NIMCET 2021 — Mathematics PYQ

1. Identify the Quadratic Equation

The given conditions ${\alpha }^2=5\alpha -3$ and ${\beta }^2=5\beta -3$ imply that $\alpha$ and $\beta$ are the roots of the quadratic equation:

From the properties of quadratic equations ($a{x}^2+bx+c=0$):

• Sum of roots: $\alpha +\beta =-\frac{b}{a}=5$

• Product of roots: $\alpha \beta =\frac{c}{a}=3$

2. Determine the Sum and Product of the New Roots

We need to find the equation whose roots are $r_1=\frac{\alpha }{\beta }$ and $r_2=\frac{\beta }{\alpha }$.

• Sum of new roots ($S$):

  $$S=\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }$$

  Using the identity ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$:

  $$S=\frac{(5{)}^2-2(3)}{3}=\frac{25-6}{3}=\frac{19}{3}$$

• Product of new roots ($P$):

  $$P=\frac{\alpha }{\beta }\cdot \frac{\beta }{\alpha }=1$$

3. Construct the New Equation

A quadratic equation is formed using the formula:

Substituting our values:

Multiply the entire equation by $3$ to remove the fraction:

Final Answer

The required quadratic equation is $3x^2-19x+3=0$.

Correct Option: (D) $3x^2-19x+3=0$