In the diagram, \(AC\) is a diameter of the circle centred at \(O\), and \(OA = AB\).
Find the value of \(\theta\). (3 marks)
Aussie Maths & Science Teachers: Save your time with SmarterEd
In the diagram, \(AC\) is a diameter of the circle centred at \(O\), and \(OA = AB\).
Find the value of \(\theta\). (3 marks)
\(\theta = 30^{\circ}\)
\(\angle ABC=90^{\circ}\ \ \text{(angle in semi-circle)}\)
\(OA=OB\ \ \text{(radii)} \)
\( \angle OAB=60^{\circ}\ \ ( \Delta OAB\ \text{is equilateral}) \)
\(\theta\) | \(= 180-(90+60)\ \ (180^{\circ}\ \text{in}\ \Delta) \) | |
\(= 30^{\circ} \) |
In the diagram, a line from the centre of the circle meets a chord at its midpoint.
Find the value of \(\theta\). (2 marks)
\(\theta = 47^{\circ}\)
\(\text{Line from centre bisects chord}\ \ \Rightarrow\ \ \text{Line is ⊥ to chord}\)
\(\theta\) | \(= 180-(90+43)\ \ (180^{\circ}\ \text{in}\ \Delta) \) | |
\(= 47^{\circ} \) |
In the circle centred at \(O\), the chord \(AC\) has length 15 and \(OB\) meets the chord \(AC\) at right angles.
Find the length of \(BC\). (1 mark)
\(BC = 7.5\)
\(BC\) | \(= \dfrac{1}{2} \times 15\ \ \text{(perpendicular from centre to chord bisects chord)}\) | |
\(= 7.5 \) |
In the diagram, two chords of a circle intersect.
Find \(x\). (2 marks)
\(x=8\)
\(3 \times x\) | \(=6 \times 4\ \ \text{(intercepts of intersecting chords)}\) | |
\(x\) | \(= \dfrac{24}{3} \) | |
\(=8\) |
In the diagram, two chords of a circle intersect.
Find \(x\). (2 marks)
\(x=6\)
\(7 \times x\) | \(=3 \times 14\ \ \text{(intercepts of intersecting chords)}\) | |
\(x\) | \(= \dfrac{42}{7} \) | |
\(=6\) |
In the diagram, the vertices of \(\Delta ABC\) lie on the circle with centre \(O\). The point \(D\) lies on \(BC\) such that \(\Delta ABD\) is isosceles and \(\angle ABC = x\).
Explain why \(\angle AOC = 2x\). (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
\(\text{See Worked Solutions}\)
The line \(AT\) is the tangent to the circle at \(A\), and \(BT\) is a secant meeting the circle at \(B\) and \(C\).
Given that \(AT = 12\), \(BC = 7\) and \(CT = x\), find the value of \(x\). (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
\(x = 9\)
\(\text{Property: square of tangent = product of secant intercepts}\)
\(AT^2\) | \(= CT \times BT\) |
\(12^2\) | \(= x(x + 7)\) |
\(144\) | \(= x^2 + 7x\) |
\(x^2 + 7x-144\) | \(= 0\) |
\((x + 16)(x-9)\) | \(= 0\) |
\(\therefore x = 9,\ (x \gt 0) \)
In the circle centred at \(O\) the chord \(AB\) has length 7. The point \(E\) lies on \(AB\) and \(AE\) has length 4. The chord \(CD\) passes through \(E\).
Let the length of \(CD\) be \(\ell\) and the length of \(DE\) be \(x\).
Show that \(x^2-\ell x + 12 = 0\). (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
\(\text{See Worked Solutions}\)
Moses finds that for a Froghead eel, its mass is directly proportional to the square of its length.
An eel of this species has a length of 72 cm and a mass of 8250 grams.
What is the expected length of a Froghead eel with a mass of 10.2 kg? Give your answer to one decimal place. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`80.1\ text{cm}`
`text(Mass) prop text(length)^2`
`m = kl^2`
`text(Find)\ k:`
`8250` | `= k xx 72^2` |
`k` | `= 8250/72^2` |
`= 1.591…` |
`text(When)\ \ l\ \ text(when)\ \ m = 10\ 200:`
`10\ 200` | `= 1.591… xx l^2` |
`l^2` | `= (10\ 200)/(1.591…)` |
`:. l` | `= 80.069…` |
`= 80.1\ text{cm (to 1 d.p.)}` |
The number of trees that can be planted along the fence line of a paddock varies inversely with the distance between each tree.
There will be 108 trees if the distance between them is 5 metres.
--- 4 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
i. `t prop 1/d`
`t` | `= k/d` |
`108` | `= k/5` |
`k` | `= 540` |
`text(Find)\ t\ text(when)\ d = 6:`
`t` | `= 540/6` |
`= 90` |
ii. `text(Find)\ d\ text(when)\ t = 120:`
`120` | `= 540/d` |
`d` | `= 540/120` |
`= 4.5\ text(metres)` |
It is known that the quantity of steel produced in tonnes `(S)`, is directly proportional to the tonnes of iron ore used in the process `(I)`.
If 16 tonnes or iron ore produces 10 tonnes of steel, calculate the tonnes of iron ore required to produce 48 tonnes of steel. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`76.8\ text{tonnes}`
`S prop I\ \ =>\ \ S=kI`
`text(Find)\ k\ text{given}\ S=10\ text{when}\ I=16:`
`10` | `=k xx 16` |
`k` | `=10/16` |
`=0.625` |
`text{Find}\ I\ text{when}\ S=48:`
`48` | `=0.625 xx I` |
`:. I` | `=48/0.625` |
`=76.8\ text{tonnes}` |
It is known that a quantity `P` kgs is proportional to the reciprocal of another quantity `Q` kgs such that `P prop 1/Q`.
If `P=12` when `Q=0.05`, calculate the estimated quantity of `Q` when `P=45` kgs, to the nearest gram. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`5333\ text{g}`
`P prop 1/Q\ \ =>\ \ P=k/Q`
`text(Find)\ k\ text{given}\ P=12\ text{when}\ Q=0.05:`
`12` | `=k/0.05` |
`:. k` | `=12 xx 20` |
`=240` |
`text{Find}\ Q\ text{when}\ P=45:`
`45` | `=240/Q` |
`:. Q` | `=240/45` |
`=5.3333\ text{kg}` | |
`=5333\ text{g}` |
The stopping distance of a car on a certain road, once the brakes are applied, is directly proportional to the square of the speed of the car when the brakes are first applied.
A car travelling at 70 km/h takes 58.8 metres to stop.
How far does it take to stop if it is travelling at 105 km/h? (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`132.3\ text(metres)`
`text(Let)\ \ d\ text(= stopping distance)`
`d prop s^2`
`d = ks^2`
`text(Find)\ k,`
`58.8` | `= k xx 70^2` |
`k` | `= 58.8/(70^2)` |
`= 0.012` |
`text(Find)\ d\ \ text(when)\ s = 105:`
`d` | `= 0.012 xx 105^2` |
`= 132.3\ text(metres)` |
Fuifui finds that for Giant moray eels, the mass of an eel `(M)` is directly proportional to the cube of its length `(l)`.
An eel of this species has a length of 15 cm and a mass of 675 grams.
What is the expected length of a Giant moray eel with a mass of 3.125 kg? Give your answer to one decimal place. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`25\ text{cm}`
`M prop l^3`
`M = kl^3`
`text(Find)\ k:`
`675` | `= k xx 15^3` |
`k` | `= 675/15^3` |
`= 0.2` |
`text(Find)\ \ l\ \ text(when)\ \ M = 3125:`
`3125` | `= 0.2 xx l^3` |
`l^3` | `= 3125/0.2` |
`:. l` | `= root3(15\ 625)` |
`= 25\ text{cm}` |
Jacques is a marine biologist and finds that the mass of a crab `(M)` is directly proportional to the cube of the diameter of its shell `(d)`.
If a crab with a shell diameter of 15 cm weighs 680 grams, what will be the diameter of a crab that weighs 1.1 kilograms? Give your answer to 1 decimal place. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`17.6\ text(cm)`
`M` | `prop d^3` | |
`M` | `= kd^3` |
`text(When)\ \ M=680, \ d=15`
`680` | `=k xx 15^3` | |
`k` | `=0.201481…` |
`text(Find)\ \ d\ \ text(when)\ \ M=1100:`
`1100` | `=0.20148… xx d^3` | |
`d` | `=root3(1100/(0.20148…))` | |
`=17.608…` | ||
`=17.6\ text{cm (to 1 d.p.)}` |
The current of an electrical circuit, measured in amps (A), varies inversely with its resistance, measured in ohms (R).
When the resistance of a circuit is 28 ohms, the current is 3 amps.
What is the current when the resistance is 8 ohms? (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`10.5`
`A` | `prop 1/R` | |
`A` | `= k/R` |
`text(When)\ \ A=3, \ R=28`
`3` | `=k/28` | |
`k` | `=84` |
`text(Find)\ \ A\ \ when\ \ R=8:`
`A` | `=84/8` | |
`=10.5` |
It is known that at a constant speed, the distance travelled in kilometres `(d)` is directly proportional to the time of travel in hours `(t)`, or `d prop t`.
--- 4 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
i. `k=15`
ii. `text{Speed}`
i. `d prop t`
`d=kt`
`text(Find)\ k\ text{given}\ d=75\ text{when}\ t=5:`
`75` | `=k xx 5` |
`:. k` | `=75/15` |
`=5` |
ii. `k\ text{represents the speed.}`
It is known that a quantity `y` is inversely proportional to another quantity `x`.
If `y=3` when `x=1.8`, calculate the constant of variation `k`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`k=5.4`
`y prop 1/x`
`y=k/x`
`text(Find)\ k\ text{given}\ y=3\ text{when}\ x=1.8:`
`3` | `=k/1.8` |
`:. k` | `=3xx1.8` |
`=5.4` |
Shade the region defined by `y+3x>3` on the graph below and verify your result. (3 marks)
--- 2 WORK AREA LINES (style=lined) ---
Shade the region defined by `x/2-y>0` on the graph below and verify your result. (2 marks)
--- 2 WORK AREA LINES (style=lined) ---
i. `y=3/5x+3`
ii.
`text{Test}\ (0,0):`
`5(0)-3(0)>15\ \ =>\ \ 0<15\ \ text{(incorrect)}`
i. | `5y-3x` | `=15` |
`5y` | `=3x+15` | |
`y` | `=3/5x+3` |
ii. `xtext{-intercept occurs when}\ y=0:`
`5(0)-3x=15\ \ =>\ \ x=-5`
`ytext{-intercept at}\ \ y=3`
`text{Test}\ (0,0):`
`5(0)-3(0)>15\ \ =>\ \ 0>15\ \ text{(Incorrect – not in shaded area.)}`
Shade the region defined by `3x-4y<12` on the graph below and verify your result. (3 marks)
--- 2 WORK AREA LINES (style=lined) ---
State the inequality that defines the domain of the function `g(x) = 2/sqrt(5-x)` ? (2 marks)
`text(Domain)\ g(x):\ \ x<5`
`g(x) = 2/sqrt(5-x)\ \ text{exists for:}`
`5-x` | `> 0\ \ \ (5-x!=0)` |
`-x` | `> -5` |
`x` | `<5` |
`:.\ text(Domain)\ g(x):\ \ x<5`
What is the domain of the function `g(x) = log_2(x^2-3)`? (2 marks)
`text{Domain:}\ x>sqrt3\ \ ∪\ \ x<-sqrt3`
`g(x)\ text{exists when:}`
`x^2-3` | `> 0` |
`x^2` | `> 3` |
`x>sqrt3\ \ or\ \ x<-sqrt3`
`:.\ text{Domain:}\ x>sqrt3\ \ ∪\ \ x<-sqrt3`
--- 1 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
i. `x^2-x-6=(x-3)(x+2)`
ii. `text{Domain:}\ x< -2 \ ∪ \ x>3`
i. `x^2-x-6=(x-3)(x+2)`
ii. `f(x) = log_2(x^2-x-6)=log_2(x-3)(x+2)`
`f(x)\ text{exists when:}`
`(x-3)(x+2)>0`
`x< -2 and x>3`
`text{Domain:}\ x< -2 \ ∪ \ x>3`
What is the domain of the function `f(x) = log_10(3-2x)`? (2 marks)
`text{Domain:}\ x<3/2`
`f(x)\ text{exists when:}`
`3-2x` | `> 0` |
`-2x` | `> -3` |
`2x` | `< 3` |
`x` | `< 3/2` |
What is the domain of the function `g(x) = log_2(x+1)`? (2 marks)
`text{Domain:}\ x> -1`
`g(x)\ text{exists when:}`
`x+1` | `> 0` |
`x` | `> -1` |
A function has the equation `h(x)=-1-(x-3)^2`.
State the domain and range of `h(x)`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`text{Domain}\ h(x):\ text{all}\ x`
`text{Range}\ h(x):\ y<=-1`
`h(x)\ text{exists for all}\ x`
`text{Domain}\ h(x):\ text{all}\ x`
`text{Consider the function transformation:}`
`y=x^2\ text{translated 3 units right}\ \ =>\ \ y=(x-3)^2`
`y=(x-3)^2\ text{reflected in the}\ xtext{-axis}\ =>\ \ y=-(x-3)^2`
`y=-(x-3)^2\ text{translated 1 unit down}\ =>\ \ y=-1-(x-3)^2`
`:.\ text{Range}\ f(x): \ y<=-1`
A function has the equation `f(x)=2x^2+1`.
State the range of `f(x)`. (2 marks)
`text{Range}\ f(x): \ y>=1`
`text{Consider the function transformation:}`
`y=2x^2\ text{translated 1 unit up}\ \ =>\ \ y=2x^2+1`
`(x^2>0\ text{for all}\ x)`
`:.\ text{Range}\ f(x): \ y>=1`
A function has the equation `f(x)=4-(x+1)^2`.
State the domain and range of `f(x)`. (3 marks)
`text{Domain}\ f(x): \ text{all}\ x`
`text{Range}\ f(x): \ y<=4`
`f(x)\ text{exists for all}\ x`
`text{Domain}\ f(x): \ text{all}\ x`
`text{Consider the function transformation:}`
`y=x^2\ text{translated 1 unit left}\ =>\ \ y=(x+1)^2`
`y=(x+1)^2\ text{reflected in the}\ xtext{-axis}\ =>\ \ y=-(x+1)^2`
`y=-(x+1)^2\ text{translated 4 units up}\ \ =>\ \ y=4-(x+1)^2`
`:.\ text{Range}\ f(x): \ y<=4`
A function has the equation `g(x)=x^2-1`.
State the range of `g(x)`. (2 marks)
`text{Range}\ g(x): \ y>=-1`
`text{Consider the function transformation:}`
`y=x^2\ text{translated 1 unit down}\ \ =>\ \ y=x^2-1`
`:.\ text{Range}\ g(x): \ y>=-1`
State the domain of the function `f(x) = x^2 + log_10(x)`. (2 marks)
`x>0`
`f(x) = x^2 + log_10(x)`
`x^2 \ text(is defined for all)\ x`
`log_10 x \ text(is defined for)\ \ x > 0`
`:. \ text(Domain)\ f(x): x>0`
The domain of the function `f (x) = log_2 (2x + 1)` is
`C`
`text(Domain exists for:)`
`2x + 1` | `> 0` |
`2x` | `> -1` |
`x` | `> -1/2` |
`=> C`
What is the domain of the function `f(x) = log_10(4-x)`?
`A`
`4-x` | `> 0` |
`-x` | `> -4` |
`x` | `< 4` |
`=> A`
State the domain and range of `y = -sqrt(12-x^2)`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`text(Domain:)\ -sqrt12<=x<= sqrt12`
`text(Range:)\ -sqrt12<=y<= 0`
`y = -sqrt(12-x^2)`
`12-x^2>=0\ \ =>\ \ x^2<=12`
`:.\ text(Domain:)\ -sqrt12<=x<= sqrt12`
`y_max =0`
`y_min = -sqrt12\ \ (text{when}\ x=0)`
`:.\ text(Range:)\ -sqrt12<=y<= 0`
A function has the equation `f(x)=(x-2)^2-5`.
State the range of `f(x)`. (2 marks)
`text{Range}\ f(x): \ y>=-5`
`text{Consider the function transformation:}`
`y=x^2\ \ text{translated 2 units to the right}\ \ =>\ \ y=(x-2)^2`
`y=(x-2)^2\ text{translated 5 units down}\ \ =>\ \ y=(x-2)^2-5`
`:.\ text{Range}\ f(x): \ y>=-5`
`f(x)` is defined by the equation `f(x)=3-x^2`.
--- 2 WORK AREA LINES (style=lined) ---
--- 3 WORK AREA LINES (style=lined) ---
i. `(sqrt3,0) and (-sqrt3, 0)`
ii. `text{Domain: all}\ x`
`text{Range}\ f(x): \ y<=3`
i. `xtext{-intercepts occur when}\ y=0`
`3-x^2` | `=0` | |
`x^2` | `=3` | |
`x` | `=+-sqrt3` |
`:. xtext{-intercepts at} (sqrt3,0) and (-sqrt3, 0)`
ii. `text{Domain: all}\ x`
`text{Find range}\ f(x):`
`x^2>=0\ text{for all}\ x \ \ => \ \ 3-x^2<=3\ text{for all}\ x`
`:.\ text{Range}\ f(x): \ y<=3`
`h(x)=x^3+3x^2+x-5`.
--- 1 WORK AREA LINES (style=lined) ---
--- 5 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
i. `text{Proof (See worked solutions)}`
ii. `h(x)=(x-1)(x^2+4x+5)`
iii. `text{Proof (See worked solutions)}`
`g(x)=(x-1)(x^2-2x+8)`.
Justify that `g(x)` only has one zero. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`text{Proof (See worked solutions)}`
`g(x)=(x-1)(x^2-2x+8)`.
`g(1)=0\ \ =>\ \ text{one zero at}\ \ x=1`
`text{Consider the roots of}\ \ y=x^2-2x+8`
`Δ = b^2-4ac=(-2)^2-4*1*8=-28<0`
`text{Since}\ \ Δ<0\ \ =>\ \ text{No zeros (roots)}`
`:. g(x)\ text{only has 1 zero}`
`P(x)` is a monic polynomial of degree 4.
The maximum number of zeros that `P(x)` can have is
`D`
`text{A polynomial of degree 4 has a leading term}\ ax^4`
`text{A monic polynomial of degree 4 has a leading term}\ x^4`
`:.\ text{Maximum number of zeroes}\ = 4`
`=>D`
Let `P(x) = x^3+5x^2+2x-8`.
--- 2 WORK AREA LINES (style=lined) ---
--- 5 WORK AREA LINES (style=lined) ---
Consider the polynomial `P(x) = 2x^4+3x^3-12x^2-7x+6`.
Fully factorised, `P(x) = (2x-1)(x+3)(x+a)(x-b)`
Find the value of `a` and `b` where `a,b>0`. (3 marks)
`a=1, b=2`
`text{Test for factors (by trial and error):}`
`P(1) = 2+3-12-7+6 = -8`
`P(-1) = 2-3-12+7+6 = 0\ \ =>\ \ (x+1)\ \ text{is a factor}`
`P(2) = 32+24-48-14+6 = 0\ \ =>\ \ (x-2)\ \ text{is a factor}`
`:. a=1, b=2`
Consider the polynomial `P(x) = 3x^3+x^2-10x-8`.
Is `(x+2)` a factor of `P(x)`? Justify your answer. (2 marks)
`P(-2) = -24+4+20-8=-8`
`:. (x+2)\ \ text(is not a factor of)\ P(x)`
`P(-2) = -24+4+20-8=-8`
`:. (x+2)\ \ text(is not a factor of)\ P(x)`
Consider the polynomial `P(x) = 2x^3-7x^2-7x+12`.
--- 2 WORK AREA LINES (style=lined) ---
--- 5 WORK AREA LINES (style=lined) ---
Consider the polynomial `P(x) = x^3-4x^2+x+6`.
--- 1 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
i. `P(-1) = -1-4-1+6 = 0`
`:. x=-1\ \ text(is a zero)`
ii. `text{Using part (i)} \ => (x+1)\ text{is a factor of}\ P(x)`
`P(x) = (x+1)*Q(x)`
`text(By long division:)`
`P(x)` | `= (x+1) (x^2-5x+6)` |
`= (x+1)(x-2)(x-3)` |
`:.\ text(Other zeroes are:)`
`x = 2 and x = 3`
Let `p(x)=x^{3}-2 a x^{2}+x-1`. When `p(x)` is divided by `(x+2)`, the remainder is 5.
Find the value of `a`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`-2`
`text{Since}\ \ p(x) -: (x+2)\ \ text{has a remainder of 5:}`
`P(-2)` | `=5` | |
`5` | `=(-2)^3-2a(-2)^2-2-1` | |
`5` | `=-8-8a-2-1` | |
`8a` | `=-16` | |
`:.a` | `=-2` |
If `x + a` is a factor of `8x^3-14x^2-a^2 x`, then the value of `a` is
`D`
`f(-a)` | `= 8(-a)^3-14(-a)^2-a^2(-a)` |
`0` | `= -8a^3-14a^2 + a^3` |
`0` | `= -7a^3-14a^2` |
`0` | `= -7a^2 (a + 2)` |
`a` | `= -2` |
`=>D`
If `P(x)=3x^3+2x^2-4x+2`, evaluate `P(-1)`. (1 mark)
--- 3 WORK AREA LINES (style=lined) ---
`5`
`P(x)` | `=3x^3+2x^2-4x+2` | |
`P(2)` | `=3(-1)^3+2(-1)^2-4(-1)+2` | |
`=-3+2+4+2` | ||
`=5` |
If `P(x)=2x^3+x^2-4x+5`, evaluate `P(2)`. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
`17`
`P(x)` | `=2x^3+x^2-4x+5` | |
`P(2)` | `=2xx2^3+2^2-4xx2+5` | |
`=16+4-8+5` | ||
`=17` |
Solve `log_9 27=x` for `x`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`3/2`
`log_9 27` | `=x` | |
`9^x` | `=27\ \ text{(by log definition)}` | |
`log_10 9^x` | `=log_10 27` | |
`x log_10 9` | `=log_10 27` | |
`x` | `=(log_10 27)/(log_10 9)` | |
`=3/2` |
Solve `log_16 2=x` for `x`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`0.25`
`log_16 2` | `=x` | |
`16^x` | `=2` | |
`log_10 16^x` | `=log_10 2` | |
`x log_10 16` | `=log_10 2` | |
`x` | `=(log_10 2)/(log_10 16)` | |
`=0.25` |
Solve `4^(x-1)=84` for `x`, giving your answer correct to 1 decimal place. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`4.2`
`4^(x-1)` | `=84` | |
`log_10 4^(x-1)` | `=log_10 84` | |
`(x-1)log_10 4` | `=log_10 84` | |
`x-1` | `=(log_10 84)/(log_10 4)` | |
`x` | `=(log_10 84)/(log_10 4)+1` | |
`=4.196…` | ||
`=4.2\ text{(to 1 d.p.)}` |
Solve `3^a=28` for `a`, giving your answer correct to 2 decimal places. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`3.03`
`3^a` | `=28` | |
`log_10 3^a` | `=log_10 28` | |
`a xx log_10 3` | `=log_10 28` | |
`a` | `=(log_10 28)/(log_10 3)` | |
`=3.033…` | ||
`=3.03` |
Solve `4^(x+1)=32` for `x`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`3/2`
`4^(x+1)` | `=32` | |
`log_10 4^(x+1)` | `=log_10 32` | |
`(x+1) log_10 4` | `=log_10 32` | |
`x+1` | `=(log_10 32)/(log_10 4)` | |
`x` | `=5/2-1` | |
`=3/2` |
Solve `2^t=16` . (2 marks)
`4`
`2^t` | `=16` | |
`log_10 2^t` | `=log_10 16` | |
`t xx log_10 2` | `=log_10 16` | |
`t` | `=(log_10 16)/(log_10 2)` | |
`=4` |