Quantitative Aptitude · Mathematics

Squares and Square Roots

196 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 fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

A square is inscribed in the circle $x^2+y^2-10x- 6y +30=0$. One side of the square is parallel to $y=x+3$. Then which of the following can be a vertex of the square

  1. $(3, 3)$
  2. $(7, 3)$
  3. $(5, 5)$
  4. $(1, 1)$
Reveal answer Fill a bubble to check yourself
A,B Correct answer
Explanation

The circle equation x^2+y^2-10x-6y+30=0 simplifies to (x-5)^2+(y-3)^2=4, so the center is (5,3) and radius is 2. A square inscribed in this circle has a diagonal of 4. Since the side is parallel to y=x+3 (slope 1), the vertices are at distance sqrt(2) from the center along lines with slopes 1 and -1. Calculating these points yields (3,3) and (7,3) as valid vertices.

Multiple choice maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

What is the least number that must be added to $594$ to make sum a perfect square?

  1. $13$
  2. $29$
  3. $31$
  4. $33$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

First calculate the square-root of $594$

$\sqrt{594}\approx 24.37$

The whole number larger than $24.37$ is $25$

and $(25)^{2}=625$

Now, $625$ is a perfect square.

So, the least number that must be added to $594$ to make sum a perfect square is $=625-594=31$

Multiple choice maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

If a number of $n$-digits is perfect square and $n$ is an odd number, then which of the following is the number of digits of its square root?

  1. $\cfrac{n-1}{2}$
  2. $\cfrac{n}{2}$
  3. $\cfrac{n+1}{2}$
  4. $2n$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

no of digits in a perfect square is $n$

 If $n$ is odd then no of digits in its square roots is $\dfrac{n+1}{2}$

Multiple choice maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

If the sum $1+2+3 +....+ K$ is a perfect square N$^{2}$ and if N is less than 100, then the possible values for K are: 

  1. only 1

  2. 1 and 8

  3. only 8

  4. 8 and 49

  5. 1,8, and 49
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation
S= K(K + l)/2 = $N^{2}$. The possible values for N$^{2}$ are $1^{2}, 2^{2}, 3^{2}.... 99^{2}$;
For K to be integral, the discriminant 1 + 8N$^{2}$ of the equation $K^{2} + K- 2N$= 0 must be a perfect square. This fact reduces the possible values for $N^{2}$ to $1^{2}, 6^{2}, and 35^{2}$. Hence the values of K are 1, 8, and 49.
Note: There are ways of shortening the number of trials for N2 still further.but these involve a knowledge of number-theoretic theorems. The shortest way to do this problem is by testing the choices given.
Multiple choice maths ways to multiply and divide mental multiplication multiplication methods multiplication of numbers

Find the least number in which multiplied by $1800$ given a perfect cube, then find the sum of the digits of that number.

  1. $2$
  2. $3$
  3. $6$
  4. $8$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
factor of $ 1800 - 10 \times 10\times 18 = 3\times 3\times 2\times 2\times 5\times 2\times 5 $

$ = 2^{3}\times 3^{2}\times 5^{2}$

for perfect cube we need $  3\times 5 = 15 $

So, sum of digits $(15) = 1+5=6$

so option (c) is right