Solutions and explanations to the math questions in this site are presented.
Solution to Question 1Rewrite the logarithmic equations using exponential equationx^{ 3 / 2} = 1 / 8 The above equation may be rewritten as x^{  3 / 2} = 1 / 2^{3} = 2^{3} (√x)^{3} = 2^{3} The above equation gives √x = 2 or x = 4
Solution to Question 220 % of 2 = (20 / 100) * 2 = 0.4
Solution to Question 3If Log_{ 4} (x) = 12, thenx = 4^{12} x / 4 = 4^{11} Hence log_{ 2} (x / 4) = log_{ 2} (4^{11}) = log_{ 2} ((2^{2})^{11}) = log_{ 2} (2^{22}) = 22
Solution to Question 4A population of size P increasing at the rate of 2% may be modelled as followsP = P_{0} e^{0.02 t} , where t is the number of years after t = 0 and P_{0} 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 = P_{0} e^{0.02*4} Solve the above for P_{0} P_{0} = 2,000,000 / e^{0.02*4} = 1 846 000 (rounded to the nearest thousand) P_{0} 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 theng(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 asy = 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 linesa 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 forx  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 for2x = 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 byC(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 by3! = 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 havef(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 by2 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*2^{2}*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*r^{3} = 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 byC(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 Exponentialsmath questions and problems with detailed solutions in this site.
