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State whether the following statement is true or false.

If one root of $5 x^{2} + 13 x + k = 0$ is the reciprocal of the other root then the value of k is 5.

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False

True

answer is A.

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Detailed Solution

Given quadratic equation is,

5 x^{2} + 13 x + k = 0

.
And one root is the reciprocal of another root.
The standard form of a quadratic equation is

a x^{2} + bx + c = 0,

where

a, b c

are real numbers and

a \neq 0

.
And, the sum of the roots is given by

- \frac{b}{a}

and product of the roots is given by

\frac{c}{a}

.
After comparing

5 x^{2} + 13 x + k = 0

with the standard form, we get,

a = 5, b = 13, c = k

.
Let’s assume one root of the equation is

m

and according to the given condition, the other one is

\frac{1}{m}

.
Product of the roots

= \frac{c}{a}

⟹ m \times \frac{1}{m} = \frac{k}{5}

⟹ \frac{m}{m} = \frac{k}{5}

⟹ 1 = \frac{k}{5}

∴ k = 5

∴

If one root of

5 x^{2} + 13 x + k = 0

is the reciprocal of the other root then the value of k is 5.
Hence, the given statement is true.
Therefore, option (1) is correct.

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