Home
Class 12
MATHS
Let f(x) = x^4 + ax^3 + bx^2 + cx + d be...

Let` f(x) = x^4 + ax^3 + bx^2 + cx + d` be a polynomial with real coefficients and real roots. If |f(i)|=1where `i=sqrt(-1)`, then the value of a +b+c+d is

A

`-1`

B

`1`

C

`0`

D

can't be determined

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )` Let `f(x)=(x-x_(1))(x-x_(2))(x-x_(3))(x-x_(4))`
`implies|f(i)|=sqrt(1+x_(1)^(2))sqrt(1+x_(2)^(2))sqrt(1+x_(3)^(2))sqrt(1+x_(4)^(2))=1`
`impliesx_(1)=x_(2)=x_(3)=x_(4)=0`
`implies` All four roots are zero.
`implies f(x)=x^(4)impliesa+b+c+d=0`
Promotional Banner

Topper's Solved these Questions

  • THEORY OF EQUATIONS

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer|6 Videos

  • THEORY OF EQUATIONS

    CENGAGE PUBLICATION|Exercise ILLUSTRATION|121 Videos

  • THEORY OF EQUATIONS

    CENGAGE PUBLICATION|Exercise JEE ADVANCED (Numerical Value Type )|1 Videos

  • STRAIGHT LINES

    CENGAGE PUBLICATION|Exercise ARCHIVES (NUMERICAL VALUE TYPE)|1 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE PUBLICATION|Exercise All Questions|291 Videos

Similar Questions

Explore conceptually related problems

Let P(x)=x^3-8x^2+c x-d be a polynomial with real coefficients and with all it roots being distinct positive integers. Then number of possible value of c is___________.

Find the quadratic equation with real coefficients which has 2±3i as a root (i=sqrt-1)

The polynomial f(x)=x^4+a x^3+b x^3+c x+d has real coefficients and f(2i)=f(2+i)=0. Find the value of (a+b+c+d)dot

A nonzero polynomial with real coefficients has the property that f(x)=f'(x)cdotf''(x). If a is the leading coefficient of f(x), then the value of 1//a is -

If f(x)=(ax-b)/(bx-a) , then find the value of f(1/x) .

If f(x)= x^(3) + ax^(2) + bx + c is an increasing function for all real values of x then-

Let f(x)=x^(3) + bx^(2) +cx +d, 0 lt b^(2) lt c . Then f(x)-

f(x) is a polynomial of degree 4 with real coefficients such that f(x)=0 is satisfied by x=1,2,3 only then f'(1).f'(2).f'(3) is equal to :

Let f(x) = (ax+b) cos x+ (cx+ d) sin x and f' (x) = x cos x be an identity in x then

Let f(x) be a polynomial of degree 5 such that f(|x|)=0 has 8 real distinct , Then number of real roots of f(x)=0 is ________.

CENGAGE PUBLICATION-THEORY OF EQUATIONS-Single correct Answer
  1. The difference of maximum and minimum value of (x^(2)+4x+9)/(x^(2)+9) ...

    Text Solution

    |

  2. If a gt 1 , then the roots of the equation (1-a)x^(2)+3ax-1=0 are

    Text Solution

    |

  3. All the values of ' a ' for which the quadratic expression a x^2+(a-2)...

    Text Solution

    |

  4. If a gt 1 , then the roots of the equation (1-a)x^(2)+3ax-1=0 are

    Text Solution

    |

  5. The equation ax^(4)-2x^(2)-(a-1)=0 will have real and unequal roots if

    Text Solution

    |

  6. If ax^(2)+bx+c=0, a ne 0, a, b, c in R has distinct real roots in (1,2...

    Text Solution

    |

  7. If c lt a lt b lt d, then roots of the equation bx^(2)+(1-b(c+d)x+bcd-...

    Text Solution

    |

  8. If 2a, b, 2c are in A.P. where a, b, c are R^(+), then the expression ...

    Text Solution

    |

  9. If a, b, c are positive numbers such that a gt b gt c and the equation...

    Text Solution

    |

  10. If the quadratic equation x^(2)-36x+lambda=0 has roots alpha and beta ...

    Text Solution

    |

  11. If the equation 2^(2x)+a*2^(x+1)+a+1=0 has roots of opposite sign, the...

    Text Solution

    |

  12. Let a, b, c ne three distinct non-zero real numbers satisfying the sys...

    Text Solution

    |

  13. In the given figure graph of y=p(x)=x^(4)+ax^(3)+bx^(2)+cx+d is given ...

    Text Solution

    |

  14. If a^(3)-3a^(2)+5a-17=0 and b^(3)-3b^(2)+5b+11=0 are such that a+b is ...

    Text Solution

    |

  15. Let f(x) = x^4 + ax^3 + bx^2 + cx + d be a polynomial with real coeffi...

    Text Solution

    |

  16. If f(x) is a polynomial of degree four with leading coefficient one s...

    Text Solution

    |

  17. Let P (x)=x ^(6) -x ^(5) -x ^(3) -x ^(2) -x and alpha, beta, gamma, de...

    Text Solution

    |

  18. The line y=mx+1 touches the curves y=-x^(4)+2x^(2)+x at two points P(x...

    Text Solution

    |

  19. If a+2b+3c=4, then find the least value of a^2+b^2+c^2dot

    Text Solution

    |

  20. If the roots of x^(4)+qx^(2)+kx+225=0 are in arthmetic progression, th...

    Text Solution

    |