Assertion and Reason Questions
-
Assertion (A):
Every integer is a rational number.
Reason (R): A rational number can be expressed in the form \(\frac{p}{q}\), where p and q are integers and q≠0.-
A) Both A and R are true, and R is the correct explanation of
A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
Every integer can be expressed as a fraction with a denominator of 1, making it
a rational number.
-
A) Both A and R are true, and R is the correct explanation of
A.
-
Assertion (A):
The decimal representation of \(\frac{1}{3}\) is non-terminating.
Reason (R): The denominator of \(\frac{1}{3}\) contains prime factors other than 2 and 5.-
A) Both A and R are true, and R is the correct explanation of
A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
\(\frac{1}{3}\) has a decimal representation of 0.333..., which is
non-terminating because the prime factorization of the denominator includes 3.
-
A) Both A and R are true, and R is the correct explanation of
A.
-
Assertion (A):
The number \(\sqrt{2}\) is an irrational number.
Reason (R): The square root of any prime number is irrational.-
A) Both A and R are true, and R is the correct explanation of
A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
\(\sqrt{2}\) is irrational because it cannot be expressed as a fraction of two
integers.
-
A) Both A and R are true, and R is the correct explanation of
A.
-
Assertion (A):
The sum of two rational numbers is always a rational number.
Reason (R): Rational numbers are closed under addition.-
A) Both A and R are true, and R is the correct explanation of
A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
The closure property confirms that the sum of any two rational numbers remains
rational.
-
A) Both A and R are true, and R is the correct explanation of
A.
-
Assertion (A):
Every rational number can be expressed as a terminating or non-terminating
decimal.
Reason (R): Only irrational numbers have non-terminating decimals.-
A) Both A and R are true, and R is the correct explanation of
A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
C) A is true, but R is false.
-
Explanation:
Rational numbers can be expressed as terminating or repeating (non-terminating)
decimals, while irrational numbers have non-terminating, non-repeating decimals.
-
A) Both A and R are true, and R is the correct explanation of
A.
-
Assertion (A):
The product of two irrational numbers is always irrational.
Reason (R): The product of two rational numbers is always rational.-
A) Both A and R are true, and R is the correct explanation of
A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
D) A is false, but R is true.
-
Explanation:
The product of two irrational numbers can be rational (e.g., \(\sqrt{2}
\times \sqrt{2} =2 \), while the statement about rational products is
true.
-
A) Both A and R are true, and R is the correct explanation of
A.
-
Assertion (A):
The number 0 is a rational number.
Reason (R): 0 can be expressed as \(\frac{0}{1}\).-
A) Both A and R are true, and R is the correct explanation of
A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
Since 0 can be expressed as a fraction, it qualifies as a rational number.
-
A) Both A and R are true, and R is the correct explanation of
A.
-
Assertion (A):
The set of rational numbers is dense in the real numbers.
Reason (R): Between any two real numbers, there exists a rational number.-
A) Both A and R are true, and R is the correct explanation of
A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
This property confirms that rational numbers can be found between any two real
numbers, demonstrating their density.
-
A) Both A and R are true, and R is the correct explanation of
A.
-
Assertion (A):
\(\frac{7}{11}\) is a rational number.
Reason (R): Rational numbers can be expressed as fractions where both numerator and denominator are integers.-
A) Both A and R are true, and R is the correct explanation of
A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
Since \(\frac{7}{11}\) is a fraction of two integers, it is a rational
number.
-
A) Both A and R are true, and R is the correct explanation of
A.
-
Assertion (A):
The number \(\sqrt{3}\) is a rational number.
Reason (R): The square root of a non-perfect square is always rational.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
D) A is false, but R is true.
-
Explanation:
\(\sqrt{3}\) is irrational because it cannot be expressed as a fraction of two
integers.
-
Assertion (A):
The square root of a negative number is a real number.
Reason (R): The square root of a negative number is defined in the real number system.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
D) A is false, but R is true.
-
Explanation:
The square root of a negative number is not a real number; it is an imaginary
number.
-
Assertion (A):
The set of real numbers includes both rational and irrational numbers.
Reason (R): Rational numbers can be expressed as fractions, while irrational numbers cannot.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
This statement correctly describes the composition of the set of real numbers.
-
Assertion (A):
The product of two even integers is an even integer.
Reason (R): An even integer can be expressed as 2n, where nnn is an integer.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
The product of two even integers results in another even integer, which can be
expressed in the form 2n.
-
Assertion (A):
The number 1 is a rational number.
Reason (R): The number 1 can be expressed as \(\frac{1}{1}\).
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
Since 1 can be expressed as a fraction, it qualifies as a rational number.
-
Assertion (A):
The square root of a perfect square is always an integer.
Reason (R): Perfect squares are defined as the squares of integers.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
By definition, the square root of a perfect square is an integer.
-
Assertion (A):
The number \(\frac{5}{0}\) is defined in the real number system.
Reason (R): Division by zero is allowed in mathematics.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
D) A is false, but R is true.
-
Explanation:
Division by zero is undefined in mathematics, making \(\frac{5}{0}\) undefined.
-
Assertion (A):
The sum of two irrational numbers is always irrational.
Reason (R): The sum of two rational numbers is always rational.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
D) A is false, but R is true.
-
Explanation:
The sum of two irrational numbers can sometimes be rational (e.g.,
\(\sqrt{2} + (2 - \sqrt{2})=2\).
-
Assertion (A):
The decimal representation of \(\frac{1}{8}\) is terminating.
Reason (R): The denominator of \(\frac{1}{8}\) has prime factors only of 2.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
The fraction \(\frac{1}{8}\) can be expressed as 0.125, a terminating decimal.
-
Assertion (A):
The square of any odd integer is odd.
Reason (R): An odd integer can be expressed as 2n +1, where nnn is an integer.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
The square of 2n + 1 results in another odd number.
-
Assertion (A):
The number \(\sqrt{4}\) is an integer.
Reason (R): The square root of a perfect square is always an integer.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
Since 4 is a perfect square, \(\sqrt{4} = 2\), which is an integer.
-
Assertion (A):
The number −2 is a rational number.
Reason (R): Any integer can be expressed as a fraction.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
−2 can be expressed as \(\frac{-2}{1}\), which makes it a rational
number.
-
Assertion (A):
The number \(\pi\) is a rational number.
Reason (R): \(\pi\)can be expressed as a fraction of two integers.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
D) A is false, but R is true.
-
Explanation:
\(\pi\) is an irrational number and cannot be expressed as a fraction of two
integers.
-
Assertion (A):
The number \(\frac{9}{12}\) can be simplified to \(\frac{3}{4}\).
Reason (R): The GCD of 9 and 12 is 3.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
A) Both A and R are true, and R is the correct explanation of A.
-
Explanation:
Dividing both numerator and denominator by their GCD (3) yields
\(\frac{3}{4}\).
-
Assertion (A):
The number \(\sqrt{5} + \sqrt{7}\) is an irrational number.
Reason (R): The sum of two irrational numbers is always irrational.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
C) A is true, but R is false.
-
Explanation:
While \(\sqrt{5} + \sqrt{7}\) is indeed irrational, the sum of two
irrational numbers can sometimes be rational (e.g., \(\sqrt{2} + (2 -
\sqrt{2})\).
-
Assertion (A):
The number \(\frac{22}{7}\) is an approximation of π\piπ.
Reason (R): \(\frac{22}{7}\) is a rational number.
-
A) Both A and R are true, and R is the correct
explanation of A.
-
B) Both A and R are true, but R is not the correct
explanation of A.
-
C) A is true, but R is false.
-
D) A is false, but R is true.
-
Answer:
B) Both A and R are true, but R is not the correct explanation of A.
-
Explanation:
While \(\frac{22}{7}\) is a rational approximation of π\piπ, the reason
provided does not explain the assertion accurately.
