model 5 simplifying roots of roots Practice Questions Answers Test with Solutions & More Shortcuts

Answer: (a)

Let x = $√{1+√{1+√{1 +...}}}$

On squaring both sides

$x^2 =1+√{1+√{1+√{1 +...}}}$

$x^2$ = 1 + x

$x^2$ - x - 1 = 0

$x = {+1± √{1+4}}/2 = {+1 ±√ 5}/2$

But sum of + ve numbers can't be negative.

$x={1+ √5}/2 ={1+ 2.236}/2$

= ${3.236}/2 =1.618$

Thus 1 < 1.618 < 2

Answer: (b)

Let x = $√{12+√{12+√{12 +...}}}$

On squaring both sides,

$x^2 =12+√{12+√{12+√{12 +...}}}$

$x^2$ = 12 + x

$x^2$ - x - 12 = 0

$x^2$ - 4x + 3x - 12 = 0

x (x - 4) + 3 (x - 4) = 0

(x - 4) (x + 3) = 0

x = 4, - 3

The given expression is positive.

x = 4

$√{12+√{12+√{12}}}$=4

It is because

12 = 3 × 4 = n (n + 1)

Answer: (c)

x = $√{12+√{12+√{12 +...}}}$

On squaring both sides,

$x^2 = 12 + √{12+√{12+√{12 +...}}}$

$x^2$ = 12 + x

$x^2$ - x - 12 = 0

$x^2$ - 4x + 3x - 12 = 0

x (x - 4) + 3 (x - 4) = 0

(x - 4) (x + 3) = 0

x = 4 because x ≠ - 3

Answer: (b)

${√{10+√{ 25+√{ 108+√{ 154+√{ 225}}}}}}/√^3{8}$= ?

?=${√{10+√{ 25+√{ 108+√{ 154+15}}}}}/√^3{2×2×2}$

=${√{10+√{ 25+√{ 108+√{169}}}}}/2$

=${√{10+√{ 25+√{ 108+13}}}}/2$

=${√{10+√{ 25+√{121}}}}/2$

=${√{10+√{ 25+11}}}/2$

=${√{10+√{36}}}/2={√{10+6}}/2$

=${√{16}}/2=4/2=2$

Answer: (d)

x=$√{2^3√{4√{2^3√{4√{2^3√{4…}}}}}}$

On squaring

$x^2 = 2 √{2^3√{4√{2^3√{4√{2^3√{4…}}}}}}$

On cubing,

$x^6$= 8 × 4x

$x^5 = 32 = 2^5$ ⇒ x = 2