt(6)-2)` | `=(2sqrt(3))/(sqrt(6)-2) xx (sqrt(6)+2)/(sqrt(6)+2)` | |
`=(2sqrt(3)(sqrt(6)+2))/((sqrt(6))^2-2^2)` | ||
`=(2sqrt(18)+4sqrt(3))/(2)` | ||
`=(6sqrt(2)+4sqrt(3))/(2)` | ||
`=3sqrt(2)+2sqrt(3)` |
Rationalise the denominator of the surd fraction `(8-2sqrt(6))/(3sqrt(2)+2sqrt(3))`. (3 marks)
`6sqrt(2)-14/3sqrt(3)`
`(8-2sqrt(6))/(3sqrt(2)+2sqrt(3))`
`=(8-2sqrt(6))/(3sqrt(2)+2sqrt(3))xx(3sqrt(2)-2sqrt(3))/(3sqrt(2)-2sqrt(3))`
`=((8-2sqrt(6))(3sqrt(2)-2sqrt(3)))/((3sqrt(2))^2-(2sqrt(3))^2)`
`=(24sqrt(2)-16sqrt(3)-6sqrt(12)+4sqrt(18))/(18-12)`
`=(24sqrt(2)-16sqrt(3)-12sqrt(3)+12sqrt(2))/6`
`=(36sqrt(2)-28sqrt(3))/6`
`=6sqrt(2)-14/3sqrt(3)`
Expand and simplify `(4sqrt(3)+sqrt(2))(4sqrt(8)-sqrt(12))`. (2 marks)
`30sqrt(6)-8`
`(4sqrt(3)+sqrt(2))(4sqrt(8)-sqrt(12))`
`=16sqrt(24)-4sqrt(36)+4sqrt(16)-sqrt(24)`
`=15sqrt(4xx6)-24+16`
`=30sqrt(6)-8`
Expand and simplify `(sqrt(20)+2sqrt(10))(3sqrt(6)-sqrt(3))`. (2 marks)
`4sqrt(30)+10sqrt(15)`
`(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)`
Simplify `(p/q)^3 ÷ (pq^(-2))`. (2 marks)
`(p^2)/q`
`(p/q)^3 ÷ (pq^(-2))` | `= (p^3)/(q^3) ÷ p/(q^2)` |
`= (p^3)/(q^3) xx (q^2)/p` | |
`= (p^2)/q` |
Worker A picks a bucket of blueberries in `a` hours. Worker B picks a bucket of blueberries in `b` hours.
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i. `(a + b)/(ab)`
ii. `text(The reciprocal represents the number of hours it would)`
`text(take to fill one bucket, with A and B working together.)`
i. `text(In one hour:)`
`text(Worker A picks)\ 1/a\ text(bucket.)`
`text(Worker B picks)\ 1/b\ text(bucket.)`
`:.\ text(Fraction picked in 1 hour working together)`
`= 1/a + 1/b`
`= (a + b)/(ab)`
ii. `text(The reciprocal represents the number of hours it would)`
`text(take to fill one bucket, with A and B working together.)`
Simplify `(4p-12p^2)/3 xx (6p)/(3p^2-p)`. (3 marks)
`-8p`
`(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` |
Simplify `(9x^2)/(x+3) -: (3x)/(x^2-9)`. (3 marks)
`3x(x-3)`
`(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)` |
Find the reciprocal of `1/a + 1/b -c/(ab)`. (3 marks)
`(ab)/(a+b-c)`
`1/a + 1/b -c/(ab)` | `=b/(ab)+a/(ab)-c/(ab)` |
`=(b+a-c)/(ab)` |
`text(Reciprocal of)\ \ x = x^(-1)`
`:.\ text(Reciprocal of)\ \ (b+a-c)/(ab)=>((b+a-c)/(ab))^(-1)=(ab)/(a+b-c)`
Simplify `(a(b^2)^3)/(a^2b)`. (2 marks)
`b^5/a`
`(a(b^2)^3)/(a^2b)` | `= (ab^6)/(a^2b)` |
`= b^5/a` |
Find `a` and `b` such that `a,b` are real numbers and
`(6sqrt3-sqrt5)/(2sqrt5)= a + b sqrt15`. (2 marks)
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`a= -1/2, \ b=3/5`
`(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`
Show working to find `a` and `b` such that `a,b` are real numbers and
`(sqrt32-6)/(3sqrt2) = a + bsqrt2`. (2 marks)
`:. a = 4/3, \ b = -1`
`(sqrt32-6)/(3sqrt2) xx (sqrt2)/(sqrt2)` | `= (sqrt2(4sqrt2-6))/6` |
`= (8-6sqrt2)/6` | |
`= 4/3-sqrt2` |
`:. a = 4/3, \ b = -1`
Show working to simplify `a` and `b` such that `a, b` are real numbers and
`(8-sqrt27)/(2sqrt3) = a + bsqrt3`. (2 marks)
`:. a =-3/2, \ b = 4/3`
`(8-sqrt27)/(2sqrt3) xx (sqrt3)/(sqrt3)` | `=(sqrt3(8-3sqrt3))/(2xx3)` |
`= (8sqrt3-9)/6` | |
`= -3/2 + 4/3sqrt3` |
`:. a = -3/2, \ b = 4/3`
Rationalise the denominator of `1/(4sqrt 3)`. (2 marks)
`sqrt 3/12`
`1/(4sqrt 3) xx (sqrt 3)/(sqrt 3)` | `= (sqrt 3)/(4xx3)` | |
`= sqrt 3/12` |
Simplify `((4p^2)/8)^(-3)` and express as a fraction involving non-negative indices only. (3 marks)
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`8/(p^6)`
`((4p^2)/8)^(-3)` | `= ((p^2)/2)^(-3)` |
`=p^(2xx -3)/(2^(-3))` | |
`= (p^(-6))/(2^(-3))= (2^(3))/(p^(6))` | |
`= 8/(p^6)` |
Simplify `((3y^3)/2)^(-2)` and express as a fraction involving non-negative indices only. (3 marks)
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`4/(9y^6)`
`((3y^3)/2)^(-2)` | `=(3^(-2)xxy^((3xx-2)))/(2^(-2))` |
`= (2^(2)xxy^(-6))/(3^(2))` | |
`= 4/(9y^6)` |
Express `4a^(-5) -: 12a^4` as a fraction involving non-negative indices only. (1 mark)
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`1/(3a^9)`
`4a^(-5) -: 12a^4` | `=(4a^(-5))/(12a^4)` |
`= a^((-5-4))/3` | |
`= (a^(-9))/3` | |
`=1/(3a^9)` |
Express `(24p^(-3)q^4)/(8pq^(2))` as a fraction involving non-negative indices only. (1 mark)
`(3q^2)/p^4`
`(24p^(-3)q^4)/(8pq^(2))` | `=3p^((-3-1))q^((4-2))` |
`= 3p^(-4)q^2` | |
`= (3q^2)/p^4` |
Simplify the expression `(36t^4)/(9t^(-2))`. (1 mark)
`4t^6`
`(36t^4)/(9t^(-2))` | `= 4t^(4-(-2))` |
`= 4t^6` |
Simplify the expression `(16x^5)/(2x^(-3))`. (1 mark)
`8x^8`
`(16x^5)/(2x^(-3))` | `= 8x^(5-(-3))` |
`= 8x^8` |
Simplify the expression `(12x)^0/4 xx (5x)/3`. (1 mark)
`(5x)/12`
`(12x)^0/4 xx (5x)/3` | `= 1/4 xx (5x)/3` |
`= (5x)/12` |
Simplify the expression `(20m^0)/n xx n^2/(5m)`. (1 mark)
`(4n)/m`
`(20m^0)/n xx n^2/(5m)` | `= 20/n xx n^2/(5m)` |
`= (4n)/m` |