Home
Class 12
MATHS
If the quadratic polynomials defined on ...

If the quadratic polynomials defined on real coefficient
`P(x)=a_(1)x^(2)+2b_(1)x+c_(1)` and `Q(x)=a_(2)x^(2)+2b_(2)x+c_(2)` take positive values `AA x in R`, what can we say for the trinomial `g(x)=a_(1)a_(2)x^(2)+b_(1)b_(2)x+c_(1)c_(2)` ?

A

`g(x)` takes positive values only.

B

`g(x)` takes negative values only.

C

`g(x)` can takes positive as well as negative values.

D

Nothing definite can be said about `g(x)`.

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `D_(1)=4b_(1)^(2)-4a_(1)c_(1) lt 0` and `a_(1) gt 0`
i.e. `a_(1)c_(1) gt b_(1)^(2)` ……..`(i)`
`D_(2)=4b(2)^(2)-4a_(2)c_(2) lt 0` and `a_(2) gt 0`
Hence `a_(2)c_(2) gt b_(2)^(2)` and `a_(2) gt 0`………`(ii)`
Multiplying `(i)` and `(ii)`, we get
`a_(1)a_(2)c_(1)c_(2) gt b_(1)^(2)b_(2)^(2)`
Now consider `g(x)`
`D=b_(1)^(2)b_(2)^(2)-4a_(1)a_(2)c_(1)c_(2) lt b_(1)^(2)b_(2)^(2)-4b_(1)^(2)b_(2)^(2)`
=-3b_(1)^(2)b_(2)^(2)`
`:. D lt 0`
`implies g(x) gt 0AA x in R (:' a_(1),a_(2) gt 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

The straight lines (x)/(a_(1))=(y)/(b_(1))=(z)/(c_(1)) and (x-2)/(a_(2))=(y-3)/(b_(2))=(z)/(c_(2)) will be parallel if -

The straight line a_(1)x+b_(1)y+c_(1)=0 anda_(2)x+b_(2)y+c_(2)=0 are parallel to each other if -

If b_(1)b_(2)=2(c_(1)+c_(2))andb_(1),b_(2),c_(1),c_(2) are all real numbers, then at least one of the equations x^(2)+b_(1)x+c_(1)=0andx^(2)+b_(2)x+c_(2)=0 has

Represent the following equations in matrix form: a_(1)x+b_(1)y+c_(1)=0 a_(2)x+b_(2)y+c_(2)=0

Repeated roots : If equation f(x) = 0, where f(x) is a polyno- mial function, has roots alpha,alpha,beta,… or alpha root is repreated root, then f(x) = 0 is equivalent to (x-alpha)^(2)(x-beta)…=0, from which we can conclude that f(x)=0 or 2(x-alpha) [(x-beta)...]+(x-alpha)^(2)[(x-beta)...]'=0 or (x-alpha) [2 {(x-beta)...}+(x-alpha){(x-beta)...}']=0 has root alpha . Thus, if alpha root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if alpha root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x)=0, and f'''(x)=0. If a_(1)x^(3)+b_(1)x^(2)+c_(1)x+d_(1)=0 and a_(2)x^(3)+b_(2)x^(2)+c_(2)x+d_(2)=0 have a pair of repeated roots common, then |{:(3a_(1),2b_(1),c_(1)),(3a_(2),2b_(2),c_(2)),(a_(2)b_(1)-a_(1)b_(2),c_(1)a_(2)-c_(2)a_(1),d_(1)a_(2)-d_(2)a_(1)):}|=

The asymptotes of the hyperbola (x^(2))/(a_(1)^(2))-(y^(2))/(b_(1)^(2))=1 and (x^(2))/(a_(2)^(2))-(y^(2))/(b_(2)^(2))=1 are perpendicular to each other. Then, (a) a_(1)/a_(2)=b_(1)/b_(2) (b) a_(1)a_(2)=b_(1)b_(2) (c) a_(1)a_(2)+b_(1)b_(2)=0 (d) a_(1)-a_(2)=b_(1)-b_(2)

If the simultaneous linear equations a_(1)x+b_(1)y+c_(1)=0 " and " a_(2)x+b_(2)y+c_(2)=0 have only one solution, then the required condition is -

a_(1), a_(2),a_(3) in R - {0} and a_(1)+ a_(2)cos2x+ a_(3)sin^(2)x=0 " for all " x in R then

If the expansion in power of x of the function (1)/(( 1 - ax)(1 - bx)) is a_(0) + a_(1) x + a_(2) x^(2) + a_(3) x^(3) + …, then a_(n) is

Find the coordinates of the centriod of the triangle whose vertices are ( a_(1), b_(1), c_(1)) , (a_(2), b_(2), c_(2)) and (a_(3), b_(3), c_(3)) .

CENGAGE PUBLICATION-THEORY OF EQUATIONS-Single correct Answer
  1. If the equation |x^2-5x + 6|-lambda x+7 lambda=0 has exactly 3 distin...

    Text Solution

    |

  2. Let alpha, beta (a lt b) be the roots of the equation ax^(2)+bx+c=0. I...

    Text Solution

    |

  3. If the quadratic polynomials defined on real coefficient P(x)=a(1)x^...

    Text Solution

    |

  4. For which of the following graphs the quadratic expression y=ax^(2)+bx...

    Text Solution

    |

  5. The difference of maximum and minimum value of (x^(2)+4x+9)/(x^(2)+9) ...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |