54 If a+b+c=5 , a2+b2+c2=12 and a3+b3+c3=25 then the value of a4+b4+c4 is (a)2516 (b) 2536 - Maths - Relations and Functions

Question

54   I f   a + b + c = 5   ,   a 2 + b 2 + c 2
= 12   a n d   a 3 + b 3 + c 3 = 25   t h e n  
t h e   v a l u e   o f   a 4 + b 4 + c 4   i s
  ( a ) 251 6       ( b )   253 6  
  ( c ) 255 6       ( d ) 257 6

Neha Sethi · Accepted Answer

Given:a+b+c=5   
(A)a2+b2+c2=12    
(B)a3+b3+c3=25     
(C)Now, we take whole square of eq(A),we geta+b+c2=25a2+b2+c2+2ab+2bc+2ca=2512+2ab+bc+ca=25ab+bc+ca=132     
(1)Now, (a+b+c)a2+b2+c2=a3+b3+c3+a2b+b2a+b2c+c2b+c2a+a2c       
(2)And(a+b+c)(ab+bc+ca)=a2b+b2a+b2c+c2b+c2a+a2c+3abcSo, a2b+b2a+b2c+c2b+c2a+a2c=(a+b+c)(ab+bc+ca)-3abcPut this value in eq(2), we get(a+b+c)a2+b2+c2=a3+b3+c3+(a+b+c)(ab+bc+ca)-3abca3+b3+c3-3abc=(a+b+c)a2+b2+c2-(ab+bc+ca)Now, we put values from equations A,B,C and 1 and ger25-3abc=512-13225-3abc=60-6523abc=-52abc=-56       
(3)Now we take whole square of eq(2),and geta2+b2+c22=122a4+b4+c4+2a2b2+b2c2+c2a2=144a4+b4+c4=144-2a2b2+b2c2+c2a2        
(4)And,ab+bc+ca2=a2b2+b2c2+c2a2+2a2bc+ab2c+abc2a2b2+b2c2+c2a2=(ab+bc+ca)2-2a2bc+ab2c+abc2a2b2+b2c2+c2a2=(ab+bc+ca)2-2abca+b+ca2b2+b2c2+c2a2=1322-2-565a2b2+b2c2+c2a2=1694+506a2b2+b2c2+c2a2=60712Put this value in (4), we geta4+b4+c4=144-260712  a4+b4+c4=2576

54 . I f a + b + c = 5 , a 2 + b 2 + c 2 = 12 a n d a 3 + b 3 + c 3 = 25 t h e n t h e v a l u e o f a 4 + b 4 + c 4 i s ( a ) 251 6 ( b ) 253 6 ( c ) 255 6 ( d ) 257 6

$54 . I f a + b + c = 5, a^{2} + b^{2} + c^{2} = 12 a n d a^{3} + b^{3} + c^{3} = 25 t h e n t h e v a l u e o f a^{4} + b^{4} + c^{4} i s (a) \frac{251}{6} (b) \frac{253}{6} (c) \frac{255}{6} (d) \frac{257}{6}$