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NDA2023MathematicsQuadratic EquationActual

Consider the equation (1-x)^4+(5-x)^4=82 What is the number of real roots of the equation?

Options

  1. A0
  2. B2
  3. C4
  4. D8

Correct answer

B. 2

Step-by-step solution

aligned & Given, (1- x )^4+(5- x )^4=82 & Let, 3- x = y & (( y -2)^2 )^2+ (( y +2)^2 )^2=82 & ( y ^2-4 y +4 )^2+ ( y ^2+4 y +4 )^2=82 & 2 [ ( y ^2+4 )^2+(4 y )^2 ]=82 & y ^4+8 y ^2+16+16 y ^2=41 & y ^4+24 y ^2-25=0 & ( y ^2+25 ) ( y ^2-1 )=0 & y = 1, 5 i aligned If y=1 x=2 If y=-1 x=4 If y=5 i x=3-5 i If y=-5 i x=3+5 i So, number of real roots =2

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