Real Numbers

Which statement correctly summarizes the chapter "Real Numbers"?

1. Every decimal expansion is terminating.
2. Every real number is rational.
3. Real numbers include both rational and irrational numbers, and concepts like Euclid's Division Lemma, Fundamental Theorem of Arithmetic, HCF, LCM, and decimal expansions help understand their properties.
4. Prime numbers are irrational numbers.

Which of the following is a real number?

1. √25
2. √−9
3. Undefined quantity
4. None of these

Which pair is coprime?

1. (12,18)
2. (15,25)
3. (16,21)
4. (24,36)

Which of the following represents the prime factorization of 180?

1. 2² × 3² × 5
2. 2 × 3 × 5²
3. 2³ × 3 × 5
4. 2² × 3 × 5²

Which statement about irrational numbers is correct?

1. They can be written as p/q.
2. They have terminating decimal expansions.
3. Their decimal expansions are non-terminating and non-repeating.
4. They are all integers.

The HCF of two coprime numbers is:

1. 0
2. 1
3. 2
4. Equal to their LCM

Which of the following numbers has a non-terminating repeating decimal expansion?

1. 3/16
2. 7/20
3. 5/12
4. 9/40

If HCF = 6 and LCM = 180 for two numbers, and one number is 30, the other number is:

1. 24
2. 36
3. 42
4. 48

Which denominator gives a terminating decimal expansion after simplification?

1. 15
2. 14
3. 40
4. 21

The decimal expansion of a rational number is:

1. Always terminating
2. Always non-terminating and non-repeating
3. Either terminating or non-terminating repeating
4. Always repeating after one digit

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as:

1. Sum of prime numbers only
2. Product of prime numbers only
3. Difference of two primes
4. Square of a natural number

What is the HCF of 72 and 120?

1. 12
2. 24
3. 36
4. 48

According to Euclid's Division Lemma, for any positive integers a and b, the relation is:

1. a = bq + r, where 0 ≤ r < b
2. a = b + q + r
3. a = br + q
4. a = qr

Euclid's algorithm is used to find:

1. LCM
2. HCF
3. Square root
4. Prime factorization

If p is a prime number, then √p is:

1. Rational
2. Irrational
3. Integer
4. Natural number

Which number is not rational?

1. 0.25
2. 0.121212...
3. √9
4. √5

The HCF of 12, 15, and 21 is:

1. 1
2. 2
3. 3
4. 6

A rational number has a terminating decimal expansion if its denominator has prime factors only of:

1. 2 and 3
2. 2 and 5
3. 3 and 7
4. 2, 3, and 5

The number 2 × 3 × 5 × 7 + 7 is:

1. Prime
2. Composite
3. Irrational
4. Even

A rational number with decimal expansion 0.3333... is equal to:

1. 1/2
2. 1/3
3. 2/3
4. 3/4

HCF of consecutive even numbers is always:

1. 1
2. 2
3. 3
4. 4

Euclid's division lemma states:

1. a = bq + r, 0 ≤ r
2. a = b + r
3. a = bq
4. a + b = q

Which of the following is irrational?

1. √25
2. √49
3. √2
4. √81

If LCM = 180, HCF = 6, and one number is 30, the other is:

1. 36
2. 42
3. 60
4. 72

A number is rational if its decimal expansion is:

1. Non-terminating, non-repeating
2. Terminating or repeating
3. Only terminating
4. Always infinite

The decimal expansion of 13/3125 terminates after:

1. 3 places
2. 4 places
3. 5 places
4. 6 places

The HCF of 96 and 404 is:

1. 2
2. 4
3. 6
4. 8

Every positive integer can be written as a product of primes. This is called:

1. Fundamental theorem of algebra
2. Fundamental theorem of arithmetic
3. Euclid's lemma
4. Irrational theorem

Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. This statement is known as

1. Remainder theorem
2. Factor theorem
3. Fermat theorem
4. Euclid's Division Lemma

Find the LCM and HCF of 336 and 54.

1. 3824, 8
2. 3024, 6
3. 3085, 9
4. 3824, 6

The HCF of two numbers is 145 and their LCM is 2175. If one of the numbers is 725, find the other number.

1. 425
2. 435
3. 485
4. 475

The smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively is

1. 204
2. 208
3. 211
4. 213

The LCM of 84, 90 and 120 by applying the prime factorization method is

1. 2620
2. 2520
3. 2780
4. 2790

The prime factorization of 13915 is

1. 5 × 7² × 35 × 8²
2. 5³ × 7² × 9²
3. 5 × 11² × 23
4. 5 × 11³ × 23²

If the HCF of 408 and 1032 is expressible in the form 1032m – 408 × 5. Find m.

1. 5
2. 3
3. 2
4. 7

Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively.

1. 155
2. 154
3. 158
4. 160

The HCF of two numbers is 16 and their product is 3072. Find their LCM.

1. 195
2. 193
3. 198
4. 192

The LCM and HCF of two numbers are 180 and 6 respectively. If one of the numbers is 30, find the other number.

1. 35
2. 39
3. 36
4. 40

There is a circular path around a sports field. Priya takes 18 minutes and Ravish takes 12 minutes per round. Starting together, after how many minutes will they meet again at the starting point?

1. 38
2. 36
3. 40
4. 37

The largest positive integer that will divide 398, 436 and 542 leaving a remainder of 7, 11 and 15 respectively is

1. 19
2. 17
3. 20
4. 22

The HCF and LCM of 90 and 144 by the prime factorization method respectively are

1. 20, 540
2. 30, 742
3. 18, 720
4. 19, 648

The prime factors of positive integer 7429 are

1. 16 × 15 × 14
2. 17 × 19 × 23
3. 16 × 20 × 21
4. 16 × 14 × 20

The HCF of 867 and 225 is

1. 5
2. 4
3. 2
4. 3

The prime factorization of 16380 is

1. 2⁴ × 3³ × 5 × 7
2. 2² × 3² × 5 × 7 × 13
3. 2³ × 3³ × 5² × 7² × 11
4. 2² × 3⁴ × 5³ × 7² × 13

The HCF of 4052 and 12576 is

1. 5
2. 7
3. 6
4. 4

The following rational number 17/8 when expressed as a decimal is

1. 4.225
2. 2.125
3. 3.225
4. 3.125

Find the HCF of 867 and 255 by using Euclid's Division Algorithm.

1. 51
2. 45
3. 50
4. 55

If the sum of two numbers is 75 and their H.C.F. and L.C.M. are 5 and 240 respectively, then the sum of the reciprocals of the numbers is equal to

1. 1/16
2. 1/16
3. 5/240
4. 1/240

In a school, section A has 32 students and section B has 36 students. Determine the minimum number of books required so that they can be distributed equally among students of either section.

1. 300
2. 296
3. 288
4. 278

Which of the following is a non-terminating repeating decimal?

1. 17/8
2. 15/1600
3. 23/2³×5²
4. 77/2²×5×7

Find the HCF of 96 and 404 by prime factorization method. Hence find their LCM.

1. 9595
2. 9696
3. 9292
4. 9393

Find the greatest number of 6 digits exactly divisible by 24, 15 and 36.

1. 999999
2. 999789
3. 999000
4. 999720

Given H.C.F (306, 657) = 9, find L.C.M. (306, 657).

1. 22222
2. 22328
3. 22302
4. 22338

Find the H.C.F and L.C.M. of 25152 and 12156 by using the Fundamental Theorem of Arithmetic.

1. 24457576
2. 25478976
3. 25478679
4. 24456567

Find the HCF and LCM of 90 and 144 by the prime factorization method.

1. 15, 20
2. 15, 720
3. 18, 720
4. 18, 520

Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.

1. 48
2. 64
3. 20
4. 16

The length, breadth and height of a room are 8m 25cm, 6m 75cm and 4m 50cm respectively. Determine the longest rod which can measure the three dimensions exactly.

1. 65 cm
2. 77 cm
3. 75 cm
4. 80 cm

Find the HCF of 65 and 117 and express it in the form 65m + 117n.

1. m = –2, n = –1
2. m = 2, n = –1
3. m = 3, n = –1
4. m = 2, n = 1

In a seminar, the number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. Find the maximum number of rooms required if each room seats the same number of participants from the same subject.

1. 17
2. 21
3. 27
4. 19

If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y, find y.

1. 19
2. 15
3. –19
4. –21

Priya takes 18 minutes and Ravish takes 12 minutes to drive one round of a circular sports field. Starting at the same point and time in the same direction, after how many minutes will they meet again at the starting point?

1. 30
2. 36
3. 40
4. 26

Find the HCF of 96 and 404 by prime factorization method. Hence find the LCM.

1. 1000
2. 9600
3. 9640
4. 9696

Find the largest number that will divide 2053 and 967 and leave a remainder of 5 and 7 respectively.

1. 128
2. 54
3. 256
4. 64

In a problem involving division, the divisor is eight times the quotient and four times the remainder. If the remainder is 12, then the dividend is

1. 400
2. 342
3. 300
4. 450

The G.C.D of two numbers is 16 and the first 4 quotients obtained in the division are equal to 2. Then the numbers are

1. 342, 126
2. 464, 192
3. 232, 90
4. 768, 336

Three numbers which are co-primes to each other are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is

1. 75
2. 81
3. 85
4. 89

The largest number which divides 62, 132 and 237 and leaves the same remainder in each case is

1. 34
2. 33
3. 35
4. 36

The G.C.D of (2002, k) = 4, then the value of k is

1. All even values
2. 4 only
3. All odd values
4. For all values of k, it is not possible

H.C.F of two numbers is 18 and the first 4 quotients obtained in the division are 2, 1, 2, 2. Then the two numbers are

1. 342, 126
2. 343, 126
3. 342, 125
4. 343, 125

Euclid's division Lemma states that if a and b are any two positive integers, then there exist unique integers q and r such that

1. a = bq + r, 0 < r ≤ b
2. a = bq + r, 0 ≤ q < b
3. a = bq + r, 0 ≤ r < b
4. a = bq + r, 0 < q ≤ b

Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.

1. 135
2. 160
3. 170
4. 210

H.C.F. of (x³ – 3x + 2) and (x² – 4x + 3) is

1. (x – 1)
2. (x – 2)²
3. (x – 1)(x + 2)
4. (x – 1)(x – 3)

The greatest number of five digits which on being divided by 56, 72, 84 and 96 leaves 50, 66, 78 and 90 as remainders respectively is

1. 98784
2. 98778
3. 98790
4. 97778

The least perfect square number which is divisible by 8, 15, 20, 22 is

1. 435600
2. 43560
3. 39600
4. 465660

The greatest number of five digits exactly divisible by 279 is

1. 99603
2. 99837
3. 99882
4. 99881

The nearest integer to 58701 which is divisible by 567 is

1. 58968
2. 58434
3. 58401
4. 58989