NCERT Class 9 Mathematics Chapter 2 Exercise 2.1 Solutions:

 NCERT Class 9 Mathematics Chapter 2, "Polynomials," introduces students to the concept of polynomials, their degrees, coefficients, and classification based on the number of terms and degrees. Exercise 2.1 focuses on identifying polynomials, determining their degrees, and classifying them accordingly.

Exercise 2.1 Solutions:

1. Identifying Polynomials in One Variable:

   Determine which of the following expressions are polynomials in one variable:

   a) $4x^2 - 3x + 7$

   b) $y^2 + \sqrt{2}$
   

   c) $3\sqrt{t} + t\sqrt{2}$
   

   d) $y + \frac{2}{y}$

   e) $x^2 + \frac{1}{x}$

   Solution:

   • (a) $4x^2 - 3x + 7$: This is a polynomial in one variable $x$ with whole number exponents (2, 1, and 0).

   • (b) $y^2 + \sqrt{2}$: This is a polynomial in one variable $y$ with whole number exponents (2 and 0).

   • (c) $3\sqrt{t} + t\sqrt{2}$: The term $3\sqrt{t}$ can be written as $3t^{1/2}$, which has a fractional exponent. Therefore, it's not a polynomial.

   • (d) $y + \frac{2}{y}$ : The term $\frac{2}{y}$ can be written as $2y^{-\mathrm{1 }}$

   • which has a negative exponent. Thus, it's not a polynomial.

   • (e) $x^2 + \frac{1}{x}$: The term $\frac{1}{x}$ can be written as $x^{-1}$ which has a negative exponent. Therefore, it's not a polynomial.

2. Coefficients of $x^2$:

   Write the coefficients of $x^2$ in each of the following:

   a) $2+x^2+\mathrm{<span\; class="base"><span\; class="strut"></span><span\; class="mord\; mathnormal">x</span></span>}$

   b) $2 - x^2 + x^3$
   

   c) $\frac{\pi}{2} x^2 + x$
   

   d) $\sqrt{2}x$

   Solution:

   • (a) $2+x^2$: Coefficient of $x^2$ is 1.

   • (b) $2 - x^2 + x^3$: Coefficient of $x^2$ is -1.

   • (c) $\frac{\pi}{2} x^2 + x$ : Coefficient of $x^2$ is $\frac{\pi}{2}$.

   • (d) $\sqrt{2}x-$1 : There is no $x^2$ term, so the coefficient is 0.

3. Examples of Specific Polynomials:

   Provide one example each of:

   a) A binomial of degree 35.

   b) A monomial of degree 100.

   Solution:

   • (a): $3x^{35} - 4$ is a binomial (two terms) with the highest exponent 35.

   • (b): $\sqrt{2}y^{100}$ is a monomial (one term) with the exponent 100.

4. Degree of Polynomials:

   Write the degree of each of the following polynomials:

   a) $5x^3+4x^2+7x$

   b) $4 - y^2$

   c) $5t - \sqrt{7}$
   

   d) $3$
   

   Solution:

   • (a): Degree is 3 (highest power of $x$).

   • (b): Degree is 2 (highest power of $y$).

   • (c): Degree is 1 (highest power of $\mathrm{t).}$

   • (d): Degree is 0 (constant term).

5. Classification of Polynomials:

   Classify the following as linear, quadratic, and cubic polynomials:

   a) $x^2 + x$
   

   b) $x - x^3$
   

   c) $y+y^2+4$

   d) $1 + x$
   

   e) $3t$
   

   f) $r^2$

   g) $7x^3$

   Solution:

• (a) $x^2 + x$: Quadratic polynomial (degree 2).

• (b) $x - x^3$: Cubic polynomial (degree 3).

• (c) $y + y^2 + 4$: Quadratic polynomial (degree 2).

• (d) $1 + x$: Linear polynomial (degree 1).

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