Questions & Answers

Question

Answers

A. $\dfrac{{ - 3}}{4}$

B. $\dfrac{3}{4}$

C. $\dfrac{{ - 4}}{3}$

D. $\dfrac{4}{3}$

E. $\dfrac{3}{2}$

Answer
Verified

Hint: In this question, a quadratic equation is given whose roots are $\alpha {\text{ and }}\beta {\text{ }}$and we have to find the value of a expression given in terms of $\alpha {\text{ and }}\beta {\text{ }}$. First write the given expression in terms of $\alpha + \beta {\text{ and }}\alpha \beta $ and then use the relationship between the sum and product of roots and the coefficient of the quadratic equation.

Complete step-by-step answer:

In the question, it is given that $\alpha {\text{ and }}\beta {\text{ }}$are roots of the quadratic equation ${{\text{x}}^2} + 3{\text{x - 4 = 0}}$ .

We have to find the value of the expression $\dfrac{1}{\alpha } + \dfrac{1}{\beta }$ .

We know that the relationship between the sum and product of roots and the coefficients of the quadratic equation ${\text{a}}{{\text{x}}^2} + {\text{bx + c = 0}}$ is given as:

$

\alpha + \beta = \dfrac{{ - {\text{b}}}}{{\text{a}}}. \\

\alpha \beta = \dfrac{{\text{c}}}{a} \\

$

On comparing the given quadratic equation with the general expression of the quadratic equation, we get:

a = 1 , b = 3 and c = -4.

Now, we will write the expression $\dfrac{1}{\alpha } + \dfrac{1}{\beta }$ in terms of $\alpha + \beta {\text{ and }}\alpha \beta $ .

$\dfrac{1}{\alpha } + \dfrac{1}{\beta } = \dfrac{{\alpha + \beta }}{{\alpha \beta }}$

Also

$

\alpha + \beta = \dfrac{{ - {\text{b}}}}{{\text{a}}} = \dfrac{{ - 3}}{1} = - 3 \\

\alpha \beta = \dfrac{{\text{c}}}{a} = \dfrac{{ - 4}}{1} = - 4 \\

$

Putting the values of $\alpha + \beta {\text{ and }}\alpha \beta $ in above expression, we get:

$\dfrac{1}{\alpha } + \dfrac{1}{\beta } = \dfrac{{\alpha + \beta }}{{\alpha \beta }} = \dfrac{{ - 3}}{{ - 4}} = \dfrac{3}{4}$ .

So option B is correct.

Note: In this type of question, we should know the general expression of a quadratic equation in terms of a, b and c. We should also remember the relation between the sum and product of roots and the coefficients of the quadratic equation ${\text{a}}{{\text{x}}^2} + {\text{bx + c = 0}}$. Finally convert the given expression in the terms of $\alpha + \beta {\text{ and }}\alpha \beta $.

Complete step-by-step answer:

In the question, it is given that $\alpha {\text{ and }}\beta {\text{ }}$are roots of the quadratic equation ${{\text{x}}^2} + 3{\text{x - 4 = 0}}$ .

We have to find the value of the expression $\dfrac{1}{\alpha } + \dfrac{1}{\beta }$ .

We know that the relationship between the sum and product of roots and the coefficients of the quadratic equation ${\text{a}}{{\text{x}}^2} + {\text{bx + c = 0}}$ is given as:

$

\alpha + \beta = \dfrac{{ - {\text{b}}}}{{\text{a}}}. \\

\alpha \beta = \dfrac{{\text{c}}}{a} \\

$

On comparing the given quadratic equation with the general expression of the quadratic equation, we get:

a = 1 , b = 3 and c = -4.

Now, we will write the expression $\dfrac{1}{\alpha } + \dfrac{1}{\beta }$ in terms of $\alpha + \beta {\text{ and }}\alpha \beta $ .

$\dfrac{1}{\alpha } + \dfrac{1}{\beta } = \dfrac{{\alpha + \beta }}{{\alpha \beta }}$

Also

$

\alpha + \beta = \dfrac{{ - {\text{b}}}}{{\text{a}}} = \dfrac{{ - 3}}{1} = - 3 \\

\alpha \beta = \dfrac{{\text{c}}}{a} = \dfrac{{ - 4}}{1} = - 4 \\

$

Putting the values of $\alpha + \beta {\text{ and }}\alpha \beta $ in above expression, we get:

$\dfrac{1}{\alpha } + \dfrac{1}{\beta } = \dfrac{{\alpha + \beta }}{{\alpha \beta }} = \dfrac{{ - 3}}{{ - 4}} = \dfrac{3}{4}$ .

So option B is correct.

Note: In this type of question, we should know the general expression of a quadratic equation in terms of a, b and c. We should also remember the relation between the sum and product of roots and the coefficients of the quadratic equation ${\text{a}}{{\text{x}}^2} + {\text{bx + c = 0}}$. Finally convert the given expression in the terms of $\alpha + \beta {\text{ and }}\alpha \beta $.

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