NCERT Solutions For Class 9 Math | Exercise 2.1

Detailed solution to all exercise questions of NCERT book for CBSE Class 9 Math in Chapter 2 - Polynomials - Exercise 2.1. The exercise comprises 5 questions.

Key concepts covered in this exercise include polynomials in one variable, coefficients of terms in a polynomial, classification of polynomials as monomial, binomial, and classification of polynomials as linear, quadratic, and cubic polynomials.

1. Question 1

   Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
   (i) 4x² - 3x + 1
   (ii) y² + √2
   (iii) 3√t + t√2
   (iv) y + \\frac{2}{y})
   (v) x¹⁰ + y³ + t⁵⁰

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   Solution

   (i) 4x² - 3x + 1
   It is a polynomial in one variable.
   There is only one variable x and all the exponents are whole numbers.

   (ii) y² + √2
   It is a polynomial in one variable.
   There is only one variable y and its power is a whole number.

   (iii) 3√t + t√2
   It is NOT a polynomial in one variable.
   The exponent of ‘t’ in √t is \\frac{1}{2}) which is not a whole number.

   (iv) y + \\frac{2}{y})
   It is NOT a polynomial in one variable.
   The exponent of y in \\frac{2}{y}) is -1, which is not a whole number.

   (v) x¹⁰ + y³ + t⁵⁰
   It is NOT a polynomial in one variable.
   There are multiple variables involved i.e., x, y, and t

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2. Question 2

   Write the coefficients of x² in each of the following :
   (i) 2 + x² + x
   (ii) 2 - x² + x³
   (iii) \\frac{π}{2})x² + x
   (iv) √2x - 1

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   Solution

   (i) 2 + x² + x
   The coefficient of x² is 1.

   (ii) 2 - x² + x³
   The coefficient of x² is -1.

   (iii) \\frac{π}{2})x² + x
   The coefficient of x² is \\frac{π}{2}).

   (iv) √2x - 1
   The coefficient of x² is 0.

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3. Question 3

   Give one example each of a binomial of degree 35, and of a monomial of degree 100.

   Binomial of degree 35 :- P(x) = x³⁵ + x³⁴

   Monomial of degree 100 :- Q(y) = y¹⁰⁰

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4. Question 4

   Write the degree of each of the following polynomials:
   (i) 5x³ + 4x² + 7x
   (ii) 4 – y²
   (iii) 5t - √7
   (iv) 3

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   Solution

   (i) 5x³ + 4x² + 7x
   The highest power of x in the polynomial is 3.
   The degree of the polynomial is 3.

   (ii) 4 – y²
   The highest power of y in the polynomial is 2.
   The degree of the polynomial is 2.

   (iii) 5t - √7
   The highest power of t in the polynomial is 1.
   The degree of the polynomial is 1.

   (iv) 3
   There is no variable in the polynomial.
   The degree of the polynomial is 0.

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5. Question 5

   Classify the following as linear, quadratic and cubic polynomials:
   (i) x² + x
   (ii) x – x³
   (iii) y + y² + 4
   (iv) 1 + x
   (v) 3t
   (vi) r²
   (vii) 7x³

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   Solution

   (i) x² + x
   The highest power of x in the polynomial is 2.
   Therefore the following polynomial is quadratic.

   (ii) x – x³
   The highest power of x in the polynomial is 3.
   Therefore the following polynomial is cubic.

   (iii) y + y² + 4
   The highest power of y in the polynomial is 2.
   Therefore the following polynomial is quadratic.

   (iv) 1 + x
   The highest power of x in the polynomial is 1.
   Therefore the following polynomial is linear.

   (v) 3t
   The highest power of t in the polynomial is 1.
   Therefore the following polynomial is linear.

   (vi) r²
   The highest power of r in the polynomial is 2.
   Therefore the following polynomial is quadratic.

   (vii) 7x³
   The highest power of x in the polynomial is 3.
   Therefore the following polynomial is cubic.