Fundamental Concepts of Algebra — Assertion-Reason Questions

Assertion (A): When we add a monomial and a trinomial, then the answer can never be a monomial.

Reason (R): For adding algebraic expressions, we add their like terms only.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

3. Assertion (A) is true but Reason (R) is false.

4. Assertion (A) is false but Reason (R) is true.

Answer

A trinomial has 3 unlike terms. A monomial can be like at most one of these three terms, so on adding, at least two unlike terms will always remain. Hence, the sum can never be a monomial.

So, Assertion (A) is true.

While adding algebraic expressions, only like terms are added.

So, Reason (R) is true.

It is precisely because only like terms are added that the unlike terms of the trinomial remain in the sum. So, Reason (R) is the correct explanation of Assertion (A).

Hence, option 1 is the correct option.

Assertion (A): In $1 + 2a + 2a^2 + 4a^3$, the coefficient of $a^2$ is 2.

Reason (R): In an algebraic expression, the term which does not contain any literal is called a constant term.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

3. Assertion (A) is true but Reason (R) is false.

4. Assertion (A) is false but Reason (R) is true.

Answer

In the expression, the term containing $a^2$ is $2a^2 = a^2 \times 2$. So, the coefficient of $a^2$ is 2.

So, Assertion (A) is true.

By definition, the term which does not contain any literal is called a constant term.

So, Reason (R) is true.

Reason (R) only defines a constant term and says nothing about how the coefficient of a term is obtained. So, Reason (R) is not the correct explanation of Assertion (A).

Hence, option 2 is the correct option.

Assertion (A): The value of the polynomial $m^2n^2 + n^2p^2 + p^2m^2$ at $m = 1, n = -1, p = 2$ is 9.

Reason (R): In case of a polynomial in more than one variable, we cannot substitute the values of the variables simultaneously.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

3. Assertion (A) is true but Reason (R) is false.

4. Assertion (A) is false but Reason (R) is true.

Answer

Substituting $m = 1, n = -1$ and $p = 2$, we get:

$\Rightarrow m^2n^2 + n^2p^2 + p^2m^2\\[1em] \Rightarrow (1)^2(-1)^2 + (-1)^2(2)^2 + (2)^2(1)^2\\[1em] \Rightarrow 1 \times 1 + 1 \times 4 + 4 \times 1\\[1em] \Rightarrow 1 + 4 + 4\\[1em] \Rightarrow 9$

So, Assertion (A) is true.

In a polynomial of more than one variable, the values of all the variables can be substituted simultaneously.

So, Reason (R) is false.

Hence, option 3 is the correct option.