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Polynomials : Exercise 2.1 (Mathematics NCERT Class 9th)


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Q.1     Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
              (i) 4{x^2} - 3x + 7        
          
(ii) {y^2} + \sqrt 2
              (iii) 3\sqrt t + t\sqrt 2    
          
(iv) y + {2 \over y}
              (v) {x^{10}} + {y^3} + {t^{50}}
Sol.

(i) In 4{x^2} - 3x + 7, all the indices of x are whole numbers so it is a polynomial in one variable x.
(ii) In {y^2} + \sqrt 2 , the index of y is a whole number so it is a polynomial in one variable y.
(iii) 3\sqrt t + t\sqrt 2 = 3{t^{{1 \over 2}}} + \sqrt 2 t, here the exponent of the first term is {1 \over 2}, which is not a whole number.  Therefore it is not a polynomial.
(iv) y + {2 \over y} = y + 2{y^{ - 1}}, here the exponent of the second term is –1, which is not a whole number and so it is not a polynomial.
(v) {x^{10}} + {y^3} + {t^{50}} is not a polynomial in one variable as three variables x, y, t occur in it.


Q.2       Write the coefficients of {x^2} in each of the following :
            
(i) 2 + {x^2} + x                   
            
(ii) 2 - {x^2} + {x^3}
            
(iii) {\pi \over 2}{x^2} + x   
            
(iv) \sqrt {2x} - 1
Sol.

Coefficient of {x^2}:
(i)  in\,2 + {x^2} + x\,is\,1
(ii) in\,2 - {x^2} + {x^3}\,is\, - 1
(iii) in {\pi \over 2}{x^2} + x\,\,is\,{\pi \over 2}
(iv)  in\,\sqrt {2x} - 1\,\,is\,\,0.


Q.3      Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Sol.

Example :
Binomial of degree 35 may be taken as  {x^{35}} + 4x
Monomial of degree 100 may be taken as 5{x^{100}}


 

Q.4       Write the degree of each of the following polynomials :
            
(i) 5{x^3} + 4{x^2} + 7x    
            
(ii) 4 - {y^2}
            
(iii) 5t - \sqrt 7                    
            
(iv) 3
Sol.

(i) The highest power term is 5{x^3} and the exponent is 3. So, the degree is 3.
(ii) The highest power term is  - {y^2} and the exponent is 2. So, the degree is 2.
(iii) The highest power term is 5t and the exponent is 1. So, the degree is 1.
(iv) The only term here is 3 which can be written as 3{x^0} and so the exponent is 0. Therefore the degree is 0.


Q.5      Classify the following as linear, quadratic and cubic polynomials :
           
(i) {x^2} + x    
           
(ii) x - {x^3}     
           
(iii) y + {y^2} + 4
           
(iv) 1+ x                   
           
(v) 3t                         
           
(vi) {r^2}
           
(vii) 7{x^3}
Sol.

(i) The highest degree of {x^2} + x\, is 2, so it is a quadratic polynomial .
(ii) The highest degree of x - {x^3}\, is 3, so it is a cubic polynomial .
(iii) The highest degree of y + {y^2} + 4 is 2, so it is a quadratic polynomial .
(iv) The highest degree of x in (1 + x) is 1.  So it is a linear polynomial.
(v) The highest degree of t in 3t is 1. So it is a linear polynomial.
(vi) The highest degree of r in {r^2} is 2. So, it is a quadratic polynomial.
(vii) The highest degree of x in 7{x^3} is 3. So, it is cubic polynomial.



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