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Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

$\sqrt{1\, +\, \sqrt{1\, +\, \sqrt{1\, +\, ..........}}}\, =\, ..........$   

  1. Equals 1

  2. Lies between 0 and 1

  3. Lies between 1 and 2

  4. Is greater than 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let 


$x=\sqrt{1+\sqrt{1-----}}$

Squaring both sides

$x^2=1+\sqrt{1+\sqrt{1+\sqrt{1------}}}$

$x^2=1+x$                $(\because x=\sqrt{1+\sqrt{1------}})$

$x^2-x-1=0$

finding roots, we get

$\dfrac{1\pm\sqrt{1+4}}{2}$

$=\dfrac{1\pm\sqrt{5}}{2}$

$=-0.615$ and $1.615$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

If $x\, \ast\, y\, =\, \sqrt{x^2\, +\, y^2}$, then the value of $(1^{\ast}\, 2\, \sqrt{2})(1^{\ast}\, - 2\, \sqrt{2})$ is:  

    • 7

  2. 0

  3. 2

  4. 9
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$1^{\ast} 2 \sqrt{2}\, =\, \sqrt{(1)^2\, +\, (2 \sqrt{2}^2}\, =\, \sqrt{1\, +\, 8}\, =\, 3$

$1^{\ast} -2 \sqrt{2}\, =\, \sqrt{(1)^2\, +\, (-2 \sqrt)^2}\, =\, \sqrt{1\, +\, 8}\, =\, 3$
$(1\, \ast\, 2 \sqrt{2})(1\, \ast\, -2 \sqrt{2})\, =\, (3)(3)\, =\, 9$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

$\displaystyle \frac{\sqrt{32}\, +\, \sqrt{48}}{\sqrt{8}\, +\, \sqrt{12}}\, =\, ?$

  1. $\sqrt{2}$

  2. 2

  3. 4

  4. 8
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$ {\cfrac{\sqrt{32} + \sqrt{48}}{\sqrt{8} + \sqrt{12}} = \cfrac{\sqrt{16 \times 2} + \sqrt{16 \times 3}}{\sqrt{4 \times 2} + \sqrt{4 \times 3}}}$

$= \cfrac{4\sqrt{2} + 4\sqrt{3}}{2\sqrt{2} + 2\sqrt{3}}$

$ = \cfrac{4 \left (\sqrt{2} + \sqrt{3} \right )}{2 \left (\sqrt{2} + \sqrt{3} \right )}$

$ = 2$
Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

$\sqrt{1\, +\, \sqrt{1\, +\, \sqrt{1\, +\, .....}}}$ = ........

  1. Equals $1$
  2. Lies between $0$ and $1$
  3. Lies between $1$ and $2$
  4. Is greater than $2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let $x=\sqrt { 1+\sqrt { 1+\sqrt { 1+....... }  }  }$

$\therefore { x }^{ 2 }=1+\sqrt { 1+\sqrt { 1+\sqrt { 1+....... }  }  }$
$\Longrightarrow { x }^{ 2 }=1+x$
$\Longrightarrow { x }^{ 2 }-x-1=0$
$\Longrightarrow x=\cfrac { 1\pm \sqrt { 1+4 }  }{ 2 }$
$ \Longrightarrow x=\cfrac { 1\pm \sqrt { 5 }  }{ 2 }$
$ \Longrightarrow \cfrac { 1\pm 2.236 }{ 2 } =\left( -0.618, 1.618 \right) $
We reject the negative value because, from the given expression,

$x$ is positive.
So, $x=1.618$ approximately.
$\therefore 1<x<2$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

$\sqrt{(12\, +\, \sqrt{12\, +\, \sqrt{12\, +\, ........}})}\, =\, ?$

  1. 3

  2. 4

  3. 6

  4. Greater than 6
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let $\sqrt{(12 + \sqrt{12 + \sqrt{12 + ........}})} = x$
Then, $\sqrt{12 + x} = x$
$ \Rightarrow 12 + x = x^2$
$\Rightarrow x^2 - x - 12 = 0$ 
$\Rightarrow (x - 4) (x + 3) = 0$
$\Rightarrow x = 4$       ...(neglecting $x = -3$)
Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the square root of each of the following correct to three places of decimal.
$17$
$1.7$
$2.5$
$\displaystyle\frac{7}{8}$

  1. $4.153\;;\;1.304\;;\;1.581\;;\;0.935$
  2. $4.123\;;\;1.304\;;\;1.581\;;\;0.995$
  3. $4.123\;;\;1.304\;;\;1.581\;;\;0.935$
  4. $4.123\;;\;1.304\;;\;1.501\;;\;0.935$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$17$

$4.123$
$4$$+4$ $17$$16$
$822$$+2$ $100$$81$
$8243$ $1900$$1644$
$25600$$24729$                 $871$

$\therefore$ The square root of $17=4.123$

$1.7$

$0$ $1.303$
$1$$+1$ $1.7$$1$
$23$$+3$ $70$$69$
$2603$ $10000$$7809$
$1191$

$\therefore$ The square root of $1.7=1.303$

$2.5$

$0$ $1.581$
$1$$+1$ $2.5$$1$
$25$$+5$ $150$$125$
$308$$+8$ $2500$$2464$
$3161$ $3600$$3161$                 $439$

$\therefore$ The square root of $2.5=1.581$

$\frac{7}{8}$

$8$ $70$$64$ $0.875$
$60$$56$
$40$$40$             $0$
$0.935$
$0$ $0.875$$0$
$9$$9$ $87$$81$
$183$$+3$ $650$$549$
$1865$ $10100$$9325$
$925$

$\therefore$ The square root of $\frac{7}{8}=0.935$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square Also find the square root of the perfect square so obtained 
$(i) 402, (ii) 1989, (iii) 3250, (iv) 825, (v) 4000$

  1. <span class="block ng-binding">Least number which must be subtracted:$(i) 2, (ii) 22, (iii) 1, (iv) 21, (v) 52$

    Square root of the perfect square:
    $(i) 20, (ii) 34 ,(iii) 55, (iv) 26, (v) 67$
  2. <span class="block ng-binding">Least number which must be subtracted:$(i) 2, (ii) 53, (iii) 1, (iv) 41, (v) 31$

    Square root of the perfect square:
    $(i) 20, (ii) 44, (iii) 57, (iv) 28, (v) 63$
  3. <span class="block ng-binding">Least number which must be subtracted:$(i) 6, (ii) 22, (iii) 50, (iv) 31, (v) 40$

    Square root of the perfect square:
    $(i) 19, (ii) 41, (iii) 49 ,(iv) 27 ,(v) 65$
  4. <span class="block ng-binding">Least number which must be subtracted:$(i) 8, (ii) 41, (iii) 12, (iv) 56, (v) 4$

    Square root of the perfect square:
    $(i) 19 ,(ii) 22, (iii) 37, (iv) 26, (v) 61$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For $402$

Nearest perfect square is $400$
$\sqrt{400}$ $=$ $20$
$402$ $-$ $400$ $=$ $2$
For $1989$
Nearest perfect square is $1936$

$\sqrt{1936}$ $=$ $44$
$1936$ $-$ $1989$ $=$ $53$

For $3250$
Nearest perfect square is $3249$

$\sqrt{3249}$ $=$ $57$
$3250$ $-$ $3249$ $=$ $1$
For $825$
Nearest perfect square is $784$
$\sqrt{784}$ $=$ $28$
$825$ $-$ $784$ $=$ $41$
For $4000$
Nearest perfect square is $3969$
$\sqrt{3969}$ $=$ $63$
$4000$ $-$ $3969$ $=$ $31$
From this, Option B is correct answer.