Negative Questions MCQs for Sub-Topics of Topic 1: Numbers & Numeriacal Applications

Question 1. Which of the following is NOT an integer?

(A) -7

(B) 0

(C) 15

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

Question 2. Which statement about natural numbers is FALSE?

(A) They are used for counting.

(B) The smallest natural number is 1.

(C) The set of natural numbers is infinite.

(D) Zero is a natural number.

Question 3. Which of the following is NOT a rational number?

(A) $\sqrt{2}$

(B) 0.75

(C) -4

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

Question 4. Which statement about integers is FALSE?

(A) All natural numbers are integers.

(B) All whole numbers are integers.

(C) Every integer is either positive, negative, or zero.

(D) Every fraction is an integer.

Question 5. Which of the following is NOT a composite number?

(A) 4

(B) 9

(C) 1

(D) 15

Question 6. Which statement about real numbers is FALSE?

(A) Every rational number is a real number.

(B) Every irrational number is a real number.

(C) The set of real numbers is the union of rational and irrational numbers.

(D) Every real number is a rational number.

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

(A) 0

(B) 5

(C) -3

(D) 100

Question 8. Which statement about irrational numbers is FALSE?

(A) Their decimal expansion is non-terminating and non-recurring.

(B) They can be expressed in the form $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$.

(C) $\pi$ is an irrational number.

(D) $\sqrt{7}$ is an irrational number.

Question 9. Which of the following is NOT a prime number?

(A) 2

(B) 7

(C) 13

(D) 21

Question 10. Which statement is FALSE?

(A) Every integer is a real number.

(B) Every whole number is a rational number.

(C) Every rational number is an integer.

(D) Every natural number is an integer.

Question 11. Which of the following is NOT a positive integer?

(A) 5

(B) 1

(C) 0

(D) 100

Question 12. Which statement about the number 0 is FALSE?

(A) It is a whole number.

(B) It is an integer.

(C) It is a rational number.

(D) It is a natural number.

Question 1. Which of the following is NOT a correct way to write a number in the Indian System of Numeration?

(A) 1,23,456

(B) 12,34,56,789

(C) 56,7890

(D) 1,00,00,000

Question 2. Which statement about place value in the decimal system is FALSE?

(A) The place value of a digit increases by a factor of 10 as it moves one position to the left.

(B) The place value of a digit is its face value multiplied by the position value.

(C) The place value of the units digit is 1.

(D) The face value of a digit changes based on its position.

Question 3. Which of the following comparisons is NOT correct?

(A) 1 Lakh = 100 Thousand

(B) 1 Crore = 10 Million

(C) 1 Million = 10 Lakh

(D) 1 Crore = 100 Lakh

Question 4. Which of the following is NOT a correct Roman numeral representation?

(A) IV (4)

(B) IX (9)

(C) XL (40)

(D) VL (45)

Question 5. Which is NOT the general form of a 3-digit number $abc$ (where $a, b, c$ are digits and $a \neq 0$)?

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

(B) $a \times 10^2 + b \times 10^1 + c \times 10^0$

(C) $a+b+c$

(D) $a00 + b0 + c$

Question 6. Which statement about Roman numerals is FALSE?

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

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

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

(D) A symbol of smaller value placed after a symbol of larger value is added to the larger value.

Question 7. What is NOT the place value of the digit 5 in the number 1,23,456?

(A) Tens

(B) 50

(C) $5 \times 10^1$

(D) Hundreds

Question 8. Which of the following is NOT a correct representation of the number 'seven lakh eighty-five thousand two hundred'?

(A) 7,85,200 (Indian)

(B) 785,200 (International)

(C) $7 \times 10^5 + 8 \times 10^4 + 5 \times 10^3 + 2 \times 10^2$

(D) 785,2000

Question 9. Which statement is FALSE about the decimal point?

(A) Digits to the left of the decimal point represent whole number parts.

(B) Digits to the right of the decimal point represent fractional parts.

(C) The place value of digits decreases by a factor of 10 as you move right from the decimal point.

(D) The decimal point is always placed after the units digit.

Question 10. Which of the following Roman numeral rules is NOT correct?

(A) If a symbol is repeated, its value is multiplied as many times as it occurs.

(B) A symbol is not repeated more than three times.

(C) V, L, and D are never subtracted.

(D) V, L, and D are never repeated.

Question 1. Which type of number is NOT typically represented by discrete points (dots) on the number line?

(A) Natural numbers

(B) Integers

(C) Rational numbers

(D) Real numbers

Question 2. Which statement about the number line is FALSE?

(A) Zero is the origin.

(B) Positive numbers are to the right of zero.

(C) Negative numbers are to the left of zero.

(D) The distance between 1 and 3 is -2.

Question 3. Which method is NOT a standard way to represent irrational numbers like $\sqrt{2}$ or $\sqrt{3}$ on the number line?

(A) Using a compass and constructing a right-angled triangle based on the Pythagorean theorem.

(B) Using successive magnification to locate the decimal approximation.

(C) Dividing the segment between two integers into equal parts based on the denominator (as done for fractions).

(D) Using a semicircle construction on a diameter of length $x+1$ to represent $\sqrt{x}$.

Question 4. Which fraction is NOT located between 0 and 1 on the number line?

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

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

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

(D) $\frac{9}{10}$

Question 5. Which number is NOT represented by a point on the standard number line?

(A) -10

(B) 3.14

(C) $\sqrt{-4}$

(D) $\pi$

Question 6. Which statement about representing numbers using successive magnification is FALSE?

(A) It helps visualize terminating decimals.

(B) It helps visualize non-terminating recurring decimals.

(C) It helps visualize integers.

(D) It involves magnifying smaller and smaller segments of the number line.

Question 7. Which point is NOT to the left of -1 on the number line?

(A) -1.5

(B) -2

(C) -0.5

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

Question 8. Which number is NOT located between 2 and 3 on the number line?

(A) 2.5

(B) $\frac{7}{3}$

(C) $\sqrt{5}$

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

Question 9. Which statement about the density of numbers on the number line is FALSE?

(A) Between any two distinct rational numbers, there are infinitely many rational numbers.

(B) Between any two distinct irrational numbers, there are infinitely many irrational numbers.

(C) Between any two distinct real numbers, there is always an integer.

(D) Between any two distinct rational numbers, there is always an irrational number.

Question 10. If you use a geometric construction involving a right triangle on the number line starting from 0, which number can NOT be represented easily as the length of the hypotenuse or a leg using simple integer lengths for the other sides?

(A) $\sqrt{13}$ (from $2^2+3^2$)

(B) $\sqrt{17}$ (from $1^2+4^2$)

(C) $\sqrt{20}$ (from $2^2+4^2$)

(D) $\sqrt{11}$

Question 1. Which inequality is NOT correct?

(A) $7 > 5$

(B) $-7 > -5$

(C) $0 < 3$

(D) $-2 < 0$

Question 2. Which statement about the absolute value of a number is FALSE?

(A) The absolute value of a number is always non-negative.

(B) $|-5| = 5$.

(C) $|x| = x$ for all real numbers $x$.

(D) The absolute value represents distance from zero.

Question 3. Which number is NOT between $\frac{1}{3}$ and $\frac{2}{3}$?

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

(B) $\frac{5}{9}$

(C) 0.4

(D) 0.7

Question 4. Which of the following is NOT a correct way to compare $\frac{3}{5}$ and $\frac{2}{3}$?

(A) Compare their decimal equivalents: 0.6 and 0.666...

(B) Find a common denominator: $\frac{9}{15}$ and $\frac{10}{15}$.

(C) Cross-multiply: $3 \times 3$ and $5 \times 2$.

(D) Compare their numerators directly.

Question 5. Which statement about comparing very large or very small positive numbers in scientific notation is FALSE?

(A) If the exponents are different, the number with the larger exponent is greater.

(B) If the exponents are the same, compare the base numbers ($a$ in $a \times 10^n$).

(C) $2 \times 10^5$ is greater than $5 \times 10^4$.

(D) $3 \times 10^{-3}$ is smaller than $5 \times 10^{-4}$.

Question 6. Which of the following numbers is NOT between $\sqrt{2}$ and $\sqrt{3}$?

(A) 1.5

(B) 1.6

(C) 1.7

(D) 1.8

Question 7. Which statement about ordering integers is FALSE?

(A) As you move to the right on the number line, integers increase.

(B) A positive integer is always greater than a negative integer.

(C) Zero is greater than any negative integer.

(D) For two negative integers, the one with the smaller absolute value is smaller.

Question 8. Which inequality represents numbers whose absolute value is NOT greater than or equal to 3?

(A) $|x| < 3$

(B) $|x| > 3$

(C) $|x| \geq 3$

(D) $-3 < x < 3$

Question 9. Which statement about comparing fractions is FALSE?

(A) If denominators are equal, compare numerators.

(B) If numerators are equal, compare denominators (the one with the smaller denominator is larger, for positive fractions).

(C) Cross-multiplication can be used.

(D) You can only compare fractions with the same numerator and denominator.

Question 10. Which of the following is NOT a number between 0 and 0.1?

(A) 0.05

(B) 0.001

(C) 0.11

(D) 0.099

Question 1. Which statement about fundamental operations on whole numbers is FALSE?

(A) Addition is commutative and associative.

(B) Multiplication is commutative and associative.

(C) Subtraction is commutative.

(D) Division by zero is undefined.

Question 2. Which of the following expressions involving integers does NOT result in a positive value?

(A) $(-5) \times (-3)$

(B) $10 + (-4)$

(C) $-8 \div (-2)$

(D) $-6 - (-10)$

Question 3. Which calculation involving fractions is NOT correct?

(A) $\frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$

(B) $\frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3}$

(C) $\frac{3}{5} - \frac{1}{5} = \frac{2}{5}$

(D) $\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$

Question 4. Which calculation involving decimal numbers is NOT correct?

(A) $0.5 + 0.3 = 0.8$

(B) $1.2 \times 0.1 = 0.12$

(C) $2.5 - 1.0 = 1.5$

(D) $4.4 \div 0.4 = 1.1$

Question 5. Which statement about operations on rational numbers is FALSE?

(A) The sum of two rational numbers is always rational.

(B) The product of two rational numbers is always rational.

(C) The difference of two rational numbers is always rational.

(D) The division of a non-zero rational number by a non-zero rational number is always irrational.

Question 6. According to BODMAS/PEMDAS, which operation should be performed LAST in the expression $10 + 5 \times (6 - 2)$?

(A) Addition

(B) Multiplication

(C) Subtraction

(D) Parentheses

Question 7. Which statement about operations on real numbers is FALSE?

(A) Addition of real numbers is commutative.

(B) Multiplication of real numbers is associative.

(C) The product of a non-zero rational number and an irrational number is always irrational.

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

Question 8. Which simplification is NOT correct?

(A) $2 \times 0 + 3 = 3$

(B) $10 \div 2 - 1 = 4$

(C) $5 + 5 \times 5 = 50$

(D) $(1 + 2) \times 3 = 9$

Question 9. Which expression involving negative integers is NOT evaluated correctly?

(A) $-4 + (-5) = -9$

(B) $-6 - (-2) = -4$

(C) $(-7) \times 2 = -14$

(D) $12 \div (-3) = 4$

Question 10. Which simplification of an expression involving various number types is NOT correct?

(A) $0.5 + \frac{1}{2} = 1$

(B) $\sqrt{4} \times 3 = 6$

(C) $\frac{3}{4} \div 0.25 = 3$

(D) $5 - \sqrt{9} = \sqrt{2}$

Question 1. Which of the following is NOT a property of addition of real numbers?

(A) Commutativity

(B) Associativity

(C) Distributivity over multiplication

(D) Existence of additive inverse

Question 2. Which is NOT the identity element for the given operation and set?

(A) Addition on Real Numbers: 0

(B) Multiplication on Real Numbers: 1

(C) Addition on Natural Numbers: 0

(D) Multiplication on Natural Numbers: 1

Question 3. Which statement about properties of operations is FALSE?

(A) Subtraction is not commutative for real numbers.

(B) Division is not associative for real numbers (excluding division by zero).

(C) Multiplication is distributive over addition for real numbers.

(D) Division is distributive over addition for real numbers (e.g., $(a+b)/c = a/c + b/c$).

Question 4. Which is NOT a correct divisibility test?

(A) A number is divisible by 2 if its units digit is even.

(B) A number is divisible by 5 if its units digit is 0 or 5.

(C) A number is divisible by 10 if its units digit is 0.

(D) A number is divisible by 3 if its units digit is 3, 6, or 9.

Question 5. Which statement about divisibility is FALSE?

(A) If a number is divisible by 6, it is divisible by both 2 and 3.

(B) If a number is divisible by 10, it is divisible by both 2 and 5.

(C) If a number is divisible by 12, it is divisible by both 3 and 4.

(D) If a number is divisible by 8, it is divisible by both 2 and 4.

Question 6. Which of the following is NOT a perfect square?

(A) 1

(B) 16

(C) 121

(D) 150

Question 7. Which is NOT a triangular number?

(A) 1

(B) 3

(C) 5

(D) 6

Question 8. Which statement about the closure property is FALSE?

(A) Natural numbers are closed under addition.

(B) Integers are closed under subtraction.

(C) Rational numbers are closed under multiplication.

(D) Whole numbers are closed under division.

Question 9. Which is NOT an identity related to real numbers?

(A) $a+0 = a$

(B) $a \times 1 = a$

(C) $a \times 0 = a$

(D) $a + (-a) = 0$

Question 10. Which number is NOT divisible by 9?

(A) 27

(B) 108

(C) 216

(D) 345

Question 1. Which is NOT a type of fraction?

(A) Proper fraction

(B) Improper fraction

(C) Negative fraction

(D) Unit fraction

Question 2. Which fraction is NOT equivalent to $\frac{1}{3}$?

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

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

(C) $\frac{4}{12}$

(D) $\frac{5}{18}$

Question 3. Which statement about reducing a fraction is FALSE?

(A) Divide the numerator and denominator by a common factor.

(B) Divide the numerator and denominator by their HCF to get the simplest form.

(C) The value of the fraction changes when it is reduced.

(D) Reduction is also called simplifying the fraction.

Question 4. Which decimal is NOT a terminating decimal?

(A) 0.25

(B) 0.75

(C) 0.333...

(D) 1.5

Question 5. Which fraction-to-decimal conversion is NOT correct?

(A) $\frac{1}{2} = 0.5$

(B) $\frac{1}{4} = 0.25$

(C) $\frac{3}{4} = 0.75$

(D) $\frac{1}{5} = 0.15$

Question 6. Which decimal-to-fraction conversion is NOT correct?

(A) $0.1 = \frac{1}{10}$

(B) $0.2 = \frac{1}{5}$

(C) $0.5 = \frac{1}{2}$

(D) $0.8 = \frac{8}{100}$

Question 7. Which of the following is NOT an improper fraction?

(A) $\frac{7}{5}$

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

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

(D) $\frac{12}{11}$

Question 8. Which of the following is NOT equivalent to $1.5$?

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

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

(C) $\frac{15}{10}$

(D) $1 \frac{5}{100}$

Question 9. Which pair are NOT like decimals?

(A) 0.23, 1.45

(B) 5.1, 0.9

(C) 10.01, 3.50

(D) 0.7, 0.07

Question 10. Which method is NOT generally used to convert a mixed number to an improper fraction?

(A) Multiply the whole number by the denominator and add the numerator.

(B) Keep the same denominator.

(C) Add the whole number to the numerator and keep the same denominator.

(D) Write the whole number as a fraction with the same denominator and add it to the fraction part.

Question 1. Which of the following numbers does NOT have a terminating decimal expansion?

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

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

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

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

Question 2. Which statement about the decimal expansion of a rational number is FALSE?

(A) It can be terminating.

(B) It can be non-terminating and recurring.

(C) It can be non-terminating and non-recurring.

(D) The prime factors of the denominator (in simplest form) determine if it terminates.

Question 3. Which of the following numbers is NOT rational?

(A) $0.\overline{1}$

(B) $0.25$

(C) $\sqrt{9}$

(D) $\sqrt{7}$

Question 4. Which $\frac{p}{q}$ form conversion is NOT correct?

(A) $0.\overline{5} = \frac{5}{9}$

(B) $0.\overline{14} = \frac{14}{99}$

(C) $0.\overline{3} = \frac{1}{3}$

(D) $0.\overline{6} = \frac{6}{10}$

Question 5. Which statement about irrational numbers is FALSE?

(A) They are real numbers.

(B) Their decimal expansions do not repeat or terminate.

(C) They can be written as a ratio of two integers.

(D) Examples include $\sqrt{p}$ where $p$ is a non-square prime.

Question 6. Which expression does NOT require rationalization of the denominator to remove a radical?

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

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

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

(D) $\frac{5}{\sqrt{7}-\sqrt{3}}$

Question 7. Which rationalization is NOT correct?

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

(B) $\frac{1}{3+\sqrt{2}} = \frac{3-\sqrt{2}}{7}$

(C) $\frac{1}{\sqrt{5}-1} = \frac{\sqrt{5}+1}{4}$

(D) $\frac{\sqrt{3}}{\sqrt{7}} = \frac{3}{\sqrt{21}}$

Question 8. Which denominator (in simplest form) does NOT result in a terminating decimal expansion?

(A) 4

(B) 10

(C) 16

(D) 21

Question 9. Which conversion from recurring decimal to fraction is NOT correct?

(A) $0.\overline{7} = \frac{7}{9}$

(B) $0.\overline{25} = \frac{25}{99}$

(C) $0.\overline{123} = \frac{123}{999}$

(D) $0.1\overline{6} = \frac{16-1}{99} = \frac{15}{99} = \frac{5}{33}$

Question 10. Which of the following is NOT a proof of irrationality discussed in number systems?

(A) Proof by contradiction (assuming rational and reaching a contradiction)

(B) Using the properties of prime factors in perfect squares (Fundamental Theorem of Arithmetic)

(C) Using long division to show the decimal doesn't terminate or repeat

(D) Direct proof by showing it cannot be written as p/q

Question 1. Which of the following is NOT a factor of 42?

(A) 6

(B) 7

(C) 8

(D) 21

Question 2. Which of the following is NOT a multiple of 5?

(A) 15

(B) 40

(C) 51

(D) 100

Question 3. Which number is NOT divisible by 3?

(A) 12

(B) 51

(C) 101

(D) 345

Question 4. Which statement about prime and composite numbers is FALSE?

(A) 2 is the smallest prime number.

(B) Every prime number is odd.

(C) 4 is the smallest composite number.

(D) 1 is neither prime nor composite.

Question 5. Which of the following is NOT the prime factorization of the given number?

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

(B) $30 = 2 \times 3 \times 5$

(C) $48 = 2^3 \times 6$

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

Question 6. Which divisibility test is NOT correctly stated?

(A) By 4: The number formed by the last two digits is divisible by 4.

(B) By 8: The number formed by the last three digits is divisible by 8.

(C) By 9: The sum of the digits is divisible by 9.

(D) By 11: The number is divisible by 1 and 11.

Question 7. Which statement about divisibility is FALSE?

(A) If a number is divisible by 6, it is divisible by both 2 and 3.

(B) If a number is divisible by 10, it is divisible by both 2 and 5.

(C) If a number is divisible by 12, it is divisible by both 3 and 4.

(D) If a number is divisible by 8, it is divisible by both 2 and 4.

Question 8. Which of the following numbers is NOT divisible by both 2 and 5?

(A) 10

(B) 40

(C) 100

(D) 150

Question 9. How many factors does a prime number NOT have?

(A) 1

(B) 2

(C) More than 2

(D) It has exactly 2 factors.

Question 10. Which pair of numbers are NOT co-prime?

(A) (2, 3)

(B) (4, 5)

(C) (8, 9)

(D) (6, 9)

Question 11. Which test is NOT sufficient to determine if a number is divisible by 6?

(A) Check if it's divisible by both 2 and 3.

(B) Check if its last digit is even and the sum of its digits is divisible by 3.

(C) Check if it's divisible by 2 and its units digit is 0 or 5.

(D) Check if it's an even number and divisible by 3.

Question 1. Which of the following is NOT the HCF of 24 and 36?

(A) A common factor of 24 and 36.

(B) The largest number that divides both 24 and 36.

(C) 12.

(D) 6.

Question 2. Which of the following is NOT the LCM of 8 and 12?

(A) A common multiple of 8 and 12.

(B) The smallest positive number that is a multiple of both 8 and 12.

(C) 24.

(D) 48.

Question 3. Which statement about the relationship between HCF and LCM of two positive integers $a$ and $b$ is FALSE?

(A) HCF(a,b) divides LCM(a,b).

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

(C) HCF(a,b) is always less than or equal to LCM(a,b).

(D) LCM(a,b) is always less than or equal to $a \times b$.

Question 4. Which is NOT a correct method for finding the HCF of two numbers?

(A) Listing all common factors and choosing the largest.

(B) Using prime factorization: product of common prime factors with lowest powers.

(C) Using the division method (Euclid's algorithm).

(D) Listing all common multiples and choosing the smallest.

Question 5. Which is NOT a correct method for finding the LCM of two numbers?

(A) Listing the first few multiples of each number and finding the smallest common one.

(B) Using prime factorization: product of all prime factors with highest powers.

(C) Using the formula $LCM(a, b) = \frac{a \times b}{HCF(a, b)}$.

(D) Using the division method (Euclid's algorithm).

Question 6. If the product of two numbers is 150 and their LCM is 30, which is NOT their HCF?

(A) 5

(B) $150/30$

(C) $30/5$

(D) HCF = 5

Question 7. Which application does NOT typically involve finding the LCM?

(A) Finding the smallest number of objects that can be arranged in groups of different sizes.

(B) Finding when events that repeat at different intervals will occur together.

(C) Adding or subtracting fractions with different denominators.

(D) Finding the largest size of squares that can tile a rectangular area.

Question 8. If two numbers are co-prime, which statement is FALSE?

(A) Their HCF is 1.

(B) Their LCM is the product of the numbers.

(C) They have no common prime factors.

(D) Both numbers must be prime.

Question 9. Which HCF/LCM calculation is NOT correct based on prime factorizations $a=2^3 \times 3 \times 5^2$ and $b=2^2 \times 3^2 \times 7$?

(A) $HCF(a, b) = 2^2 \times 3^1 = 12$

(B) $LCM(a, b) = 2^3 \times 3^2 \times 5^2 \times 7^1 = 8 \times 9 \times 25 \times 7 = 12600$

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

(D) Common prime factors are 2, 3, 5, and 7.

Question 10. Which application does NOT typically involve finding the HCF?

(A) Dividing two different types of items into the largest possible equal groups.

(B) Finding the maximum capacity of a container that can measure different quantities exactly.

(C) Simplifying a fraction to its lowest terms.

(D) Finding the smallest number of students needed for different class sizes.

Question 1. In Euclid's Division Lemma, $a = bq + r$, where $a$ and $b$ are positive integers, which is NOT a correct condition for $r$?

(A) $r \geq 0$

(B) $r < b$

(C) $r$ is an integer

(D) $r > 0$

Question 2. Which statement about Euclid's Division Algorithm is FALSE?

(A) It is based on Euclid's Division Lemma.

(B) It is used to find the HCF of two positive integers.

(C) The process terminates when the remainder is 0.

(D) The HCF is the last non-zero remainder.

Question 3. Which statement about the Fundamental Theorem of Arithmetic is FALSE?

(A) Every composite number can be factored into primes.

(B) The prime factorization of a composite number is unique (ignoring the order of factors).

(C) The theorem applies to prime numbers as well.

(D) The theorem is also known as the Unique Sum Theorem.

Question 4. Which of the following is NOT an application of the Fundamental Theorem of Arithmetic?

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

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

(C) Proving the irrationality of certain numbers like $\sqrt{3}$.

(D) Performing addition of integers.

Question 5. When finding the HCF of 100 and 190 using Euclid's algorithm, which step is NOT correct?

(A) $190 = 100 \times 1 + 90$

(B) $100 = 90 \times 1 + 10$

(C) $90 = 10 \times 9 + 0$

(D) The HCF is 90.

Question 6. Which number is NOT correctly expressed by its prime factorization?

(A) $18 = 2 \times 3^2$

(B) $56 = 2^3 \times 7$

(C) $72 = 2^3 \times 3^2$

(D) $105 = 3 \times 5 \times 7 \times 1$

Question 7. In the equation $a = bq + r$, which term is NOT necessarily a positive integer?

(A) $a$

(B) $b$

(C) $q$

(D) $r$

Question 8. The property of prime factor exponents in a perfect square is crucial for proving $\sqrt{2}$ is irrational. Which statement about this property is FALSE?

(A) In the prime factorization of a perfect square, all exponents are even.

(B) If a number is a perfect square, its prime factors' exponents are divisible by 2.

(C) $(\sqrt{2})^2 = 2$, and the exponent of 2 in its prime factorization is 1, which is odd.

(D) This property applies to all positive integers, whether they are perfect squares or not.

Question 9. Which statement about Euclid's Division Lemma is FALSE?

(A) It states that for any two positive integers $a$ and $b$, there exist unique integers $q$ and $r$.

(B) The remainder $r$ must satisfy $0 \leq r < b$.

(C) It is a fundamental result in number theory.

(D) It can be used to directly find the LCM of two numbers.

Question 10. Which of the following numbers does NOT fit the property from Euclid's Lemma? If $a=10, b=3$, then $10 = 3 \times 3 + 1$. Which combination of $q$ and $r$ for $a=10, b=3$ is NOT unique?

(A) $q=3, r=1$

(B) $q=2, r=4$

(C) $q=4, r=-2$

(D) All are unique.

Question 1. In the expression $a^n$, which term is NOT correctly identified?

(A) $a$ is the base.

(B) $n$ is the exponent.

(C) $a^n$ is the power.

(D) $n$ indicates how many times the base is added to itself.

Question 2. Which of the following is NOT a correct law of exponents?

(A) $a^m \times a^n = a^{m+n}$

(B) $(a^m)^n = a^{m \times n}$

(C) $\frac{a^m}{a^n} = a^{m-n}$ ($a \neq 0$)

(D) $(a+b)^m = a^m + b^m$

Question 3. Which statement about zero or negative exponents is FALSE?

(A) Any non-zero number raised to the power 0 is 1.

(B) $0^0$ is undefined.

(C) $a^{-n} = \frac{1}{a^n}$ for $a \neq 0$.

(D) $a^{-n} = -a^n$ for $a \neq 0$.

Question 4. Which calculation is NOT correct?

(A) $2^3 = 8$

(B) $(-3)^2 = 9$

(C) $10^{-2} = 0.01$

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

Question 5. Which of the following numbers is NOT written in correct standard form ($a \times 10^n$ with $1 \leq |a| < 10$)?

(A) $5.6 \times 10^7$

(B) $0.8 \times 10^{-4}$

(C) $1.0 \times 10^2$

(D) $9.99 \times 10^1$

Question 6. Which statement about comparing numbers in scientific notation is FALSE?

(A) $3 \times 10^5 > 2 \times 10^5$

(B) $1.5 \times 10^{-3} < 2.0 \times 10^{-3}$

(C) $4 \times 10^8 < 5 \times 10^7$

(D) $6 \times 10^{-6} < 7 \times 10^{-5}$

Question 7. Which simplification using laws of exponents is NOT correct?

(A) $a^5 \div a^2 = a^3$

(B) $(a^3)^4 = a^{12}$

(C) $(ab)^5 = a^5 b^5$

(D) $a^2 + a^3 = a^5$

Question 8. Which expression is NOT equivalent to $(\frac{2}{3})^{-2}$?

(A) $\frac{1}{(\frac{2}{3})^2}$

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

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

(D) $-\frac{4}{9}$

Question 9. Which application does NOT typically involve exponents or standard form?

(A) Representing very large numbers (like astronomical distances).

(B) Representing very small numbers (like the size of atoms).

(C) Simplifying complex multiplication and division (historically using logarithms, which are related to exponents).

(D) Finding the sum of a series of numbers.

Question 10. Which equation based on laws of exponents is FALSE?

(A) $a^m \div a^m = a^0 = 1$ (for $a \neq 0$)

(B) $(a \times b)^m = a^m \times b^m$

(C) $(\frac{a}{b})^m = \frac{a^m}{b^m}$ (for $b \neq 0$)

(D) $a^m + a^n = a^{m+n}$

Question 1. Which of the following is NOT a perfect square?

(A) 4

(B) 25

(C) 64

(D) 72

Question 2. Which statement about perfect squares is FALSE?

(A) A number ending in 0, 1, 4, 5, 6, or 9 can be a perfect square.

(B) A perfect square can end in 2, 3, 7, or 8.

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

(D) The square of an even number is even.

Question 3. Which is NOT a correct square root calculation?

(A) $\sqrt{100} = 10$

(B) $\sqrt{1.44} = 1.2$

(C) $\sqrt{0.09} = 0.03$

(D) $\sqrt{\frac{16}{25}} = \frac{4}{5}$

Question 4. Which is NOT a Pythagorean triplet?

(A) (3, 4, 5)

(B) (5, 12, 13)

(C) (8, 15, 17)

(D) (6, 7, 8)

Question 5. Which method is NOT a standard way to find the square root of a perfect square?

(A) Repeated subtraction of consecutive odd numbers.

(B) Prime factorization method.

(C) Long division method.

(D) Adding up consecutive integers.

Question 6. Which statement about estimating square roots is FALSE?

(A) $\sqrt{50}$ is between $\sqrt{49}=7$ and $\sqrt{64}=8$.

(B) $\sqrt{10}$ is between $\sqrt{9}=3$ and $\sqrt{16}=4$.

(C) Estimation gives an exact value for the square root.

(D) You can use perfect squares to estimate the value of other square roots.

Question 7. Which calculation using prime factorization to find a square root is NOT correct?

(A) $\sqrt{36} = \sqrt{2^2 \times 3^2} = 2 \times 3 = 6$

(B) $\sqrt{100} = \sqrt{2^2 \times 5^2} = 2 \times 5 = 10$

(C) $\sqrt{144} = \sqrt{2^4 \times 3^2} = 2^2 \times 3 = 12$

(D) $\sqrt{81} = \sqrt{3^4} = 3$

Question 8. Which statement about the square root of a number is FALSE?

(A) Every positive number has a unique positive square root.

(B) $\sqrt{x^2} = x$ for all real numbers $x$.

(C) $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ for non-negative $a$ and $b$.

(D) $\sqrt{a/b} = \sqrt{a} / \sqrt{b}$ for non-negative $a$ and positive $b$.

Question 9. Which number, when squared, does NOT result in the given value?

(A) $0.1 \to 0.01$

(B) $0.5 \to 0.25$

(C) $1.1 \to 1.21$

(D) $0.2 \to 0.4$

Question 10. Which is NOT a correct observation about the units digit of a perfect square?

(A) If a number ends in 1, its square ends in 1.

(B) If a number ends in 4, its square ends in 6.

(C) If a number ends in 5, its square ends in 25.

(D) If a number ends in 8, its square ends in 4.

Question 1. Which of the following is NOT a perfect cube?

(A) 8

(B) 27

(C) 64

(D) 100

Question 2. Which statement about cubes and cube roots is FALSE?

(A) The cube of a positive number is positive.

(B) The cube of a negative number is negative.

(C) The cube root of a negative number is negative.

(D) The cube root of any real number is always positive.

Question 3. Which is NOT a correct cube root calculation?

(A) $\sqrt[3]{1} = 1$

(B) $\sqrt[3]{125} = 5$

(C) $\sqrt[3]{-8} = -2$

(D) $\sqrt[3]{0.027} = 0.03$

Question 4. Which calculation using prime factorization to find a cube root is NOT correct?

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

(B) $\sqrt[3]{1000} = \sqrt[3]{2^3 \times 5^3} = 2 \times 5 = 10$

(C) $\sqrt[3]{729} = \sqrt[3]{3^6} = 3^2 = 9$

(D) $\sqrt[3]{1728} = \sqrt[3]{2^4 \times 3^3} = 2 \times 3 = 6$

Question 5. Which statement about the units digit of a cube is FALSE?

(A) If a number ends in 0, its cube ends in 0.

(B) If a number ends in 1, its cube ends in 1.

(C) If a number ends in 2, its cube ends in 4.

(D) If a number ends in 7, its cube ends in 3.

Question 6. Which method is NOT typically used to find the cube root of a number?

(A) Prime factorization method.

(B) Estimation method (for large numbers).

(C) Long division method.

(D) Using cube root tables or calculators.

Question 7. Which number, when cubed, does NOT result in the given value?

(A) $0.2 \to 0.008$

(B) $-1 \to -1$

(C) $\frac{1}{3} \to \frac{1}{27}$

(D) $10 \to 100$

Question 8. Which number is NOT a perfect cube?

(A) 1

(B) 1000

(C) 2744

(D) 2500

Question 9. Which statement about making a number a perfect cube is FALSE?

(A) To make a number a perfect cube by multiplication, each prime factor exponent must become a multiple of 3.

(B) To make a number a perfect cube by division, divide by the prime factors whose exponents are not multiples of 3.

(C) If the prime factorization of a number is $2^2 \times 3^1$, multiply by $2^1 \times 3^2$ to make it a perfect cube.

(D) If the prime factorization is $2^4 \times 5^2$, divide by $2^1 \times 5^2$ to make it a perfect cube.

Question 10. Which is NOT a correct property of cube roots?

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

(B) $\sqrt[3]{a / b} = \sqrt[3]{a} / \sqrt[3]{b}$ (for $b \neq 0$)

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

(D) $\sqrt[3]{a^3} = a$

Question 1. Which of the following roundings is NOT correct?

(A) 78 rounded to the nearest ten is 80.

(B) 143 rounded to the nearest hundred is 100.

(C) 567 rounded to the nearest hundred is 600.

(D) 95 rounded to the nearest ten is 100.

Question 2. Which statement about rounding is FALSE?

(A) If the digit to be rounded is less than 5, the previous digit remains unchanged.

(B) If the digit to be rounded is 5 or greater, the previous digit is increased by one.

(C) Digits to the right of the rounding place are replaced by zeros (for whole numbers).

(D) Digits to the right of the rounding place are discarded (for decimal places).

Question 3. Which rounding of a decimal number is NOT correct?

(A) 0.78 rounded to nearest tenth is 0.8.

(B) 1.23 rounded to nearest tenth is 1.2.

(C) 5.05 rounded to nearest tenth is 5.0.

(D) 0.96 rounded to nearest tenth is 1.0.

Question 4. Which statement about estimation is FALSE?

(A) Estimation provides an approximate value.

(B) Estimation is useful for checking the reasonableness of a calculated answer.

(C) Estimation is always more accurate than exact calculation.

(D) Rounding is often used as a method for estimation.

Question 5. Which estimation is NOT correct?

(A) $19 \times 32 \approx 20 \times 30 = 600$

(B) $145 + 278 \approx 100 + 300 = 400$

(C) $512 \div 24 \approx 500 \div 20 = 25$

(D) $\textsf{₹} 98.75 \times 5 \approx \textsf{₹} 100 \times 5 = \textsf{₹} 500$

Question 6. Which rounding to the nearest Rupee is NOT correct?

(A) $\textsf{₹} 45.60 \to \textsf{₹} 46$

(B) $\textsf{₹} 123.40 \to \textsf{₹} 123$

(C) $\textsf{₹} 99.50 \to \textsf{₹} 99$

(D) $\textsf{₹} 5.90 \to \textsf{₹} 6$

Question 7. Which rounding to the nearest thousand (Indian System) is NOT correct?

(A) 1,23,456 $\to$ 1,23,000

(B) 5,67,890 $\to$ 5,68,000

(C) 10,00,499 $\to$ 10,00,000

(D) 99,500 $\to$ 99,000

Question 8. Which rounding to the nearest million (International System) is NOT correct?

(A) 1,456,789 $\to$ 1,000,000

(B) 5,678,901 $\to$ 6,000,000

(C) 10,000,000 $\to$ 10,000,000

(D) 999,999 $\to$ 1,000,000

Question 9. Which statement about rounding to different place values is FALSE?

(A) Rounding 12345 to the nearest hundred gives 12300.

(B) Rounding 12345 to the nearest thousand gives 12000.

(C) Rounding 12345 to the nearest ten thousand gives 10000.

(D) Rounding 12345 to the nearest ten gives 12350.

Question 10. Which is NOT a typical scenario where estimation is commonly used?

(A) Quickly calculating approximate costs while shopping.

(B) Checking if the result of a precise calculation is reasonable.

(C) Solving complex mathematical equations requiring high accuracy.

(D) Estimating distances or quantities in everyday life.

Question 1. Which statement about logarithms is FALSE?

(A) $\log_b A = C$ means $b^C = A$.

(B) The base $b$ must be a positive number not equal to 1.

(C) The number $A$ (whose logarithm is taken) must be positive.

(D) The logarithm of a negative number is always negative.

Question 2. Which is NOT a correct law of logarithms (assuming a valid base $b$)?

(A) $\log_b (M \times N) = \log_b M + \log_b N$

(B) $\log_b (M / N) = \log_b M - \log_b N$

(C) $\log_b M^k = k \times \log_b M$

(D) $\log_b (M + N) = \log_b M + \log_b N$

Question 3. Which value is NOT correct?

(A) $\log_{10} 1 = 0$

(B) $\log_5 5 = 1$

(C) $\log_2 8 = 3$

(D) $\log_{10} 0 = 1$

Question 4. Which statement about Antilogarithms is FALSE?

(A) Antilogarithm is the inverse operation of logarithm.

(B) If $\log_b A = C$, then Antilog$_b (C) = A$.

(C) Finding the antilog of $C$ to base $b$ is equivalent to calculating $b^C$.

(D) Antilogarithm is only defined for positive numbers.

Question 5. Which characteristic calculation (base 10) is NOT correct?

(A) Characteristic of $\log_{10} 500$ is 2.

(B) Characteristic of $\log_{10} 50.5$ is 1.

(C) Characteristic of $\log_{10} 5.005$ is 0.

(D) Characteristic of $\log_{10} 0.5$ is 0.

Question 6. Which statement about the mantissa of a logarithm (base 10) is FALSE?

(A) It is the fractional part of the logarithm.

(B) It is always non-negative.

(C) It is usually found using logarithm tables.

(D) The mantissa depends on the position of the decimal point in the number.

Question 7. Which logarithm calculation using laws is NOT correct (assuming $\log_{10} 2 = 0.3010$, $\log_{10} 3 = 0.4771$)?

(A) $\log_{10} 4 = 2 \times \log_{10} 2 = 0.6020$

(B) $\log_{10} 6 = \log_{10} 2 + \log_{10} 3 = 0.7781$

(C) $\log_{10} 9 = 3 \times \log_{10} 3 = 1.4313$

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

Question 8. Which statement is FALSE about the bases of logarithms?

(A) The common logarithm uses base 10.

(B) The natural logarithm uses base $e$ (approximately 2.718).

(C) Any positive number other than 1 can be used as a base.

(D) The base of a logarithm can be a negative number.

Question 9. Which calculation using antilogarithms is NOT correct?

(A) Antilog$_{10}(1) = 10$

(B) Antilog$_{10}(0) = 1$

(C) Antilog$_{10}(2) = 100$

(D) Antilog$_{10}(-1) = -10$

Question 10. Which is NOT a typical application where logarithms are used to simplify calculations?

(A) Calculating complex products like $123.45 \times 67.89$ by adding their logarithms.

(B) Calculating complex divisions like $987.6 \div 54.3$ by subtracting their logarithms.

(C) Calculating powers like $(1.05)^{20}$ by multiplying the logarithm and finding the antilog.

(D) Finding the sum of a large number of terms in an arithmetic series.

Question 1. Which of the following modulo operations is NOT correct?

(A) $10 \pmod 3 = 1$

(B) $15 \pmod 4 = 3$

(C) $20 \pmod 5 = 0$

(D) $22 \pmod 7 = 3$

Question 2. Which statement about congruence modulo $m$ ($a \equiv b \pmod m$) is FALSE?

(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) $a \equiv b \pmod m$ implies $a$ and $b$ are equal.

Question 3. Which is NOT a property of congruence modulo $m$?

(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) Division property (If $ka \equiv kb \pmod m$, then $a \equiv b \pmod m$)

Question 4. Which calculation of the last digit using modulo 10 is NOT correct?

(A) Last digit of $2^3$ is $2^3 \pmod{10} = 8 \pmod{10} = 8$.

(B) Last digit of $4^2$ is $4^2 \pmod{10} = 16 \pmod{10} = 6$.

(C) Last digit of $5^3$ is $5^3 \pmod{10} = 125 \pmod{10} = 5$.

(D) Last digit of $7^2$ is $7^2 \pmod{10} = 49 \pmod{10} = 4$.

Question 5. Which number is NOT congruent to 5 modulo 6?

(A) 11

(B) -1

(C) 17

(D) 4

Question 6. Which statement is FALSE about modulo arithmetic operations?

(A) If $a \equiv b \pmod m$ and $c \equiv d \pmod m$, then $a+c \equiv b+d \pmod m$.

(B) If $a \equiv b \pmod m$ and $c \equiv d \pmod m$, then $a-c \equiv b-d \pmod m$.

(C) If $a \equiv b \pmod m$ and $c \equiv d \pmod m$, then $ac \equiv bd \pmod m$.

(D) If $a \equiv b \pmod m$ and $c \equiv d \pmod m$, then $a/c \equiv b/d \pmod m$ (always, for non-zero $c, d$).

Question 7. Which of the following is NOT a valid application of modulo arithmetic?

(A) Determining the day of the week after a certain number of days.

(B) Calculating the time on a clock after a certain number of hours.

(C) Checking divisibility rules (e.g., a number is divisible by 10 if it's congruent to 0 mod 10).

(D) Finding the exact square root of a number.

Question 8. If $x \equiv 3 \pmod 7$, which of the following can NOT be a value of $x$?

(A) 10

(B) 17

(C) -4

(D) 21

Question 9. Which statement about congruence classes is FALSE?

(A) The integers are partitioned into $m$ distinct congruence classes modulo $m$.

(B) The set of congruence classes modulo $m$ is denoted by $\mathbb{Z}_m$ or $\mathbb{Z}/m\mathbb{Z}$.

(C) Two integers are in the same congruence class modulo $m$ if and only if they are congruent modulo $m$.

(D) Each congruence class modulo $m$ contains only one integer.

Question 10. Which calculation using properties of congruence is NOT correct?

(A) $10 \equiv 1 \pmod 3$, so $10^2 \equiv 1^2 \pmod 3 \implies 100 \equiv 1 \pmod 3$.

(B) $7 \equiv 3 \pmod 4$, so $7 \times 2 \equiv 3 \times 2 \pmod 4 \implies 14 \equiv 6 \pmod 4 \implies 14 \equiv 2 \pmod 4$.

(C) $5 \equiv 5 \pmod {10}$.

(D) $12 \equiv 2 \pmod 5$ and $18 \equiv 3 \pmod 5$, so $12+18 \equiv 2+3 \pmod 5 \implies 30 \equiv 5 \pmod 5 \implies 30 \equiv 1 \pmod 5$.

Question 1. Which unit conversion is NOT correct?

(A) 1 meter = 100 cm

(B) 1 km = 1000 m

(C) 1 litre = 1000 ml

(D) 1 kg = 100 g

Question 2. Which operation is NOT typically used to solve word problems involving combining quantities?

(A) Addition

(B) Multiplication

(C) Subtraction

(D) Finding HCF

Question 3. A shopkeeper had 500 kg of rice. He sold 325.5 kg. Which calculation does NOT represent the amount of rice left?

(A) $500 - 325.5$ kg

(B) 174.5 kg

(C) $500 + 325.5$ kg

(D) (Original quantity) - (Quantity sold)

Question 4. The cost of 1 meter of wire is $\textsf{₹} 15.75$. Which calculation does NOT represent the cost of 10 meters of wire?

(A) $10 \times \textsf{₹} 15.75$

(B) $\textsf{₹} 157.50$

(C) $\textsf{₹} 15.75 + \textsf{₹} 10$

(D) Sum of $\textsf{₹} 15.75$ ten times.

Question 5. A car travels 120 km in 2 hours. Which calculation does NOT represent its average speed?

(A) Distance / Time

(B) 120 km / 2 hours = 60 km/hour

(C) $120 \times 2$ km/hour

(D) 60 km/hour

Question 6. If a recipe requires $\frac{1}{4}$ cup of sugar, and you make 3 times the recipe, which calculation does NOT represent the amount of sugar needed?

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

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

(C) $0.25 \times 3$ cups

(D) $\frac{1}{4} + 3$ cups

Question 7. A sum of $\textsf{₹} 500$ is divided into 5 equal parts. Which calculation does NOT represent the amount in each part?

(A) $500 \div 5$

(B) $\textsf{₹} 100$

(C) $500 \times \frac{1}{5}$

(D) $500 - 5$

Question 8. A worker is paid $\textsf{₹} 800$ per day. How much is NOT earned in 7 days?

(A) $800 \times 7$ Rupees

(B) $\textsf{₹} 5600$

(C) Sum of $\textsf{₹} 800$ seven times.

(D) $\textsf{₹} 800 + 7$ Rupees.

Question 9. A rectangular field has length 15 m and width 10 m. Which calculation does NOT represent its perimeter?

(A) $2 \times (15 + 10)$ m

(B) $2 \times 15 + 2 \times 10$ m

(C) $30 + 20 = 50$ m

(D) $15 \times 10 = 150$ sq m

Question 10. A two-digit number has digits $t$ (tens) and $u$ (units). Which expression does NOT represent the number?

(A) $10t + u$

(B) $t \times 10 + u \times 1$

(C) $t+u$

(D) $t0 + u$ (in expanded form concept)