Class 9, Maths, Chapter 2, Exercise 2.1, Solutions
Q.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) y2 + $\sqrt{2}$
(iii) 3 $\sqrt{t}$+ t $\sqrt{2}$
(iv) y + $\frac{2}{y}$
(v) x10 + y3 + t50
Ans:
(i) In 4x2 – 3x + 7,
All the indices of x are whole numbers so it is a polynomial in one variable x.
(ii) y2 + $\sqrt{2}$
The index of y is a whole number so it is a polynomial in one variable y.
(iii) 3 + t or 3 + t 3$\sqrt{t}$+ t$\sqrt{2}$ or 3${{t}^{\frac{1}{2}}}$+ t${{2}^{\frac{1}{2}}}$
Here the exponent of the first term is , which is not a whole number. Therefore, it is not a polynomial
(iv) y + $\frac{2}{y}$ or y + 2y-1
Here the exponent of the second term is – 1, which is not a whole number and so it is not a polynomial.
(v) x10 + y3 + t50,
It is a polynomial in one variable as three variables x, y, t occur in it.
Q.2. Write the coefficients of x2 in each of the following:
(i) 2 + x2 + x
(ii) 2 – x2 + x3
(iii) $\frac{\pi }{2}{{x}^{2}}$+ x
(iv) $\sqrt{2}$-1
Ans: (i) in 2 + x2 + x,the coefficients of x2 is 1
(ii) In 2 – x2 + x3, The coefficients of x2 is -1
(iii) In $\frac{\pi }{2}{{x}^{2}}$+ x, The coefficients of x2 is $\frac{\pi }{2}$
(iv) In $\sqrt{2}$ – 1 or 0x2 + $\sqrt{2}$x – 1, The coefficients of x2 is 0
Q.3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Ans: (i) Binomial of degree 35 may be taken as x35 + 2.
(ii) Monomial of degree 100 may be taken as x100.
Q.4. Write the degree of each of the following polynomials:
(i) 5x3 + 4x2 + 7x
(ii) 4 – y2
(iii) 5t – $\sqrt{7}$
(iv) 3
Ans: (i) The highest degree (power) term is 5x3 and the exponent is 3. So, the degree is 3.
(ii) The highest degree (power) term is – y2 and the exponent is 2. So, the degree is 2.
(iii) The highest degree (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 3x0 and so the exponent is 0. Therefore, the degree (power) is 0.
Q.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
Ans: (i) The highest degree of x2 + x is 2, so it is a quadratic.
(ii) The highest degree of x – x3 is 3, so it is a cubic.
(iii) The highest degree of y + y2 + 4 is 2, so it is a quadratic.
(iv) The highest degree of x in 1 + x in 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 r2 is 2. So, it is a quadratic polynomial. –
(vii) The highest degree of x in 7x3 is 3. So, it is cubic polynomial.