Question

1. From a square with sides of length 5, triangular pieces from the four corners are removed
to form a regular octagon. Find the area removed to the nearest integer?
2. Let f(x) = x2 + ax + b. If for all nonzero real x
s(x+2) = f(x) + f (2)
and the roots of f(x) = 0 are integers, what is the value of a2 + b2 ?
3. Let mi be a positive real number and for every integer n > 1 let In+1 = 1 + 21 22 ... In-117. If
35 = 43, what is the sum of digits of the largest prime factor of 36?
4. An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a
160° turn to the right and walks 4 more feet. It then makes another 160° turn to the right and
valks 4 more feet. If the ant continues this pattern until it reaches the anthill again, what is
he distance in feet it would have walked?
. Five persons wearing badges with numbers 1,2,3,4,5 are seated on 5 chairs around a cir-
ular table. In how many ways can they be seated so that no two persons whose badges have
onsecutive numbers are seated next to each other? (Two arrangements obtained by rotation​

Answer 1

Answer:

4

Step-by-step explanation:

From a square with sides of length 5, triangular pieces from the four corners are removed

to form a regular octagon. Find the area removed to the nearest integer?

Let say Equal Sides of Triangle cut = b

Then side of Hexagon = 5 - 2b

Side of Hexagon = √b² + b² = b√2

5 - 2b = b√2

=> b(2 + √2) = 5

=> b = 5/(2 + √2)

Area of Triangle = (1/2)b * b = (1/2)  25/(4 + 2 + 4√2)   = 25/(12 + 8√2)

Area of 4 Triangles = 25/(3 + 2√2)

= 25(3 - 2√2)/(9 - 8)

= 25 ( 3 - 2√2)

= 4.3

= 4  (to the nearest integer)