ISI Maths

Question

For any real number x, let [x] denote the largest integer less than or equal to x and <x>=x-[x], that is, the fractional part of x. For arbitrary real numbers x,y and z only one of the following statement is correct. Which one is it?

• A. [x+y+z]=[x]+[y]+[z]
• B. [x=y=z]=[x+y]+[z]=[x]+[y+z]=[x+z]+[y]
• C. <x+y+z> = y+z – [y+z]+<x>
• D. [x+y+z]=[x+y]+[z+<y+x>]

Answer 1

Note: these type of problems can be easily solved by one example :)

Let x=0.9, y=1.9, z=2.9

a) [x+y+z] =[0.9+1.9+2.9] = [5.7]= 5

[x]+[y]+[z] =[0.9]+[1.9]+[2.9] = 0+1+2 =3≠5             // A is wrong

b) [x+y]+[z]= [2.9]+[2.9]= 2+2 =4 ≠5                      // B is wrong

c)<x+y+z> =5.7 - 5 =0.7

y+z -[y+z]+<x> = 1.9+2.9 -[1.9+2.9]+ 0.9 -[0.9] =

4.8 -4+0.9 -0 =1.7 ≠ 0.7                                        //C is wrong

d)[x+y+z] =5

[x+y]+ [z+<y+x>] = [0.9+1.9]+[2.9 + 1.9+0.9 -[1.9+0.9]]

= 2 + 3  = 5                                    // D is correct

ans D