Tag: finding the square of a number

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Questions Related to finding the square of a number

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Find the missing term in the following problem.
$\left (\dfrac {3x}{4} - \dfrac {4y}{3}\right )^{2} = \dfrac {9x^{2}}{16} + \dfrac {16y^{2}}{9} + ?$.

  1. $2xy$
  2. $-2xy$
  3. $12xy$
  4. $-12xy$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
We know that $(a-b)^2=a^2+b^2-2ab$
$\left (\dfrac{3x}{4}-\dfrac{4y}{3}\right)^2=\left (\dfrac{3x}{4}\right)^2+\left (\dfrac{4y}{3}\right)^2-2\left(\dfrac{3x}{4}\right)\left (\dfrac{4y}{3}\right)$
$=\dfrac{9x^2}{16}+\dfrac{16y^2}{9}-2xy$
Therefore, the missing term is $-2xy$.

Option (B) is correct
Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

$(9p - 5q)^{2} + 180 pq$ is equivalent to _______.

  1. $(5p + 9q)^{2}$
  2. $(5p - 9q)^{2}$
  3. $(9p + 5q)^{2}$
  4. $(9p - 5q)^{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
We know that $a^2+b^2+2ab$
$(a-b)^2=a^2+b^2-2ab$
$\Rightarrow (a-b)^2+4ab=a^2+b^2-2ab+4ab=(a+b)^2$    $(\because a=9p$ and $b=5q)$
$\Rightarrow 4ab=4(9p)(5q)=180pq$
Therefore, $ (9p-5q)^2+180pq=(9p+5q)^2$

Option (C) is correct.
Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

If $\sqrt{\left(12+\sqrt{12+\sqrt{12+....}}\right)}=x$, then the value of x is ____________.

  1. $3$
  2. $4$
  3. $6$
  4. Greater than $6$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\sqrt{12+\sqrt{12+\sqrt{12+....}}} = x$.........................(1)

$\sqrt{12+x}=x$
$(12+x)=x^2$
$x^2-x-12=0$
$(x-4)(x+3)=0$
$ x=4$ or $x=-3$
since x is a positive number (eq 1)
$x=4$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

$\sqrt { 3+2\sqrt { 2 }  } +\sqrt { 3-2\sqrt { 2 }  } =...$ ?

  1. $2+2\sqrt {2}$
  2. $2\sqrt {2}$
  3. $1$
  4. $0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\sqrt { 3+2\sqrt { 2 }  } +\sqrt { 3-2\sqrt { 2 }  }$ 


$\Rightarrow$  $\left(\sqrt { 3+2\sqrt { 2 }  } +\sqrt { 3-2\sqrt { 2 }  }\right)^2$          [ Squaring both  sides ]

$\Rightarrow$  $(\sqrt{3+2\sqrt{2}})^2+(\sqrt{3-2\sqrt{2}})^2+2(\sqrt{3+2\sqrt{2}})(\sqrt{3-2\sqrt{2}})$

$\Rightarrow$  $3+2\sqrt{2}+3-2\sqrt{2}+2\sqrt{(3)^2-(2\sqrt{2})^2}$

$\Rightarrow$  $6+2\sqrt{9-8}$

$\Rightarrow$  $8$
Taking square root
$\Rightarrow$  $2\sqrt{2}$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

if $y = 500{e^{7x}} + 600{e^{-7x}}$ . Then $\dfrac{{d^2y}}{dx^2} = 49y$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$y=500e^{7x}+600e^{-7x}$


Differentiationg w.r.t. $x$

$\Rightarrow$  $\dfrac{dy}{dx}=500\dfrac{d(e^{7x})}{dx}+600\dfrac{d(e^{-7x})}{dx}$

$\Rightarrow$  $500\times e^{7x}\times \dfrac{d(7x)}{dx}+600\times e^{-7x}\times \dfrac{d(-7x)}{dx}$

$\Rightarrow$  $\dfrac{dy}{dx}=500\times e^{7x}\times 7+600\times e^{-7x}\times (-7)$

$\Rightarrow$  $\dfrac{dy}{dx}=500\times 7\times e^{7x}-600\times 7\times e^{-7x}$

Again differentiating w.r.t. $x$,

$\Rightarrow$  $\dfrac{d^2y}{dx^2}=500\times 7\times \dfrac{d(e^{7x})}{dx}-600\times 7\times \dfrac{d(e^{-7x})}{dx}$

$\Rightarrow$  $\dfrac{d^2y}{dx^2}=500\times 7\times e^{7x}\times 7-600\times 7\times (-7)\times e^{-7x}$

$\Rightarrow$  $\dfrac{d^2y}{dx^2}=500\times 7\times 7e^{7x}+600\times 7\times 7\times e^{-7x}$

$\Rightarrow$  $\dfrac{d^2y}{dx^2}=7\times 7(500e^{7x}+600e^{-7x})$

$\Rightarrow$  $\dfrac{d^2y}{dx^x}=49y$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Evaluate each of the following using identities :
i) $(399)^2$
ii) $(0.98)^2$
iii) $991 \times 1009$

  1. i) 159876ii) 0.91
    iii) 876590

  2. i) 135879ii) 0.87
    iii) 896750

  3. i) 159201ii) 0.9604
    iii) 999919

  4. i) 138760ii) 0.9
    iii) 999999
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(i)$

$(399)^2=(400-1)^2$
             $=(400)^2+(1)^2-2\times 400\times 1$                [ $(a-b)=a^2+b^2-2ab$ ]
             $=160000+1-800$
             $=159201$
$\therefore$  $(399)^2=159201$

$(ii)$
$(0.98)^2=(1-0.02)^2$
              $=(1)^2+(0.02)^2-2\times 1\times 0.02$                 [ $(a-b)=a^2+b^2-2ab$ ]
              $=1+0.0004-0.04$
              $=0.9604$

$(iii)$ 
$991\times 1009=(1000-9)(1000+9)$
                       $=(1000)^2-(9)^2$                               [ $(a+b)(a-b)=a^2-b^2$ ]     
                       $=1000000-81$ 
                       $=999919$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

If $x^{2}+2(a-1)x+a+5=0$ has real roots to the interval $(1,3)$, then complete set of value of $'a'$ is

  1. $\left(-\infty,-\dfrac {8}{7}\right)$
  2. $(4,\infty)$
  3. $\left(-\infty,-\dfrac {48}{3}\right)$
  4. $a\ \epsilon \left(-\dfrac {8}{7},-1 \right]$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given $f(x)=x^{2}+2(a-1)x+a+5=0$ has real roots $\implies b^{2}-4{a}{c}\geq 0 $ in $a{x^{2}}+b{x}+c=0$

                     $\implies (a-1)^{2}-(a+5)\geq 0\implies a\in (-\infty,-1]\cup[4,\infty)\cdots\cdots(1)$
 the roots lies in $(1,3)\implies f(1).f(3)> 0\implies (3{a}+4)(7{a}+8)>0$
                              $\implies a\in (-\infty,-\dfrac{4}{3})\cup(-\dfrac{8}{7},\infty)\cdots\cdots(2)$
from $(1),(2)$    $a\in\bigg(-\dfrac{8}{7},-1\bigg]$

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

If $x^{2}+\dfrac{1}{^{x^2}}=18$, then the value of $\left(x+\dfrac{1}{x}\right)$ is ?

  1. $1$
  2. $3$
  3. $\sqrt {20}$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$x^2+\dfrac{1}{x^2}=18$


$\Rightarrow$  $x^2+\dfrac{1}{x^2}=20-2$                           [ Since, $20-2=18$ ]

$\Rightarrow$  $x^2+\dfrac{1}{x^2}+2=20$

$\Rightarrow$  $x^2+\dfrac{1}{x^2}+2\times x\times\dfrac{1}{x}=20$

$\Rightarrow$  $\left(x+\dfrac{1}{x}\right)^2=20$                      [ Since, $a^2+b^2+2ab=(a+b)^2$ ]

$\Rightarrow$  $\left(x+\dfrac{1}{x}\right)=\sqrt{20}$