Tag: patterns in square numbers

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Questions Related to patterns in square numbers

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

Find the square of $\displaystyle 2a - b - 3c$

  1. $\displaystyle 4a^{2} + b^{2} + 9c^{2} - 4ab + 6bc - 12ca$
  2. $\displaystyle a^{3} + b^{3} + c^{2} - 4ab + 6bc - ca$
  3. $\displaystyle a^{2} + b^{2} - c^{2} - 4ab + 6bc - 2ca$
  4. $\displaystyle 4a^{3} + b^{3} + 9c^{2} - 4ab - 6bc - 12ca$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

   $2a-b-3c$
Squaring, we get
   $(2a-b-3c)^2$
Using $(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ac$
$=(2a)^2+(-b)^2+(-3c)^2+2(2a)(-b)+2(-b)(-3c)+2(2a)(-3c)$
$=4a^{2} + b^{2} + 9c^{2} - 4ab + 6bc - 12ca$

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

Evaluate :

$\displaystyle \left ( \frac{2x}{5} + \frac{3y}{4} - \frac{4z}{7} \right )^{2}$

  1. $\displaystyle \frac{x^{2}}{25} + \frac{9y^{2}}{16} + \frac{z^{2}}{49} + \frac{3}{5} xy - \frac{6}{7}yz - \frac{16}{35}zx$
  2. $\displaystyle \frac{4x^{2}}{25} + \frac{9y^{2}}{16} + \frac{16z^{2}}{49} + \frac{1}{5} xy - \frac{6}{3}yz - \frac{36}{35}zx$
  3. $\displaystyle \frac{x^{2}}{25} + \frac{9y^{2}}{16} + \frac{5z^{2}}{49} + \frac{3}{5} xy - \frac{6}{7}yz - \frac{16}{35}zx$
  4. $\displaystyle \frac{4x^{2}}{25} + \frac{9y^{2}}{16} + \frac{16z^{2}}{49} + \frac{3}{5} xy - \frac{6}{7}yz - \frac{16}{35}zx$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

    $ \left ( \dfrac{2x}{5} + \dfrac{3y}{4} - \dfrac{4z}{7} \right )^{2}$
Squaring, we get
Using $(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ac$
$=( \dfrac{2x}{5})^2+(\dfrac{3y}{4})^2+(- \dfrac{4z}{7})^2+2( \dfrac{2x}{5})(\dfrac{3y}{4})+2(\dfrac{3y}{4})(- \dfrac{4z}{7})+2( \dfrac{2x}{5})(- \dfrac{4z}{7})$
$=\dfrac{4x^{2}}{25} + \dfrac{9y^{2}}{16} + \dfrac{16z^{2}}{49} + \dfrac{3}{5} xy - \dfrac{6}{7}yz - \dfrac{16}{35}zx$

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

Find the squares of the following numbers without actual multiplication:
$49$
$52$

  1. $2441; 2704$
  2. $2401; 2784$
  3. $2401; 2704$
  4. $2441; 2784$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$49^2=(40+9)^2=40(40+9)+9(40+9)$
$\;\;\;\;\;=(40)^2+40\times9+9\times40+9^2$
$\;\;\;\;\;=1600+360+360+81$
So, $49^2=2401$
$52^2=(50+2)^2=50(50+2)+2(50+2)$
$\;\;\;\;\;\;=50^2+50\times2+2\times50+2^2$
$\;\;\;\;\;\;=2500+100+100+4$
So, $52^2=2704$.

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

Evaluate the following
$(0.98)^{2}$

  1. $0.9664$
  2. $0.9604$
  3. $0.9864$
  4. $0.9964$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The given square $(0.98)^2$ can be evaluated as shown below:


${ (0.98) }^{ 2 }\ =(1-0.02)^{ 2 }\ =1^{ 2 }+(0.02)^{ 2 }-(2\times 1\times 0.02)\quad \quad \quad \quad \quad (\because \quad (a-b)^{ 2 }=a^{ 2 }+b^{ 2 }-2ab)\ =1+0.0004-0.04\ =1.0004-0.04\ =0.9604$

Hence, ${ (0.98) }^{ 2 }=0.9604$

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

A factor of $(3x^{4} - 12y^{4})$ is _________.

  1. $3$
  2. $x^{2} - 2y^{2}$
  3. $x^{2} + 2y^{2}$

  4. All of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given expression is $(3x^{4}-12y^{4})$
Taking $3$ common, we get
$= 3(x^{4}-4y^{4})$
$= 3(x^4-(\sqrt{2}y)^{4})$
We know the identity and applying it $a^{2}-b^{2} = (a-b)(a+b)$

$3x^{4}-12y^{4}= 3(x^{4}-(\sqrt{2}y)^{4})$

$3x^{2}-12y^{2}= 3(x^{2}-\sqrt{2}y^{2})(x^{2}+\sqrt{2}y^{2})$