Tag: squares and square roots

Quizzes (7) Questions (116)

Questions Related to squares and square roots

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

If $x - \dfrac {1}{x} = \sqrt {6}$, then $x^{2} + \dfrac {1}{x^{2}}$ is ________.

  1. $2$
  2. $4$
  3. $6$
  4. $8$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, $\dfrac {x - 1}{ x} = 6$

Multiplying and divide the above equation with $x - \dfrac {1}{x}$
Thus $ \dfrac{x-\dfrac{1}{x}\times x-\dfrac{1}{x}}{x-\dfrac{1}{x}} = \sqrt{6} $
Using $(a-b)^{2} = a^{2} + b^{2} - 2ab $
and substituting $x-\dfrac{1}{x} = \sqrt{6}$  in denominator, we get
$\dfrac{x^{2} + \dfrac{1}{x^{2}} - 2x\dfrac{1}{x}}{\sqrt{6}} = \sqrt{6}$
$\Rightarrow x^{2} + \dfrac{1}{x^{2}} - 2  =\sqrt{6}\times \sqrt{6}$
$\Rightarrow x^{2} + \dfrac{1}{x^{2}} = 6+2 = 8$

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

If $2l - 3m = -1$ and $lm = 20$, then the value of $4l^{2} + 9m^{2}$ is ________.

  1. $239$
  2. $240$
  3. $241$
  4. $361$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know the identity $a^{2}+b^{2}-2ab = (a-b)^{2}$

Given, $2l-3m= -1$

Squaring on both sides, we get 

$(2l-3m)^{2}= (-1)^{2}$

$\Rightarrow 4l^{2}+9m^{2}-12lm = 1$     .....Also given that $lm =20 $

$\Rightarrow 4l^{2}+9m^{2}-12 \times 20 = 1$

$\Rightarrow 4l^{2}+9m^{2}-240 = 1$

$\Rightarrow 4l^{2}+9m^{2}= 241$

Hence, 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

On simplification the product of given expression $\left (x - \dfrac {1}{x}\right )\left (x + \dfrac {1}{x}\right )\left (x^{2} + \dfrac {1}{x^{2}}\right )$ is ________.

  1. $x^{3} - \dfrac {1}{x^{3}}$
  2. $x^{3} + \dfrac {1}{x^{3}}$
  3. $x^{4} - \dfrac {1}{x^{4}}$
  4. $x^{4} + \dfrac {1}{x^{4}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
We need to find value of $\left (x-\dfrac{1}{x}\right)\left (x+\dfrac{1}{x}\right)\left (x^{2}+\dfrac{1}{x^{2}}\right)$
We know the identity $a^{2}-b^{2} = (a-b)(a+b)$

Then applying this to first two terms:

$\left (x^{2}-\dfrac{1}{x^{2}}\right)\left (x^{2}+\dfrac{1}{x^{2}}\right)$

Again applying same identity:

$x^{4}-\dfrac{1}{x^{4}}$

Hence, option C is correct.