Polynomials, SMB-014
Polynomials, SMB-013
Polynomials, SMB-012
`h(x)=x^3+3x^2+x-5`.
- Show `h(1)=0` (1 mark)
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- Express `h(x)` in the form `h(x)=(x-1)*g(x)` where `g(x)` is a quadratic factor. (2 marks)
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- Justify that `h(x)` only has one zero. (2 marks)
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Polynomials, SMB-011
`g(x)=(x-1)(x^2-2x+8)`.
Justify that `g(x)` only has one zero. (2 marks)
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Polynomials, SMB-010 MC
`P(x)` is a monic polynomial of degree 4.
The maximum number of zeros that `P(x)` can have is
- `0`
- `1`
- `3`
- `4`
Polynomials, SMB-009
Let `P(x) = x^3+5x^2+2x-8`.
- Show that `P(-2) = 0`. (1 mark)
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- Hence, factor the polynomial `P(x)` as `A(x)B(x)`, where `B(x)` is a quadratic polynomial. (2 marks)
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Polynomials, SMB-008
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)
Polynomials, SMB-007
Consider the polynomial `P(x) = 3x^3+x^2-10x-8`.
Is `(x+2)` a factor of `P(x)`? Justify your answer. (2 marks)
Polynomials, SMB-006
Consider the polynomial `P(x) = 2x^3-7x^2-7x+12`.
- Show that `(x-1)` is a factor of `P(x)`. (1 mark)
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- Fully factorise `P(x)`. (2 marks)
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Polynomials, SMB-005
Consider the polynomial `P(x) = x^3-4x^2+x+6`.
- Show that `x = -1` is a zero of `P(x)`. (1 mark)
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- Find the other zeros. (2 marks)
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Polynomials, SMB-004
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)
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Polynomials, SMB-003 MC
If `x + a` is a factor of `8x^3-14x^2-a^2 x`, then the value of `a` is
- 7
- 4
- 1
- –2
Polynomial, SMB-002
If `P(x)=3x^3+2x^2-4x+2`, evaluate `P(-1)`. (1 mark)
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Polynomials, SMB-001
If `P(x)=2x^3+x^2-4x+5`, evaluate `P(2)`. (1 mark)
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Logarithm, SMB-024
Solve `log_9 27=x` for `x`. (2 marks)
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Logarithm, SMB-023
Solve `log_16 2=x` for `x`. (2 marks)
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Logarithm, SMB-022
Solve `4^(x-1)=84` for `x`, giving your answer correct to 1 decimal place. (2 marks)
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Logarithm, SMB-021
Solve `3^a=28` for `a`, giving your answer correct to 2 decimal places. (2 marks)
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Logarithm, SMB-020
Solve `4^(x+1)=32` for `x`. (2 marks)
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Logarithm, SMB-019
Solve `2^t=16` . (2 marks)
Logarithm, SMB-018
Evaluate `log_a 6` given `log_a 2=0.62` and `log_a 24=2.67`. (2 marks)
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Logarithm, SMB-017
Evaluate `log_b 2` given `log_b 6=1.47` and `log_b 12=2.18`. (2 marks)
Logarithm, SMB-016
Evaluate `log_c 12` given `log_c 3=1.02` and `log_c 4=1.35`. (2 marks)
Logarithms, SMB-015
Evaluate `log_x 20` given `log_x 2=0.458` and `log_x 5=0.726`. (2 marks)
Logarithms, SMB-014
Evaluate `log_a 15` given `log_a 3=0.378` and `log_a 5=0.591`. (2 marks)
Logarithms, SMB-013
Evaluate `log_a 18` given `log_a 2=0.431` and `log_a 3=0.683`. (2 marks)
Logarithm, SMB-012
Solve the equation `log_9 x=-3/2`. (2 marks)
Logarithms, SMB-011
Solve the equation `log_4 x=3/2`. (2 marks)
Logarithms, SMB-010
Solve the equation `2 log_2(x + 5)-log_2(x + 9) = 1`. (3 marks)
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Logarithms, SMB-009
What is the solution to the equation `log_3(a-1) = -2`? (2 marks)
L&E, 2ADV E1 2008 HSC 7a
Solve `log_2 x-3/log_2 x=2` (3 marks)
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Logarithms, SMB-008
What is the solution to the equation `log_3 x = -1`? (1 mark)
Logarithms, SMB-007
Use the change of base formula to evaluate `log_7 13`, correct to two decimal places. (1 mark)
Logarithms, SMB-006 MC
The expression
`log_c(a) + log_a(b) + log_b(c)`
is equal to
- `1/(log_c(a)) + 1/(log_a(b)) + 1/(log_b(c))`
- `1/(log_a(c)) + 1/(log_b(a)) + 1/(log_c(b))`
- `-1/(log_a(b))-1/(log_b(c))-1/(log_c(a))`
- `1/(log_a(a)) + 1/(log_b(b)) + 1/(log_c(c))`
Logarithms, SMB-005
Solve `log_3(t)-log_3(t^2-4) = -1` for `t`. (3 marks)
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Logarithms, SMB-004
Solve `log_2(6-x)-log_2(4-x) = 2` for `x`, where `x < 4`. (2 marks)
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Logarithms, SMB-003
Solve the equation `2 log_3(5)-log_3 (2) + log_3 (x) = 2` for `x.` (2 marks)
Logarithms, SMB-002
Solve the equation `log_3(3x + 5) + log_3(2) = 2`, for `x`. (2 marks)
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Logarithms, SMB-001 MC
It is given that `log_10 a = log_10 b-log_10 c`, where `a, b, c > 0.`
Which statement is true?
- `a = b-c`
- `a = b/c`
- `log_10 a = b/c`
- `log_10 a = (log_10 b)/(log_10 c)`
Numbers of Any Magnitude, SMB-026
The distance between the Earth and the moon is 384 712 303 metres.
How far is that to the nearest million? (1 mark)
Numbers of Any Magnitude, SMB-025 MC
Which is the correct expression for 96.4851 rounded to 2 decimal places?
- 96.00
- 96.48
- 96.49
- 96.50
Numbers of Any Magnitude, SMB-024 MC
The first three days of the Brisbane cricket test had the following attendances:
\begin{array} {|l|c|}
\hline
\rule{0pt}{2.5ex}\text{Day 1}\rule[-1ex]{0pt}{0pt} & 20\ 156\\
\hline
\rule{0pt}{2.5ex}\text{Day 2}\rule[-1ex]{0pt}{0pt} & 18\ 397\\
\hline
\rule{0pt}{2.5ex}\text{Day 3}\rule[-1ex]{0pt}{0pt} & 29\ 981\\
\hline
\end{array}
What was the total crowd over the first 3 days, to the nearest 1000?
- `68\ 000`
- `69\ 000`
- `70\ 000`
- `71\ 000`
Numbers of Any Magnitude, SMB-023
A red blood cell and a white blood cell have a combined total mass of `4.2 xx 10^-11` grams.
If the red blood cell has a mass of `2 xx 10^-12` grams, what is the mass of the white blood cell? (2 marks)
Numbers of Any Magnitude, SMB-022 MC
The age of the Universe is estimated to be 14 000 000 000 years old.
This number can also be written in scientific notation as
- `1.4 xx 10^9`
- `1.4 xx 10^10`
- `14 xx 10^10`
- `14 xx 10^11`
Numbers of Any Magnitude, SMB-021
An insect expert estimates that an anthill 30 cm high contains `10^8` ants.
He also estimates that a 25 cm high anthill contains `10^7` ants.
If his estimations are correct, how many more ants live in the 30 cm high anthill than the 25 cm high one? (2 marks)
Numbers of Any Magnitude, SMB-020
The diameter of a pinhead is 0.0023 mm.
What is this measurement written in scientific notation?
Numbers of Any Magnitude, SMB-019 MC
The diameter of the earth is approximately 13 000 kilometres.
Which of these shows 13 000 in scientific notation?
- `1.3 xx 10^3`
- `1.3 xx 10^4`
- `1.3 xx 10^5`
- `1.3 xx 10^6`
Numbers of Any Magnitude, SMB-018 MC
One billion is one thousand million.
Which of the following is 300 billion?
- `3.0 xx 10^14`
- `3.0 xx 10^11`
- `3.0 xx 10^8`
- `3.0 xx 10^7`
Numbers of Any Magnitude, SMB-017
A tennis racquet length is measured at 68.58 centimetres.
Express this measurement in metres, rounded to two decimal places. (2 marks)
Numbers of Any Magnitude, SMB-016
A solution of acid is measures 12 982 millilitres.
Express this measurement in litres, correct to one decimal place. (2 marks)
Numbers of Any Magnitude, SMB-015
The width of red blood cell is 8 nanometres or `8 xx 10^{-9}` metres.
How many red blood cells, lined up side by side in straight line, would it take to form a 1 cm distance. (2 marks)
Numbers of Any Magnitude, SMB-014
The moon is 384 400 kilometres from the Earth.
Express this distance in scientific notation. (1 mark)
Measurement, STD2 M1 2009 HSC 25b
The mass of a sample of microbes is 50 mg. There are approximately `2.5 × 10^6` microbes in the sample.
In scientific notation, what is the approximate mass in grams of one microbe? (2 marks)
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Numbers of Any Magnitude, SMB-013
What is 0.000709 expressed in scientific notation? (1 mark)
Numbers of Any Magnitude, SMB-012 MC
What is 5.6582 rounded to two decimal places?
- `5.6`
- `5.7`
- `5.65`
- `5.66`
Numbers of Any Magnitude, SMB-011 MC
What is 0.002073 expressed in scientific notation with two significant figures?
- `2.07 xx 10^(-2)`
- `2.1 xx 10^(-2)`
- `2.07 xx 10^(-3)`
- `2.1 xx 10^(-3)`
Numbers of Any Magnitude, SMB-010 MC
What is 5.4782 correct to two significant figures?
- 5.0
- 5.5
- 5.47
- 5.48
Numbers of Any Magnitude, SMB-009
Noel's suitcase is weighed before his plane flight at 14 kilograms, to the nearest kilogram.
What is the absolute error of this measurement? (1 mark)
Numbers of Any Magnitude, SMB-008 MC
A person's height is measured as 1.7 metres.
What is the absolute error of this measurement?
- 1 centimetre
- 5 centimetres
- 10 centimetres
- 50 centimetres
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