Fundamental Concepts of Algebra — Assertion Reason Type Questions

Assertion (A): Degree of

$3y^3 + \dfrac{1}{2}x^{\frac{5}{2}} \sqrt{y} + xy - 2xy^2$ is 3.

Reason (R): The highest sum of powers of all variables in a term of a polynomial is the degree of the polynomial.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

In the term $\dfrac{1}{2}x^{\frac{5}{2}}\sqrt{y} = \dfrac{1}{2}x^{\frac{5}{2}}y^{\frac{1}{2}}$, the power of $y$ is not a whole number. So the given expression is not a polynomial and hence its degree cannot be stated as 3. Thus, Assertion (A) is false.

Reason (R) is true, because the degree of a term of a polynomial is the sum of the exponents of the variables occurring in that term, and the degree of the polynomial is the greatest of the degrees of its terms. For example, in $4x^2y^3 + 7xy$, the degrees of the terms are (2 + 3) = 5 and (1 + 1) = 2, so the degree of the polynomial is 5.

Hence, option 2 is the correct option.

Assertion (A): $3xyz + 2x^2y + z^2x$ is a cubic polynomial in three variables.

Reason (R): A polynomial with degree 3 may have one variable only.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

In $3xyz + 2x^2y + z^2x$, the degree of each term is (1 + 1 + 1) = 3, (2 + 1) = 3 and (2 + 1) = 3. So it is a polynomial of degree 3 (a cubic polynomial) in the three variables $x, y$ and $z$. Thus, Assertion (A) is true.

A polynomial of degree 3 can also be in a single variable, for example $x^3 + 2x + 1$. So Reason (R) is also true.

Hence, option 3 is the correct option.

Assertion (A): $\dfrac{x}{2} + \dfrac{y^2}{5} + \dfrac{yz}{7}, \dfrac{xy}{2} + \dfrac{z^2}{5} + \dfrac{xz}{7}$ are polynomials of degree 2.

Reason (R): A polynomial with degree 2 may be a quadratic polynomial.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

In $\dfrac{x}{2} + \dfrac{y^2}{5} + \dfrac{yz}{7}$, the degrees of the terms are 1, 2 and 2, so its degree is 2.

In $\dfrac{xy}{2} + \dfrac{z^2}{5} + \dfrac{xz}{7}$, the degrees of the terms are 2, 2 and 2, so its degree is 2.

Thus, Assertion (A) is true. A polynomial of degree 2 is called a quadratic polynomial, so Reason (R) is also true.

Hence, option 3 is the correct option.