Question

$\alpha$ and $\beta$ are zeroes of polynomial $x^2-2x+1,$ then product of zeroes of a polynomial having zeroes $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ is

• A. $\alpha \beta$
• B. $\frac{1}{\alpha \beta }$
• C. $0$
• D. $1$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $\alpha \beta$
B $\frac{1}{\alpha \beta }$
D $1$

Given: Polynomial $x^2-2x+1$ having $\alpha ,\beta$ as the zeroes.

To find product of zeroes of a polynomial having zeroes $\frac{1}{\alpha }$ and $\frac{1}{\beta }$

Sol: For polynomial $x^2-2x+1$ having $\alpha ,\beta$ as the zeroes.

We know, sum of zeroes = $\alpha +\beta =2⟹\alpha =2-\beta ..........\left(i\right)$

product of zeroes = $\alpha \beta =1$

Substituting the valu from eqn(i) we get

$(2-\beta )\beta =1⟹{\beta }^2-2\beta +1=0⟹\beta =1$

and $\alpha =2-\beta =2-1=1$

Now the product of zeroes of the polynomial with $\frac{1}{\alpha },\frac{1}{\beta }$ as zeroes, is

$\frac{1}{\alpha }\times \frac{1}{\beta }=\frac{1}{\alpha \beta }=1$