Saket wrote four different algebraic expressions in his notebook. These are $f(x) = 2x^3 - \dfrac{1}{3}x,\ g(x) = x - \dfrac{1}{x} + 2,\ h(x) = x^2 - 2\sqrt{x}\ \text{and}\ k(y) = 2y - 1$.

(1) Which of these expressions is/are binomials ?

1. f(x), g(x), h(x) only
2. f(x), g(x), k(y) only
3. f(x), h(x), k(y) only
4. f(x), h(x) only

(2) Which of these expressions is/are not polynomials ?

1. f(x), g(x), h(x) only
2. f(x), h(x) only
3. g(x), h(x) only
4. h(x) only

(3) Which of these expressions is/are polynomials in two variables ?

1. f(x) only
2. k(y) only
3. f(x), k(y) only
4. None of these

(4) The degree of the polynomial f (x) is :

1. 0
2. 1
3. 2
4. 3

Given:

$f(x) = 2x^3 - \dfrac{1}{3}x$

$g(x) = x - \dfrac{1}{x} + 2$

$h(x) = x^2 - 2\sqrt{x}$

k(y) = 2y - 1

(1)

f(x): 2 terms (2x³ and $-\dfrac{1}{3}x$) → Binomial.

g(x): 3 terms (x, $-\dfrac{1}{x}$, and $2$) → Trinomial.

h(x): 2 terms (x² and $-2\sqrt{x}$) → Binomial.

k(y): 2 terms (2y and -1) → Binomial.

Binomial expressions are f(x), h(x), k(y).

Hence, option 3 is the correct option.

(2) A polynomial cannot have variables in the denominator or under a root (fractional powers).

Analysis:

$f(x) = 2x^3 - \dfrac{1}{3}x$

The powers of x are 3 and 1. Both are whole numbers. It is a polynomial.

$g(x) = x - \dfrac{1}{x} + 2$

The term $-\dfrac{1}{x}$ can be written as -x^-1, the exponent -1 which is a negative integer not a whole number. So, it is not a polynomial.

$h(x) = x^2 - 2\sqrt{x}$

The term $-2\sqrt{x}$ can be written as -2x^1/2, the exponent $\dfrac{1}{2}$ is a fraction, not a whole number. So, it is not a polynomial.

k(y) = 2y - 1

The power of y is 1, which is a whole number. So, it is a polynomial.

So, g(x) and h(x) are not polynomials.

Hence, option 3 is the correct option.

(3) A "polynomial in two variables" must contain exactly two different letters (like x and y, or a and b) throughout its terms.

Analysis:

$f(x) = 2x^3 - \dfrac{1}{3}x$

Only x appears in both terms. This is a polynomial in one variable (x).

$g(x) = x - \dfrac{1}{x} + 2$

Only x appears. This is an algebraic expression in one variable (x). It is not a polynomial.

$h(x) = x^2 - 2\sqrt{x}$

Only x appears. This is an algebraic expression in one variable (x). It is not a polynomial.

k(y) = 2y - 1

Only y appears. This is a polynomial in one variable (y).

No expression is a polynomial in two variables.

Hence, option 4 is the correct option.

(4) $f(x) = 2x^3 - \dfrac{1}{3}x$

The highest power of x in $2x^3 - \dfrac{1}{3}x$ is 3.

Hence, option 4 is the correct option.