Home
Class 12
MATHS
If a x^2+(b-c)x+a-b-c=0 has unequal real...

If `a x^2+(b-c)x+a-b-c=0` has unequal real roots for all `c in R ,then`

A

`blt0lta`

B

`altoltb`

C

`bltalt0`

D

`bgtagt0`

Text Solution

Verified by Experts

The correct Answer is:
C, D
We have `D=(b=c)^(2)-4a(a-b-c)gt0`
`impliesb^(2)+c^(2)-2bc-4a^(2)+4ab+4acgt0`
`impliesc^(2)+(4a-2b)c-4a^(2)+4ab+b^(2)gt0,AAc epsilonR`
Since `c epsionR` so we have
`(4a-2b)^(2)-4(-4a^(2)+4ab+b^(2))lt0`
`implies4a^(2)-4ab+b^(2)+4a^(2)-4ab-b^(2)lt0`
`impliesa(a-b)lt0`
If `agt0`, then `a-blt0`
i.e. `0ltaltb`
or `bgtagt0`
If `alt0`, then `a-bgt0`
i.e. `0gtagtb`
or `bltalt0`
Promotional Banner

Topper's Solved these Questions

  • THEORY OF EQUATIONS

    ARIHANT MATHS|Exercise Exercise (Passage Based Questions)|21 Videos

  • THEORY OF EQUATIONS

    ARIHANT MATHS|Exercise Exercise (Single Integer Answer Type Questions)|10 Videos

  • THEORY OF EQUATIONS

    ARIHANT MATHS|Exercise SCQ_TYPE|1 Videos

  • THE STRAIGHT LINES

    ARIHANT MATHS|Exercise The Straight Lines Exercise 8 : (Questions Asked in Previous 13 years Exams)|1 Videos

  • THREE DIMENSIONAL COORDINATE SYSTEM

    ARIHANT MATHS|Exercise Three Dimensional Coordinate System Exercise 12 : Question Asked in Previous Years Exam|2 Videos

Similar Questions

Explore conceptually related problems

Let a ,b ,c be real. If a x^2+b x+c=0 has two real roots alpha and beta ,where alpha 1 , then show that 1+c/a+|b/a|<0

Let a in R and f : R rarr R be given by f(x)=x^(5)-5x+a , then (a) f(x)=0 has three real roots if a gt 4 (b) f(x)=0 has only one real root if a gt 4 (c) f(x)=0 has three real roots if a lt -4 (d) f(x)=0 has three real roots if -4 lt a lt 4

If x^2+3x+5=0a n da x^2+b x+c=0 have common root/roots and a ,b ,c in N , then find the minimum value of a+b+ c dot

Statement 1 The equation a^(x)+b^(x)+c^(x)-d^(x)=0 has only real root, if agtbgtcgtd . Statement 2 If f(x) is either strictly increasing or decreasing function, then f(x)=0 has only real root.

f(x , y)=x^2+y^2+2a x+2b y+c=0 represents a circle. If f(x ,0)=0 has equal roots, each being 2, and f(0,y)=0 has 2 and 3 as its roots, then the center of the circle is

Let f(x)=x^(3)+ax^(2)+bx+c be the given cubic polynomial and f(x)=0 be the corresponding cubic equation, where a, b, c in R. Now, f'(x)=3x^(2)+2ax+b Let D=4a^(2)-12b=4(a^(2)-3b) be the discriminant of the equation f'(x)=0 . IF D=4(a^(2)-3b)lt0 . Then, a. f(x) has all real roots b. f(x) has one real and two imaginary roots c. f(x) has repeated roots d. None of the above

If 2a+3b+6c = 0, then show that the equation a x^2 + bx + c = 0 has atleast one real root between 0 to 1.

If ax^2+bx+c=0 and 2x^2+3x+4=0 have a common root, where a, b, c in N (set of natural numbers), the least value of a + b +c is:

If a and b ( != 0) are roots of x^(2) +ax + b = 0 , then the least value of x^(2) +ax + b ( x in R) is :

If the roots of the equation (b-c)x^2+(c-a)x+(a-b)=0 are equal then a,b,c will be in