Question

3.
If one root of 5x2 + 13x + k = 0 is reciprocals of the other, then k is
(1) 0, (b) 5, (c),1/6 (d) 6​

Answer 1

Given :-

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One root of 5x² + 13x + k = 0 is the reciprocal of the other.

${}$

Solution :-

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• Let the one root be x
• Then, other root = 1/x

We know that,

If the quadratic equation is ax² + bx + c = 0 then,

$\bf\large{ \pink{product \: of \: roots = { \dfrac{c}{a} } }}$

and here, in 5x² + 13x + k = 0

• a = 5
• b = 13
• c = k
• roots = x , 1/x

and

$\bf\large{product \: of \: roots = \frac{k}{5} } \\ \\ \\ \implies \bf\large { \cancel{x} \times \dfrac{1}{ \cancel{x}} = \dfrac{k}{5} } \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \bf\large{ \implies 1 = \dfrac{k}{5} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \bf\large{ \implies{ \purple{k = 5}}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

So, option (b) is correct i.e. k = 5 is correct

Answer 2

Answer :

(b) 5

Note :

★ The possible values of the variable which satisfy the equation are called its roots or solutions .

★ A quadratic equation can have atmost two roots .

★ The general form of a quadratic equation is given as ; ax² + bx + c = 0

★ If α and ß are the roots of the quadratic equation ax² + bx + c = 0 , then ;

• Sum of roots , (α + ß) = -b/a

• Product of roots , (αß) = c/a

Solution :

Here ,

The given quadratic equation is ;

5x² + 13x + k = 0

Now ,

Comparing the given quadratic equation with the general quadratic equation ax² + bx + c = 0 , we have ;

a = 5

b = 13

c = k

Also ,

It is given that , one root of the given quadratic equation is reciprocal of the other .

Thus ,

Let α and 1/α be the roots of the given quadratic equation .

Now ,

The product of the roots of the given quadratic equation will be given as ;

=> α•(1/α) = c/a

=> 1 = k/5

=> 5 = k

=> k = 5

Hence , k = 5 .