ncert solutions for class 9 maths ex 2.1

Question 1: 

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

(i)$$ 4x² – 3x + 7

ii)${\displaystyle y^2+\sqrt{2}}$

iii)${\displaystyle 3\sqrt{t}+t\sqrt{2}}$

iv)${\displaystyle y+{\displaystyle \frac{2}{y}}}$

(v) ${\displaystyle x^{10}+y^3+t^{50}}$

Solution 1:

(i)4x² – 3x + 7

One variable is involved in given polynomial which is $x$

Therefore, it is a polynomial in one variable $x$.

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

One variable is involved in given polynomial which is $y$

Therefore, it is a polynomial in one variable '$y$'.

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

No. It can be observed that the exponent of variable $t$ in term

$3\sqrt{t}$ is ${\displaystyle \frac{1}{2}}$, which is not a whole number.

Therefore, this expression is not a polynomial.

iv)$y+{\displaystyle \frac{2}{y}}=y+2y^{-1}$

The power of variable ‘$y$’ is $-1$ which is not a whole number.

Therefore, it is not a polynomial in one variable

No. It can be observed that the exponent of variable $y$ in term

${\displaystyle \frac{2}{y}}$ is $-1$, which is not a whole number.

Therefore, this expression is not a polynomial.

(v) $x^{10}+y^3+t^{50}$

In the given expression there are $3$ variables which are ‘$x,y,t$’ involved.

Therefore, it is not a polynomial in one variable.

Question 2:

Write the coefficients of x² in each of the following:

(i) $2+x^2+x$

(ii) $2-x^2+x^3$

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

(iv) $\surd 2x-1$

Solution 2:

(i) $2+x^2+x$

= $2+1(x^2)+x$

The coefficient of $x^2$ is $1$.

(ii) $2-x^2+x^3$

$2-1(x^2)+x^3$

The coefficient of $x^2$ is $-1$.

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

The coefficient $x^2$ is ${\displaystyle \frac{\pi }{2}}$.

(iv) $\sqrt{2}x-1=0x^2+\sqrt{2}x-1$

The coefficient of $x^2$ is $0$.

Question 3:

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

Solution 3 :

Binomial of degree $35$ means a polynomial is having

1.Two terms

2.Highest degree is $35$

Example: $x^{35}+x^{34}$

Monomial of degree $100$ means a polynomial is having

1.One term

2.Highest degree is $100$

Example : $x^{100}$.

Question 4:

Write the degree of each of the following polynomials:

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

(ii) $4-y^2$

(iii) $5t-\sqrt{7}$

(iv) $3$

Solution 4:
Degree of a polynomial is the highest power of the variable in the polynomial.

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

Highest power of variable ‘$x$’ is $3$. Therefore, the degree of this polynomial is $3$

(ii) $4-y^2$

Highest power of variable ‘$y$’ is $2$. Therefore, the degree of this polynomial is $2$.

(iii) $5t-\sqrt{7}$

Highest power of variable ‘$t$’ is $1$. Therefore, the degree of this polynomial is $1$.

(iv) $3$

This is a constant polynomial. Degree of a constant polynomial is always $0$.

Question 5: 

Classify the following as linear, quadratic and cubic polynomial:

(i)  x² + x 

(ii)  x - x³ 

(iii)  y + y² + 4  

(iv)  1 + x 

(v)  3t 

(vi) r² 

(vii) 7x² + 7x³

Solution 5:

Linear polynomial – whose variable power is ‘1’

Quadratic polynomial - whose variable highest power is ‘2’

Cubic polynomial- whose variable highest power is ‘3’

(i) x² + x is a quadratic polynomial as its highest degree is 2.

(ii) x - x³  is a cubic polynomial as its highest degree is 3.

(iii) y + y² + 4 is a quadratic polynomial as its highest degree is 2.

(iv) 1 + x is a linear polynomial as its degree is 1.

(v) 3t is a linear polynomial as its degree is 1.

(vi) r² is a quadratic polynomial as its degree is 2.

(vii) 7x² + 7x³ is a cubic polynomial as highest its degree is 3.