(a) Show that the equation can be written as . [3]
(b) Hence solve the equation , giving your answer correct to two decimal places. [2]
Question 3
A curve has equation .
Find the equation of the tangent to the curve at the point .
Give your answer in the form , where , and are integers. [6]
Question 4
Solve the equation .
Give your answers in the form , where and are real. [6]
Question 5
Find the exact value of . [7]
Question 6
(a) On a sketch of an Argand diagram, shade the region which represents complex numbers satisfying both the inequalities and . [5]
(b) Calculate the maximum value of for points in this region. [3]
Question 7
(a) Prove the identity . [3]
(b) Hence, find the exact value of . [5]
Question 8
(a) Express in partial fractions. [5]
(b) The variables and satisfy the differential equation
and when .
Solve the differential equation and obtain an expression for in terms of . [5]
Question 9
The points , and have position vectors, relative to the origin , given by
The line passes through the points and .
(a) Find the length of the perpendicular from to . [5]
The line has vector equation , where is an integer.
The acute angle between the directions of lines and is equal to .
(b) Find the value of . [5]
Question 10
(a) It is given that . Using the substitution , where , form an equation in and hence show that . [4]
(b) Given that there is exactly one real value of satisfying the equation , verify by calculation that this value lies between and . [2]
(c) Use the iterative formula , together with initial value , to find the value of correct to decimal places. Give the result of each iteration to decimal places. [3]
(d) Using the value of found in part (c), solve the equation for .
Give your answers correct to three significant figures. [2]