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 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 fundamental concept of ratio and proportion division problem dividing a quantity in a given ratio problems on ratios

Divide $16$ into two parts such that the twice of the square of the greater part exceeds, the square of the smaller part by $164$. Then, the greater part is  

  1. $58$
  2. $10$
  3. $6$
  4. $15$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the greater part be x, so the smaller part is 16 - x. According to the problem, 2x^2 - (16 - x)^2 = 164. Expanding and simplifying gives 2x^2 - (256 - 32x + x^2) = 164, which leads to x^2 + 32x - 420 = 0. Factoring gives (x - 10)(x + 42) = 0, so the positive valid solution for the greater part is 10.

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
Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

If it is possible to form a number with the second, the fifth and the eighth digits of the number 31549786, which is the perfect square of a two digit even number, which of the following will be the second digit of that even number?

  1. 1

  2. 4

  3. 6

  4. No such number can be formed
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The 2nd , 5th and 8th digit of the number 31549786 are 196 respectively.
196 is a perfect square of 14. 
Therefore, the even number required is 14.
Second digit of that number is 4.
Answer is 4