Question 1
SLPaper 2Let , , and .
Find .
Let be a point on the graph of . The tangent to the graph of at is parallel to the graph of . Find the -coordinate of .
Question 2
SLPaper 1The function is defined for all . The line with equation is the tangent to the graph of at .
The function is defined for all where and .
Write down the value of .
Find .
Find .
Hence find the equation of the tangent to the graph of at .
Question 3
SLPaper 2SpeedWay airline flies from city to city . The flight time is normally distributed with a mean of 260 minutes and a standard deviation of 15 minutes. A flight is considered late if it takes longer than 275 minutes.
The flight is considered to be on time if it takes between and 275 minutes. The probability that a flight is on time is 0.830.
During a week, SpeedWay has 12 flights from city to city . The time taken for any flight is independent of the time taken by any other flight.
Calculate the probability a flight is not late.
Find the value of .
Calculate the probability that at least of these flights are on time.
Given that at least of these flights are on time, find the probability that exactly flights are on time.
SpeedWay increases the number of flights from city to city to flights each week, and improves their efficiency so that more flights are on time. The probability that at least flights are on time is . A flight is chosen at random. Calculate the probability that it is on time.
Question 4
SLPaper 2Let and .
The graphs of and intersect at and , where .
Question 5
SLPaper 1The function is defined by , where , .
Write down the equation of
Find the coordinates where the graph of crosses
the vertical asymptote of the graph of .
the horizontal asymptote of the graph of .
the -axis.
the -axis.
Sketch the graph of on the axes below.
Question 6
SLPaper 1A function is defined by , where , .
The graph of has a vertical asymptote and a horizontal asymptote.
Write down the equation of the vertical asymptote.
Write down the equation of the horizontal asymptote.
On the set of axes below, sketch the graph of .
On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
Hence, solve the inequality .
Question 7
HLPaper 1The cubic equation where has roots and .
Given that , find the value of .
Question 8
SLPaper 1The functions and are defined for by and , where .
Given that and , find the value of and the value of .
Question 9
SLPaper 2The functions and are defined for by and , where .
Find the range of .
Given that for all , determine the set of possible values for .
Question 10
HLPaper 3This question asks you to explore properties of a family of curves of the type for various values of and , where .
On the same set of axes, sketch the following curves for and , clearly indicating any points of intersection with the coordinate axes.
Now, consider curves of the form , for , where .
Next, consider the curve , for .
The curve has two points of inflection. Due to the symmetry of the curve these points have the same -coordinate.
is defined to be a rational point on a curve if and are rational numbers. The tangent to the curve at a rational point intersects the curve at another rational point . Let be the curve , for . The rational point lies on .
,
,
Write down the coordinates of the two points of inflexion on the curve
By considering each curve from part (a), identify two key features that would distinguish one curve from the other.
By varying the value of , suggest two key features common to these curves.
Show that , for .
Hence deduce that the curve has no local minimum or maximum points.
Find the value of this -coordinate, giving your answer in the form , where , , and are integers.
Find the equation of the tangent to at .
Hence, find the coordinates of the rational point where this tangent intersects , expressing each coordinate as a fraction.