Tag: squares and square roots

Questions Related to squares and square roots

Use identities to evaluate $\displaystyle \left ( 97 \right )^{2}$

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

  1. $\displaystyle 9,109$

  2. $\displaystyle 9,409$

  3. $\displaystyle 9,909$

  4. $\displaystyle 9,209$

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Correct Option: B
Explanation:

$\left ( 97 \right )^{2}$
Using,
$(a-b)^2=a^2-2ab+b^2$
$=(100-3)^2$
$=(100)^2-2(100)(3)+(3)^2$
$=10000-600+9$
$=9,409$

Use identities to evaluate $\displaystyle \left ( 101 \right )^{2}$

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

  1. $\displaystyle 10,301$

  2. $\displaystyle 12,301$

  3. $\displaystyle 10,201$

  4. $\displaystyle 12,201$

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Correct Option: C
Explanation:

$ \left ( 101 \right )^{2}$
Using,
$(a+b)^2=a^2+2ab+b^2$
$=(100+1)^2$
$=(100)^2+2(100)(1)+(1)^2$
$=10000+200+1$
$=10,201$

Use identities to evaluate :$\displaystyle \left ( 502 \right )^{2}$

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

  1. $\displaystyle 1,62,004$

  2. $\displaystyle 1,22,004$

  3. $\displaystyle 2,12,004$

  4. $\displaystyle 2,52,004$

Show answer
Correct Option: D
Explanation:

$\left ( 502 \right )^{2}$
Using,
$(a+b)^2=a^2+2ab+b^2$
$=(500+2)^2$
$=(500)^2+2(500)(2)+(2)^2$
$=2,50000+2000+4$
$=252,004$

The simplified value of $\displaystyle \left ( \sqrt{3}+1 \right )^{2}-2\left ( 2+\sqrt{3} \right )$ is

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

  1. 2

  2. -1

  3. 1

  4. 0

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Correct Option: D
Explanation:
$(\sqrt{3}+1)^{2}-2(2+\sqrt{3})$
=$(3+1+2\sqrt{3})-2(2+\sqrt{3})$
=$4+2\sqrt{3}-4-2\sqrt{3}=0$

Evaluate the following
$(105)^{2}$

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

  1. $10025$

  2. $11025$

  3. $11125$

  4. $12025$

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Correct Option: B
Explanation:
$ (105)^{2} = (100+5)^{2} = 100^{2}+5^{2}+2(100)(5) $
$ = 10000+25+1000 = \boxed {11025} $ 
Evaluate the following 
$(97)^{2}$
maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

  1. $9409$

  2. $9049$

  3. $9949$

  4. $4949$

Show answer
Correct Option: A
Explanation:
$ (97)^{2} = (100-3)^{2} = 100^{2}+3^{2}-2(400)(3) $
$ = 10000+9-600 = \boxed {9409} $ 

If $3x - \dfrac {1}{2x} = 6$, then the value of $9x^{2} + \dfrac {1}{4x^{2}}$

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

  1. $36$

  2. $33$

  3. $30$

  4. $39$

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Correct Option: D
Explanation:
Given that $3x-\dfrac{1}{2x} = 6$
Squaring both sides, we get
$9x^{2}+\dfrac{1}{4x^{2}}-3 = 36$
We know the identity $a^{2}+b^{2}-2ab = (a-b)^{2}$
$9x^{2}+\dfrac{1}{4x^{2}}= 39$     

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

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

  1. $2$

  2. $4$

  3. $6$

  4. $8$

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Correct Option: D
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
$\dfrac{x^{2} + \dfrac{1}{x^{2}} - 2x\dfrac{1}{x}}{\sqrt{6}} = \sqrt{6}$
$\Rightarrow x^{2} + \dfrac{1}{x^{2}} - 2  =\sqrt{6}\times \sqrt{6}$
$\Rightarrow x^{2} + \dfrac{1}{x^{2}} = 6+2 = 8$

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

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

  1. $239$

  2. $240$

  3. $241$

  4. $361$

Show answer
Correct Option: C
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.

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 ________.

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

  1. $x^{3} - \dfrac {1}{x^{3}}$

  2. $x^{3} + \dfrac {1}{x^{3}}$

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

  4. $x^{4} + \dfrac {1}{x^{4}}$

Show answer
Correct Option: C
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.