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Q.

State whether the following statement is true or false.

If $α and β$ be the zeroes of the polynomial $p x = 2 x ² − 5 x + 7,$ then $2 α + 3 β$ and $2 β + 3 α$ are the zeroes of the polynomial $x 2 − 252 x + 41 .$

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a

True

b

False

answer is A.

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Detailed Solution

Given polynomial is,

p x = 2 x 2 − 5 x + 7

Since

α and β

are the zeroes of the polynomial

p x = 2 x ² − 5 x + 7,

then we have,

α + β = 52 & α β = 72

If

2 α + 3 β

and

2 β + 3 α

are the zeroes of any polynomial, then we have,

q (x) = x 2 − 2 α + 3 β + 3 α + 2 β x + 2 α + 3 β 3 α + 2 β ⇒ q (x) = x 2 − 5 (α + β) x + 6 α 2 + 9 α β + 4 α β + 6 β 2 ⇒ q (x) = x 2 − 5 (α + β) x + 6 α 2 + β 2 + 13 α β (1)

Now we know that,

α 2 + β 2 = (α + β) 2 − 2 α β ⇒ α 2 + β 2 = 522 − 272

⇒ α 2 + β 2 = 254 − 7 ⇒ α 2 + β 2 = − 34

Therefore, from equation (1), we have,

q (x) = x 2 − 552 x + 6 − 34 + 1372 ​ = 0 ⇒ q (x) = x 2 − 252 x − 92 + 912 ⇒ q (x) = x 2 − 252 x + 41

Therefore if

α and β

be the zeroes of the polynomial

p x = 2 x ² − 5 x + 7,

then

2 α + 3 β

and

2 β + 3 α

are the zeroes of the polynomial

x 2 − 252 x + 41 .

Thus, the given statement is true.
Hence, option (1) is correct.

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