Single Best Answer MCQs for Sub-Topics of Topic 1: Numbers & Numeriacal Applications

Question 1. Which of the following collections represents the set of natural numbers?

(A) $\lbrace 0, 1, 2, 3, \dots \rbrace$

(B) $\lbrace 1, 2, 3, 4, \dots \rbrace$

(C) $\lbrace \dots, -2, -1, 0, 1, 2, \dots \rbrace$

(D) $\lbrace \frac{p}{q} \mid p, q \in \mathbb{Z}, q \neq 0 \rbrace$

Question 2. The smallest whole number is:

(A) 1

(B) 0

(C) -1

(D) Any positive integer

Question 3. Which statement is TRUE about integers?

(A) All natural numbers are integers.

(B) All integers are whole numbers.

(C) All fractions are integers.

(D) Integers only include positive numbers.

Question 4. A rational number can always be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and:

(A) $q = 0$

(B) $p = 0$

(C) $q \neq 0$

(D) $p \neq 0$

Question 5. Which of the following is an irrational number?

(A) $\sqrt{9}$

(B) $\frac{22}{7}$

(C) $\pi$

(D) $0.333\dots$

Question 6. The set of real numbers is the union of:

(A) Natural numbers and Integers

(B) Whole numbers and Rational numbers

(C) Rational numbers and Irrational numbers

(D) Integers and Irrational numbers

Question 7. Which of the following numbers is a composite number?

(A) 7

(B) 13

(C) 19

(D) 21

Question 8. The number 0 is a:

(A) Natural number

(B) Whole number

(C) Positive integer

(D) Both (A) and (B)

Question 9. Which of these numbers is neither prime nor composite?

(A) 0

(B) 1

(C) 2

(D) 3

Question 10. A terminating decimal is always a:

(A) Natural number

(B) Integer

(C) Rational number

(D) Irrational number

Question 11. Which set includes all other sets listed?

(A) Natural numbers

(B) Integers

(C) Rational numbers

(D) Real numbers

Question 12. The number $\sqrt[3]{8}$ is a:

(A) Irrational number

(B) Rational number

(C) Whole number

(D) Both (B) and (C)

Question 13. Which of the following is a property of natural numbers?

(A) Closure under subtraction

(B) Closure under division

(C) Associativity of addition

(D) Existence of additive inverse

Question 14. The number $-5$ is a:

(A) Natural number

(B) Whole number

(C) Integer

(D) All of the above

Question 15. Which statement is FALSE?

(A) Every integer is a rational number.

(B) Every rational number is a real number.

(C) Every real number is either rational or irrational.

(D) Every rational number is an integer.

Question 16. The decimal expansion of an irrational number is always:

(A) Terminating

(B) Non-terminating and recurring

(C) Non-terminating and non-recurring

(D) Terminating or non-terminating recurring

Question 17. Which of the following is a pair of twin primes?

(A) (2, 3)

(B) (3, 5)

(C) (4, 5)

(D) (6, 8)

Question 18. A number that is both positive and an integer but not a natural number is:

(A) -1

(B) 0

(C) 0.5

(D) Such a number does not exist.

Question 19. The sum of a rational and an irrational number is always:

(A) Rational

(B) Irrational

(C) An integer

(D) A whole number

Question 20. Which of the following is a property of whole numbers but not natural numbers?

(A) Closure under addition

(B) Existence of additive identity

(C) Commutativity of multiplication

(D) Associativity of addition

Question 1. In the Indian System of Numeration, the number 7,56,43,210 is read as:

(A) Seventy-five million four hundred thirty-two thousand one hundred ten

(B) Seven crore fifty-six lakh forty-three thousand two hundred ten

(C) Seven hundred fifty-six lakh four hundred thirty-two thousand one hundred ten

(D) Seven crore fifty-six million four hundred thirty-two thousand one hundred ten

Question 2. What is the place value of the digit 8 in the number 98,765 using the International System of Numeration?

(A) Thousands

(B) Ten thousands

(C) Hundred thousands

(D) Millions

Question 3. The number 'fifty-two crore sixty-five lakh nine thousand seven hundred' in figures (Indian System) is:

(A) 52,65,09,700

(B) 52,65,970,000

(C) 52,65,90,700

(D) 5,265,097,00

Question 4. How many millions make one crore?

(A) 10

(B) 100

(C) 1000

(D) 10000

Question 5. In the decimal number system, the place value of a digit increases by how many times as it moves one place to the left?

(A) 1 time

(B) 5 times

(C) 10 times

(D) 100 times

Question 6. The expanded form of the number 45,607 is:

(A) $4 \times 1000 + 5 \times 100 + 6 \times 10 + 0 \times 1 + 7 \times 0$

(B) $4 \times 10000 + 5 \times 1000 + 6 \times 100 + 0 \times 10 + 7 \times 1$

(C) $4 \times 100000 + 5 \times 10000 + 6 \times 1000 + 7 \times 10$

(D) $4000 + 500 + 60 + 7$

Question 7. The value of the digit 3 in the number 1.234 is:

(A) 3 tenths

(B) 3 hundredths

(C) 3 thousandths

(D) 3 units

Question 8. Which of the following is the Roman numeral for 99?

(A) IC

(B) XCIX

(C) XCIXI

(D) LXLIX

Question 9. How is the number 50,000 written in general form in terms of its digits $a_4, a_3, a_2, a_1, a_0$?

(A) $a_4 \times 10^4 + a_3 \times 10^3 + a_2 \times 10^2 + a_1 \times 10^1 + a_0 \times 10^0$ where $a_4=5, a_3=0, \dots, a_0=0$

(B) $a_4 \times 10^5 + a_3 \times 10^4 + a_2 \times 10^3 + a_1 \times 10^2 + a_0 \times 10^1$ where $a_4=5, a_3=0, \dots, a_0=0$

(C) $a_4 \times 10000 + a_3 \times 1000 + a_2 \times 100 + a_1 \times 10 + a_0 \times 1$ where $a_4=5, a_3=0, \dots, a_0=0$

(D) $a_5 \times 10^5 + a_4 \times 10^4 + \dots + a_0 \times 10^0$ where $a_5=5, \dots, a_0=0$

Question 10. In the number 3,456,789, the value of the digit 4 is how many times the value of the digit 5?

(A) 10 times

(B) 100 times

(C) 1000 times

(D) Not a fixed multiple

Question 11. The number 1,000,000 in the International System is equal to how many lakhs in the Indian System?

(A) 1 lakh

(B) 10 lakhs

(C) 100 lakhs

(D) 1000 lakhs

Question 12. What is the smallest 5-digit number formed using the digits 8, 0, 5, 3, 1, each exactly once?

(A) 01358

(B) 10358

(C) 10385

(D) 13058

Question 13. The Roman numeral XL represents:

(A) 40

(B) 60

(C) 90

(D) 110

Question 14. The difference between the place value and the face value of the digit 6 in the number 76,853 is:

(A) 6000

(B) 5994

(C) 6

(D) 6006

Question 15. How is 'one hundred twenty-five million, three hundred thousand, fifty' written in figures (International System)?

(A) 125,300,050

(B) 125,030,050

(C) 125,300,500

(D) 125,350,000

Question 16. The general form of a 3-digit number $abc$ is:

(A) $a + b + c$

(B) $a \times b \times c$

(C) $100a + 10b + c$

(D) $a^3 + b^2 + c^1$

Question 17. How many thousands make one lakh?

(A) 10

(B) 100

(C) 1000

(D) 10000

Question 18. The largest 4-digit number using the digits 9, 0, 2, 5 exactly once is:

(A) 9025

(B) 9502

(C) 9520

(D) 9250

Question 19. The Roman numeral CM represents:

(A) 900

(B) 1100

(C) 400

(D) 600

Question 20. What is the difference between the place value of 7 and 3 in the number 7,832?

(A) $7000 - 30 = 6970$

(B) $700 - 30 = 670$

(C) $7000 - 3 = 6997$

(D) $700 - 3 = 697$

Question 1. On a number line, where is the number -3 located relative to 0?

(A) To the right of 0

(B) To the left of 0

(C) At 0

(D) It depends on the scale

Question 2. To represent $\frac{3}{4}$ on a number line between 0 and 1, you need to divide the segment into how many equal parts?

(A) 3

(B) 4

(C) 7

(D) 12

Question 3. Which point on the number line is likely to represent 2.5?

(A) A point exactly midway between 0 and 1

(B) A point exactly midway between 1 and 2

(C) A point exactly midway between 2 and 3

(D) A point exactly midway between 3 and 4

Question 4. To represent $\frac{7}{3}$ on the number line, it should be located between:

(A) 0 and 1

(B) 1 and 2

(C) 2 and 3

(D) 3 and 4

Question 5. To represent $\sqrt{2}$ on the number line, we typically construct a right-angled triangle with legs of length:

(A) 1 and 1

(B) 1 and 2

(C) 2 and 2

(D) 1 and $\sqrt{2}$

Question 6. The point representing $-1 \frac{1}{2}$ on the number line is:

(A) To the right of -1

(B) To the left of -1

(C) At -1

(D) At 1.5

Question 7. Successive magnification is a technique used to visualize the representation of numbers with:

(A) Terminating decimal expansions

(B) Non-terminating recurring decimal expansions

(C) Non-terminating non-recurring decimal expansions

(D) All of the above, to locate them more precisely

Question 8. To represent $\sqrt{5}$ on the number line, one method is to construct a right triangle with a base of length 2 and a perpendicular of length:

(A) 1

(B) $\sqrt{2}$

(C) $\sqrt{3}$

(D) 2

Question 9. On the number line, as you move towards the right, the numbers:

(A) Decrease

(B) Increase

(C) Remain the same

(D) Alternate between increasing and decreasing

Question 10. To represent $3.765$ on the number line using successive magnification, you would first locate it between:

(A) 3 and 4

(B) 3.7 and 3.8

(C) 3.76 and 3.77

(D) 3.765 and 3.766

Question 11. The distance between two points representing numbers $a$ and $b$ on the number line is given by:

(A) $a+b$

(B) $a-b$

(C) $|a-b|$

(D) $|a+b|$

Question 12. Which of the following is NOT a rational number representation?

(A) A terminating decimal

(B) A non-terminating recurring decimal

(C) A point obtained by constructing $\sqrt{3}$

(D) An integer

Question 13. To represent $\sqrt{x}$ for any positive number $x$ on the number line, one common method involves constructing a semicircle on a diameter of length:

(A) $x$

(B) $x+1$

(C) $x-1$

(D) $2x$

Question 14. The number line provides a visual representation for which set of numbers?

(A) Natural numbers

(B) Integers

(C) Rational numbers

(D) Real numbers

Question 15. Between any two distinct rational numbers on the number line, there exists:

(A) Exactly one rational number

(B) Exactly one irrational number

(C) Infinitely many rational numbers and infinitely many irrational numbers

(D) No other numbers

Question 16. Which point would be closest to -2 on the number line?

(A) -1.9

(B) -2.1

(C) -2.01

(D) -1.99

Question 17. The point on the number line representing $\frac{-5}{2}$ is equivalent to the point representing:

(A) -2.5

(B) -0.4

(C) 2.5

(D) 0.4

Question 18. To accurately locate a number like $5.3\overline{7}$ on the number line using successive magnification, you would magnify the segment between:

(A) 5 and 6

(B) 5.3 and 5.4

(C) 5.37 and 5.38

(D) 5.377 and 5.378

Question 19. Which geometric construction on the number line can be used to represent square roots of positive integers?

(A) Drawing circles

(B) Constructing perpendicular bisectors

(C) Using the Pythagorean theorem

(D) Drawing parallel lines

Question 20. The representation of integers on the number line is:

(A) A continuous line

(B) A set of discrete points at equal intervals

(C) A shaded region

(D) A set of clustered points

Question 1. Which of the following inequalities is correct?

(A) $-5 > -3$

(B) $0 > -2$

(C) $-10 < -12$

(D) $4 < -6$

Question 2. The absolute value of -15 is:

(A) -15

(B) 15

(C) 0

(D) $-|15|$

Question 3. Which method is generally used to compare two fractions, say $\frac{a}{b}$ and $\frac{c}{d}$?

(A) Comparing numerators

(B) Comparing denominators

(C) Finding a common denominator

(D) Finding the sum of numerators and denominators

Question 4. How many rational numbers are there between 0 and 1?

(A) None

(B) One

(C) A finite number

(D) Infinitely many

Question 5. To compare 0.45 and 0.450, you should consider:

(A) The number of digits after the decimal point

(B) The place value of the last digit

(C) That trailing zeros after the decimal point do not change the value

(D) The sum of the digits

Question 6. Which of the following numbers is the smallest?

(A) -0.1

(B) -0.01

(C) -0.101

(D) -0.001

Question 7. The value of $|-2.7|$ is:

(A) -2.7

(B) 2.7

(C) $\frac{1}{2.7}$

(D) $-\frac{1}{2.7}$

Question 8. A number between $\frac{1}{2}$ and $\frac{3}{4}$ is:

(A) $\frac{1}{4}$

(B) $\frac{2}{3}$

(C) $\frac{4}{5}$

(D) $\frac{7}{8}$

Question 9. Which number is greater: $10^{100}$ or $100^{10}$?

(A) $10^{100}$

(B) $100^{10}$

(C) They are equal

(D) Cannot be determined

Question 10. The absolute value of 0 is:

(A) -1

(B) 1

(C) 0

(D) Undefined

Question 11. To find rational numbers between two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$, one method is to find their average:

(A) $\frac{a+c}{b+d}$

(B) $\frac{a/b + c/d}{2}$

(C) $\frac{a+b}{c+d}$

(D) $\frac{ac}{bd}$

Question 12. Which is the correct ascending order for $\frac{1}{3}, 0.3, 33\%$?

(A) $0.3, \frac{1}{3}, 33\%$

(B) $33\%, 0.3, \frac{1}{3}$

(C) $\frac{1}{3}, 0.3, 33\%$

(D) $0.3, 33\%, \frac{1}{3}$

Question 13. How many irrational numbers are there between 2 and 3?

(A) None

(B) A finite number

(C) Infinitely many

(D) Exactly one

Question 14. Comparing $-1.5$ and $-1.05$, which is smaller?

(A) -1.5

(B) -1.05

(C) They are equal

(D) Cannot be determined

Question 15. The distance of a number from zero on the number line is its:

(A) Value

(B) Sign

(C) Absolute value

(D) Reciprocal

Question 16. Which of the following is a number between $\sqrt{2}$ and $\sqrt{3}$?

(A) 1.4

(B) 1.5

(C) 1.7

(D) 1.8

Question 17. Which inequality represents numbers whose absolute value is less than 4?

(A) $|x| > 4$

(B) $|x| \geq 4$

(C) $|x| < 4$

(D) $|x| \leq 4$

Question 18. To compare $3.14$ and $\pi$, you should know that:

(A) $3.14 > \pi$

(B) $3.14 < \pi$

(C) $3.14 = \pi$

(D) They are irrational numbers, so cannot be compared easily.

Question 19. When comparing two negative integers, the integer with the greater absolute value is the:

(A) Larger integer

(B) Smaller integer

(C) Positive integer

(D) Integer closer to zero

Question 20. Which of the following statements is true about comparing very large and very small positive numbers?

(A) A number with more digits before the decimal point is generally larger.

(B) For numbers less than 1, a number with more zeros immediately after the decimal point is larger.

(C) Scientific notation makes comparison more difficult.

(D) The sign of the exponent in scientific notation is irrelevant for comparison.

Question 1. The sum of 1234 and 5678 is:

(A) 6802

(B) 6912

(C) 7912

(D) 7802

Question 2. $15 \times (-4)$ equals:

(A) -60

(B) 60

(C) -11

(D) 11

Question 3. Simplify: $\frac{2}{3} + \frac{1}{6}$

(A) $\frac{3}{9}$

(B) $\frac{3}{18}$

(C) $\frac{5}{6}$

(D) $\frac{3}{6}$

Question 4. Subtract 3.15 from 7.8:

(A) 4.65

(B) 4.75

(C) 5.35

(D) 5.65

Question 5. The product of $\frac{-3}{5}$ and $\frac{10}{9}$ is:

(A) $\frac{-30}{45}$

(B) $\frac{-2}{3}$

(C) $\frac{-1}{3}$

(D) $\frac{2}{3}$

Question 6. Simplify using BODMAS: $10 + 2 \times 5 - (8 \div 2)$

(A) 16

(B) 10

(C) 20

(D) 12

Question 7. What is $-10 - (-12)$?

(A) -22

(B) 22

(C) -2

(D) 2

Question 8. Divide $\frac{5}{8}$ by $\frac{15}{16}$:

(A) $\frac{75}{128}$

(B) $\frac{3}{2}$

(C) $\frac{2}{3}$

(D) $\frac{5}{16}$

Question 9. The value of $5.25 \div 0.5$ is:

(A) 10.5

(B) 1.05

(C) 2.625

(D) 0.105

Question 10. If you multiply an irrational number by a non-zero rational number, the result is always:

(A) Rational

(B) Irrational

(C) An integer

(D) A real number (could be rational or irrational)

Question 11. Simplify: $(-2) \times (-3) \times (-4)$

(A) -24

(B) 24

(C) -9

(D) 9

Question 12. Calculate: $10 - [5 - (3 - (2-1))]$

(A) 7

(B) 8

(C) 9

(D) 10

Question 13. The reciprocal of $-\frac{7}{12}$ is:

(A) $\frac{7}{12}$

(B) $\frac{12}{7}$

(C) $-\frac{12}{7}$

(D) $\frac{1}{-\frac{7}{12}}$

Question 14. Find the value of $5.6 + 2.31 - 1.025$:

(A) 6.885

(B) 6.985

(C) 7.885

(D) 7.985

Question 15. Simplify: $\sqrt{2} \times \sqrt{8}$

(A) $\sqrt{16}$

(B) 4

(C) $\sqrt{10}$

(D) 16

Question 16. What is $45 \div (-9)$?

(A) -5

(B) 5

(C) -405

(D) 405

Question 17. Calculate: $(\frac{1}{2} + \frac{1}{3}) \div \frac{5}{6}$

(A) 1

(B) $\frac{25}{36}$

(C) $\frac{1}{6}$

(D) 5

Question 18. Evaluate $0.1 \times 0.1 \times 0.1$:

(A) 0.1

(B) 0.01

(C) 0.001

(D) 0.0001

Question 19. Simplify: $5 \times 0 + 10 \div 2 - 3$

(A) 0

(B) 2

(C) 5

(D) 7

Question 20. The result of $(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})$ is a:

(A) Rational number

(B) Irrational number

(C) Negative integer

(D) Prime number

Question 1. The property $(a+b)+c = a+(b+c)$ is known as the:

(A) Commutative property of addition

(B) Associative property of addition

(C) Distributive property

(D) Identity property of addition

Question 2. Which property is illustrated by $5 \times 1 = 5$?

(A) Commutative property of multiplication

(B) Associative property of multiplication

(C) Multiplicative identity

(D) Multiplicative inverse

Question 3. If a number is divisible by both 2 and 3, it must also be divisible by:

(A) 5

(B) 6

(C) 8

(D) 9

Question 4. For any real number $a$, $a + (-a) = 0$. This illustrates the existence of the:

(A) Additive identity

(B) Additive inverse

(C) Multiplicative identity

(D) Multiplicative inverse

Question 5. The property $a(b+c) = ab + ac$ is the:

(A) Commutative property

(B) Associative property

(C) Distributive property

(D) Closure property

Question 6. A number is divisible by 9 if the sum of its digits is divisible by:

(A) 3

(B) 6

(C) 9

(D) 18

Question 7. Which of the following is NOT a square number?

(A) 121

(B) 144

(C) 169

(D) 196

Question 8. The sum of any two odd numbers is always an:

(A) Odd number

(B) Even number

(C) Prime number

(D) Composite number

Question 9. If $a \times b = 1$, then $a$ and $b$ are called:

(A) Additive inverses

(B) Multiplicative identities

(C) Additive identities

(D) Multiplicative inverses (Reciprocals)

Question 10. A number is divisible by 4 if the number formed by its last two digits is divisible by:

(A) 2

(B) 4

(C) 8

(D) 12

Question 11. The sum of the first $n$ odd natural numbers is:

(A) $n$

(B) $2n$

(C) $n^2$

(D) $n(n+1)$

Question 12. For real numbers, which operation is NOT commutative?

(A) Addition

(B) Subtraction

(C) Multiplication

(D) None of the above

Question 13. A number is divisible by 8 if the number formed by its last three digits is divisible by:

(A) 4

(B) 6

(C) 8

(D) 16

Question 14. The identity element for addition in the set of real numbers is:

(A) 0

(B) 1

(C) -1

(D) Undefined

Question 15. The square of an even number is always:

(A) Odd

(B) Even

(C) Prime

(D) Perfect cube

Question 16. If a number ends with the digit 5, for it to be divisible by 5, the digit in the units place must be:

(A) 0

(B) 5

(C) 0 or 5

(D) Any digit

Question 17. The sum of two irrational numbers can be:

(A) Always rational

(B) Always irrational

(C) Either rational or irrational

(D) Always an integer

Question 18. What is the pattern for triangular numbers?

(A) Sum of consecutive integers starting from 1

(B) Numbers obtained by squaring integers

(C) Numbers that are multiples of 3

(D) Numbers that are perfect cubes

Question 19. The property $a \times \frac{1}{a} = 1$ (for $a \neq 0$) illustrates the existence of the:

(A) Multiplicative identity

(B) Multiplicative inverse

(C) Additive inverse

(D) Commutative property

Question 20. A number is divisible by 10 if its units digit is:

(A) 0

(B) 5

(C) 0 or 5

(D) Any even number

Question 1. In the fraction $\frac{3}{7}$, the numerator is:

(A) 3

(B) 7

(C) 10

(D) 4

Question 2. Which of the following is an improper fraction?

(A) $\frac{2}{5}$

(B) $\frac{7}{7}$

(C) $\frac{9}{8}$

(D) $\frac{1}{10}$

Question 3. The decimal equivalent of $\frac{1}{4}$ is:

(A) 0.14

(B) 0.25

(C) 0.5

(D) 0.4

Question 4. Converting $0.6$ to a fraction in simplest form gives:

(A) $\frac{6}{10}$

(B) $\frac{3}{5}$

(C) $\frac{6}{100}$

(D) $\frac{1}{6}$

Question 5. Which of the following fractions is equivalent to $\frac{2}{5}$?

(A) $\frac{4}{10}$

(B) $\frac{20}{50}$

(C) $\frac{6}{15}$

(D) All of the above

Question 6. The mixed number $3 \frac{1}{2}$ is equivalent to the improper fraction:

(A) $\frac{3}{2}$

(B) $\frac{7}{2}$

(C) $\frac{4}{2}$

(D) $\frac{6}{2}$

Question 7. Which of the following is a pair of like decimals?

(A) 0.2, 0.20

(B) 1.5, 2.34

(C) 0.05, 0.500

(D) 10.1, 1.01

Question 8. The fraction $\frac{18}{24}$ reduced to its simplest form is:

(A) $\frac{9}{12}$

(B) $\frac{3}{4}$

(C) $\frac{6}{8}$

(D) $\frac{2}{3}$

Question 9. Convert the decimal $0.125$ to a fraction in simplest form:

(A) $\frac{125}{1000}$

(B) $\frac{1}{8}$

(C) $\frac{1}{4}$

(D) $\frac{25}{200}$

Question 10. Which is greater: $\frac{3}{5}$ or 0.7?

(A) $\frac{3}{5}$

(B) 0.7

(C) They are equal

(D) Cannot compare

Question 11. The decimal representation of $\frac{5}{8}$ is:

(A) 0.625

(B) 0.58

(C) 0.85

(D) 0.588...

Question 12. A unit fraction is a fraction where the numerator is:

(A) 0

(B) 1

(C) Equal to the denominator

(D) Less than the denominator

Question 13. The fraction form of $2.3$ is:

(A) $\frac{23}{100}$

(B) $\frac{23}{10}$

(C) $\frac{2}{3}$

(D) $\frac{3}{2}$

Question 14. Which of the following is a proper fraction?

(A) $\frac{5}{4}$

(B) $\frac{10}{10}$

(C) $\frac{3}{8}$

(D) $\frac{12}{5}$

Question 15. The decimal equivalent of $1 \frac{1}{5}$ is:

(A) 1.15

(B) 1.2

(C) 1.5

(D) 1.05

Question 16. To reduce a fraction to its lowest terms, you divide the numerator and denominator by their:

(A) LCM

(B) Product

(C) HCF

(D) Sum

Question 17. Which type of decimal has a finite number of digits after the decimal point?

(A) Recurring decimal

(B) Terminating decimal

(C) Non-terminating decimal

(D) Irrational decimal

Question 18. The fraction $\frac{7}{100}$ written as a decimal is:

(A) 0.7

(B) 0.07

(C) 7.00

(D) 0.007

Question 19. The value of $0.5 + \frac{1}{2}$ is:

(A) 0.5

(B) 1

(C) $\frac{1}{4}$

(D) 0.25

Question 20. To convert a fraction to a decimal, you divide the:

(A) Denominator by the numerator

(B) Numerator by the denominator

(C) Numerator by 100

(D) Denominator by 10

Question 1. The decimal expansion of $\frac{1}{7}$ is:

(A) Terminating

(B) Non-terminating and recurring

(C) Non-terminating and non-recurring

(D) An integer

Question 2. Which of the following numbers has a terminating decimal expansion?

(A) $\frac{1}{3}$

(B) $\frac{2}{7}$

(C) $\frac{3}{8}$

(D) $\frac{5}{6}$

Question 3. The decimal expansion $0.1010010001\dots$ is:

(A) Terminating

(B) Non-terminating and recurring

(C) Non-terminating and non-recurring

(D) Cannot be determined

Question 4. Express $0.\overline{6}$ in $\frac{p}{q}$ form:

(A) $\frac{6}{10}$

(B) $\frac{2}{3}$

(C) $\frac{6}{99}$

(D) $\frac{6}{100}$

Question 5. The decimal expansion of an irrational number is always:

(A) Terminating

(B) Non-terminating recurring

(C) Non-terminating non-recurring

(D) An integer

Question 6. The process of converting a non-terminating recurring decimal to a fraction $\frac{p}{q}$ involves:

(A) Division

(B) Multiplication by powers of $10$ and subtraction

(C) Taking square roots

(D) Prime factorisation

Question 7. Rationalize the denominator of $\frac{1}{\sqrt{2}}$:

(A) $\frac{1}{2}$

(B) $\sqrt{2}$

(C) $\frac{\sqrt{2}}{2}$

(D) $\frac{2}{\sqrt{2}}$

Question 8. Which of the following cannot be represented as a terminating decimal?

(A) $\frac{1}{5}$

(B) $\frac{3}{20}$

(C) $\frac{7}{16}$

(D) $\frac{1}{6}$

Question 9. The decimal expansion of a rational number is either terminating or:

(A) Non-terminating and non-recurring

(B) Terminating and recurring

(C) Non-terminating and recurring

(D) An integer

Question 10. Express $0.\overline{23}$ in $\frac{p}{q}$ form:

(A) $\frac{23}{100}$

(B) $\frac{23}{99}$

(C) $\frac{23}{90}$

(D) $\frac{23}{990}$

Question 11. To prove that $\sqrt{2}$ is irrational, a common method is to assume it is rational and use:

(A) Direct proof

(B) Proof by contradiction

(C) Proof by induction

(D) Proof by construction

Question 12. Rationalize the denominator of $\frac{1}{2+\sqrt{3}}$:

(A) $2+\sqrt{3}$

(B) $2-\sqrt{3}$

(C) $\frac{2+\sqrt{3}}{7}$

(D) $\frac{2-\sqrt{3}}{-1}$

Question 13. The decimal expansion of $\frac{1}{8}$ is:

(A) $0.125$

(B) $0.12\overline{5}$

(C) $0.\overline{125}$

(D) $0.8$

Question 14. Which of the following is an irrational number whose decimal expansion is $1.41421356\dots$?

(A) $\sqrt{1.4}$

(B) $\sqrt{2}$

(C) $\sqrt{3}$

(D) $\sqrt{4}$

Question 15. Express $0.4\overline{7}$ in $\frac{p}{q}$ form:

(A) $\frac{47}{100}$

(B) $\frac{43}{90}$

(C) $\frac{47}{99}$

(D) $\frac{43}{99}$

Question 16. Rationalize the denominator of $\frac{5}{\sqrt{7} - \sqrt{5}}$:

(A) $\frac{5(\sqrt{7}+\sqrt{5})}{2}$

(B) $\frac{5(\sqrt{7}-\sqrt{5})}{2}$

(C) $\frac{5\sqrt{7}+5\sqrt{5}}{12}$

(D) $\frac{5\sqrt{7}-5\sqrt{5}}{12}$

Question 17. The prime factors of the denominator of a fraction, when in its simplest form, determine if its decimal expansion is terminating. For a terminating decimal, the prime factors must only be:

(A) $2$ and $3$

(B) $2$ and $5$

(C) $3$ and $5$

(D) Any prime number

Question 18. Which of the following is NOT a rational number?

(A) $3.14$

(B) $3.\overline{14}$

(C) $\sqrt{16}$

(D) $\sqrt{10}$

Question 19. The reciprocal of $\frac{1}{2-\sqrt{3}}$ after rationalizing the denominator is:

(A) $2+\sqrt{3}$

(B) $2-\sqrt{3}$

(C) $\frac{1}{2+\sqrt{3}}$

(D) $\frac{1}{2-\sqrt{3}}$

Question 20. The number $0.202002000\dots$ is:

(A) Rational

(B) Irrational

(C) An integer

(D) A terminating decimal

Question 1. A factor of a number is a number that divides it completely, leaving a remainder of:

(A) $1$

(B) The number itself

(C) $0$

(D) Any number

Question 2. Which of the following is a multiple of 7?

(A) 14

(B) 21

(C) 35

(D) All of the above

Question 3. A number is divisible by 5 if its unit digit is:

(A) 0

(B) 5

(C) 0 or 5

(D) Any even digit

Question 4. Which of the following is a prime number?

(A) 4

(B) 9

(C) 11

(D) 15

Question 5. The number 1 is:

(A) A prime number

(B) A composite number

(C) Neither prime nor composite

(D) An even number

Question 6. To check if a number is prime, you only need to test divisibility by prime numbers up to the:

(A) Number itself

(B) Number minus 1

(C) Square root of the number

(D) Half of the number

Question 7. The prime factorization of 36 is:

(A) $4 \times 9$

(B) $2 \times 18$

(C) $2^2 \times 3^2$

(D) $3 \times 12$

Question 8. Which of the following numbers is divisible by 3?

(A) 124

(B) 356

(C) 789

(D) 901

Question 9. The smallest prime number is:

(A) 0

(B) 1

(C) 2

(D) 3

Question 10. A number is divisible by 6 if it is divisible by both:

(A) 2 and 4

(B) 2 and 3

(C) 3 and 4

(D) 2 and 8

Question 11. The factors of 12 are:

(A) 1, 2, 3, 4, 6, 12

(B) 1, 2, 3, 4, 12

(C) 1, 2, 3, 6, 12

(D) 12, 24, 36, ...

Question 12. Which of the following is a composite number?

(A) 2

(B) 3

(C) 5

(D) 6

Question 13. A number is divisible by 11 if the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) is either 0 or a multiple of:

(A) 2

(B) 5

(C) 11

(D) 9

Question 14. The multiples of 4 are:

(A) 1, 2, 4

(B) 4, 8, 12, 16, ...

(C) All even numbers

(D) All numbers divisible by 2

Question 15. The number of factors of a prime number is always:

(A) 1

(B) 2

(C) 3

(D) More than 2

Question 16. The prime factorization of 100 is:

(A) $10 \times 10$

(B) $2 \times 50$

(C) $2^2 \times 5^2$

(D) $4 \times 25$

Question 17. Which of the following numbers is divisible by 10?

(A) 105

(B) 120

(C) 234

(D) 567

Question 18. The smallest composite number is:

(A) 1

(B) 2

(C) 3

(D) 4

Question 19. If a number is divisible by 9, it is also divisible by:

(A) 3

(B) 6

(C) 18

(D) 2

Question 20. How many prime numbers are there between 1 and 10?

(A) 3

(B) 4

(C) 5

(D) 6

Question 1. The HCF of 12 and 18 is:

(A) 36

(B) 6

(C) 3

(D) 2

Question 2. The LCM of 8 and 12 is:

(A) 4

(B) 24

(C) 96

(D) 12

Question 3. If the product of two numbers is $120$ and their HCF is $6$, their LCM is:

(A) 20

(B) 720

(C) 114

(D) 126

Question 4. The HCF of two prime numbers is always:

(A) The product of the numbers

(B) The smaller number

(C) 1

(D) The larger number

Question 5. The LCM of two co-prime numbers is always:

(A) Their HCF

(B) Their product

(C) 1

(D) The sum of the numbers

Question 6. Find the HCF of 42 and 98 using prime factorization:

(A) $2 \times 7 = 14$

(B) $2 \times 3 \times 7^2 = 294$

(C) $2 \times 3 \times 7 = 42$

(D) $7$

Question 7. Find the LCM of 15, 20, and 25:

(A) 5

(B) 100

(C) 300

(D) 600

Question 8. Two numbers are in the ratio 3:4. If their HCF is 4, their LCM is:

(A) 12

(B) 16

(C) 48

(D) 64

Question 9. What is the HCF of $2^3 \times 3^2 \times 5$ and $2^2 \times 3^3 \times 7$?

(A) $2^2 \times 3^2$

(B) $2^3 \times 3^3 \times 5 \times 7$

(C) $2^3 \times 3^3$

(D) $2^2 \times 3^2 \times 5 \times 7$

Question 10. The LCM of $2^3 \times 3^2 \times 5$ and $2^2 \times 3^3 \times 7$ is:

(A) $2^2 \times 3^2$

(B) $2^3 \times 3^3 \times 5 \times 7$

(C) $2^3 \times 3^3$

(D) $2^2 \times 3^2 \times 5 \times 7$

Question 11. Find the greatest number that divides 60 and 96 exactly.

(A) 6

(B) 12

(C) 10

(D) 15

Question 12. Find the smallest number that is divisible by 18, 24, and 30.

(A) 6

(B) 120

(C) 360

(D) 720

Question 13. The HCF of two numbers is 16 and their product is 3072. Their LCM is:

(A) 192

(B) 49152

(C) 16

(D) 3056

Question 14. Three bells ring at intervals of 12, 15, and 18 minutes respectively. If they start ringing together at 10:00 AM, they will next ring together at:

(A) 10:18 AM

(B) 11:00 AM

(C) 11:48 AM

(D) 12:00 PM

Question 15. Find the largest number that divides 24 and 36 leaving no remainder.

(A) 6

(B) 12

(C) 8

(D) 4

Question 16. The LCM of two numbers is 180 and their HCF is 5. If one number is 45, the other number is:

(A) 20

(B) 25

(C) 30

(D) 40

Question 17. The HCF of 7 and 11 is:

(A) 77

(B) 1

(C) 7

(D) 11

Question 18. What is the relationship between HCF and LCM of two positive integers $a$ and $b$?

(A) $HCF(a,b) \times LCM(a,b) = a+b$

(B) $HCF(a,b) + LCM(a,b) = a \times b$

(C) $HCF(a,b) \times LCM(a,b) = a \times b$

(D) $HCF(a,b) - LCM(a,b) = a \times b$

Question 19. Find the largest number that divides 70 and 105 leaving remainders 0 and 0 respectively.

(A) 5

(B) 7

(C) 10

(D) 35

Question 20. Two tanker contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum capacity of a container that can measure the kerosene oil of both the tankers when used an exact number of times.

(A) 170 litres

(B) 10 litres

(C) 85 litres

(D) 20 litres

Question 1. According to Euclid's Division Lemma, for any two positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a = bq + r$, where:

(A) $0 < r < b$

(B) $0 \leq r < b$

(C) $0 < r \leq b$

(D) $0 \leq r \leq b$

Question 2. Applying Euclid's division lemma to 27 and 6, we get $27 = 6 \times 4 + 3$. Here, the remainder is:

(A) 27

(B) 6

(C) 4

(D) 3

Question 3. Euclid's Division Algorithm is a method to calculate the:

(A) LCM of two numbers

(B) HCF of two numbers

(C) Prime factors of a number

(D) Sum of two numbers

Question 4. Using Euclid's algorithm to find the HCF of 135 and 225, the first step is $225 = 135 \times 1 + 90$. The next step involves 135 and:

(A) 225

(B) 1

(C) 90

(D) 135

Question 5. The HCF of 135 and 225 using Euclid's algorithm is:

(A) 45

(B) 90

(C) 135

(D) 15

Question 6. The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur. This factorization is called the:

(A) Divisibility test

(B) Euclidean division

(C) Prime factorization

(D) HCF and LCM

Question 7. The prime factorization of 140 is:

(A) $2 \times 70$

(B) $2^2 \times 5 \times 7$

(C) $4 \times 5 \times 7$

(D) $2 \times 5 \times 14$

Question 8. Euclid's Division Lemma is a restatement of:

(A) The concept of prime numbers

(B) The standard long division process

(C) The properties of exponents

(D) The definition of rational numbers

Question 9. The Fundamental Theorem of Arithmetic is also known as the:

(A) Division Algorithm

(B) Unique Factorization Theorem

(C) Remainder Theorem

(D) Prime Number Theorem

Question 10. What is the HCF of 4052 and 12576 using Euclid's algorithm? (First step: $12576 = 4052 \times 3 + 420$)

(A) 4

(B) 12

(C) 420

(D) 4052

Question 11. The prime factorization of 17 is:

(A) $1 \times 17$

(B) $17$

(C) It cannot be prime factorized

(D) $17^1$

Question 12. If $a = bq + r$, where $0 \leq r < b$, then $HCF(a, b) =$

(A) $HCF(q, r)$

(B) $HCF(b, r)$

(C) $HCF(a, q)$

(D) $HCF(b, q)$

Question 13. The fundamental theorem of arithmetic can be used to prove the irrationality of numbers like $\sqrt{2}$ by considering the prime factorization of squares. What property of squares is crucial here?

(A) The prime factors of a square number always occur with even exponents.

(B) The prime factors of a square number are always prime.

(C) A square number is always composite.

(D) A square number is always positive.

Question 14. Which of the following is an application of the Fundamental Theorem of Arithmetic?

(A) Finding the sum of two numbers

(B) Checking the divisibility of a number by another

(C) Finding the HCF and LCM of numbers

(D) Converting fractions to decimals

Question 15. In the equation $a = bq + r$, $q$ is the:

(A) Dividend

(B) Divisor

(C) Quotient

(D) Remainder

Question 16. Find the HCF of 18 and 24 using repeated division (Euclid's algorithm):

(A) 3

(B) 6

(C) 12

(D) 18

Question 17. According to the Fundamental Theorem of Arithmetic, the prime factorization of a number like 50 is unique. Which of these represents the unique prime factorization?

(A) $5 \times 10$

(B) $2 \times 25$

(C) $2 \times 5 \times 5$

(D) $50 \times 1$

Question 18. If the remainder $r = 0$ in Euclid's Division Lemma $a = bq + r$, then $b$ is the:

(A) LCM of $a$ and $b$ (if $b \neq 0$)

(B) HCF of $a$ and $b$ (if $b \neq 0$)

(C) Quotient

(D) Remainder is always 0

Question 19. Consider the numbers 6 and 20. Their prime factorizations are $6 = 2 \times 3$ and $20 = 2^2 \times 5$. Using these, their HCF is $2$ and LCM is $2^2 \times 3 \times 5 = 60$. This illustrates the use of which theorem?

(A) Euclid's Division Lemma

(B) Euclid's Division Algorithm

(C) Fundamental Theorem of Arithmetic

(D) Remainder Theorem

Question 20. The step-by-step procedure to obtain the HCF of two positive integers using Euclid's Division Algorithm terminates when the remainder is:

(A) 1

(B) A prime number

(C) The HCF itself

(D) 0

Question 1. In the expression $5^3$, the base is:

(A) 3

(B) 5

(C) 15

(D) 125

Question 2. According to the law of exponents, $a^m \times a^n =$

(A) $a^{m+n}$

(B) $a^{m-n}$

(C) $a^{m \times n}$

(D) $a^{m/n}$

Question 3. The value of $10^0$ is:

(A) 0

(B) 1

(C) 10

(D) Undefined

Question 4. $(a^m)^n =$

(A) $a^{m+n}$

(B) $a^{m-n}$

(C) $a^{m \times n}$

(D) $a^{m/n}$

Question 5. The value of $2^{-3}$ is:

(A) -8

(B) 8

(C) $\frac{1}{8}$

(D) $-\frac{1}{8}$

Question 6. The standard form of a number is expressed as $a \times 10^n$, where $a$ is:

(A) An integer

(B) A real number

(C) $1 \leq a < 10$

(D) Any positive number

Question 7. Write 3,45,000,000 in standard form:

(A) $3.45 \times 10^8$

(B) $3.45 \times 10^9$

(C) $34.5 \times 10^7$

(D) $0.345 \times 10^9$

Question 8. The value of $(-1)^{101}$ is:

(A) 1

(B) -1

(C) 0

(D) 101

Question 9. Simplify: $\frac{a^m}{a^n}$

(A) $a^{m+n}$

(B) $a^{m-n}$

(C) $a^{n-m}$

(D) $a^{m/n}$

Question 10. Write 0.0000056 in standard form:

(A) $5.6 \times 10^{-5}$

(B) $5.6 \times 10^{-6}$

(C) $5.6 \times 10^5$

(D) $5.6 \times 10^6$

Question 11. Compare $2.5 \times 10^{10}$ and $5 \times 10^9$. Which is larger?

(A) $2.5 \times 10^{10}$

(B) $5 \times 10^9$

(C) They are equal

(D) Cannot compare

Question 12. $(ab)^m =$

(A) $a^m + b^m$

(B) $a^m b^m$

(C) $ab^m$

(D) $a^m b$

Question 13. The value of $(\frac{1}{3})^{-2}$ is:

(A) 9

(B) $\frac{1}{9}$

(C) -9

(D) $-\frac{1}{9}$

Question 14. What is the exponential form of 10,000?

(A) $10^3$

(B) $10^4$

(C) $10^5$

(D) $10^6$

Question 15. $\frac{a^m}{b^m} =$

(A) $(\frac{a}{b})^{m+m}$

(B) $(a-b)^m$

(C) $(\frac{a}{b})^m$

(D) $(\frac{b}{a})^m$

Question 16. Simplify: $(2^0 + 3^0) \times 4^0$

(A) 0

(B) 1

(C) 2

(D) 4

Question 17. The expanded form of $a^{-n}$ is:

(A) $a \times (-n)$

(B) $-a^n$

(C) $\frac{1}{a^n}$

(D) $a - n$

Question 18. Which is smaller: $3 \times 10^{-5}$ or $5 \times 10^{-4}$?

(A) $3 \times 10^{-5}$

(B) $5 \times 10^{-4}$

(C) They are equal

(D) Cannot compare

Question 19. Calculate: $5^2 \times 5^{-1}$

(A) $5^1 = 5$

(B) $5^{-2}$

(C) $5^3$

(D) $5^{-3}$

Question 20. The value of $(2/3)^3$ is:

(A) $\frac{6}{9}$

(B) $\frac{8}{27}$

(C) $\frac{4}{9}$

(D) $\frac{27}{8}$

Question 1. Which of the following numbers is a perfect square?

(A) 32

(B) 64

(C) 75

(D) 90

Question 2. The square of an odd number is always:

(A) Even

(B) Odd

(C) Prime

(D) Composite

Question 3. The square root of 144 is:

(A) 10

(B) 12

(C) 14

(D) 16

Question 4. A Pythagorean triplet is a set of three positive integers $a, b, c$ such that:

(A) $a+b=c$

(B) $a^2 + b^2 = c^2$

(C) $a \times b = c$

(D) $a+b+c = 180$

Question 5. Find the square root of 400 by prime factorization:

(A) $2^2 \times 5^2 = 100$

(B) $2^2 \times 5 = 20$

(C) $2 \times 5 = 10$

(D) $2 \times 2 \times 5 \times 5 = 100$

Question 6. The square root of 0.09 is:

(A) 0.3

(B) 0.03

(C) 0.003

(D) 0.9

Question 7. Estimate the square root of 50. It lies between:

(A) 6 and 7

(B) 7 and 8

(C) 8 and 9

(D) 5 and 6

Question 8. The square root of $\frac{9}{16}$ is:

(A) $\frac{3}{4}$

(B) $\frac{9}{16}$

(C) $\frac{81}{256}$

(D) $\frac{4}{3}$

Question 9. Which of the following is a Pythagorean triplet?

(A) (1, 2, 3)

(B) (2, 3, 4)

(C) (3, 4, 5)

(D) (4, 5, 6)

Question 10. The number of zeros at the end of a perfect square is always:

(A) Odd

(B) Even

(C) Zero

(D) Cannot be determined

Question 11. Which digit is NOT possible in the units place of a perfect square?

(A) 1

(B) 4

(C) 8

(D) 9

Question 12. The method of finding square root by repeated subtraction works for numbers that are:

(A) Perfect squares

(B) Prime numbers

(C) Any positive numbers

(D) Numbers ending in 0, 1, 4, 5, 6, 9

Question 13. Find the square root of 64 using the method of repeated subtraction.

Step 1: $64 - 1 = 63$

Step 2: $63 - 3 = 60$

Step 3: $60 - 5 = 55$

Step 4: $55 - 7 = 48$

Step 5: $48 - 9 = 39$

Step 6: $39 - 11 = 28$

Step 7: $28 - 13 = 15$

Step 8: $15 - 15 = 0$

The square root is the number of steps taken.

(A) 6

(B) 7

(C) 8

(D) 9

Question 14. The square root of a number is $x$. The number is:

(A) $\sqrt{x}$

(B) $2x$

(C) $x^2$

(D) $x/2$

Question 15. Which of the following is the smallest perfect square greater than 100?

(A) 101

(B) 121

(C) 110

(D) 105

Question 16. The square of -5 is:

(A) -25

(B) 25

(C) -10

(D) 10

Question 17. The square root of 2.25 is:

(A) 1.5

(B) 0.15

(C) 15

(D) 0.015

Question 18. Which method is most suitable for finding the square root of a large number like 1024?

(A) Repeated subtraction

(B) Prime factorization

(C) Long division method

(D) Estimation

Question 19. The value of $\sqrt{196/289}$ is:

(A) $\frac{14}{17}$

(B) $\frac{19.6}{28.9}$

(C) $\frac{13}{16}$

(D) $\frac{14}{16}$

Question 20. If a number ends with the digit 6, its square root can end with the digit:

(A) 4 or 6

(B) 3 or 7

(C) 1 or 9

(D) 5 or 0

Question 1. Which of the following is a perfect cube?

(A) 8

(B) 16

(C) 25

(D) 50

Question 2. The cube of a negative integer is always:

(A) Positive

(B) Negative

(C) Zero

(D) An even number

Question 3. The cube root of 27 is:

(A) 3

(B) 9

(C) $\sqrt{27}$

(D) $27^3$

Question 4. The cube of $\frac{1}{2}$ is:

(A) $\frac{1}{4}$

(B) $\frac{1}{6}$

(C) $\frac{1}{8}$

(D) $\frac{1}{16}$

Question 5. Find the cube root of 125 by prime factorization:

(A) $5^3 = 125$

(B) 5

(C) $5 \times 5 = 25$

(D) $\sqrt[3]{125}$

Question 6. The cube root of -64 is:

(A) 4

(B) -4

(C) -8

(D) 8

Question 7. Estimate the cube root of 200. It lies between:

(A) 5 and 6

(B) 6 and 7

(C) 7 and 8

(D) 8 and 9

Question 8. The cube root of $\frac{8}{27}$ is:

(A) $\frac{2}{3}$

(B) $\frac{4}{9}$

(C) $\frac{2}{9}$

(D) $\frac{16}{81}$

Question 9. Which of the following is NOT a perfect cube?

(A) 1000

(B) 1728

(C) 2197

(D) 3456

Question 10. The units digit of the cube of a number ending in 7 is:

(A) 1

(B) 3

(C) 7

(D) 9

Question 11. The units digit of the cube root of a number ending in 8 is:

(A) 2

(B) 8

(C) 4

(D) 6

Question 12. Find the cube root of 729 using prime factorization:

(A) $3^2 \times 3^2 \times 3^2 = 9 \times 9 \times 9$

(B) $3^6$

(C) 9

(D) 27

Question 13. The smallest number by which 256 must be multiplied to get a perfect cube is:

(A) 2

(B) 4

(C) 8

(D) 16

Question 14. The cube root of $a \times b$ is equal to:

(A) $\sqrt[3]{a} + \sqrt[3]{b}$

(B) $\sqrt[3]{a} \times \sqrt[3]{b}$

(C) $\sqrt[3]{a} - \sqrt[3]{b}$

(D) $\sqrt[3]{a} \div \sqrt[3]{b}$

Question 15. The cube of a rational number $\frac{p}{q}$ is:

(A) $\frac{p^3}{q}$

(B) $\frac{p}{q^3}$

(C) $\frac{p^3}{q^3}$

(D) $\frac{p+q}{p-q}$

Question 16. The cube root of a perfect cube with $n$ digits (where $n$ is a multiple of 3) will have how many digits?

(A) $n/3$

(B) $n/3 + 1$

(C) $3n$

(D) $n-2$

Question 17. Find the cube root of $0.001$:

(A) 0.1

(B) 0.01

(C) 0.001

(D) 1

Question 18. If a number ends in the digit 2, its cube ends in the digit:

(A) 4

(B) 6

(C) 8

(D) 0

Question 19. The smallest number by which 135 must be divided to get a perfect cube is:

(A) 3

(B) 5

(C) 9

(D) 15

Question 20. The cube of an even number is always:

(A) Odd

(B) Even

(C) Prime

(D) Perfect square

Question 1. Round 4567 to the nearest hundred.

(A) 4500

(B) 4600

(C) 4000

(D) 5000

Question 2. Round 12.345 to the nearest tenth.

(A) 12.3

(B) 12.4

(C) 12.35

(D) 12.34

Question 3. Estimate the product of 23 and 48 by rounding each number to the nearest ten.

(A) $20 \times 50 = 1000$

(B) $20 \times 40 = 800$

(C) $25 \times 50 = 1250$

(D) $23 \times 50 = 1150$

Question 4. Round 999 to the nearest ten.

(A) 990

(B) 1000

(C) 900

(D) 999

Question 5. Estimate the sum of 345 and 589 by rounding to the nearest hundred.

(A) $300 + 600 = 900$

(B) $300 + 500 = 800$

(C) $350 + 590 = 940$

(D) $400 + 600 = 1000$

Question 6. Round $\textsf{₹} 149.75$ to the nearest Rupee.

(A) $\textsf{₹} 149$

(B) $\textsf{₹} 150$

(C) $\textsf{₹} 149.80$

(D) $\textsf{₹} 140$

Question 7. Estimate the quotient of $815 \div 19$ by rounding both numbers to the nearest ten.

(A) $810 \div 20 = 40.5$

(B) $820 \div 20 = 41$

(C) $800 \div 20 = 40$

(D) $820 \div 19 = 43.16$

Question 8. Round 0.0078 to three decimal places.

(A) 0.007

(B) 0.008

(C) 0.00780

(D) 0.010

Question 9. Estimation is useful for:

(A) Getting an exact answer

(B) Quickly checking if an answer is reasonable

(C) Finding prime factors

(D) Solving complex equations

Question 10. Round 23,456 to the nearest thousand.

(A) 23,000

(B) 24,000

(C) 23,500

(D) 20,000

Question 11. Round 5.99 to the nearest tenth.

(A) 5.9

(B) 6.0

(C) 5.90

(D) 5.99

Question 12. Estimate the difference between 712 and 390 by rounding to the nearest hundred.

(A) $700 - 400 = 300$

(B) $700 - 300 = 400$

(C) $710 - 390 = 320$

(D) $712 - 390 = 322$

Question 13. Round 0.5 to the nearest whole number.

(A) 0

(B) 1

(C) 0.5

(D) Cannot be rounded

Question 14. In rounding, if the digit to be rounded is 5 or greater, the previous digit is:

(A) Kept the same

(B) Increased by one

(C) Decreased by one

(D) Replaced by zero

Question 15. A shopkeeper sells an item for $\textsf{₹} 487$. Estimate the total sales if 9 such items are sold, by rounding the price to the nearest hundred.

(A) $\textsf{₹} 400 \times 9 = \textsf{₹} 3600$

(B) $\textsf{₹} 500 \times 9 = \textsf{₹} 4500$

(C) $\textsf{₹} 490 \times 9 = \textsf{₹} 4410$

(D) $\textsf{₹} 487 \times 9 = \textsf{₹} 4383$

Question 16. Round 1,23,45,678 to the nearest lakh (Indian System).

(A) 1,23,00,000

(B) 1,24,00,000

(C) 1,23,46,000

(D) 1,23,45,000

Question 17. When rounding to a specific place value, all digits to the right of that place are replaced by:

(A) Their original values

(B) Zeros

(C) Ones

(D) The digit being rounded

Question 18. Estimate the sum of 2675 and 3225 by rounding to the nearest thousand.

(A) $3000 + 3000 = 6000$

(B) $2000 + 3000 = 5000$

(C) $2700 + 3200 = 5900$

(D) $2600 + 3200 = 5800$

Question 19. Round 0.99 to the nearest tenth.

(A) 0.9

(B) 1.0

(C) 0.99

(D) 0.90

Question 20. The purpose of estimation is primarily to get a value that is:

(A) Exact

(B) Approximate

(C) Larger than the actual value

(D) Smaller than the actual value

Question 1. The statement $10^2 = 100$ is equivalent to the logarithmic statement:

(A) $\log_{100} 10 = 2$

(B) $\log_{2} 10 = 100$

(C) $\log_{10} 100 = 2$

(D) $\log_{10} 2 = 100$

Question 2. $\log_b 1 =$

(A) 0

(B) 1

(C) $b$

(D) Undefined

Question 3. $\log_b b =$

(A) 0

(B) 1

(C) $b$

(D) Undefined

Question 4. $\log (MN) =$ (assume base $b$)

(A) $\log M + \log N$

(B) $\log M - \log N$

(C) $\log M \times \log N$

(D) $\log (M+N)$

Question 5. $\log (M/N) =$ (assume base $b$)

(A) $\log M + \log N$

(B) $\log M - \log N$

(C) $\log M \times \log N$

(D) $\log (M-N)$

Question 6. $\log M^k =$ (assume base $b$)

(A) $k + \log M$

(B) $k \times \log M$

(C) $\log (M+k)$

(D) $M^k$

Question 7. If $\log_{10} x = 3$, then $x =$

(A) 30

(B) 100

(C) 1000

(D) $\frac{1}{3}$

Question 8. Antilogarithm is the inverse operation of:

(A) Addition

(B) Multiplication

(C) Logarithm

(D) Exponentiation

Question 9. If $\log_{10} x = 0.3010$, then antilog(0.3010) base 10 is approximately:

(A) 1

(B) 2

(C) 3

(D) 10

Question 10. The characteristic of $\log_{10} 245.67$ is:

(A) 0

(B) 1

(C) 2

(D) 3

Question 11. The characteristic of $\log_{10} 0.0078$ is:

(A) -3

(B) -2

(C) 2

(D) 3

Question 12. $\log_b (b^k) =$

(A) $b^k$

(B) $k$

(C) $b$

(D) 1

Question 13. If $\log_2 8 = x$, then $x =$

(A) 2

(B) 3

(C) 4

(D) 8

Question 14. The base of the common logarithm is:

(A) $e$

(B) 2

(C) 10

(D) 1

Question 15. $\log_{10} 1000 =$

(A) 2

(B) 3

(C) 4

(D) 10

Question 16. If $\log_x 16 = 2$, then $x =$

(A) 2

(B) 4

(C) 8

(D) 16

Question 17. The base of the natural logarithm is:

(A) 2

(B) 10

(C) $e$

(D) 1

Question 18. If $\log 2 = 0.3010$, then $\log 4 =$

(A) 0.6020

(B) 0.0906

(C) 0.3010

(D) 0.5010

Question 19. The mantissa of $\log_{10} 245.67$ is:

(A) The integer part (2)

(B) The decimal part

(C) The whole number

(D) The base (10)

Question 20. Which of the following is a use of logarithms?

(A) To calculate square roots easily

(B) To convert addition problems into multiplication problems

(C) To simplify complex calculations involving multiplication and division

(D) To determine if a number is prime

Question 1. What is $15 \pmod{4}$?

(A) 3

(B) 4

(C) 11

(D) 15

Question 2. The remainder when 25 is divided by 7 is:

(A) 3

(B) 4

(C) 5

(D) 6

Question 3. $a \equiv b \pmod{m}$ means that $a$ and $b$ have the same remainder when divided by $m$. This is equivalent to saying that $m$ divides:

(A) $a+b$

(B) $a-b$

(C) $ab$

(D) $a/b$

Question 4. Is $10 \equiv 22 \pmod{6}$?

(A) Yes

(B) No

(C) Cannot be determined

(D) Only if the numbers are positive

Question 5. What is $-5 \pmod{3}$?

(A) -2

(B) 1

(C) 2

(D) -5

Question 6. If $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$, then $a+c \equiv$?

(A) $b+d \pmod{m}$

(B) $bd \pmod{m}$

(C) $a-c \pmod{m}$

(D) $a+d \pmod{m}$

Question 7. If $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$, then $ac \equiv$?

(A) $b+d \pmod{m}$

(B) $bd \pmod{m}$

(C) $a-c \pmod{m}$

(D) $a+d \pmod{m}$

Question 8. What is the last digit of $7^{10}$?

Hint: Use modulo arithmetic. The last digit is $7^{10} \pmod{10}$. Observe the pattern of powers of 7 modulo 10.

(A) 1

(B) 3

(C) 7

(D) 9

Question 9. Congruence modulo $m$ defines a relation between integers. This relation is:

(A) Reflexive only

(B) Symmetric only

(C) Transitive only

(D) An equivalence relation (Reflexive, Symmetric, and Transitive)

Question 10. If today is Wednesday, what day of the week will it be 100 days from now?

Hint: Use modulo arithmetic with modulo 7.

(A) Saturday

(B) Sunday

(C) Monday

(D) Tuesday

Question 11. The set of remainders when integers are divided by $m$ is $\lbrace 0, 1, 2, \dots, m-1 \rbrace$. This set is called the set of:

(A) Integers modulo $m$

(B) Rational numbers modulo $m$

(C) Prime numbers modulo $m$

(D) Factors modulo $m$

Question 12. What is $2 \times 3 \pmod{5}$?

(A) 6

(B) 1

(C) 0

(D) 2

Question 13. If $a \equiv b \pmod{m}$, then for any integer $k$, $ka \equiv kb \pmod{m}$. This is a property of:

(A) Division

(B) Subtraction

(C) Addition

(D) Multiplication

Question 14. What is $2^{5} \pmod{3}$?

Hint: Calculate $2^1 \pmod 3, 2^2 \pmod 3, \dots$ or use $(a^k)^n = a^{kn}$.

(A) 1

(B) 2

(C) 0

(D) 32

Question 15. The concept of congruence modulo $m$ is fundamental in studying the properties of:

(A) Real numbers

(B) Integers

(C) Irrational numbers

(D) Fractions

Question 16. Which of the following is NOT a valid property of modular arithmetic?

(A) If $a \equiv b \pmod{m}$, then $a+k \equiv b+k \pmod{m}$ for any integer $k$.

(B) If $a \equiv b \pmod{m}$, then $ak \equiv bk \pmod{m}$ for any integer $k$.

(C) If $ak \equiv bk \pmod{m}$, then $a \equiv b \pmod{m}$ for any integer $k$.

(D) If $a \equiv b \pmod{m}$ and $b \equiv c \pmod{m}$, then $a \equiv c \pmod{m}$.

Question 17. What is the remainder when $3^{20}$ is divided by 4?

Hint: $3 \equiv -1 \pmod 4$.

(A) 0

(B) 1

(C) 2

(D) 3

Question 18. Modulo arithmetic is often used in fields like:

(A) Geometry

(B) Trigonometry

(C) Cryptography

(D) Calculus

Question 19. If $x \equiv 5 \pmod{8}$, which of the following could be a value of $x$?

(A) 13

(B) 3

(C) -3

(D) All of the above

Question 20. The relation $a \equiv b \pmod{m}$ partitions the set of integers into $m$ distinct sets called:

(A) Modulo sets

(B) Remainder sets

(C) Equivalence classes (or congruence classes)

(D) Quotient sets

Question 1. A factory produced 1,50,000 bolts in January and 1,85,500 bolts in February. What is the total production in these two months?

(A) 3,35,500

(B) 3,00,000

(C) 3,85,500

(D) 3,15,500

Question 2. A box contains 5.5 kg of apples. If one apple weighs 100 grams, how many apples are there approximately in the box?

(A) 55

(B) 550

(C) 5500

(D) 55000

Question 3. A recipe requires $\frac{3}{4}$ cup of flour. If you want to make half the recipe, how much flour do you need?

(A) $\frac{3}{8}$ cup

(B) $\frac{6}{8}$ cup

(C) $\frac{1}{4}$ cup

(D) $\frac{3}{2}$ cups

Question 4. A vehicle covers a distance of 16.8 km in 2.4 litres of petrol. How much distance will it cover in 1 litre of petrol?

(A) 7 km

(B) 14 km

(C) 4 km

(D) 8 km

Question 5. The population of a city is 10 lakh more than the population of another city. If the population of the second city is 45 lakh, what is the population of the first city?

(A) 55 lakh

(B) 35 lakh

(C) 46 lakh

(D) 50 lakh

Question 6. A sum of $\textsf{₹} 500$ is distributed among three friends. If the first friend gets $\frac{1}{4}$ of the total amount and the second friend gets $\frac{2}{5}$ of the total amount, how much does the third friend get?

(A) $\textsf{₹} 125$

(B) $\textsf{₹} 200$

(C) $\textsf{₹} 175$

(D) $\textsf{₹} 325$

Question 7. Convert 5 kilometers to meters.

(A) 50 meters

(B) 500 meters

(C) 5000 meters

(D) 50000 meters

Question 8. A tailor needs 1.25 meters of cloth to stitch one shirt. How much cloth is needed to stitch 8 such shirts?

(A) 8 meters

(B) 9 meters

(C) 10 meters

(D) 12 meters

Question 9. If you reverse the digits of a two-digit number, the new number is 18 more than the original number. The sum of the digits is 8. What is the original number?

(A) 26

(B) 35

(C) 44

(D) 53

Question 10. A carton contains 24 boxes, and each box contains 100 pencils. The total number of pencils in 5 such cartons is:

(A) 1200

(B) 2400

(C) 12000

(D) 24000

Question 11. Ram spends $\frac{1}{3}$ of his salary on food and $\frac{1}{4}$ on rent. What fraction of his salary is left?

(A) $\frac{1}{2}$

(B) $\frac{5}{12}$

(C) $\frac{7}{12}$

(D) $\frac{1}{12}$

Question 12. If the sum of the digits of a two-digit number is 9, and if you subtract 27 from the number, the digits are reversed. What is the number?

(A) 36

(B) 63

(C) 45

(D) 54

Question 13. A car travels at a speed of 60 km/hour. How many meters does it travel in one minute?

(A) 1000 meters

(B) 6000 meters

(C) 100 meters

(D) 600 meters

Question 14. The cost of 1 kg of sugar is $\textsf{₹} 42.50$. What is the cost of 3.5 kg of sugar?

(A) $\textsf{₹} 148.75$

(B) $\textsf{₹} 120$

(C) $\textsf{₹} 150$

(D) $\textsf{₹} 145.25$

Question 15. The price of a scooter is $\textsf{₹} 45,000$. If the price increases by $\textsf{₹} 5,500$, what is the new price?

(A) $\textsf{₹} 50,000$

(B) $\textsf{₹} 49,500$

(C) $\textsf{₹} 50,500$

(D) $\textsf{₹} 40,000$

Question 16. A rectangle has a length of $5 \frac{1}{4}$ cm and a width of $3 \frac{1}{2}$ cm. Find its perimeter.

(A) $17.5$ cm

(B) $8.75$ cm

(C) $18$ cm

(D) $10.75$ cm

Question 17. The cost of painting a wall is $\textsf{₹} 15$ per square meter. If a wall is 4.5 meters long and 3.2 meters high, what is the total cost of painting?

(A) $\textsf{₹} 216$

(B) $\textsf{₹} 200$

(C) $\textsf{₹} 180$

(D) $\textsf{₹} 250$

Question 18. A number is such that the sum of its digits is 11. If 45 is added to the number, the digits are reversed. What is the number?

(A) 29

(B) 92

(C) 38

(D) 83

Question 19. A milkman sold 15.5 litres of milk on Monday, 18.75 litres on Tuesday, and 20.2 litres on Wednesday. What is the total quantity of milk sold over these three days?

(A) 54.45 litres

(B) 54.00 litres

(C) 54.25 litres

(D) 53.45 litres

Question 20. A worker earns $\textsf{₹} 1200$ per day. If they work for 20 days, how much will they earn in total?

(A) $\textsf{₹} 2400$

(B) $\textsf{₹} 14000$

(C) $\textsf{₹} 24000$

(D) $\textsf{₹} 12200$