Exercise-2.1, Class 9th, Maths, Chapter 2, NCERT

1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

(i) $4x^2-3x+7$
Answer. Yes — a polynomial in one variable $x$. All powers of $x$(2,1,0) are whole numbers and only $x$ appears.

(ii) $y^2+\sqrt{2}$
Answer. Yes — a polynomial in one variable $y$. $\sqrt{2}$ is a constant; powers of $y$ are whole numbers (2 and 0).

(iii) $3\sqrt{t}+t\sqrt{2}$ (i.e. $3t^{1\mathrm{/}2}+\sqrt{2}t$)
Answer. Not a polynomial in one variable, because $3t^{1\mathrm{/}2}$ has exponent $1\mathrm{/}2$ (not a whole number).

(iv) $y+{\displaystyle \frac{2}{y}}$ (i.e. $y+2y^{-1}$)
Answer. Not a polynomial — it contains $y^{-1}$ (exponent $-1$), not a whole number.

(v) $x^{10}+y^3+t^{50}$
Answer. This is a polynomial but in three variables $x,y,t$. Not a polynomial in one variable.

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2. Write the coefficient of $x^2$ in each of the following:

(i) $2+x^2+x$
Answer. $1$.

(ii) $2-x^2+x^3$
Answer. $-1$.

(iii) ${\displaystyle \frac{\pi }{2}}{x}^2+x$
Answer. ${\displaystyle \frac{\pi }{2}}$.

(iv) $\sqrt{2}x-1$
Answer. $0$ (there is no $x^2$ term).

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3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Answer. Examples (many possible):

• Binomial of degree 35: $x^{35}+1$.

• Monomial of degree 100: $7y^{100}$.

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4. Write the degree of each of the following polynomials:

(i) $5x^3+4x^2+7x$
Answer. Degree $3$.

(ii) $4-y^2$
Answer. Degree $2$.

(iii) $5t-\sqrt{7}$
Answer. Degree $1$ (since $\sqrt{7}$ is a constant).

(iv) $3$
Answer. Degree $0$ (constant polynomial).

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5. Classify the following as linear, quadratic or cubic polynomials:

(i) $x^2+x$ — Quadratic.
(ii) $x-x^3$ — Cubic.
(iii) $y+y^2+4$ — Quadratic.
(iv) $1+x$ — Linear.
(v) $3t$ — Linear.
(vi) $r^2$ — Quadratic.
(vii) $7x^3$ — Cubic.