## Chapter 1 Real Numbers Bits for 10th Class AP Board

Multiple Choice Questions (MCQs) :

Question 1.

The HCF of 95 and 152 is ……………… .

A) 19

B) 38

C) 95

D) 152

Answer:

A) 19

Question 2.

LCM of 24 and 96 is ……………….. .

A) 24

B) 96

C) 48

D) 72

Answer:

B) 96

Question 3.

Set of rational and irrational numbers is called ……………….. .

A) Real numbers

B) Natural numbers

C) Whole numbers

D) Integers

Answer:

A) Real numbers

Question 4.

The prime factorization of 729 is

A) 3^{4}

B) 3^{5}

C) 3^{6}

D) 3^{7}

Answer:

C) 3^{6}

Question 5.

If a and b are two prime numbers, then their HCF is ……………….. .

A) 0

B) 1

C) ab

D) \(\frac{a}{b}\)

Answer:

B) 1

Question 6.

HCF of two numbers is 9 and their LCM is 108 and one of them is 54. Then the second number is ……………….. .

A) 12

B) 6

C) 24

D) 18

Answer:

D) 18

Question 7.

The last digit of 6^{60} is ………………… .

A) 6

B) 3

C) 2

D) 4

Answer:

A) 6

Question 8.

Which of the following is a terminating decimal ?

A) \(\frac{8}{125}\)

B) \(\frac{3}{14}\)

C) \(\frac{41}{75}\)

D) \(\frac{10}{81}\)

Answer:

A) \(\frac{8}{125}\)

Question 9.

The number of prime factors of 36 is

A) 1

B) 2

C) 3

D) 4

Answer:

B) 2

Question 10.

Which of the following is true for ail irrational number ?

i) Which can be written in the form of \(\frac{\mathrm{p}}{\mathrm{q}}\), where p, q ∈ Z, q ≠ 0.

ii) Which cannot be written in the form of \(\frac{\mathrm{p}}{\mathrm{q}}\), where p, q ∈ Z, q ≠ 0.

iii) Non-terminating repeating decimals.

iv) Non-terminating, non-repeating decimals.

A) (i), (ii)

B) (ii), (iv)

C) (i) only

D) (iii) only

Answer:

B) (ii), (iv)

Question 11.

LCM of 24 and 36 is ……………….. .

A) 144

B) 72

C) 24

D) 36

Answer:

B) 72

Question 12.

Which one of the following rational numbers has terminating decimal expression ?

A) \(\frac{11}{7000}\)

B) \(\frac{91}{21000}\)

C) \(\frac{20}{9000}\)

D) \(\frac{343}{2^3 \times 5^3 \times 7^3}\)

Answer:

D) \(\frac{343}{2^3 \times 5^3 \times 7^3}\)

Question 13.

If P_{1} and p_{2} are two odd prime numbers such that p_{1} > p_{2}, then

P_{1}^{2} – P_{2}^{2} is ……………… .

A) an even

B) an odd

C) a prime

D) an odd prime

Answer:

A) an even

Question 14.

The number of digits in the fractional part of the decimal form of \(\frac{7}{40}\) is ……………….. .

A) 1

B) 2

C) 3

D) 4

Answer:

C) 3

Question 15.

The prime factorization of 72 is

A) 4^{2} × 3^{2}

B) 2^{5} × 3^{3}

C) 2^{3} × 3^{2}

D) 2^{4} × 3^{3}

Answer:

C) 2^{3} × 3^{2}

Question 16.

The decimal expansion of 0.225 in its rational form is ……………….

A) 325

B) \(\frac{225}{10^4}\)

C) \(\frac{325}{10^4}\)

D) \(\frac{9}{40}\)

Answer:

D) \(\frac{9}{40}\)

Question 17.

What is the LCM of greatest 3-digit number and the greatest 2-digit number?

A) 999

B) 9999

C) 9 × 11 × 111

D) 99 × 11 × 111

Answer:

C) 9 × 11 × 111

Question 18.

What is the LCM of least prime and the least composite number ?

A) 2

B) 4

C) 6

D) 9

Answer:

B) 4

Question 19.

What is the HCF of least prime and the least composite number ?

A) 2

B) 4

C) 6

D) 8

Answer:

A) 2

Question 20.

What is the HCF of least odd prime and the least odd composite number?

A) 2

B) 6

C) 4

D) 3

Answer:

D) 3

Question 21.

If n is a natural number, then which of the following expression ends in zero ?

A) (3 × 5)^{n}

B) (5 × 7)^{n}

C) (9 × 5)^{n}

D) (2 × 5)^{n}

Answer:

D) (2 × 5)^{n}

Question 22.

The decimal expansion of \(\frac{5}{19}\) is ………………… .

A) terminating

B) non-terminating and non-repeating

C) non-terminating and repeating

D) none

Answer:

C) non-terminating and repeating

Question 23.

7.4589 \(\overline{3}\) is a ……………. number.

A) prime

B) composite

C) irrational

D) rational

Answer:

D) rational

Question 24.

0.15115111511115… is ……………….. number.

A) rational

B) irrational

C) composite

D) prime

Answer:

B) irrational

Question 25.

5^{n} can end in ……………… (for n is any natural number)

A) 0

B) 2

C) 5

D) 6

Answer:

C) 5

Question 26.

Which of the following is correct ?

A) N ⊂ W ⊂ Z

B) Z ⊂ Q ⊂ R

C) W ⊂ Z ⊂ Q

D) All of the above

Answer:

B) Z ⊂ Q ⊂ R

Question 27.

2^{x} + 5^{x} = 7, then x = …………………. .

A) 1

B) 0

C) 2

D) 3

Answer:

A) 1

Question 28.

7^{n} can end with ……………. (for n is any natural number)

A) 3

B) 9

C) 1

D) A and B

Answer:

D) A and B

Question 29.

3 × 5 × 7 × 11 + 35 is ……………… number.

A) prime

B) composite

C) negative

D) irrational

Answer:

B) composite

Question 30.

2 × 7 × 13 × 17 + 21 is divisible by ……………….. .

A) 3

B) 7

C) 21

D) 5

Answer:

B) 7

Question 31.

The smallest number by which \(\sqrt{125}\) should be multiplied so as to get a rational number is …………………. .

A) √5

B) \(\sqrt{125}\)

C) √3

D) 5

Answer:

A) √5

Question 32.

If n is any natural number, then 6^{n} – 5^{n} always ends with ………………. .

A) 2

B) 5

C) 1

D) 3

Answer:

C) 1

Question 33.

The remainder when the square of any prime number greater than 3 is divided by 6 is ………………. .

A) 4

B) 3

C) 2

D) 1

Answer:

D) 1

Question 34.

The LCM and HCF of two rational numbers are equal, then the numbers must be ………………. .

A) co-prime

B) composite

C) prime

D) equal

Answer:

D) equal

Question 35.

If two positive integers p and q are expressible in the form of p = ab^{2} and q = a^{3}b, a, b being prime numbers, then LCM of p, q is

A) ab

B) a^{3}b^{3}

C) a^{2}b^{2}

D) a^{3}b^{2}

Answer:

D) a^{3}b^{2}

Question 36.

If two positive integers p and q are

expressible in the form of p = ab^{2} and q = a^{3}b, a, b being prime numbers, then HCF of a, b is

A) a^{3}b^{3}

B) a^{2}b^{2}

C) ab

D) a^{3}b^{2}

Answer:

C) ab

Question 37.

If p and q are co-primes, then p^{2} and q^{2} are

A) odd

B) even

C) co-prime

D) none

Answer:

C) co-prime

Question 38.

If n is a natural number, then 7^{2n} – 4^{2n} is always divisible by ……………… .

A) 3

B) 11

C) 2

D) A and B

Answer:

D) A and B

Question 39.

The exponent of 2 in the prime factorization of 36 is ………………. .

A) 2

B) 3

C) 4

D) 5

Answer:

A) 2

Question 40.

The sum of the exponents of the prime factors in the prime factorization of 144 is ……………….. .

A) 5

B) 6

C) 7

D) 8

Answer:

B) 6

Question 41.

The remainder when square of any number greater than 1 is divided by 4 is ………………. .

A) 1

B) 0

C) either 0 (or) 1

D) neither 0 nor 1

Answer:

C) either 0 (or) 1

Question 42.

If n is odd natural number, 3^{2n} + 2^{2n} is always divisible by ……………… .

A) 9

B) 4

C) 13

D) 5

Answer:

C) 13

Question 43.

For what values of n, 2^{n} × 5^{n} ends in 5?

A) 2

B) 1

C) 0

D) No value of n

Answer:

D) No value of n

Question 44.

The remainder when the square of any prime number greater than 3 is divided by 6 is ………………. .

A) 0

B) 1

C) 2

D) 3

Answer:

B) 1

Question 45.

The value of (12)^{3n} + (18)^{3n} ends in ……………… (a ∈ N)

A) 0

B) 6

C) 4

D) 2

Answer:

A) 0

Question 46.

If we apply Euclid’s division lemma for two numbers 15 and 4, then we get ……………… .

A) 15 = 4 × 1 + 11

B) 15 = 4 × 4 + (-1)

C) 15 = 4 × 3 + 3

D) 15 = 4 × 5 + (-5)

Answer:

C) 15 = 4 × 3 + 3

Question 47.

The sum of HCF and LCM of two numbers is 1260. If their LCM is 900 more than their HCF. Find the product of two numbers is …………………. .

A) 204400

B) 194400

C) 184400

D) 174400

Answer:

B) 194400

Question 48.

Aditya has two vessels which contain 720 ml and 405 ml of milk respectively. Milk in each vessel is poured into glasses of equal capacity to their brim. Find the minimum number of glasses which can be filled with milk.

A) 25

B) 35

C) 20

D) 30

Answer:

A) 25

Question 49.

Find the HCF of 432 and 504 using prime factorization method.

A) 24

B) 36

C) 48

D) 72

Answer:

D) 72

Question 50.

Decimal notation is invented by

A) John Napier

B) Ramanujan

C) George Canter

D) Aditya

Answer:

A) John Napier

Fill in the blanks :

1. \(\frac{3^3}{5^3}\) ________ is decimal.

Answer:

Terminating

2. \(\frac{7^4}{3^3 \times 5^2}\) ________ is decimal.

Answer:

Non-terminating

3. If 0.\(\overline{3}\) = \(\frac{p}{q}\), then p + q = ________ .

Answer:

\(\frac{1}{3}\); (1 + 3 = 4)

4. If a = 3 + √5 and b = 3 – √5, then a – b ________ .

Answer:

2√5

5. 3 × 7 × 11 × 13 + 22 is a ________ number.

Answer:

composite

6. The HCF of polynomials (x^{3} – 4x + 4) (x + 3) and (x^{3} + 2x – 3) (x – 2) is _________.

Answer:

(x – 2)

7. The LCM and HCF of two numbers are equal, then the numbers must be __________.

Answer:

equal

8. HCF of two co-primes a and b is __________.

Answer:

1

9. LCM of two co-primes a and b is __________.

Answer:

a.b

10. Sum of exponents of prime factors in the prime factorization of 98 is __________.

Answer:

3(2^{1} × 7^{2} → 1 + 2 = 3)

11. The exponent of 3 in the prime factorization of 144 is __________.

Answer:

2(2^{4} × 3^{2})

12. HCF of 60 and 100 is __________.

Answer:

20

13. 5.\(\overline{23}\) is __________ number.

Answer:

rational

14. The smallest rational number by which \(\frac{1}{3}\) should be multiplied, so that its

decimal expansion terminates after one place of decimal is __________.

Answer:

\(\frac{3}{10}\)

15. A positive integer a is prime, if it is not divisible by any prime less than or equal to __________.

Answer:

√n

Match the following :

Question 1.

(A) | (B) |

1. Terminating decimal | A) Euclid’s division lemma |

2. Non-terminating decimal | B) p^{2} |

3. Every composite number can be expressed as a product of primes. | C) √p |

4. a = bq + r, 0 ≤ r < b, a, b are integers, q, r are whole numbers. | D) \(\frac{3}{10}\) |

5. If p is a positive prime, then irrational number is | E) Fundamental theorem of Arithmetic |

F) \(\frac{3}{7}\) |

Answer:

(A) | (B) |

1. Terminating decimal | D) \(\frac{3}{10}\) |

2. Non-terminating decimal | F) \(\frac{3}{7}\) |

3. Every composite number can be expressed as a product of primes. | E) Fundamental theorem of Arithmetic |

4. a = bq + r, 0 ≤ r < b, a, b are integers, q, r are whole numbers. | A) Euclid’s division lemma |

5. If p is a positive prime, then irrational number is | C) √p |

Question 2.

(A) | (B) |

1. If n is natural number a^{2n} – b^{2n} is always divisible by ……….. |
A) 4 (or) 6 |

2. If n is natural number a^{2n} + b^{2n} is always divisible by ……….. |
B) 24 |

3. If n is any natural number, then 6^{n} – 5^{n} ends with ………… |
C) a – b |

4. The value of (37)^{3n} -(33)^{3n} ends in ……… (n ∈ N) ……… |
D) 625 |

5. HCF of 72 and 264 is ………. | E) 1 |

6. LCM of 25 and 625 is …….. | F) a + b |

Answer:

(A) | (B) |

1. If n is natural number a^{2n} – b^{2n} is always divisible by ……….. |
C) a – b |

2. If n is natural number a^{2n} + b^{2n} is always divisible by ……….. |
F) a + b |

3. If n is any natural number, then 6^{n} – 5^{n} ends with ………… |
E) 1 |

4. The value of (37)^{3n} -(33)^{3n} ends in ……… (n ∈ N) ………… |
A) 4 (or) 6 |

5. HCF of 72 and 264 is ………. | B) 24 |

6. LCM of 25 and 625 is ……….. | D) 625 |