Tag: squares and square roots

Questions Related to squares and square roots

Find the square of the following number without multiplication.

46

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

  1. 2116

  2. 2002

  3. 2424

  4. 1988

Show answer
Correct Option: A
Explanation:

${46}^{2} = 46\times46 = 2116$

If sin$\theta -cosec  \theta =\sqrt{5},$ then the value of sin  $\theta  + cosec  \theta$ is:

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

  1. $\sqrt{3}$

  2. 1

  3. 3

  4. 9

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Correct Option: C
Explanation:

$\Rightarrow \sin\theta-cosec\theta=\sqrt{5}$


$\Rightarrow \sin\theta-\dfrac{1}{\sin\theta}=\sqrt{5}$      $(\because cosec\theta=\dfrac{1}{\sin\theta})$

$\Rightarrow \sin^2\theta-\sqrt{5}\sin\theta-1=0$

Solving equation to get roots.

$\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{\pm3+\sqrt{5}}{2}$ (substitute values to get roots)

To find:-

$\sin\theta+cosec\theta$

Using $\dfrac{3+\sqrt{5}}{2}$

$=\dfrac{3+\sqrt{5}}{2}+\dfrac{2}{3+\sqrt{5}}$

$=\dfrac{(3+\sqrt{5})^2+4}{2(3+\sqrt{5})}$

$=\dfrac{9+4+5+6\sqrt{5}}{2(3+\sqrt{5})}$

$=\dfrac{6(3+\sqrt{5})}{2(3+\sqrt{5})}$

$=3$


Using $\dfrac{-3+\sqrt{5}}{2}$

$=\dfrac{-3+\sqrt{5}}{2}+\dfrac{2}{-3+\sqrt{5}}$

$=\dfrac{(-3+\sqrt{5})^2+4}{2(-3+\sqrt{5})}$

$=\dfrac{9+4+5-6\sqrt{5}}{2(-3+\sqrt{5})}$

$=\dfrac{-6(-3+\sqrt{5})}{2(-3+\sqrt{5})}$

$=-3$


According to option answer is $3$

If $a+b+c=6$ and $ ab+bc+ca = 11 $
Find $\left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } \right)$ ?
maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

  1. $14$

  2. $25$

  3. $36$

  4. $47$

Show answer
Correct Option: A
Explanation:
Given $a+b+c=6$ 

$(a+b+c)^2=36$

$(a^2+b^2+c^2)+2(ab+ba+ca)=36$

$a^2+b^2+c^2+(2\ast 11)=36$

$a^2+b^2+c^2=36-22=14$

Evaluating the following :
$(3+\sqrt{2})^{5}-(3-\sqrt{2})^{5}$

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

  1. $1718\sqrt 3$

  2. $1718\sqrt 2$

  3. $1178\sqrt 3$

  4. $1178\sqrt 2$

Show answer
Correct Option: D
Explanation:

Given term is $(3+\sqrt{2})^{5}-(3-\sqrt{2})^{5}$


$\Rightarrow 2\left[\ ^{5}C _{1}\times 3^{4}\times (\sqrt{2})^{1}+\ ^{5}C _{3}\times 3^{2}\times (\sqrt{2})^{3}+\ ^{5}C _{5}\times 3^{0}\times (\sqrt{2})^{5}\right]$

$\Rightarrow 2\left[5\times 81\times\sqrt{2}+10\times 9\times 2\sqrt{2}+4\sqrt{2}\right]$

$\Rightarrow 2\sqrt{2}(405+180+4)$

$\Rightarrow 1178\sqrt{2}$

Evaluating the following :
$(1+2\sqrt{x})^{5}+(1-2\sqrt{x})^{5}$

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

  1. $2(1+40x^2+80x)$

  2. $2(1-40x+81x^2)$

  3. $2(1+40x+80x^2)$

  4. None of these

Show answer
Correct Option: C
Explanation:
Given to evaluate is $(1+2\sqrt x)^5 +(1-2\sqrt x)^5$

$\Rightarrow 2[^{5}C _{0} (2\sqrt x)^0 +^5C _2 (2\sqrt x)^2 +^5C _4 (2\sqrt x)^4]$

$\Rightarrow 2[1+10\times 4x+5\times 16x^2]$

$\Rightarrow 2[1+40x+80x^2]$

Find the square root by guessing the units and tens digit:
$2116$
maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

  1. $46$

  2. $50$

  3. $52$

  4. $58$

Show answer
Correct Option: A
Explanation:

Its unit digit is 6 and tens digit is 1


For unit digit 6 , there are two possibilities either the square root end with 6 or 4

And according to options $46$ is the correct option

The number which exceeds its positive square root by $12$ is

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

  1. $9$

  2. $16$

  3. $25$

  4. None of these

Show answer
Correct Option: B
Explanation:
Let the positive number be x according to question,

$\sqrt{x}+12=x$

$\Rightarrow \sqrt{x}=x-12$

Squaring both sides,

$\Rightarrow x=x^{2}-24x+144$

$x^{2}-25x+144=0$

$x^{2}-16x-9x+144=0$

$x(x-16)-9(x-16)=0$

$(x-9)(x-16)=0$

 $  (x-9)=0 $ or $    (x-16)=0 $

 $ x=9 $  or $ x=16 $

So, the number is $16$.

Find the square root of which of the following numbers will be the least :

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

  1. $7\dfrac{58}{81}$

  2. $11\dfrac{14}{25}$

  3. $10\dfrac{1}{36}$

  4. $0.3481$

Show answer
Correct Option: D
Explanation:

$A.$


$7\dfrac{58}{81}=\dfrac{625}{81}$


$\Rightarrow$  $\sqrt{\dfrac{625}{81}}=\dfrac{25}{9}=2.77$

$B.$

$11\dfrac{14}{25}=\dfrac{289}{25}$

$\Rightarrow$  $\sqrt{\dfrac{289}{25}}=\dfrac{17}{5}=3.4$

$C.$

$10\dfrac{1}{36}=\dfrac{361}{36}$

$\Rightarrow$  $\sqrt{\dfrac{361}{36}}=\dfrac{19}{6}=3.16$

$D.$

$0.3481=\dfrac{3481}{10000}$

$\Rightarrow$  $\sqrt{\dfrac{3481}{10000}}=\dfrac{59}{100}=0.59$

$\therefore$  We can see, $0.3481$  has least square root.

Simlify: $\sqrt{\dfrac{-17}{144}-i}$

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

  1. $ \pm \left( {\dfrac{3}{2} - \dfrac{i}{3}} \right)$

  2. $ \pm \left( {\dfrac{3}{4} - \dfrac{{2i}}{3}} \right)$

  3. $ \pm \left( {\dfrac{3}{5} - \dfrac{{5i}}{6}} \right)$

  4. $ \pm \left( {\dfrac{2}{3} - \dfrac{{3i}}{4}} \right)$

Show answer
Correct Option: D
Explanation:

We know that,

${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$

 

Now, let,

$ -2ab=-i $

$ ab=i $

 

Now, consider $\dfrac{-17}{144}$. We can write it as,

$\dfrac{-17}{144}=\dfrac{64-81}{9\times 16}=\dfrac{4}{9}-\dfrac{9}{16}$

 

Thus,

$ \sqrt{\dfrac{-17}{144}-i}=\sqrt{\dfrac{4}{9}-\dfrac{9}{16}-i} $

$ =\sqrt{{{\left( \dfrac{2}{3} \right)}^{2}}+{{\left( i \right)}^{2}}{{\left( \dfrac{3}{4} \right)}^{2}}-2\times \left( \dfrac{2}{3} \right)\times \left( \dfrac{3i}{4} \right)} $

$ =\sqrt{{{\left( \dfrac{2}{3}-\dfrac{3i}{4} \right)}^{2}}} $

$ =\pm \left( \dfrac{2}{3}-\dfrac{3i}{4} \right) $

 

Hence, this is the required result.

The approximate value of$\sqrt { { \left( 1.97 \right)  }^{ 2 }{ \left( 4.02 \right)  }^{ 2 }{ \left( 3.98 \right)  }^{ 2 } }$

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

  1. $31.59 $

  2. $5.099$

  3. $5.009$

  4. $5.734$

Show answer
Correct Option: A
Explanation:

We have,

$ \sqrt{{{\left( 1.97 \right)}^{2}}{{\left( 4.02 \right)}^{2}}{{\left( 3.98 \right)}^{2}}} $

$ =1.97\times 4.02\times 3.98 $

$ =31.59 $

Hence, this is the answer.