Question

Value of $(\sqrt{2}+1{)}^6+(\sqrt{2}-1{)}^6$ is

• A. 90
• B. 70
• C. 142
• D. 198

Answer 1

The correct option is D 198

$1^2{+}^6{C}_3.x^3.1^3{+}^6{C}_4.x^2$
$1^4{+}^6{C}_5.x{.1}^5{+}^6{C}_6.x^0.1^6$
$=x^6+6x^5+{15}^4+20x^3+{15}^2+6x+1$ ....... (i)
$(x-1{)}^6=x^6{+}^6{C}_1.x^5.(-1){+}^6{C}_2{x}^4.(-1{)}^2{+}^6{C}_3.x^3.(-1{)}^3{+}^6{C}_4.x^2.(-1{)}^4{+}^6{C}_5.x.(-1{)}^5{+}^6{C}_6.x.(-1{)}^6$
$=x^6-6x^5+15x^4-20x^3+15x^2-6x+1$ .... (ii)
Adding (i) and (ii)
$(x+1{)}^6+(x-1{)}^6=2[x^6+15x^4+15x^4+15x^2+1]$
Putting x $=\sqrt{2}$
$(\sqrt{2}+1{)}^6+(\sqrt{2-1}{)}^6=2\left[\right(\sqrt{2}{)}^6+15(\sqrt{2}{)}^4+15(\sqrt{2}{)}^2+1]$
$=2[8+60+30+1]=2\times 99=198$