Question:17. Simplify :
    (i) `896 xx 896 - 204 xx 204 =?`
    
   (ii) `57 xx 57 + 43 xx 43 + 2 xx 57 xx 43 =?`
    
   (iii) `81 xx 81 + 68 xx 68 - 2 xx 81 xx 68=?` 

Sol. (i) Given Exp.=`(896)^2-(204)^2`
                          
                          =`(896+204)^2(896-204)`
                           
                          =`1100 xx 692`
                           
                          =761200.
       
 (ii) Gieven Exp. =`(57)^2+(43)^2+2 xx 57 xx 43`
                              
                        =`a^2+b^2+2ab`,where a = 57b=43
                              
                        =`(a+b)^2`=`(57+43)^2 = (100)^2`
                              
                        =10000.
      
 (iii)Given exp.=`(81)^2+(68)^2-2xx81xx68`
                          
                     =`a^2+b^2-2ab`,where a=81&b=68
                          
                     =`(a-b)^2`=`(81-68)^2`=`(13)^2`=169.

Question:18.Evaluate :`(313 xx 313 + 287 xx 287).` 

`(a^2+b^2)=``1/2``[(a+b)^2`+`(a^2-b)^2`]
   
  :. `(313)^2+(287)^2`
         
   =`1/2``[(313+287)^2`+`(313-287)^2`]
         
   =`1/2``[(600)^2`+`(26)^2]`
          
   =`1/2``(360000+676)`
           
   =180338

Question:2o.What least number must be subtracted from 2000 to get a number exactly divisible by 17? 

On dividing 2000 by 17,we get 11 as remainder.
:.Required number to be subtracted =11

Question:21.What least number be added to 3000 to obtain a number exactly divisible by 19? 

On dividing 3000 by 19, we get 17 remainder.
:. Number to be added =(19-17)=2.

Question:22.Find the number which is nearest to 3105 and exactly divisible by 21. 

On dividing 3105 by 21,we get 18 as remainder.
:. number to added to 3105 is (21-18)=3
:. 3108 the required number.

Question:A number when divided by 342 gives a remainder 47.when the same number is divided by 19,what would be the remainder ? 

On dividing the given number by 342,let k be the quotient and 47 as remainder.
Then, number `=342k+47`
                     `=(19x18k+19x2+9)=19(18k+2)+9`
      `:.`The given number when divided by 19,gives (18k+2) as quotient and 9 as remainder.

Question:24.(i) `? ÷ 147 = 29`   (ii) `? xx 144 = 12528` 

(i) Let`x/147`=29.Then,`x = (147 xx 29) = 4263.`
  
 :.Missing number is 4263.

 (ii) Let`xx 144 =12528.`Then,x=`12528/144`=87.

Question:How many numbers between `11` and `90` are divisible by `7`? 

The required numbers are `14,21,28,35....77,84.`
This is an A.P. with `a=14` and `d=(21-14)=7`.
Let it contain n terms.
Then,`T~n=84`
`⇒ a+(n-1)d=84`
`⇒ 14+(n-1) xx 7`
`=84 or n=11`
`:.` Required number of terms `=11`

Question:26.Find the sum of all odd numbers upto 100. 

The given numbers are `1,3,5,7.....99,`
This an A.P with a=1 and d =2.
Let it contain n terms.Then,
`1+(n-1) xx 2=99 or n=50.`
`:.`Required sum =`n/2 ("first terms" + "last terms")`
                       =`50/2 xx (1+99)=2500`.

Question:Find the sum of all `2` digit numbers divisible by `3`. 

All 2 digit numbers divisible by `3` are :

 `12,15,18,21,......99`.

 This is an A.P. with `a =12` and `d=3`.

 Let it contain n terms.Then,

 `12+(n-1) xx 3=99` or `n=30`.

 Required sum =`30/2 xx (12+99) = 1665`.