Tag: patterns in square numbers

Questions Related to patterns in square numbers

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

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

  1. $2xy$

  2. $-2xy$

  3. $12xy$

  4. $-12xy$

Show answer
Correct Option: B
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

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

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

  1. $(5p + 9q)^{2}$

  2. $(5p - 9q)^{2}$

  3. $(9p + 5q)^{2}$

  4. $(9p - 5q)^{2}$

Show answer
Correct Option: C
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.

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

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

  1. $3$

  2. $4$

  3. $6$

  4. Greater than $6$

Show answer
Correct Option: B
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)
$x=4$

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

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

  1. $2+2\sqrt {2}$

  2. $2\sqrt {2}$

  3. $1$

  4. $0$

Show answer
Correct Option: B
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
$\Rightarrow$  $2\sqrt{2}$

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

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

  1. True

  2. False

Show answer
Correct Option: A
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. $x$,

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

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

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

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

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

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

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

  1. i) 159876

    ii) 0.91
    iii) 876590

  2. i) 135879

    ii) 0.87
    iii) 896750

  3. i) 159201

    ii) 0.9604
    iii) 999919

  4. i) 138760

    ii) 0.9
    iii) 999999

Show answer
Correct Option: C
Explanation:

$(i)$

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

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

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

If $a + b + c = 9$ and $ab + bc + ca = 26$, then the value of $a^2 + b^2 + c^2$ is

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

  1. $29$

  2. $52$

  3. $81$

  4. None

Show answer
Correct Option: A
Explanation:

Given, $a + b + c = 9$......(1) and $ab + bc + ca = 26$.......(2).


Now we have,


$\Rightarrow$$a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)$

$\Rightarrow$$a^2+b^2+c^2=(9)^2-2\times26=81-52=29$.

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

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

  1. $\left(-\infty,-\dfrac {8}{7}\right)$

  2. $(4,\infty)$

  3. $\left(-\infty,-\dfrac {48}{3}\right)$

  4. $a\ \epsilon \left(-\dfrac {8}{7},-1 \right]$

Show answer
Correct Option: D
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 $(1),(2)$    $a\in\bigg(-\dfrac{8}{7},-1\bigg]$

Find the square of $83$ without actual multiplication.

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

  1. $6880$

  2. $3881$

  3. $3889$

  4. $6889$

Show answer
Correct Option: D
Explanation:

$\ 83^2\(80+3)^2\= 80^2+2\cdot 80\cdot 3+3^2\= 6400+480+9\= 6889$

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

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

  1. $1$

  2. $3$

  3. $\sqrt {20}$

  4. $6$

Show answer
Correct Option: C
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, $a^2+b^2+2ab=(a+b)^2$ ]

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