Solution to Question 1Rewrite the logarithmic equations using exponential equation
x- 3 / 2 = 1 / 8
The above equation may be rewritten as
x - 3 / 2 = 1 / 23 = 2-3
(√x)-3 = 2-3
The above equation gives
√x = 2 or x = 4
Solution to Question 2
20 % of 2 = (20 / 100) * 2 = 0.4
Solution to Question 3If Log 4 (x) = 12, then
x = 412
x / 4 = 411
log 2 (x / 4) = log 2 (411)
= log 2 ((22)11)
= log 2 (222)
Solution to Question 4A population of size P increasing at the rate of 2% may be modelled as follows
P = P0 e0.02 t , where t is the number of years after t = 0 and P0 is the population at t = 0.
Let t = 0 corresponds to January 1, 2000 and therefore t = 4 corresponds to December 31. But P is 2,000,000 when t = 4. Hence
2,000,000 = P0 e0.02*4
Solve the above for P0
P0 = 2,000,000 / e0.02*4 = 1 846 000 (rounded to the nearest thousand)
P0 is the population at t = 0 or on January 1, 2000.
Solution to Question 5If the graph of f is a parabola with vertex on x axis and opening up, then the range of f is given by the interval
[0 , + infinity)
The graph of f(x - 5) is that of f(x) shifted 5 units to the right and therefore no change to the range. However since the graph of - f(x - 5) is that of f(x - 5) reflected on the x axis then the range of - f(x - 5) is given by
(- infinity , 0]
The graph g(x) = 2 - f(x - 5) is that of - f(x - 5) shifted up by 2 units; hence the range of g(x) is given by
(- infinity , 2]
Solution to Question 6Note that since f(x) < 0 then
g(x) = | f(x) | = - f(x)
The graph of g(x) = - f(x) is therefore the reflection of the graph of f on the x axis.
Solution to Question 7We first solve 2y - 6 = - 4 f(x - 3) for y.
y = - 2 f(x - 3) + 3
The graph of y = - 2 f(x - 3) + 3 is that of y = f(x) shifted 3 units to the right, stretched vertically by a factor of 2, reflected on the x axis and shifted up by 3 units. A point of y = f(x) will undergo the same transforamtions. Hence
Point (a , b) on the graph of y = f(x)
Becomes (a + 3 , b) on the graph of y = f(x - 3) : shifted 3 units to the right
Becomes ( a + 3 , 2 b) on the graph of y = 2 f(x - 3) : stretched vertically by 2
Becomes ( a + 3 , - 2 b) on the graph of y = - 2 f(x - 3): reflected on x axis
Becomes ( a + 3 , - 2 b + 3) on the graph of y = - 2 f(x - 3) + 3 : shifted up 3 units
Solution to Question 8First rewrite y - 2x 2 = 8 x + 5 as
y = 2x 2 + 8 x + 5
Complete square and determine vertex.
y = 2(x 2 + 4x + 4) - 8 + 5
= 2(x + 2) 2 - 3
vertex at (- 2 , - 3)
If parabola is translated 3 units to the left and 2 units up its vertex is also translated 3 units to the right and 2 units up .
vertex after translations is at: (-2 - 3 , - 3 + 2) = (-5 , -1)
Solution to Question 9let us find the slopes of the two lines
a x + b y = c , slope m1 = - a / b
b x - a y = c , slope m2 = b / a
m1*m2 = (- a / b)(b / a) = - 1
The two lines are perpendicular
Solution to Question 10Since a and A have different signs the graphs of the two equations are parabolas opening in diffent directions: If one opens up the other opens down. Also since b 2 - 4 a c and B 2 - 4 A C are both negative, none of the parabola cuts the x axis. This means that each one of these parabolas is either above the x axis or below the x axis and therefore do not intersect.
Solution to Question 11sin(0) = 0 and cos(0) = 1, and from x = 0 to x = pi/2, sin(x) increases from 0 to 1 and cos(x) decreases from 1 to 0. From x = pi/2 to x = pi, sin(x) decreases from 1 to 0 and cos(x) decreases from 0 to -1. Hence both sin(x) and cos(x) decreases on the interval (pi/2 , pi)
Solution to Question 12If f(x) = 0 at x = -2, 0 and 3 then f(x - 2) = 0 for
x - 2 = -2 , x - 2 = 0 and x - 2 = 3
Solve the above equations to find
x = 0 , x = 2 and x = 5
Solution to Question 13If f(x) = 0 at x = - 4, 8 and 11 then f(2x) = 0 for
2x = -4 , 2x = 8 and 2x = 11
Solve the above equations to find
x = -2 , x = 4 and x = 11/2
Solution to Question 14There are C(5,2) ways to select 2 teachers from 5 and C(10,4) ways to select 4 students from 10 where C(n,r) is the combinations of n items taken r at the time. Using the multiplication counting principle,the number of different committees that can be formed is given by
C(5,2)*C(10,4) = 2100
Solution to Question 15Since books C and D are arranged first and second, only books A, B and E will change order. Therefore it is an arrangement problem involving 3 items and the number of different order is given by
3! = 6
Solution to Question 16Since 5 is added to all data values, the mean will also increase by 5 and becomes 15. But the standard deviation which measure the "distance" between the mean and the data values does not change.
Solution to Question 17Let m be the mean and s be the standard deviation and find the z score.
z = (x - m) /s = (0.8 s + m - m) / s = 0.8
The percentage of student who scored above Jane is (from table of normal distribution).
1 - 0.7881 = 0.2119 = 21.19%
The number of student who scored above Jane is (from table of normal distribution).
21.19% 0f 500 = 106
Solution to Question 18Since f(x) is
odd, we have
f(-x) = - f(x)
Let g(x) = |f(x)|.
g(- x) = |f(- x)| = |- f(x)) = |f(x)| = g(x)
|f(x)| is even.
Solution to Question 19The period of sin (3x) is given by
2 pi / 3
In order to graph |sin(3x)| given the graph of sin(3x), we need to reflect all neagtive parts of the graph of sin(3x) which halves the period. Hence the peiod of |sin(3x)| is given by
(2 pi / 3) / 2 = pi / 3
Solution to Question 20When the ball was put in water, the volume of water increased by.
pi*22*0.6 = 2.4 pi cubic cm
The above volume is equal to the volume of the ball with radius r to find. Hence
(4/3)*pi*r3 = 2.4 pi
Solve the above for r
r = 1.2 cm
Solution to Question 21According to
trigonometric identities, 2 sin x cos x = sin(2x) and the period is given by.
2 pi / 2 = pi
Solution to Question 22Let q being the probability a device does not function properly and p = 1 - q = 0.9 the probability that a device functions properly. Since there are only two possible results, it is a binomial distribution. The P probability that 7 out of 10 devices function properly is given by
C(10,7)*0.9 7*0.1 3 , where C(10,7) is the number of ways that 7 items are selected from 10 and is given by
P = C(10,7) = 10! / (7!3!)
A calculation of P gives
P = 0.057
More References and LinksConvert Logarithms and Exponentials
math questions and problems with detailed solutions in this site.