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Perfect Square

1. A number is called a

**perfect square**if it is expressed as the square of a number.
2. E.g. 1, 4, 9, 16, 25, ... are called perfect squares (1x1 = 1, 2x2 = 4, 3 x 3 = 9...)

3. In square numbers, the digits at the unit’s place are always 0, 1, 4, 5, 6 or 9.

4. The numbers having 2, 3, 7 or 8 at its units' place are not perfect square numbers.

**Q1: Which one of the following number is a perfect square:**

a) 622

b) 393

c) 5778

d) 625

**Answer: d.**

5. If a number ends with odd number of zeros then it is not a perfect square.

**Q2: Check which of the following is a not a perfect square.**

a) 81000

b) 8100

c) 900

d) 6250000

**Answer: a) 81000 (= 92 x 102 x 10)**

6. The square of an even number is an even number while the square of an odd number is an odd number.

7. If n is a positive whole number then (n+1)2 - n2 = 2n + 1

or 2n numbers in between the squares of the numbers n and (n + 1)

**Q3: Which of the following perfect square numbers, is the square of an odd number?**

289, 400, 900, 1600

a) 289

b) 400

c) 900

d) 1600

**Answer: a) 289**

**Q5: Which of the following perfect square numbers, is the square of a even number?**

361, 625, 4096, 65536

a) 361

b) 625

c) 4096

d) 2601

**Answer: c)**

Q6: How many natural numbers lie between squares of 12 and 13.

a) 22

b) 23

c) 24

d) 25

**Answer: c) 24 (2x12 = 24)**

**Q7: A yoga instructor wants to arrange maximum possible number of 6000 students in a ground so that the number of rows is same as the number of columns. How many rows will be there if 71 students were left out after the arrangement.**

a) 80

b) 88

c) 77

d) 78

**Answer: c) 77. (Hint: Remaining students = 6000 - 71 = 5929 = 11 x 11 x 7 x 7 = 11 x 7 = 77)**

**Q8: The perfect square number between 30 and 40 is:**

a) 32

b) 35

c) 36

d) 39

**Answer: c) 36**

**Q9: Can a prime number be a perfect square?**

a) True

b) False

**Answer: b) False**

8. Unit digits of x

^{n}where x, n ∈ WUnits Digit of Number (x) | Units Digit of the number (x ^{n}) | No. of possibilities |
---|---|---|

0 | 0 | 1 |

1 | 1 | 1 |

2 | 2,4,6,8 | 4 |

3 | 3,9,7,1 | 4 |

4 | 4,6 | 2 |

5 | 5 | 1 |

6 | 6 | 1 |

7 | 7,9,3,1 | 4 |

8 | 8,4,2,6 | 4 |

9 | 9,1 | 2 |

**Q10: Find the units digit of (564)64 .**

a. 2

b. 4

c. 6

d. 8

**Answer: (c) 6.**

Hint: Option a and d can be easily eliminated, since 2 and 8 never comes in units place in 4x If you see pattern (4 x 4 = 42 = 16 i.e. 42x = 6 at units place) and (4 x 4 x 4 = 43 = 64 i.e. 42x+1 = 4 at units place). Similarly, (4)64 = 6.

9. The sum of first m odd natural numbers is a perfect square and is equal to m

^{2}
10. 1

^{2}+ 2^{2}+ 3^{2}+ ... + n^{2}= n(n+1)(2n+1)/6**Q11. 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 is a perfect square of number?**

a) 8

b) 7

c) 6

d) 9

**Answer: a) 8**

**Q12: The square of an integer is called a perfect square number. If x is a perfect square number, then its next one is**

a. x+1,

b. x2 +1,

c. x2 +2x+1,

d. x+2

√x+1.

a. 12 units

b. 12.5 units

c. 19.5 units

d. 20 units

11. The sum of 1 and the product of any four consecutive integers is a perfect square.

E.g. 3 x 4 x 5 x 6 + 1 = 360 + 1 = 361 = √361 = 19

12. To find square root of a number, we generally use following two methods:

i.e. LCM(16,20,24) = 4x4x5x3 = 240

To make 240 perfect square (multiply by 3x5) = 4

Answer:

Answer: Let the four consecutive integers are: x, x+1, x+ 2 and x+3.

Let P is the product.

⇒ P = x(x+1)(x+2)(x+3)

P = (x

Adding 1 to both sides,

P + 1 = x

Answer: as per the question, x

⇒ x

⇒ x =

**Answer: d. (Hint: if x is perfect square, then number is √x. Next perfect square will be (√x + 1)2**

i.e. x+2

√x+1.i.e. x+2

√x+1.

**Q13: As shown in figure below, the area of three squares are given. Find the perimeter of Î”ABC**.b. 12.5 units

c. 19.5 units

d. 20 units

**Answer**: a. (Hint: Length of each side of square is √25 = 5, √9 = 3 and √16 = 4. Perimeter = 5+3+4 = 12units)11. The sum of 1 and the product of any four consecutive integers is a perfect square.

E.g. 3 x 4 x 5 x 6 + 1 = 360 + 1 = 361 = √361 = 19

12. To find square root of a number, we generally use following two methods:

- Factorization Method: Finding prime factors of a number
- Long Division method.

**Q14: Find the least perfect square number which is divisible 16, 20 and 24?****Answer**: Take the LCM of 16, 20 and 24 which is the least number divisible by all three.i.e. LCM(16,20,24) = 4x4x5x3 = 240

To make 240 perfect square (multiply by 3x5) = 4

^{2}x 5^{2}x 3^{2}= 3600 ...(answer)**Q15. Compute √0.0016**Answer:

**√**(0.0016) =**√**(16/10000) =**√**(4^{2}x 10^{-4}) = 4 x 10^{-2}= 0.04**Q16: Prove that product of four consecutive integers plus one is a perfect square number.**Answer: Let the four consecutive integers are: x, x+1, x+ 2 and x+3.

Let P is the product.

⇒ P = x(x+1)(x+2)(x+3)

P = (x

^{2}+ x)(x+2)(x+3) = (x^{3}+ 3x^{2}+ 2x)(x+3) = x^{4}+ 6x^{3}+ 11x^{2}+ 6xAdding 1 to both sides,

P + 1 = x

^{4}+ 6x^{3}+ 11x^{2}+ 6x + 1 = (x^{2}+ 3x + 1)^{2}⇒ is a perfect square.**Q17: If (17)**^{2}is subtracted from a square of a number (x), the result obtained is 1232. Find the number 'x'.Answer: as per the question, x

^{2}- 17^{2}= 1232⇒ x

^{2}= 1232 + 17^{2}= 1232 + 289 = 1521⇒ x =

**√**(1521) =**39****<<Back to TET Practice Papers**

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