Evaluate `log_a 6` given `log_a 2=0.62` and `log_a 24=2.67`. (2 marks)
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Evaluate `log_a 6` given `log_a 2=0.62` and `log_a 24=2.67`. (2 marks)
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`1.43`
`log_a 6` | `=log_a (24/4)` | |
`=log_a 24-log_b 2^2` | ||
`=log_a 24-2log_a 2` | ||
`=2.67-2 xx 0.62` | ||
`=1.43` |
Evaluate `log_b 2` given `log_b 6=1.47` and `log_b 12=2.18`. (2 marks)
`0.71`
`log_b 2` | `=log_b (12/6)` | |
`=log_b 12-log_b 6` | ||
`=2.18-1.47` | ||
`=0.71` |
Evaluate `log_c 12` given `log_c 3=1.02` and `log_c 4=1.35`. (2 marks)
`2.37`
`log_c 12` | `=log_c (3xx4)` | |
`=log_c 3+log_c 4` | ||
`=1.02+1.35` | ||
`=2.37` |
Evaluate `log_x 20` given `log_x 2=0.458` and `log_x 5=0.726`. (2 marks)
`1.642`
`log_x 20` | `=log_x (4xx5)` | |
`=log_x (2^2xx 5)` | ||
`=2log_x 2+log_x 5` | ||
`=2 xx 0.458 + 0.726` | ||
`=1.642` |
Evaluate `log_a 15` given `log_a 3=0.378` and `log_a 5=0.591`. (2 marks)
`0.969`
`log_a 15` | `=log_a (3xx5)` | |
`=log_a 3+log_a 5` | ||
`=0.378 + 0.591` | ||
`=0.969` |
Evaluate `log_a 18` given `log_a 2=0.431` and `log_a 3=0.683`. (2 marks)
`1.797`
`log_a 18` | `=log_a (3^2xx2)` | |
`=log_a 3^2+log_a 2` | ||
`=2log_a 3+log_a 2` | ||
`=2 xx 0.683 + 0.431` | ||
`=1.797` |
Solve the equation `log_9 x=-3/2`. (2 marks)
`x = 1/27`
`log_9 x` | `=-3/2` | |
`x` | `=9^(-3/2)` | |
`=1/(sqrt9)^3` | ||
`=1/27` |
Solve the equation `log_4 x=3/2`. (2 marks)
`x = 8`
`log_4 x` | `=3/2` | |
`x` | `=4^(3/2)` | |
`=8` |
Solve the equation `2 log_2(x + 5)-log_2(x + 9) = 1`. (3 marks)
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`x = text{−1}`
`2 log_2(x + 5)-log_2(x + 9)` | `= 1` |
`log_2(x + 5)^2-log_2(x + 9)` | `= 1` |
`log_2(((x + 5)^2)/(x + 9))` | `= 1` |
`((x + 5)^2)/(x + 9)` | `= 2` |
`x^2 + 10x + 25` | `= 2x + 18` |
`x^2 + 8x + 7` | `= 0` |
`(x + 7)(x + 1)` | `= 0` |
`:. x = -1\ \ \ \ (x != text{−7}\ \ text(as)\ \ x > text{−5})`
What is the solution to the equation `log_3(a+1) = -2`? (2 marks)
`a=10/9`
`log_3 (a-1)` | `= -2` |
`a-1` | `= 3^{-2}` |
`a` | `=1/3^2+1` |
`= 10/9` |
Solve `log_2 x-3/log_2 x=2` (3 marks)
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`x=8\ \ text(or)\ \ -1/2`
`log_2 x-3/(log_2 x)` | `=2` |
`(log_2 x)^2-3` | `=2log_2 x` |
`(log_2 x)^2-2log_2 x-3` | `=0` |
`text(Let)\ X=log_2 x` | |
`:.\ X^2-2X-3` | `=0` |
`(X-3)(X+1)` | `=0` |
`X` | `=3` | `\ \ \ \ \ \ \ \ \ \ ` | `X` | `=-1` |
`log_2 x` | `=3` | `\ \ \ \ \ \ \ \ \ \ ` | `log_2 x` | `=-1` |
`x` | `=2^3=8` | `\ \ \ \ \ \ \ \ \ \ ` | `x` | `=2^{-1}=1/2` |
`:.x=8\ \ text(or)\ \ -1/2`
What is the solution to the equation `log_3 x = -1`? (1 mark)
`x=1/3`
`log_3 x` | `=-1` |
`x` | `=3^{-1}` |
`=1/3` |
Use the change of base formula to evaluate `log_7 13`, correct to two decimal places. (1 mark)
`1.32\ \ text{(to 2 d.p.)}`
`log_7 13` | `= (log_10 13)/(log_10 7)` |
`= 1.3181…` | |
`= 1.32\ \ text{(to 2 d.p.)}` |
The expression
`log_c(a) + log_a(b) + log_b(c)`
is equal to
`B`
`text(Solution 1)`
`text(Using Change of Base:)`
`log_c(a) + log_a(b) + log_b(c)`
`=(log_a(a))/(log_a(c)) + (log_b(b))/(log_b(a)) + (log_c(c))/(log_c(b))`
`=1/(log_a(c)) + 1/(log_b(a)) + 1/(log_c(b))`
`=> B`
Solve `log_3(t)-log_3(t^2-4) = -1` for `t`. (3 marks)
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`4 `
`log_3(t)-log_3(t^2-4)` | `= -1` |
`log_3 ({t}/{t^2-4})` | `= -1` |
`(t)/(t^2-4)` | `= (1)/(3)` |
`t^2-4` | `= 3t` |
`t^2-3t – 4` | `= 0` |
`(t-4)(t+ 1)` | `= 0` |
`:. t=4 \ \ \ (t > 0, \ t!= –1)`
Solve `log_2(6-x)-log_2(4-x) = 2` for `x`, where `x < 4`. (2 marks)
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`10/3`
`text(Simplify using log laws:)`
`log_2((6-x)/(4-x))` | `= 2` |
`2^2` | `= (6-x)/(4-x)` |
`16-4x` | `= 6-x` |
`3x` | `= 10` |
`:. x` | `= 10/3` |
Solve the equation `2 log_3(5)-log_3 (2) + log_3 (x) = 2` for `x.` (2 marks)
`18/25`
`log_3 (5)^2-log_3 (2) + log_3 (x)` | `= 2` |
`log_3 (25x)-log_3 (2)` | `=2` |
`log_3 ((25 x)/2)` | `= 2` |
`(25x)/2` | `= 3^2` |
`:. x` | `= 18/25` |
Solve the equation `log_3(3x + 5) + log_3(2) = 2`, for `x`. (2 marks)
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`x =-1/6`
`text(Simplify using log laws:)`
`log_3(2(3x + 5))` | `=2` |
`log_3(6x + 10)` | `=2` |
`6x +10` | `=9` |
`6x` | `= -1` |
`x` | `=-1/6` |
It is given that `log_10 a = log_10 b-log_10 c`, where `a, b, c > 0.`
Which statement is true?
`B`
`log_10 a` | `= log_10 b-log_10 c` |
`log_10 a` | `= log_10 (b/c)` |
`:. a` | `= b/c` |
`=> B`
A student believes that the time it takes for an ice cube to melt (`M` minutes) varies inversely with the room temperature `(T^@ text{C})`. The student observes that at a room temperature of `15^@text{C}` it takes 12 minutes for an ice cube to melt.
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\begin{array} {|c|c|c|c|}
\hline \ \ T\ \ & \ \ 5\ \ & \ 15\ & \ 30\ \\
\hline M & & & \\
\hline \end{array}
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a. `M=180/T`
b.
\begin{array} {|c|c|c|c|}
\hline \ \ T\ \ & \ \ 5\ \ & \ 15\ & \ 30\ \\
\hline M & 36 & 12 & 6 \\
\hline \end{array}
a. | `M` | `prop 1/T` |
`M` | `=k/T` | |
`12` | `=k/15` | |
`k` | `=15 xx 12` | |
`=180` |
`:.M=180/T`
b.
\begin{array} {|c|c|c|c|}
\hline \ \ T\ \ & \ \ 5\ \ & \ 15\ & \ 30\ \\
\hline M & 36 & 12 & 6 \\
\hline \end{array}
What is the remainder when `P(x) = -x^3-2x^2-3x + 8` is divided by `x + 2`?
`D`
`P(-2)` | `= -(-2)^3-2(-2)^2-3(-2) + 8` |
`= 8-8 + 6 + 8` | |
`= 14` |
`=> D`
The graph of `y = f(x)` is shown.
Which of the following could be the equation of this graph?
`C`
`text(By elimination:)`
`text(A single negative root occurs when)\ \ x =–1`
`->\ text(Eliminate A and D)`
`text(When)\ \ x = 0, \ y > 0`
`->\ text(Eliminate B)`
`=> C`
Let `P(x) = x^3 + 3x^2-13x + 6`.
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Which inequality gives the domain of `y = sqrt(2x-3)`?
`D`
`text(Domain exists when:)`
`2x-3` | `>= 0` |
`2x` | `>= 3` |
`x` | `>= 3/2` |
`=>D`
The relationship between British pounds `(p)` and Australian dollars `(d)` on a particular day is shown in the graph.
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Convert 93 100 Japanese yen to British pounds. (2 marks)
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a. `m = text(rise)/text(run) = 4/7`
`p = 4/7 d`
b. `text(Yen to Australian dollars:)`
`y` | `=76d` |
`93\ 100` | `= 76d` |
`d` | `= (93\ 100)/76` |
`= 1225` |
`text(Aust dollars to pounds:)`
`p` | `= 4/7 xx 1225` |
`= 700\ text(pounds)` |
`:. 93\ 100\ text(Yen = 700 pounds)`
The time taken for a car to travel between two towns at a constant speed varies inversely with its speed.
It takes 1.5 hours for the car to travel between the two towns at a constant speed of 80 km/h.
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Which of the following is equal to `(log_2 9)/(log_2 3)`?
`A`
`(log_2 9)/(log_2 3)` | `= (log_2 3^2)/(log_2 3)` |
`= (2 log_2 3)/(log_2 3)` | |
`= 2` |
`=> A`
Two secants from the point `C` intersect a circle as shown in the diagram.
What is the value of `x`? (2 marks)
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`4`
`text(Using formula for intercepts of intersecting secants:)`
`x (x + 2)` | `= 3 (3 + 5)` |
`x^2 + 2x` | `= 24` |
`x^2 + 2x – 24` | `= 0` |
`(x + 6) (x – 4)` | `= 0` |
`:. x` | `= 4 \ \ \ (x > 0)` |
Consider the polynomial `P(x) = x^3-2x^2-5x + 6`.
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i. `P(1) = 1-2-5 + 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-x-6)` |
`= (x-1) (x-3) (x + 2)` |
`:.\ text(Other zeroes are:)`
`x = -2 and x = 3`
Which polynomial is a factor of `x^3-5x^2 + 11x-10`?
`A`
`f(2)` | `= 2^3-5*2^2 + 11*2-10` |
`= 8-20 + 22 – 10` | |
`= 0` |
`:. (x-2)\ text(is a factor)`
`⇒ A`
Find the domain of the function `f(x) = sqrt (3-x)`. (2 marks)
`x <= 3 or (-oo,3].`
`text(Domain of)\ \ f(x) = sqrt (3-x)`
`3-x` | `>= 0` |
`x` | `<= 3` |
`text(Note domain can also be expressed as:)\ \ (-oo,3]`
The region enclosed by `y = 4 - x,\ \ y = x` and `y = 2x + 1` is shaded in the diagram below.
Which of the following defines the shaded region?
(A) | `y <= 2x + 1, qquad` | `y <= 4-x, qquad` | `y >= x` |
(B) | `y >= 2x + 1, qquad` | `y <= 4-x, qquad` | `y >= x` |
(C) | `y <= 2x + 1, qquad` | `y >= 4-x, qquad` | `y >= x` |
(D) | `y >= 2x + 1, qquad` | `y >= 4-x, qquad` | `y >= x` |
`A`
`text(Consider)\ \ y = 2x + 1,`
`text(Shading is below graph)`
`=> y <= 2x + 1`
`text(Consider)\ \ y = 4-x,`
`text(Shading is below graph)`
`=> y <= 4-x`
`=> A`
What is the remainder when `2x^3-10x^2 + 6x + 2` is divided by `x-2`?
`B`
`P(2)` | `= 2 · 2^3-10 · 2^2 + 6 · 2 + 2` |
`= -10` |
`=> B`
Write `log 2 + log 4 + log 8 + … + log 512` in the form `a log b` where `a` and `b` are integers greater than `1.` (2 marks)
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`45 log 2`
`log 2 + log 4 + log 8 + … + log 512`
`= log 2^1 + log 2^2 + log2^3 + … + log 2^9`
`= log 2 + 2 log 2 + 3 log 2 + … + 9 log 2`
`= 45 log 2`
The polynomial `P(x) = x^2 + ax + b` has a zero at `x = 2`. When `P(x)` is divided by `x + 1`, the remainder is `18`.
Find the values of `a` and `b`. (3 marks)
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`a = -7\ \ text(and)\ \ b = 10`
`P(x) = x^2 + ax + b`
`text(S)text(ince there is a zero at)\ \ x = 2,`
`P(2)` | `=0` | |
`2^2 + 2a + b` | `= 0` | |
`2a + b` | `= -4` | `…\ (1)` |
`P(-1) = 18,`
`(-1)^2-a + b` | `= 18` | |
`-a + b` | `= 17` | `…\ (2)` |
`text(Subtract)\ \ (1)-(2),`
`3a` | `= -21` |
`a` | `= -7` |
`text(Substitute)\ \ a = -7\ \ text{into (1),}`
`2(-7) + b` | `= -4` |
`b` | `= 10` |
`:.a = -7\ \ text(and)\ \ b = 10`
Consider the polynomials `P(x) = x^3-kx^2 + 5x + 12` and `A(x) = x - 3`.
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i. | `P(x)` | `= x^3-kx^2 + 5x + 12` |
`A(x)` | `= x-3` |
`text(If)\ P(x)\ text(is divisible by)\ A(x)\ \ =>\ \ P(3) = 0`
`3^3-k(3^2) + 5 xx 3 + 12` | `= 0` |
`27-9k + 15 + 12` | `= 0` |
`9k` | `= 54` |
`:.k` | `= 6\ \ …\ text(as required)` |
ii. `text(Find all roots of)\ P(x)`
`P(x)=(x-3)*Q(x)`
`text{Using long division to find}\ Q(x):`
`:.P(x)` | `= x^3-6x^2 + 5x + 12` |
`= (x-3)(x^2-3x − 4)` | |
`= (x-3)(x-4)(x + 1)` |
`:.\ text(Zeros at)\ \ \ x = -1, 3, 4`
Two secants from the point `P` intersect a circle as shown in the diagram.
What is the value of `x`?
`B`
`text{Property: products of intercepts of secants from external point are equal}`
`x(x + 3)` | `= 4(4 + 6)` |
`x^2 + 3x` | `= 40` |
`x^2 + 3x-40` | `= 0` |
`(x-5)(x + 8)` | `= 0` |
`:.x = 5,\ \ (x>0)`
`=>B`
What is the remainder when `x^3-6x` is divided by `x + 3`?
`A`
`text(Remainder)` | `= P(-3)` |
`= (-3)^3-6(-3)` | |
`= -27 + 18` | |
`= -9` |
`=> A`
Use the change of base formula to evaluate `log_3 7`, correct to two decimal places. (1 mark)
`1.77\ \ text{(to 2 d.p.)}`
`log_3 7` | `= (log_10 7)/(log_10 3)` |
`= 1.771…` | |
`= 1.77\ \ text{(to 2 d.p.)}` |
Sandy travels to Europe via the USA. She uses this graph to calculate her currency conversions.
She converts all of her money to euros.How many euros does she have to spend in Europe? (3 marks)
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`text(A$, then 1 A$ will buy more euros than)`
`text(before and the gradient used to convert)`
`text{the currencies will steepen (increase).}`
i. `text(From graph:)`
`75\ text(US$)` | `=\ text(100 A$)` |
`=> 150\ text(US$)` | `=\ text(200 A$)` |
`:.\ text(Sandy has a total of 800 A$)`
`text(Converting A$ to €:)`
`text(100 A$)` | `= 60\ €` |
`:.\ text(800 A$)` | `= 8 xx 60` |
`= 480\ €` |
ii. `text(If the value of the euro falls against the)`
`text(A$, then 1 A$ will buy more euros than)`
`text(before and the gradient used to convert)`
`text{the currencies will steepen (increase).}`
If pressure (`p`) varies inversely with volume (`V`), which formula correctly expresses `p` in terms of `V` and `k`, where `k` is a constant?
`A`
`p prop 1/V`
`p = k/V`
`=> A`
The polynomial `x^3` is divided by `x + 3`. Calculate the remainder. (2 marks)
`-27`
`P(-3)` | `= (-3)^3` |
`= -27` |
`:.\ text(Remainder when)\ x^3 -: (x + 3) = -27`
The remainder when the polynomial `P(x) = x^4-8x^3-7x^2 + 3` is divided by `x^2 + x` is `ax + 3`.
What is the value of `a`?
`C`
`P(x) = x^4-8x^3-7x^2 + 3`
`text(Given)\ \ P(x)` | `= (x^2 + x) *Q(x) + ax + 3` |
`= x (x + 1) Q(x) + ax + 3` |
`P(-1) = 1 + 8-7 + 3 = 5`
`:. -a + 3` | `= 5` |
`a` | `= -2` |
`=> C`
The points \(A\), \(B\) and \(C\) lie on a circle with centre \(O\), as shown in the diagram.
The size of \(\angle ACB\) is 40°.
What is the size of \(\angle AOB\)?
\(D\)
\(\angle AOB = 2 \times 40 = 80^{\circ}\)
\(\text{(angles at centre and circumference on arc}\ AB\text{)}\)
\(\Rightarrow D\)
The polynomial `p(x) = x^3-ax + b` has a remainder of `2` when divided by `(x-1)` and a remainder of `5` when divided by `(x + 2)`.
Find the values of `a` and `b`. (3 marks)
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`a` | `= 4` |
`b` | `= 5` |
`p(x)` | `= x^3-ax + b` |
`P(1)` | `= 2` |
`1-a + b` | `= 2` |
`b` | `= a+1\ \ \ …\ text{(1)}` |
`P (-2)` | `= 5` |
`-8 + 2a + b` | `= 5` |
`2a + b` | `= 13\ \ \ …\ text{(2)}` |
`text(Substitute)\ \ b = a+1\ \ text(into)\ \ text{(2)}`
`2a + a+1` | `= 13` |
`3a` | `= 12` |
`:. a` | `= 4` |
`:. b` | `= 5` |
What is the solution to the equation `log_2(x-1) = 8`?
`D`
`log_2 (x-1)` | `= 8` |
`x-1` | `= 2^8` |
`x` | `= 257` |
`=> D`
The weight of an object on the moon varies directly with its weight on Earth. An astronaut who weighs 84 kg on Earth weighs only 14 kg on the moon.
A lunar landing craft weighs 2449 kg when on the moon. Calculate the weight of this landing craft when on Earth. (2 marks)
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`14\ 694\ text(kg)`
`W_text(moon) prop W_text(earth)`
`=> W_text(m) = k xx W_text(e)`
`text(Find)\ k\ text{given}\ W_text(e) = 84\ text{when}\ W_text(m) = 14`
`14` | `= k xx 84` |
`k` | `= 14/84 = 1/6` |
`text(If)\ W_text(m) = 2449\ text(kg),\ text(find)\ W_text(e):`
`2449` | `= 1/6 xx W_text(e)` |
`W_text(e)` | `= 14\ 694\ text(kg)` |
`:.\ text(Landing craft weighs)\ 14\ 694\ text(kg on earth)`
The polynomial `P(x) = x^3-4x^2-6x + k` has a factor `x-2`.
What is the value of `k`?
`C`
`P(x) = x^3-4x^2-6x + k`
`text(S)text(ince)\ \ (x-2)\ \ text(is a factor,)\ \ P(2) = 0`
`2^3-4*2^2-6*2 + k` | `= 0` |
`8-16-12 + k` | `= 0` |
`k` | `= 20` |
`=> C`
Let `P(x) = (x + 1)(x-3) Q(x) + ax + b`,
where `Q(x)` is a polynomial and `a` and `b` are real numbers.
The polynomial `P(x)` has a factor of `x-3`.
When `P(x)` is divided by `x + 1` the remainder is `8`.
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i. `P(x) = (x+1)(x-3)Q(x) + ax + b`
`(x-3)\ \ text{is a factor (given)}`
`:. P (3)` | `= 0` |
`3a + b` | `= 0\ \ \ …\ text{(1)}` |
`P(x) ÷ (x+1)=8\ \ \ text{(given)}`
`:.P(-1)` | `= 8` |
`-a + b` | `= 8\ \ \ …\ text{(2)}` |
`text{Subtract (1) – (2)}`
`4a` | `= -8` |
`a` | `= -2` |
`text(Substitute)\ \ a = -2\ \ text{into (1)}`
`-6 + b` | `= 0` |
`b` | `= 6` |
`:. a= – 2, \ b=6`
ii. `P(x) -: (x + 1)(x-3)`
`= ((x+1)(x-3)Q(x)-2x + 6)/((x+1)(x-3))`
`= Q(x) + (-2x + 6)/((x+1)(x-3))`
`:.\ text(Remainder is)\ \ -2x + 6`
Let `P(x) = x^3-ax^2 + x` be a polynomial, where `a` is a real number.
When `P(x)` is divided by `x-3` the remainder is `12`.
Find the remainder when `P(x)` is divided by `x + 1`. (3 marks)
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`-4`
`P(x) = x^3 – ax^2 + x`
`text(S)text(ince)\ \ P(x) -: (x – 3)\ \ text(has remainder 12,)`
`P(3) = 3^3-a xx 3^2 + 3` | `=12` |
`27-9a + 3` | `= 12` |
`9a` | `= 18` |
`a` | `=2` |
`:.\ P(x) = x^3-2x^2 + x`
`text(Remainder)\ \ P(x) -: (x + 1)\ \ text(is)\ \ P(–1)`
`P(-1)` | `= (-1)^3-2(-1)^2-1` |
`= – 4` |
The points `A`, `B` and `P` lie on a circle centred at `O`. The tangents to the circle at `A` and `B` meet at the point `T`, and `/_ATB = theta`.
What is `/_APB` in terms of `theta`?
`B`
`/_ BOA= 2 xx /_ APB`
`text{(angles at centre and circumference on arc}\ AB text{)}`
`/_TAO = /_ TBO = 90^@\ text{(angle between radius and tangent)}`
`:.\ theta + /_BOA` | `= 180^@\ text{(angle sum of quadrilateral}\ TAOB text{)}` |
`theta + 2 xx /_APB` | `= 180^@` |
`2 xx /_APB` | `= 180^@-theta` |
`/_APB` | `= 90^@-theta/2` |
`=> B`
When the polynomial `P(x)` is divided by `(x + 1)(x-3)`, the remainder is `2x + 7`.
What is the remainder when `P(x)` is divided by `x-3`?
`D`
`text(Let)\ \ P(x) =A(x) * Q(x) + R(x)`
`text(where)\ \ A(x) = (x + 1)(x-3),\ text(and)\ \ R(x)=2x+7`
`text(When)\ \ P(x) -: (x-3),\ text(remainder) = P(3)`
`P(3)` | `= 0 + R(3)` |
`= (2 xx 3) + 7` | |
`= 13` |
`=> D`
Let `f(x) = sqrt(x-8)`. What is the domain of `f(x)`? (1 mark)
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`x >= 8`
`f(x) = sqrt(x-8)`
`text(Domain exists for:)`
`(x-8)` | `>= 0` |
`x` | `>= 8` |
Which inequality defines the domain of the function `f(x) = 1/sqrt(x+3)` ?
`A`
`text(Given)\ f(x) = 1/sqrt(x+3)`
`(x + 3)` | `> 0` |
`x` | `> -3` |
`:.\ text(The domain of)\ f(x)\ text(is)\ \ \ f(x)> -3`
`=> A`
The air pressure, `P`, in a bubble varies inversely with the volume, `V`, of the bubble.
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By finding the value of the constant, `a`, find the value of `P` when `V = 4`. (2 marks)
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Use the horizontal axis to represent volume and the vertical axis to represent air pressure. (2 marks)
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i. | `P` | `prop 1/V` |
`= a/V` |
ii. | `text(When)\ P=3,\ V = 2` |
`3` | `= a/2` |
`a` | `=6` |
`text(Need to find)\ P\ text(when)\ V = 4`
`P` | `=6/4` |
`= 1 1/2` |
iii. |
The height above the ground, in metres, of a person’s eyes varies directly with the square of the distance, in kilometres, that the person can see to the horizon.
A person whose eyes are 1.6 m above the ground can see 4.5 km out to sea.
How high above the ground, in metres, would a person’s eyes need to be to see an island that is 15 km out to sea? Give your answer correct to one decimal place. (3 marks)
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`17.8\ text(m)\ \ text{(to 1 d.p.)}`
`h prop d^2`
`h=kd^2`
`text(When)\ h = 1.6,\ d = 4.5`
`1.6` | `= k xx 4.5^2` |
`:. k` | `= 1.6/4.5^2` |
`= 0.07901` `…` |
`text(Find)\ h\ text(when)\ d = 15`
`h` | `= 0.07901… xx 15^2` |
`= 17.777…` | |
`= 17.8\ text(m)\ \ \ text{(to 1 d.p.)}` |
The time for a car to travel a certain distance varies inversely with its speed.
Which of the following graphs shows this relationship?
`A`
`T` | `prop 1/S` |
`T` | `= k/S` |
`text{By elimination:}`
`text(As S) uarr text(, T) darr => text(cannot be B or D)`
`text(C is incorrect because it graphs a linear relationship)`
`=> A`
The number of hours that it takes for a block of ice to melt varies inversely with the temperature. At 30°C it takes 8 hours for a block of ice to melt.
How long will it take the same size block of ice to melt at 12°C?
`B`
`text{Time to melt}\ (T) prop1/text(Temp) \ => \ T=k/text(Temp)`
`text(When) \ T=8, text(Temp = 30)`
`8` | `=k/30` |
`k` | `=240` |
`text{Find}\ T\ text{when Temp = 12:}`
`T` | `=240/12` |
`=20\ text(hours)` |
`=> B`
Conversion graphs can be used to convert from one currency to another.
Sarah converted 60 Australian dollars into Euros. She then converted all of these Euros
into New Zealand dollars.
How much money, in New Zealand dollars, should Sarah have?
`C`
`text(Using the graphs)`
`$60\ text(Australian)` | `=46\ text(Euro)` |
`46 \ text(Euro)` | `=$78\ text(New Zealand)` |
`=> C`