Question

If the difference between two natural numbers is 2 and the sum of their squares is 244, then the sum of the smaller number and the square of the larger number is

• A. 154
• B. 112
• C. 109
• D. 180

Answer 1

The correct option is A 154

Let the two natural numbers be x and y, x > y.
$\therefore x-y=2and{x}^2+y^2=244$
$\Rightarrow (x-y{)}^2+2xy=244[\therefore x^2+y{)}^2+2xy$
$\Rightarrow 2xy=244-4=240$
$\Rightarrow xy=120$
$\Rightarrow y=\frac{120}{x}$

Now, x - y = 2

$\Rightarrow x-\frac{120}{x}=2$
$\Rightarrow x^2-120=2x$
$\Rightarrow x^2-2x-120=0$
$\Rightarrow x^2-12x+10x-120=0$
$\Rightarrow x(x-12)+10(x-12)=0$
$\Rightarrow (x-12)(x+10)=0$
$\Rightarrow y=\frac{120}{12}=10(Neglectingx=-10,\text{as it is a natural number}$
$\therefore y+x^2=10+{12}^2=154$
Hence, the correct answer is option (1).