model 5 simplifying roots of roots Section-Wise Topic Notes With Detailed Explanation And Example Questions
MOST IMPORTANT quantitative aptitude - 5 EXERCISES
The following question based on power, indices and surds topic of quantitative aptitude
(a) 5
(b) 4
(c) 6
(d) 3
The correct answers to the above question in:
Answer: (d)
Let x = $√{6+√{6+√{6 +...}}}$
Squaring on both sides,
$x^2 = 6+√{6+√{6+√{6 +...}}}$
$x^2$ = 6 + x
$x^2$ - x - 6 = 0
$x^2$ - 3x + 2x - 6 = 0
x (x - 3) + 2 (x - 3) = 0
(x + 2) (x - 3) = 0
x = 3 because x ≠ - 2
$√{6+√{6+√{6 +...}}}$ = 3
It is because
6 = 2 × 3 = n (n + 1)
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Read more simplifying roots of roots Based Quantitative Aptitude Questions and Answers
Question : 1
$√{1+√{1+√{1 +...}}}$
a) lies between 1 and 2
b) lies between 0 and 1
c) is greater than 2
d) equals 1
Answer »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
Question : 2
$√{12+√{12+√{12 +...}}}$ is equal to
a) 6
b) 4
c) 2
d) 3
Answer »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
Using Rule 25If $√{x+√{x+√{x +...∞}}}$ where, x=n(n + 1)then $√{x+√{x+√{x +...∞}}}$ = (n + 1)
$√{12+√{12+√{12}}}$=4
It is because
12 = 3 × 4 = n (n + 1)
Question : 3
The value of the following is : $√{12+√{12+√{12 +...}}}$
a) 2
b) $2√3$
c) 4
d) $2√2$
Answer »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
Question : 4
Find the value of $√{10+√{ 25+√{ 108+√{ 154+√{ 225}}}}}$.
a) 8
b) 10
c) 4
d) 6
Answer »Answer: (c)
Expression
=$√{10+√{ 25+√{ 108+√{ 154+√{ 225}}}}}$
=$√{10+√{ 25+√{ 108+√{ 154+15}}}}$
=$√{10+√{ 25+√{ 108+√{169}}}}$
=$√{10+√{ 25+√{ 108+13}}}$
=$√{10+√{ 25+√{121}}}$
=$√{10+√{ 25+11}}$
=$√{10+6}=√16=4$
Question : 5
$√{3√{3√{3...}}}$ is equal to
a) $2√3$
b) 3
c) $3√3$
d) $√3$
Answer »Answer: (b)
Let x = $√{3√{3√{3...}}}$
Squaring both sides,
$x^2 = 3√{3√{3√{3...}}}$ = 3x
$x^2$ - 3x = 0
x (x - 3) = 0
x = 3 because x ≠ 0
Using Rule 23$√{x√{x√{x...n times}}}= x^(1-1/{x^n})$
$√{3+√{3+√{3+...∞}}}$ = 3
It is because, here
n = ∞ and x =3
$√{3+√{3+√{3+...∞}}}=3^(1-1/{3∞})$
= $3^(1 - 0)$ [${something}/∞ = 0$] = 3
Question : 6
If m = $√{5+√{5+√{5 +...}}}$ and n = $√{5-√{5-√{5-...}}}$, then among the following the relation between m and n holds is
a) m + n + 1 = 0
b) m + n - 1 = 0
c) m - n - 1 = 0
d) m - n + 1 = 0
Answer »Answer: (c)
m = $√{5+√{5+√{5 +...}}}$
On squaring both sides,
$m^2 = 5 + m ⇒ m^2$ - m = 5 ....(i)
Again,
n = $√{5-√{5-√{5-...}}}$
On squaring both sides,
$n^2$ = 5 - n
$n^2$ + n = 5 .........(ii)
$m^2$ - m = $n^2$ + n
$(m^2 - n^2)$ = m + n
(m + n) (m - n) - (m + n) = 0
(m + n) (m - n - 1) = 0
GET power, indices and surds PRACTICE TEST EXERCISES
model 1 find largest and smallest value
model 2 based on simplification
model 3 based on positive and negative exponent
model 4 simplifying roots with values
model 5 simplifying roots of roots
power, indices and surds Shortcuts and Techniques with Examples
-
model 1 find largest and smallest value
Defination & Shortcuts … -
model 2 based on simplification
Defination & Shortcuts … -
model 3 based on positive and negative exponent
Defination & Shortcuts … -
model 4 simplifying roots with values
Defination & Shortcuts … -
model 5 simplifying roots of roots
Defination & Shortcuts …
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