Quantitative Aptitude · Mathematics

Squares and Square Roots

191 Questions

Squares and square roots questions assess numerical ability through perfect squares, digit properties, and fast calculation techniques. Many competitive exams feature Vedic mathematics shortcuts to solve these quickly. Mastering these rules improves overall calculation speed.

Perfect square propertiesVedic math squaresFinding square rootsDigit replacement puzzlesLeast perfect squares

Squares and Square Roots Questions

Multiple choice maths vedic mathematics square roots using vedic maths square and square roots using vedic mathematics history of mathematics

Identify the correct representation of the square of the number $95$ using Vedic Mathematics.

  1. $(95 \times 10)5$
  2. $(9 \times 9)25$
  3. $(9 \times 10)125$
  4. $(9 \times 10)25$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation


$\underset { +5 }{ 95 }  \times \underset { +5 }{  95 } $                                                          Using base 10
                                                                       $9 = 9 \times  base$
Mutiply $5$ with $5 = 25$

Add $5$ to $95 = 100$

Multiply 9 to sum  $= 9\times 100 = 900$

 Take first 2 digits $= 90 = 9\times 10$
 
Last 2 digits $= 25$

$\therefore $   ${95 }^{ 2 } = (9\times 10)25$
Multiple choice maths vedic mathematics square roots using vedic maths square and square roots using vedic mathematics history of mathematics

Square of a $5$4 by Upsutra Yavadunam Tavadunam Vargecha Yojayet method is?

  1. $2686$
  2. $5656$
  3. $6966$
  4. $2916$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To find the square of $54$,


This is closer to $100$ (base of 10). Write it as $100-46$.

From this method, we can write,

$\dfrac{(54-46)}{46^2}$  i.e., $\dfrac{Number-deficiency}{deficiency^2}$

$=\dfrac{8}{2116}$

As we are using base $100$, digits in hundred's place and above is carry forwarded. Add $21$ to $8$.

ie., $(21+8)16=2916$

$2916$ is the square of $54$.

Multiple choice maths vedic mathematics square roots using vedic maths square and square roots using vedic mathematics history of mathematics

Square of $63$ by Sutra Urdhva-tiryagbhyam method is :

  1. $8529$
  2. $4569$
  3. $3969$
  4. $5479$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To find the square of $63$


$63 \ \times \ 63$

3 steps are there to solve this.

(i)  Multiply the unit digits
 $3 \times 3=9$

(ii) Take the units and tens digit to cross multiply and add the products
$(3\times6)+(6\times3)=18+18=36$

Keep $6$ intact and carry forward $3$ to next step.

(iii) Multiply the tens digits
$6\times6=36$

Add the carry forward $3$ to $36$.
$36+3=39$

Write the numbers obtained from step (iii) to (i) in order.
i.e., $3969$

This the square of $63$ is $3969$.

Multiple choice maths vedic mathematics square roots using vedic maths square and square roots using vedic mathematics history of mathematics

Square of $112$ by Sutra Urdhva-tiryagbhyam method is :

  1. $21844$
  2. $12544$
  3. $16544$
  4. $17644$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To find the square of $112$


$112 \ \times \ 112$

5 steps are there to solve this.

(i)  Multiply the unit digits
 $2 \times 2=4$

(ii) Take the units and tens digit to cross multiply and add the products
$(2\times1)+(1\times2)=2+2=4$

(iii) Take all three digits to cross multiply and add the products
$(2\times1)+(1\times2)+(1\times1)=2+2+1=5$

(iv) Take the hundreds and tens digit to cross multiply and add the products
$(1\times1)+(1\times1)=1+1=2$


(v) Multiply the hundred digits
$1\times1=1$


Write the numbers obtained from step (v) to (i) in order.
i.e., $12544$

This the square of $112$ is $12544$.

Multiple choice maths vedic mathematics square roots using vedic maths square and square roots using vedic mathematics history of mathematics

To find the square of $45$ by Ekadhikena Purvena method the digit $4$ should be multiplied by which number.

  1. By its previous number

  2. By zero

  3. By its next number

  4. By ten
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In ekadhikena purvena the first digit is multiplied by its next number.

$45^2\Rightarrow $ First 2 digits $=4\times 5$
and last two digits are$=5^2=25\ \Rightarrow 45^2=2025$

Multiple choice maths vedic mathematics square roots using vedic maths square and square roots using vedic mathematics history of mathematics

Find the square of the number $65$ using Vedic Mathematics.

  1. $4235$
  2. $4335$
  3. $4220$

  4. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
To find $(65)^{2}$
Let's take $a=6$. So, $a5=10a+5$ 
Now, $a5$ square can be obtained as follows
$a5=a(a+1)|25$ where $a(a+1)$ is the left side of the number and right side will always be $25$ for the numbers ending with $5$.
Right side of $65$ will be $6\times 7=42$
and left side will be $25$.
So, the number is $4225$.
Hence, $65^{2}=4225$. 
Multiple choice maths application of derivatives - iii second derivative test maxima and minima application of derivatives

Divide 64 into two parts such that the sum of the cubes of two parts is minimum.

  1. 30, 34

  2. 31, 33

  3. 32, 32.

  4. 35, 29
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let one part be x.
Hence another part will be 64-x.
Let 
$f(x)=x^{3}+(64-x)^{3}$.
$f'(x)$
$=3x^{2}-3(64-x)^{2}$
$=0$
Or 
$x^{2}=(64-x)^{2}$
Or 
$x=64-x$ or $x=-64+x$
Considering equation $x=64-x$, we get 
$x=32$.
Hence another part will also be 32.

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, $52^2=2704$.