Q1. [N08.P1]- 7 marks
Find the equation of the normal to the curve
at the point (1, 2) .
Q2. [N09.P1]- 7 marks
A certain population can be modelled by the differential equation
is the population at time
is a positive constant.
(a) Given that when , express in terms of , and .
(b) Find the ratio of the minimum size of the population to the maximum size of the population.
Q3. [M09.P1]-5 marks
The diagram below shows a curve with equation
, defined for
The point lies on the curve and B is the maximum point.
(a) Show that .
(b) Hence, find the values of and . [3 marks]
Q4. [N10.P1]- 8 marks
Consider the curve and the line (a) Let .
(l)Show that the curve and the line intersect once.
(ll)Find the angle between the tangent to the curve and the line at the point of intersection.
. Show that the line is a tangent to the curve.
Q5. [M10.P1]- 8 marks
The function f is defined by .
(a) Find .
(b) You are given that
has a local minimum at
. Find the value of
Q6. [M14.P1]- 9 marks
A curve has equation
(a) Find in terms of and .
(b) Find the gradient of the curve at the point where and
Q7. [N14.P1]- 6 marks
A tranquilizer is injected into a muscle from which it enters the bloodstream. The concentration
, of tranquilizer in the bloodstream can be modelled by the function
is the number of minutes after the injection. Find the maximum concentration of tranquilizer in the bloodstream.
Q8. [M15.P1]- 8 marks
In triangle , , and .
(a) Show that length .
(b) Given that
has a minimum value, determine the value of
for which this occurs.
Q9. [N15.P1]- 6 marks
Consider the curve , , .
(b) Determine the equation of the normal to the curve at the point
in the form
Q10. [M16.P1]- 7 marks
A curve is given by the equation
Find the coordinates of all the points on the curve for which , .