Multiple Correct Answers MCQs for Sub-Topics of Topic 1: Numbers & Numeriacal Applications

Question 1. Which of the following numbers are integers?

(A) $5$

(B) $-10$

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

(D) $0$

Question 2. Identify the rational numbers from the list below:

(A) $\sqrt{4}$

(B) $\pi$

(C) $0.5$

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

Question 3. Which statements are true about whole numbers?

(A) All natural numbers are whole numbers.

(B) The set of whole numbers includes zero.

(C) All whole numbers are positive.

(D) Whole numbers are a subset of integers.

Question 4. Which of the following numbers have non-terminating and recurring decimal expansions?

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

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

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

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

Question 5. Select the numbers that are irrational:

(A) $\sqrt{25}$

(B) $\sqrt{7}$

(C) $0.121221222\dots$

(D) $0.333\dots$

Question 6. Which of the following statements are true?

(A) Every rational number is a real number.

(B) Every irrational number is a real number.

(C) The set of real numbers includes all rational and irrational numbers.

(D) The set of real numbers is the same as the set of rational numbers.

Question 7. Identify the prime numbers from the list:

(A) $2$

(B) $15$

(C) $23$

(D) $31$

Question 8. Which of the following describe the number $0$?

(A) Natural number

(B) Whole number

(C) Integer

(D) Rational number

Question 9. Which of the following numbers are composite numbers?

(A) $1$

(B) $9$

(C) $17$

(D) $21$

Question 10. Which of these statements are true?

(A) All negative numbers are integers.

(B) All terminating decimals are rational numbers.

(C) The set of rational numbers is closed under addition.

(D) The product of two irrational numbers is always irrational.

Question 11. Select the sets of numbers that are infinite:

(A) Natural numbers

(B) Prime numbers

(C) Numbers divisible by 5 between 1 and 100

(D) Real numbers

Question 12. Which of the following numbers can be expressed in the form $\frac{p}{q}$, where $p, q$ are integers and $q \neq 0$?

(A) $0.\overline{12}$

(B) $\sqrt{3}$

(C) $-7$

(D) $0.456$

Question 13. Which of the following statements about natural numbers are correct?

(A) They are used for counting.

(B) The smallest natural number is 0.

(C) The sum of two natural numbers is always a natural number.

(D) The difference between two natural numbers is always a natural number.

Question 14. Consider the number $\frac{22}{7}$. Which descriptions apply to this number?

(A) Rational number

(B) Irrational number

(C) Real number

(D) Approximation of $\pi$

Question 15. Which pairs consist of one rational and one irrational number?

(A) $(5, \sqrt{5})$

(B) $(\frac{1}{3}, 0.\overline{3})$

(C) $(\sqrt{9}, \sqrt{10})$

(D) $(0, \pi)$

Question 1. In the number 56,789,012, identify the place value of the digit 8 in the Indian System and the International System.

(A) Indian System: Lakhs

(B) International System: Hundred Thousands

(C) Indian System: Ten Lakhs

(D) International System: Millions

Question 2. Select the correct comparisons between the Indian and International systems:

(A) 1 Lakh = 100 Thousand

(B) 1 Crore = 1 Million

(C) 10 Crore = 100 Million

(D) 10 Lakh = 1 Million

Question 3. In the decimal number system, which digits have a place value of 1000 in the number 1,234,567?

(A) $1$

(B) $2$

(C) $3$

(D) $4$

Question 4. The number 456 can be written in general form as:

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

(B) $100a + 10b + c$ where $a=4, b=5, c=6$

(C) $400 + 50 + 6$

(D) $45 \times 10 + 6$

Question 5. Which of the following are valid Roman Numerals and their corresponding values?

(A) IV = 4

(B) XL = 60

(C) XC = 90

(D) CM = 1100

Question 6. Which digits have the same face value and place value in the number 50,873?

(A) $5$

(B) $0$

(C) $8$

(D) None of the digits have the same face value and place value.

Question 7. The number 'twelve crore thirty-four lakh fifty-six thousand seven hundred eighty-nine' in figures (Indian System) can be represented as:

(A) 12,34,56,789

(B) 123,456,789

(C) 123456789

(D) One hundred twenty-three million four hundred fifty-six thousand seven hundred eighty-nine

Question 8. Which statements about Roman numerals are true?

(A) The symbol I can be subtracted from V and X.

(B) The symbol X can be subtracted from L and C.

(C) The symbol C can be subtracted from D and M.

(D) The symbol V is never repeated.

Question 9. Which of these numbers, when written in the International System, would use the term 'million'?

(A) 5,67,890 (Indian System)

(B) 1,23,45,678 (Indian System)

(C) 9,99,999 (Indian System)

(D) 10,00,000 (Indian System)

Question 10. Consider a 2-digit number $ab$, where $a$ is the tens digit and $b$ is the units digit. Which expressions represent the value of this number?

(A) $a+b$

(B) $10a + b$

(C) $a \times b$

(D) $a \times 10 + b \times 1$

Question 1. Which numbers are typically represented by distinct points on the number line without gaps?

(A) Natural numbers

(B) Integers

(C) Rational numbers

(D) Real numbers

Question 2. To represent $\frac{5}{3}$ on the number line, which segments could you divide into equal parts?

(A) The segment between 0 and 1 into 3 parts.

(B) The segment between 1 and 2 into 3 parts.

(C) The segment between 0 and 2 into 3 parts.

(D) The segment between 1 and 2 into 5 parts.

Question 3. Which numbers can be represented by constructing a right-angled triangle on the number line and using the hypotenuse or a leg?

(A) $\sqrt{2}$

(B) $\sqrt{3}$

(C) $\sqrt{4}$

(D) $\sqrt{5}$

Question 4. Successive magnification is a technique used to locate the representation of which types of numbers more precisely on the number line?

(A) Integers

(B) Terminating decimals

(C) Non-terminating recurring decimals

(D) Irrational numbers with known decimal expansions

Question 5. If you locate the number 3.7 on the number line using successive magnification, the next step might involve magnifying the segment between:

(A) 0 and 10

(B) 3 and 4

(C) 3.7 and 3.8

(D) 3.70 and 3.71

Question 6. Which points on the number line represent integers?

(A) Points at equal distance from 0

(B) The point 0

(C) The points 1, 2, 3, ...

(D) The points -1, -2, -3, ...

Question 7. To represent $\sqrt{x}$ on the number line using the semicircle method, you draw a semicircle with diameter $AOB$ on the number line, where $A$ is at -1 and $B$ is at $x$. The point $P$ on the semicircle such that $OP$ is perpendicular to the number line at the origin O will have length $\sqrt{x}$. What should be the coordinates of A and B if you start with a diameter of length $x+1$ and one endpoint at 0?

(A) A at 0, B at $x+1$ on the positive side.

(B) A at 0, B at $x$ on the positive side.

(C) A at -0.5, B at $x+0.5$ on the positive side.

(D) A at 0, B at $x+1$. The perpendicular is raised from a point $X$ on the diameter such that $OX=1$ and $XB=x$.

Question 8. Which of the following numbers are located to the left of 0 on the standard number line?

(A) Natural numbers

(B) Negative integers

(C) Negative rational numbers

(D) $\sqrt{2}$

Question 9. If a point on the number line is halfway between two integers, it could represent:

(A) A rational number

(B) A decimal number with one decimal place

(C) An irrational number

(D) A mixed number

Question 10. Which of the following are true about the representation of numbers on the number line?

(A) Every point on the number line represents a real number.

(B) Every rational number can be represented on the number line.

(C) There are points on the number line that do not represent rational numbers.

(D) The distance between two points on the number line is always positive.

Question 1. Which of the following inequalities are correct?

(A) $5 > 3$

(B) $-5 > -3$

(C) $0 < -1$

(D) $\frac{1}{2} < \frac{2}{3}$

Question 2. The absolute value of a number $x$, denoted as $|x|$, represents its distance from zero. Which statements about absolute value are true for integers and rational numbers?

(A) $|x|$ is always non-negative.

(B) $|-x| = |x|$.

(C) $|x| = x$ if $x \geq 0$.

(D) $|x| = x$ for all integer values of $x$.

Question 3. Identify numbers that lie between $\frac{1}{4}$ and $\frac{1}{2}$:

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

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

(C) $0.3$

(D) $0.6$

Question 4. Which pairs of numbers are correctly ordered in ascending order?

(A) $-2, -1, 0, 1$

(B) $\frac{1}{5}, \frac{1}{4}, \frac{1}{3}$

(C) $0.1, 0.01, 1.0$

(D) $\sqrt{2}, 1.5, \sqrt{3}$

Question 5. When comparing two numbers written in scientific notation, $a \times 10^m$ and $b \times 10^n$, which factors influence the comparison?

(A) The sign of the base $a$ and $b$.

(B) The exponent $m$ and $n$.

(C) The value of the base $a$ and $b$ (where $1 \leq |a|, |b| < 10$).

(D) The sum of the digits of the numbers.

Question 6. Which methods can be used to find a rational number between two given rational numbers $\frac{a}{b}$ and $\frac{c}{d}$?

(A) Find their sum.

(B) Find their average: $\frac{1}{2}(\frac{a}{b} + \frac{c}{d})$.

(C) Make their denominators equal and find a rational number between the new numerators.

(D) Multiply them.

Question 7. The inequality $|x| > 5$ represents numbers $x$ such that:

(A) $x > 5$

(B) $x < 5$

(C) $x < -5$

(D) $-5 < x < 5$

Question 8. Which statements are true when comparing $10^{-3}$ and $10^{-5}$?

(A) $10^{-3} = 0.001$

(B) $10^{-5} = 0.00005$

(C) $10^{-3} > 10^{-5}$

(D) $10^{-3}$ is smaller than $10^{-5}$.

Question 9. Identify numbers that lie between $\sqrt{3}$ and $\sqrt{5}$:

(A) $1.7$

(B) $2$

(C) $\sqrt{4}$

(D) $2.3$

Question 10. Which properties are useful when comparing decimal numbers?

(A) Comparing digits from left to right, starting from the leftmost digit.

(B) Comparing the number of digits after the decimal point.

(C) Adding trailing zeros to make the number of decimal places equal.

(D) Comparing the sum of the digits.

Question 1. Which of the following expressions evaluate to 10?

(A) $2 \times 5$

(B) $15 - 5$

(C) $20 \div 2$

(D) $4 + 6$

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

(A) $\frac{3}{6} + \frac{2}{6} + \frac{1}{6}$

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

(C) $1$

(D) $\frac{3+2+1}{2+3+6}$

Question 3. Evaluate: $-8 + (-3) - (-5)$

(A) $-8 - 3 + 5$

(B) $-11 + 5$

(C) $-6$

(D) $-16$

Question 4. The product of $2.5 \times 1.4$ is:

(A) $3.5$

(B) $3.50$

(C) $0.35$

(D) $35/10$

Question 5. Simplify using the order of operations: $10 \times (5-2) \div 6 + 1$

(A) $10 \times 3 \div 6 + 1$

(B) $30 \div 6 + 1$

(C) $5 + 1$

(D) $6$

Question 6. Which expressions have a negative result?

(A) $(-4) \times 5$

(B) $(-10) \div (-2)$

(C) $-3 - 7$

(D) $5 + (-8)$

Question 7. The value of $\frac{3}{4} \times \frac{8}{9}$ is:

(A) $\frac{24}{36}$

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

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

(D) $\frac{1}{2} \times \frac{4}{3}$

Question 8. Calculate: $(1.2)^2$

(A) $1.44$

(B) $1.2 \times 1.2$

(C) $2.4$

(D) $\frac{144}{100}$

Question 9. Which of the following operations on irrational numbers can result in a rational number?

(A) Addition ($\sqrt{2} + (-\sqrt{2})$)

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

(C) Multiplication ($\sqrt{5} \times \sqrt{5}$)

(D) Division ($\sqrt{8} \div \sqrt{2}$)

Question 10. The result of $5 - (3 - (2+1))$ is:

(A) $5 - (3 - 3)$

(B) $5 - 0$

(C) $5$

(D) $4$

Question 1. Which properties hold true for addition of real numbers?

(A) Commutative property ($a+b = b+a$)

(B) Associative property ($(a+b)+c = a+(b+c)$)

(C) Existence of additive identity (0)

(D) Existence of multiplicative identity (1)

Question 2. Which properties hold true for multiplication of real numbers?

(A) Commutative property ($a \times b = b \times a$)

(B) Associative property ($(a \times b) \times c = a \times (b \times c)$)

(C) Existence of multiplicative identity (1)

(D) Existence of additive inverse ($-a$)

Question 3. A number is divisible by 12 if it is divisible by:

(A) 3 and 4

(B) 2 and 6

(C) 3 and 6

(D) 4 and 6

Question 4. Which of the following numbers are perfect squares?

(A) $49$

(B) $81$

(C) $125$

(D) $100$

Question 5. Which statements illustrate the distributive property?

(A) $a \times (b+c) = ab + ac$

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

(C) $(a+b)c = ac + bc$

(D) $a(b-c) = ab - ac$

Question 6. Identify numbers that are divisible by 5:

(A) $125$

(B) $340$

(C) $502$

(D) $785$

Question 7. Which of the following are triangular numbers?

(A) $1$

(B) $3$

(C) $6$

(D) $10$

Question 8. The existence of a multiplicative inverse ($1/a$ for $a \neq 0$) is a property of which number sets?

(A) Natural numbers

(B) Integers (excluding 0)

(C) Rational numbers (excluding 0)

(D) Real numbers (excluding 0)

Question 9. Which numbers are divisible by 9?

(A) $108$

(B) $216$

(C) $345$

(D) $459$

Question 10. Which of the following are perfect cubes?

(A) $27$

(B) $64$

(C) $100$

(D) $125$

Question 1. Which of the following are proper fractions?

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

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

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

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

Question 2. Select fractions that are equivalent to $\frac{1}{2}$:

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

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

(C) $\frac{5}{10}$

(D) $\frac{10}{20}$

Question 3. Convert the mixed number $2 \frac{3}{5}$ to an improper fraction.

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

(B) $\frac{13}{5}$

(C) $\frac{10+3}{5}$

(D) $\frac{2 \times 5 + 3}{5}$

Question 4. Which of the following decimals are terminating?

(A) $0.75$

(B) $0.111...$

(C) $1.2345$

(D) $\sqrt{2}$ (as a decimal)

Question 5. Convert $\frac{3}{4}$ to a decimal.

(A) $0.75$

(B) $3 \div 4$

(C) $\frac{75}{100}$

(D) $0.34$

Question 6. Convert the decimal 0.8 to a fraction in simplest form.

(A) $\frac{8}{10}$

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

(C) $\frac{80}{100}$

(D) $\frac{2}{2.5}$

Question 7. Which of the following are types of fractions?

(A) Proper fraction

(B) Improper fraction

(C) Like fractions

(D) Integer fraction

Question 8. To reduce a fraction to its simplest form, you can:

(A) Divide the numerator and denominator by their common factors.

(B) Divide the numerator and denominator by their HCF.

(C) Multiply the numerator and denominator by the same non-zero number.

(D) Divide the numerator by the denominator.

Question 9. Convert $\frac{1}{8}$ to a decimal.

(A) $0.125$

(B) $\frac{1000}{8} \times 0.001$

(C) $1 \div 8$

(D) $0.12$

Question 10. Which of the following decimals are 'like decimals' with 1.23?

(A) $0.45$

(B) $10.67$

(C) $5.00$

(D) $0.123$

Question 1. Which of the following numbers have terminating decimal expansions?

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

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

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

(D) $\frac{7}{20}$

Question 2. Which of the following represent rational numbers?

(A) $0.555\dots$

(B) $1.234567\dots$ (non-repeating, non-terminating)

(C) $\sqrt{99}$

(D) $2.\overline{45}$

Question 3. Express $0.\overline{15}$ in $\frac{p}{q}$ form.

(A) $\frac{15}{100}$

(B) $\frac{15}{99}$

(C) $\frac{5}{33}$

(D) Let $x = 0.1515\dots$, then $100x = 15.1515\dots$, $100x - x = 15$, $99x=15$, $x=\frac{15}{99}$.

Question 4. Which of the following are irrational numbers?

(A) $\sqrt{1.6}$

(B) $0.101001000\dots$

(C) $\sqrt[3]{27}$

(D) $\pi$

Question 5. Rationalize the denominator of $\frac{1}{\sqrt{3}-1}$.

(A) Multiply numerator and denominator by $\sqrt{3}+1$.

(B) The result is $\frac{\sqrt{3}+1}{2}$.

(C) The result is $\sqrt{3}+1$.

(D) The denominator becomes $(\sqrt{3})^2 - 1^2 = 3-1 = 2$.

Question 6. A rational number can have which type of decimal expansion?

(A) Terminating

(B) Non-terminating recurring

(C) Non-terminating non-recurring

(D) Finite

Question 7. Consider the number $0.4\overline{2}$. To convert it to $\frac{p}{q}$ form, which steps are appropriate?

(A) Let $x = 0.4222\dots$

(B) Multiply $x$ by 10 to get $10x = 4.222\dots$

(C) Multiply $x$ by 100 to get $100x = 42.222\dots$

(D) Subtract $10x$ from $100x$: $100x - 10x = 42.222\dots - 4.222\dots$

Question 8. Which denominators (when the fraction is in simplest form) indicate a terminating decimal expansion?

(A) $20$

(B) $15$

(C) $16$

(D) $30$

Question 9. The decimal expansion $2.71828\dots$ which is non-terminating and non-recurring, represents which type of number?

(A) Rational

(B) Irrational

(C) Integer

(D) Real

Question 10. Rationalize the denominator of $\frac{\sqrt{2}}{\sqrt{5}}$.

(A) Multiply numerator and denominator by $\sqrt{5}$.

(B) The result is $\frac{\sqrt{10}}{5}$.

(C) The result is $\frac{2}{\sqrt{10}}$.

(D) The denominator becomes $\sqrt{25} = 5$.

Question 1. Which of the following are factors of 24?

(A) $1$

(B) $6$

(C) $8$

(D) $48$

Question 2. Which of the following are multiples of 9?

(A) $3$

(B) $18$

(C) $27$

(D) $90$

Question 3. A number is divisible by 4 if:

(A) Its last digit is 4.

(B) The number formed by its last two digits is divisible by 4.

(C) The sum of its digits is divisible by 4.

(D) It is an even number.

Question 4. Which of the following are prime numbers?

(A) $13$

(B) $29$

(C) $37$

(D) $51$

Question 5. Which statements about composite numbers are true?

(A) They have more than two factors.

(B) 4 is the smallest composite number.

(C) Every even number greater than 2 is composite.

(D) 9 is a composite number.

Question 6. The prime factorization of 60 is:

(A) $2^2 \times 3 \times 5$

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

(C) $4 \times 3 \times 5$

(D) $2 \times 30$

Question 7. A number is divisible by 3 if:

(A) Its last digit is 3 or 6 or 9.

(B) The sum of its digits is divisible by 3.

(C) It is an odd number.

(D) It is a multiple of 3.

Question 8. Which of the following are factors of both 18 and 30?

(A) $1$

(B) $2$

(C) $3$

(D) $6$

Question 9. Which of the following are multiples of both 4 and 6?

(A) $12$

(B) $24$

(C) $36$

(D) $48$

Question 10. To determine if a large number is prime, you should check for divisibility by prime numbers up to its square root. Which prime numbers would you check for divisibility to determine if 121 is prime?

Hint: $\sqrt{121} = 11$.

(A) $2$

(B) $3$

(C) $5$

(D) $7$

Question 1. The HCF of 30 and 42 is:

(A) A common factor of 30 and 42.

(B) The largest number that divides both 30 and 42.

(C) 6.

(D) Found using prime factorization as $2 \times 3$.

Question 2. The LCM of 10 and 15 is:

(A) A common multiple of 10 and 15.

(B) The smallest positive number that is a multiple of both 10 and 15.

(C) 30.

(D) 150.

Question 3. Which statements are true about the relationship between HCF and LCM of two positive integers $a$ and $b$?

(A) $HCF(a,b) \leq LCM(a,b)$

(B) $HCF(a,b)$ is a factor of $LCM(a,b)$

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

(D) $HCF(a,b)$ is a multiple of $a$ and $b$.

Question 4. Find the HCF of 36 and 48 using prime factorization.

(A) $36 = 2^2 \times 3^2$

(B) $48 = 2^4 \times 3$

(C) HCF is the product of common prime factors with the lowest power.

(D) HCF = $2^2 \times 3^1 = 12$.

Question 5. Find the LCM of 36 and 48 using prime factorization.

(A) $36 = 2^2 \times 3^2$

(B) $48 = 2^4 \times 3$

(C) LCM is the product of all prime factors with the highest power.

(D) LCM = $2^4 \times 3^2 = 16 \times 9 = 144$.

Question 6. Two numbers are 18 and 24. Their HCF is 6 and LCM is 72. Which statements are true?

(A) $HCF(18, 24) = 6$

(B) $LCM(18, 24) = 72$

(C) $18 \times 24 = 432$

(D) $HCF \times LCM = 6 \times 72 = 432$.

Question 7. HCF can be used in applications such as:

(A) Finding the largest size of tiles to cover a rectangular floor without cutting.

(B) Finding the maximum capacity of a container to measure quantities from different containers exactly.

(C) Finding the time when multiple events will occur together again.

(D) Simplifying fractions.

Question 8. LCM can be used in applications such as:

(A) Finding the smallest number of items to distribute equally among different groups.

(B) Finding when multiple events (like traffic lights changing or bells ringing) will coincide.

(C) Adding or subtracting fractions with different denominators.

(D) Finding the greatest common divisor.

Question 9. If the HCF of two numbers is 1, the numbers are:

(A) Prime numbers

(B) Composite numbers

(C) Co-prime numbers

(D) Relatively prime numbers

Question 10. Which of the following pairs are co-prime?

(A) (4, 9)

(B) (15, 21)

(C) (8, 25)

(D) (10, 15)

Question 1. In Euclid's Division Lemma, $a = bq + r$, with $a$ as the dividend and $b$ as the divisor, which conditions are correct?

(A) $a$ and $b$ are positive integers.

(B) $q$ and $r$ are unique integers.

(C) $0 \leq r < b$.

(D) $b > 0$.

Question 2. Euclid's Division Algorithm is based on the principle that $HCF(a, b) = HCF(b, r)$ where $a = bq + r$. Which of the following is true in this context?

(A) The algorithm terminates when the remainder is 1.

(B) The HCF is the non-zero remainder in the last step.

(C) The HCF is the divisor in the step where the remainder is 0.

(D) The algorithm is used to find the LCM.

Question 3. The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of prime numbers. Which parts of this statement are crucial?

(A) Every number is composite.

(B) The factorization is into prime numbers.

(C) The factorization is unique.

(D) The order of prime factors does not matter for uniqueness.

Question 4. Applying Euclid's algorithm to find HCF of 45 and 75, the steps would involve:

(A) $75 = 45 \times 1 + 30$

(B) $45 = 30 \times 1 + 15$

(C) $30 = 15 \times 2 + 0$

(D) The HCF is 30.

Question 5. The Fundamental Theorem of Arithmetic is used in which applications?

(A) Finding the HCF of two numbers using prime factorization.

(B) Finding the LCM of two numbers using prime factorization.

(C) Proving the irrationality of numbers like $\sqrt{2}$.

(D) Performing long division.

Question 6. Which of the following numbers are expressed as a product of their prime factors according to the Fundamental Theorem of Arithmetic?

(A) $12 = 2^2 \times 3$

(B) $50 = 2 \times 5 \times 5$

(C) $28 = 4 \times 7$

(D) $100 = 2 \times 5 \times 10$

Question 7. When applying Euclid's Division Algorithm to find the HCF of two numbers, say $a$ and $b$ ($a>b$), the first step is $a = bq + r$. Which of the following is always true about $r$?

(A) $r < b$

(B) $r \geq 0$

(C) $r$ is an integer

(D) $r$ is positive

Question 8. The Fundamental Theorem of Arithmetic applies to which set of numbers?

(A) All real numbers

(B) All integers

(C) All natural numbers greater than 1

(D) All composite numbers

Question 9. Consider the statement: "The prime factorization of 72 is $2^3 \times 3^2$". This statement reflects which aspect(s) of the Fundamental Theorem of Arithmetic?

(A) Every composite number has a prime factorization.

(B) The factors are indeed prime numbers.

(C) The factorization is unique (ignoring order).

(D) The number is composite.

Question 10. In the process of finding the HCF of 90 and 144 using Euclid's algorithm, the sequence of remainders will be:

(A) 54

(B) 36

(C) 18

(D) 0

Question 1. Apply the laws of exponents to simplify $x^5 \times x^3$.

(A) $x^{5+3}$

(B) $x^8$

(C) $(x \times x)^8$

(D) $x^{15}$

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

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

(B) $3^6$

(C) $9^3$

(D) $729$

Question 3. Which expressions are equivalent to $\frac{a^7}{a^4}$ (where $a \neq 0$)?

(A) $a^{7-4}$

(B) $a^3$

(C) $a \times a \times a$

(D) $\frac{1}{a^{-3}}$

Question 4. Which of the following are correct representations of $4^{-2}$?

(A) $-16$

(B) $\frac{1}{4^2}$

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

(D) $0.0625$

Question 5. The standard form of the number 567,000 is:

(A) $5.67 \times 10^5$

(B) $5.67 \times 100000$

(C) $56.7 \times 10^4$

(D) $5.67 \times 10 \times 10 \times 10 \times 10 \times 10$

Question 6. Which expressions evaluate to 1?

(A) $5^0$

(B) $(ab)^0$ (where $ab \neq 0$)

(C) $\frac{7^3}{7^3}$

(D) $10^{-1} \times 10^1$

Question 7. The number 0.000023 can be written in standard form as:

(A) $2.3 \times 10^{-5}$

(B) $2.3 \times 10^{-6}$

(C) $23 \times 10^{-6}$

(D) $2.3 \times 0.00001$

Question 8. Which inequalities are correct?

(A) $10^2 > 10^1$

(B) $2^3 > 3^2$

(C) $5^0 < 5^1$

(D) $10^{-2} < 10^{-3}$

Question 9. Simplify: $(2^3 \times 3^2)^2$

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

(B) $2^6 \times 3^4$

(C) $8 \times 9$ squared

(D) $(72)^2 = 5184$

Question 10. Which are equivalent to $(\frac{a}{b})^{-n}$?

(A) $\frac{a^{-n}}{b^{-n}}$

(B) $\frac{b^n}{a^n}$

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

(D) $-\frac{a^n}{b^n}$

Question 1. Which numbers are perfect squares?

(A) $1$

(B) $4$

(C) $9$

(D) $16$

Question 2. Which properties are true for perfect squares?

(A) They can end with the digit 2, 3, 7, or 8.

(B) The number of zeros at the end of a perfect square is always even.

(C) A number ending in 5 has its square ending in 25.

(D) The sum of the first $n$ odd natural numbers is $n^2$.

Question 3. Which are Pythagorean triplets?

(A) (3, 4, 5)

(B) (6, 8, 10)

(C) (5, 12, 13)

(D) (7, 24, 25)

Question 4. Find the square root of 324 using prime factorization.

(A) $324 = 2^2 \times 3^4$

(B) $\sqrt{324} = \sqrt{2^2 \times 3^4}$

(C) $\sqrt{324} = 2^{2/2} \times 3^{4/2} = 2^1 \times 3^2 = 2 \times 9 = 18$

(D) The square root is 18.

Question 5. The value of $\sqrt{0.25}$ is:

(A) $0.5$

(B) $\sqrt{\frac{25}{100}}$

(C) $\frac{5}{10}$

(D) $0.05$

Question 6. Estimate the square root of 90. It lies between:

(A) 9 and 10

(B) 8 and 9

(C) $\sqrt{81}$ and $\sqrt{100}$

(D) 81 and 100

Question 7. The square root of $\frac{49}{64}$ is:

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

(B) $\sqrt{\frac{49}{64}}$

(C) $\frac{\sqrt{49}}{\sqrt{64}}$

(D) $\frac{7}{64}$

Question 8. Which methods can be used to find the square root of a perfect square?

(A) Repeated subtraction of consecutive odd numbers.

(B) Prime factorization.

(C) Long division method.

(D) Adding consecutive even numbers.

Question 9. The square of a positive number is always:

(A) Positive

(B) Negative

(C) Non-negative

(D) Greater than the number itself (if the number is > 1)

Question 10. Which numbers have a square root that is an integer?

(A) $10$

(B) $25$

(C) $100$

(D) $144$

Question 1. Which numbers are perfect cubes?

(A) $1$

(B) $8$

(C) $27$

(D) $64$

Question 2. Which properties are true for perfect cubes?

(A) They can end with any digit from 0 to 9.

(B) The units digit of the cube of a number ending in 4 is 4.

(C) The units digit of the cube of a number ending in 9 is 9.

(D) A number ending in 0 has its cube ending in three zeros.

Question 3. Find the cube root of 216 by prime factorization.

(A) $216 = 2^3 \times 3^3$

(B) $\sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3}$

(C) $\sqrt[3]{216} = 2^{3/3} \times 3^{3/3} = 2^1 \times 3^1 = 6$

(D) The cube root is 6.

Question 4. The value of $\sqrt[3]{-125}$ is:

(A) $-5$

(B) $\sqrt[3]{(-5) \times (-5) \times (-5)}$

(C) $- \sqrt[3]{125}$

(D) $5$

Question 5. The cube of $\frac{2}{3}$ is:

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

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

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

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

Question 6. Estimate the cube root of 500. It lies between:

(A) 7 and 8

(B) $\sqrt[3]{343}$ and $\sqrt[3]{512}$

(C) 8 and 9

(D) 7.9 and 8.1

Question 7. The cube root of $\frac{27}{64}$ is:

(A) $\sqrt[3]{\frac{27}{64}}$

(B) $\frac{\sqrt[3]{27}}{\sqrt[3]{64}}$

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

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

Question 8. Which of the following are perfect cubes?

(A) $100$

(B) $1000$

(C) $1728$

(D) $2197$

Question 9. The smallest number by which 108 must be multiplied to get a perfect cube is:

Hint: Prime factorization of 108 is $2^2 \times 3^3$.

(A) $2$

(B) $2^1$

(C) $4$

(D) $2^2$

Question 10. The cube root of $0.125$ is:

(A) $0.5$

(B) $\sqrt[3]{\frac{125}{1000}}$

(C) $\frac{5}{10}$

(D) $0.05$

Question 1. Round 789 to the nearest hundred.

(A) 700

(B) 800

(C) The tens digit is 8, which is $\geq 5$.

(D) The result is 800.

Question 2. Round 5.67 to the nearest tenth.

(A) 5.6

(B) 5.7

(C) The hundredths digit is 7, which is $\geq 5$.

(D) The result is 5.7.

Question 3. Estimate the product of 38 and 53 by rounding each to the nearest ten.

(A) Round 38 to 40.

(B) Round 53 to 50.

(C) Estimate is $40 \times 50$.

(D) The estimated product is 2000.

Question 4. Round 0.49 to the nearest tenth.

(A) 0.4

(B) 0.5

(C) The hundredths digit is 9, which is $\geq 5$.

(D) The result is 0.5.

Question 5. Estimation is useful in:

(A) Everyday shopping calculations.

(B) Checking the reasonableness of exact calculations.

(C) Planning budgets.

(D) Precisely measuring quantities.

Question 6. Round 1,99,999 to the nearest lakh (Indian System).

(A) 1,00,000

(B) 2,00,000

(C) The ten thousands digit is 9, which is $\geq 5$.

(D) The result is 2 lakh.

Question 7. Round 9.5 to the nearest whole number.

(A) 9

(B) 10

(C) The tenths digit is 5, which is $\geq 5$.

(D) The result is 10.

Question 8. When rounding a number to a specific place, digits to the right of that place are changed to zero. This is true for:

(A) Digits between the rounded place and the decimal point.

(B) Digits after the decimal point.

(C) Only if the original number was an integer.

(D) Only if the original number was a decimal.

Question 9. Estimate the total cost of 5 items priced at $\textsf{₹} 98.50$ each, by rounding the price to the nearest ten Rupees.

(A) Round $\textsf{₹} 98.50$ to $\textsf{₹} 100$.

(B) Estimate is $5 \times \textsf{₹} 100$.

(C) The estimated total cost is $\textsf{₹} 500$.

(D) The exact cost is $\textsf{₹} 492.50$.

Question 10. Round 12,345,678 to the nearest million (International System).

(A) 12,000,000

(B) 12 million

(C) The hundred thousands digit is 3, which is $< 5$.

(D) The result is 12,000,000.

Question 1. The statement $\log_b A = C$ is equivalent to:

(A) $b^C = A$

(B) A is the logarithm of C to base b.

(C) C is the exponent to which b must be raised to get A.

(D) $C^b = A$

Question 2. Which are correct applications of the logarithm laws?

(A) $\log (A \times B) = \log A + \log B$

(B) $\log (\frac{A}{B}) = \log A - \log B$

(C) $\log (A+B) = \log A + \log B$

(D) $\log A^p = p \log A$

Question 3. The value of $\log_{10} 10000$ is:

(A) $4$

(B) $\log_{10} 10^4$

(C) $10^4$

(D) $1000$

Question 4. If $\log_{10} 2 = 0.3010$ and $\log_{10} 3 = 0.4771$, then which statements are true?

(A) $\log_{10} 6 = \log_{10} (2 \times 3) = \log_{10} 2 + \log_{10} 3$

(B) $\log_{10} 6 = 0.3010 + 0.4771 = 0.7781$

(C) $\log_{10} 5 = \log_{10} (10/2) = \log_{10} 10 - \log_{10} 2 = 1 - 0.3010 = 0.6990$

(D) $\log_{10} 9 = (\log_{10} 3)^2$

Question 5. Antilog of a number $x$ (base $b$) is the number $N$ such that $\log_b N = x$. Which statements are correct?

(A) Antilog$_{10}(2) = 10^2 = 100$

(B) Antilog$_{10}(3.4567) = 10^{3.4567}$

(C) If Antilog$_b(x) = N$, then $b^x = N$.

(D) Antilog is only defined for base 10.

Question 6. The characteristic of a logarithm $\log_{10} N$ is the integer part of the logarithm. Which statements are true about the characteristic?

(A) If $N > 1$, the characteristic is one less than the number of digits before the decimal point in $N$.

(B) If $0 < N < 1$, the characteristic is negative.

(C) The characteristic of $\log_{10} 567$ is 2.

(D) The characteristic of $\log_{10} 0.012$ is -2.

Question 7. The mantissa of a logarithm $\log_{10} N$ is the fractional part, which is always positive. Which statements are true?

(A) Mantissa is the part found using logarithm tables.

(B) The mantissa of $\log_{10} 0.012$ is the same as the mantissa of $\log_{10} 1.2$ if the digits are the same.

(C) Mantissa is always between 0 and 1 (inclusive of 0, exclusive of 1).

(D) Mantissa can be negative.

Question 8. $\log_b b^x =$

(A) $b^x$

(B) $x$

(C) $\log_b b \times x$

(D) $1$

Question 9. $\log_b 1 =$

(A) $0$

(B) This means $b^0 = 1$.

(C) This is true for any valid base $b > 0, b \neq 1$.

(D) $1$

Question 10. Which statements about logarithms are correct?

(A) $\log_{10} (0.1) = -1$

(B) $\log_2 4 = 2$

(C) $\log_5 5 = 1$

(D) $\log_b 0$ is undefined.

Question 1. What is $20 \pmod{6}$?

(A) The remainder when 20 is divided by 6.

(B) 2

(C) $20 = 6 \times 3 + 2$

(D) 3

Question 2. The congruence $a \equiv b \pmod{m}$ is true if:

(A) $a-b$ is divisible by $m$.

(B) $a$ and $b$ have the same remainder when divided by $m$.

(C) $a = b + km$ for some integer $k$.

(D) $a = b$

Question 3. If $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$, then which of the following congruences are true?

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

(B) $a-c \equiv b-d \pmod{m}$

(C) $ac \equiv bd \pmod{m}$

(D) $a/c \equiv b/d \pmod{m}$ (division is more complex in modular arithmetic)

Question 4. What is the remainder when $10 \times 15$ is divided by 7? Use modular arithmetic.

(A) $10 \equiv 3 \pmod 7$

(B) $15 \equiv 1 \pmod 7$

(C) $10 \times 15 \equiv 3 \times 1 \pmod 7$

(D) $10 \times 15 \equiv 3 \pmod 7$

Question 5. The relation of congruence modulo $m$ is an equivalence relation because it satisfies:

(A) Reflexivity ($a \equiv a \pmod m$)

(B) Symmetry (If $a \equiv b \pmod m$, then $b \equiv a \pmod m$)

(C) Transitivity (If $a \equiv b \pmod m$ and $b \equiv c \pmod m$, then $a \equiv c \pmod m$)

(D) Commutativity

Question 6. What is the last digit of $3^{5}$?

Hint: This is $3^5 \pmod{10}$. Look for a pattern in powers of 3 modulo 10.

(A) $3^1 \equiv 3 \pmod{10}$

(B) $3^2 \equiv 9 \pmod{10}$

(C) $3^3 \equiv 27 \equiv 7 \pmod{10}$

(D) $3^4 \equiv 81 \equiv 1 \pmod{10}$

Question 7. Which of the following numbers are congruent to 2 modulo 5?

(A) $7$

(B) $12$

(C) $-3$

(D) $-8$

Question 8. If $a \equiv b \pmod{m}$, then $a^k \equiv b^k \pmod{m}$ for any positive integer $k$. Which of the following calculations use this property?

(A) Calculating the last digit of a large power.

(B) Finding the remainder of a large power divided by a number.

(C) $7 \equiv 2 \pmod 5$, so $7^2 \equiv 2^2 \pmod 5$, which means $49 \equiv 4 \pmod 5$.

(D) $10 \equiv 3 \pmod 7$, so $100 \equiv 30 \pmod 7$.

Question 9. Which remainders are possible when an integer is divided by 8?

(A) $0$

(B) $7$

(C) $8$

(D) $-1$

Question 10. If the time is 9 AM, what time will it be after 25 hours? Use modulo arithmetic (modulo 24).

(A) $25 \equiv 1 \pmod{24}$

(B) It will be 1 hour past 9 AM of the next day.

(C) It will be 10 AM.

(D) It will be 9 PM.

Question 1. A company had a profit of $\textsf{₹} 15,50,000$ in one year. If the profit increased by $\textsf{₹} 3,75,000$ in the next year, which expressions represent the profit in the second year?

(A) $\textsf{₹} (15,50,000 + 3,75,000)$

(B) $\textsf{₹} 19,25,000$

(C) $\textsf{₹} 19$ Lakh $25$ Thousand

(D) $\textsf{₹} 1550000 + \textsf{₹} 375000$

Question 2. A jug contains $2 \frac{1}{2}$ litres of juice. If $0.75$ litres are poured into a glass, how much juice is left in the jug?

(A) $2.5 - 0.75$ litres

(B) $1.75$ litres

(C) $1 \frac{3}{4}$ litres

(D) $2 \frac{1}{2} - \frac{3}{4}$ litres

Question 3. A car travels 180 km in 3 hours. Its average speed in meters per second is:

(A) $180 \text{ km} / 3 \text{ hours} = 60 \text{ km/hour}$

(B) $60 \text{ km/hour} = 60 \times \frac{1000 \text{ m}}{3600 \text{ s}}$

(C) $\frac{60000}{3600} \text{ m/s}$

(D) $\frac{600}{36} = \frac{100}{6} = \frac{50}{3} \text{ m/s}$

Question 4. A recipe calls for $\frac{2}{3}$ cup of milk. If you use $\frac{1}{4}$ of that amount, how much milk did you use?

(A) $\frac{1}{4} \times \frac{2}{3}$ cup

(B) $\frac{2}{12}$ cup

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

(D) Multiply the numerators and the denominators.

Question 5. The cost of 1 meter of cloth is $\textsf{₹} 75.50$. What is the cost of 5 meters of cloth?

(A) $5 \times \textsf{₹} 75.50$

(B) $\textsf{₹} 377.50$

(C) $\textsf{₹} 377 \frac{1}{2}$

(D) Add $\textsf{₹} 75.50$ five times.

Question 6. A rectangular field is 150 meters long and 80 meters wide. What is its area in square kilometers?

(A) Area = Length $\times$ Width = $150 \times 80 = 12000$ square meters.

(B) 1 km = 1000 m, so 1 sq km = $1000 \times 1000 = 1,000,000$ sq m.

(C) Area in sq km = $\frac{12000}{1000000}$

(D) Area = 0.012 sq km.

Question 7. A number is formed by reversing the digits of a two-digit number. The sum of the digits is 10. If the original number is $10t + u$, the reversed number is $10u + t$. The difference between the original number and the reversed number is 36. What is the original number?

(A) $t+u = 10$

(B) $(10t + u) - (10u + t) = 36$

(C) $9t - 9u = 36 \implies t-u = 4$

(D) Solving $t+u=10$ and $t-u=4$ gives $2t=14 \implies t=7$, and $u=3$. The number is 73.

Question 8. A sum of $\textsf{₹} 1000$ is divided between A and B such that A gets $\frac{3}{5}$ of the total. How much does B get?

(A) A gets $\frac{3}{5} \times \textsf{₹} 1000 = \textsf{₹} 600$.

(B) B gets $1 - \frac{3}{5} = \frac{2}{5}$ of the total.

(C) B gets $\frac{2}{5} \times \textsf{₹} 1000 = \textsf{₹} 400$.

(D) B gets $\textsf{₹} 1000 - \textsf{₹} 600 = \textsf{₹} 400$.

Question 9. Which of the following represent a valid unit conversion?

(A) 1 meter = 100 centimeters

(B) 1 kilometer = 1000 meters

(C) 1 litre = 1000 milliliters

(D) 1 kilogram = 1000 grams

Question 10. The sum of two numbers is 15. If one number is $x$, the other number is $15-x$. If their product is 54, find the numbers.

(A) $x(15-x) = 54$

(B) $15x - x^2 = 54$

(C) $x^2 - 15x + 54 = 0$

(D) Factoring the quadratic equation $(x-6)(x-9)=0$ gives $x=6$ or $x=9$. The numbers are 6 and 9.