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Question : 15 of 160
Marks:
+1 ,
-0
Solution:
Given, equation
6 x 6 − 25 x 5 + 31 x 4 − 31 x 2 + 25 x − 6 = 0 which is of reciprocal equation of even degree with opposite sign.
∴ x 2 − 1 is the root of the equation.
∴ x = ± I are the zeroes of the equation.
      6 x 4 − 25 x 3 + 37 x 2 − 25 x + 6 x 2 − 1 ) ‌ 6 x 6 − 2 5 x 5 + 3 1 x 4 − 3 1 x 2 + 2 5 x − 6 ‌ ‌ 6 x 6 − 6 x 4 ‌ ‌ ‌ ‌ ( − ) ‌ ‌ ‌ ‌ ( + ) − 25 x 5 + 37 x 4 − 31 x 2 + 25 x − 6 − 25 x 5 ‌ + 25 x 3 ( + ) ( − ) 37 x 4 − 25 x 3 − 31 x 2 + 25 x − 6 37 x 4 ‌ − 37 x 2 ( − ) ( + ) − 25 x 3 + 6 x 2 + 25 x − 6 ‌ ‌ − 25 x 3 ‌ ‌ + 25 x ( + ) ( + ) 6 x 2 + 6 6 x 2 + 6 ( − ) ‌ ‌ ( + ) 0
⇒ 6 x 4 − 25 x 3 + 37 x 2 − 25 x + 6 = 0 Dividing by
x 2 , we get
‌ 6 x 2 − 25 x + 37 − ‌ + ‌ = 0 ∴ 6 t 2 − 25 t + 25 = 0 [ . Let
x + ‌ = t ] ‌ ⇒ 6 t 2 − 15 t − 10 t + 25 = 0 ∴ ‌ ‌ x + ‌ = ‌ or
x + ‌ = ‌ ‌ ⇒ ‌ ‌ 3 x 2 + 3 = 5 x or
2 x 2 + 2 = 5 x ‌ ⇒ ‌ ‌ 3 x 2 − 5 x + 3 = 0 or
2 x 2 − 5 x + 2 = 0 The roots of
3 x 2 − 5 x + 3 = 0 are not real
‌ 2 x 2 − 5 x + 2 = 0 ⇒ 2 x 2 − 4 x − x + 2 = 0 ∴ The rational roots
‌ , 2 , − 1 , ‌ , ‌ . The value of
m = − 1 and
M = 2 M − m = 2 − ( − 1 ) = 3 © examsiri.com
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