VCE Math Methods Units 3 & 4 2004
Trial Examination 1 Solutions
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Part I
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9D / C / E / D / A / C / A / D / B
10 / 11 / 12 / 13 / 14 / 15 / 16 / 17 / 18
D / D / E / B / A / D / C / A / E
19 / 20 / 21 / 22 / 23 / 24 / 25 / 26 / 27
B / A / D / B / B / E / B / E / C
(1) The cubic function has a stationary inflection point and it is a one-to-one function, it intersects the x-axis once. D.
(2) The x-intercepts are at , hence . Possibly , hence or . C.
(3) implies , if . Since , . Hence or . E.
(4) The function is the reflection of in the x-axis, . It is a translation to the left of , . D.
(5) The graph of the inverse is the reflection of the function in the line and because the function has as its line of symmetry,
the inverse is the function itself. A.
(6) From x = 9.5 to 9.9, the y-values increase rapidly. This indicates asymptotic behaviour. The one that shows this behaviour is C.
(7) Rewrite the equation in the form and consider the graph from . This section shows a cosine graph with an amplitude of and a period of 2. and , i.e. . It is translated upwards by units , and to the left by 1 unit , i.e. . A.
(8) The solutions of are the negatives of the solutions of for . Hence the sum of the solutions of the two equations must be zero. D.
(9) is undefined at the end point. is positive between the end point and .
at . B.
(10)
. D.
(11), , apply the product rule,
. D.
(12) Use graphics calculator, enter , graph, 2nd calc, , at. –0.7259 E.
(13)
4
1
Using left rectangles,
area
Exact area .
overestimated by . B.
(14)
A.
(15)
(let c = 1). D.
(16), ,
, or . C.
(17) B, C, D and E are many-to-one functions and their inverses are not functions. A.
(18) The term containing the term is which can be simplified to . E.
(19) The shaded region consists of the area under the curve minus the area bounded by the curve and the line . The former is and the latter is .
B.
(20) Since , for B, D and E , and for C when , but when , B, C, D and E are out. A.
(21) Only two dice are rolled, the possible number of odd numbers is 0, 1 or 2. , ,
. D.
(22) The 95% interval is .
,
. Since the values of X are whole numbers, the next whole number greater than 82.35 is 83 and the whole number just less than 107.65 is 107. B.
(23) The experiment has a binomial distribution for the random variable-number of tails. The mean is and the number of trial is . Since where p is the probability of getting a tail in a toss, and . B.
(24) Since the mean for a binomial distribution, increases as n increases, and the graph also tends towards bell-shape. E.
(25) A random variable in sampling without replacement from a small population has a hypergeometric distribution. A is definitely binomial, C and D can be considered as binomial, E has a normal distribution. B.
(26) measures the central position of X and the spread of X, and . E.
(27)
0.37 0.37
0.13 0.13
X
5 2 – 5
C.
Part II, next page
Part II
(1) a.
b.
OR
c., y-intercept is which is also the turning point, x-intercepts are and .
is the translation of to the left by 2 units, y-intercept is which is also one of the x-intercepts, the other one is . is the turning point.
y
f(x+2) f(x)
–4 –2 0 2 x
–16
(2) Solve and simultaneously, ,
, , where , i.e. or .
When , , when , . The intersections are and .
(3) a., , , av rate .
b. Instant. rate . Let
, where . Squaring both sides, , , .
(4) Let , and .
.
Given and
, substitute in to obtain
(5) a. Equation of the inverse of is , or , .
b. y = x
y
f(x) g(x)
0 x
, .
y-intercept: , ,
hence x-intercept: .
has asymptote x = 0, for , y = 0.
Use graphics calculator to find the coordinates of the intersections, they are and .
c. The area bounded by and is twice the area bounded by y = x and ,
= 4.503 square units.
(6) a. Graphics calculator:
b.
Graphics calculator:
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