This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Section A: Pure Mathematics 1 How many integers between 10 000 and 100 000 (inclusive) contain exactly two different digits? (23 332 contains exactly two different digits but neither of 33 333 and 12 331 does.) 2 Show, by means of a suitable change of variable, or otherwise, that Z f(( x 2 + 1) 1 / 2 + x ) d x = 1 2 Z 1 (1 + t 2 )f( t ) d t. Hence, or otherwise, show that Z (( x 2 + 1) 1 / 2 + x ) 3 d x = 3 8 . 3 Which of the following statements are true and which are false? Justify your answers. (i) a ln b = b ln a for all a, b > 0. (ii) cos(sin ) = sin(cos ) for all real . (iii) There exists a polynomial P such that  P( ) cos  6 10 6 for all real . (iv) x 4 + 3 + x 4 > 5 for all x > 0. 4 Prove that the rectangle of greatest perimeter which can be inscribed in a given circle is a square. The result changes if, instead of maximising the sum of lengths of sides of the rectangle, we seek to maximise the sum of n th powers of the lengths of those sides for n > 2. What happens if n = 2? What happens if n = 3? Justify your answers. 1 5 (i) In the Argand diagram, the points Q and A represent the complex numbers 4 + 6 i and 10 + 2 i . If A , B , C , D , E , F are the vertices, taken in clockwise order, of a regular hexagon (regular sixsided polygon) with centre Q , find the complex number which represents B . (ii) Let a , b and c be real numbers. Find a condition of the form Aa + Bb + Cc = 0, where A , B and C are integers, which ensures that a 1 + i + b 1 + 2 i + c 1 + 3 i is real....
View
Full
Document
This note was uploaded on 04/01/2012 for the course MATH 1016 taught by Professor Rotar during the Spring '12 term at Central Lancashire.
 Spring '12
 rotar
 Math, Integers

Click to edit the document details