A. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
B. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A).
C. Assertion (A) is true but reason (R) is false.
D. Assertion (A) is false but reason (R) is true.
Assertion(A): $\frac{-7}{12}$ is a rational number.
Reason (R): Every rational number is a fraction.
Assertion(A): Every integer is a rational number.
Reason (R): On a number line, rational numbers $\frac{-3}{4}$ and $\frac{3}{4}$ are at equal distance from zero.
Assertion(A): The angle bisectors of a triangle meet at its orthocentre.
Reason (R): The incentre of a triangle is equidistant from its sides.
Assertion(A): $(5,12,13)$ are pythagorean triplets.
Reason (R): In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Answer (2)
Pair 1: Both assertion and reason are true, but the reason doesn't explain the assertion → (b).
Pair 2: Both assertion and reason are true, but the reason doesn't explain the assertion → (b).
Pair 3: Assertion is false, but the reason is true → (d).
Pair 4: Both assertion and reason are true, and the reason explains the assertion → (a).
Explanation
Analyze each assertion and reason pair Let's analyze each assertion and reason pair to determine the correct option.
Pair 1: Assertion (A): 12 − 7 is a rational number. Reason (R): Every rational number is a fraction.
A rational number is any number that can be expressed as a fraction q p , where p and q are integers and q = 0 . The number 12 − 7 fits this definition, so the assertion is true. The reason is also true, as rational numbers can be expressed as fractions. However, the reason doesn't explain why 12 − 7 is a rational number; it just states a property of rational numbers. Therefore, the correct option for this pair is (b).
Pair 2: Assertion (A): Every integer is a rational number. Reason (R): On a number line, rational numbers 4 − 3 and 4 3 are at equal distance from zero.
Any integer n can be written as 1 n , which fits the definition of a rational number. So, the assertion is true. The reason is also true; 4 − 3 and 4 3 are equidistant from zero. However, the reason does not explain why every integer is a rational number. Therefore, the correct option for this pair is (b).
Pair 3: Assertion (A): The angle bisectors of a triangle meet at its orthocentre. Reason (R): The incentre of a triangle is equidistant from its sides.
The angle bisectors of a triangle meet at the incentre , not the orthocentre. So, the assertion is false. The incentre is equidistant from the sides of the triangle, so the reason is true. Therefore, the correct option for this pair is (d).
Pair 4: Assertion (A): ( 5 , 12 , 13 ) are pythagorean triplets. Reason (R): In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
To check if ( 5 , 12 , 13 ) are Pythagorean triplets, we need to verify if 5 2 + 1 2 2 = 1 3 2 .
5 2 = 25 1 2 2 = 144 1 3 2 = 169 25 + 144 = 169 , so the assertion is true. The reason is the Pythagorean theorem, which is also true and explains why ( 5 , 12 , 13 ) are Pythagorean triplets. Therefore, the correct option for this pair is (a).
State the correct option for each pair Based on the analysis:
Pair 1: (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A) Pair 2: (b) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A) Pair 3: (d) Assertion (A) is false but reason (R) is true. Pair 4: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A)
Final Answer The correct options for each assertion and reason pair are:
Pair 1: (b) Pair 2: (b) Pair 3: (d) Pair 4: (a)
Examples
Understanding the relationship between assertions and reasons is crucial in various fields, such as law, scientific research, and even everyday problem-solving. For instance, in a legal argument, an assertion might be a claim made by a lawyer, and the reason would be the evidence supporting that claim. Similarly, in scientific research, an assertion could be a hypothesis, and the reason would be the experimental data supporting that hypothesis. By carefully analyzing the validity of assertions and the relevance of reasons, we can make informed decisions and construct sound arguments.
For the pairs of assertions and reasons, the correct options are: Pair 1 (B), Pair 2 (B), Pair 3 (D), and Pair 4 (A). Each analysis clarifies the truthfulness of assertions and their respective reasons. In summary, while some reasons support their assertions, others do not directly explain them.
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