Home
Class 12
MATHS
Let a, b, c be non-zero real numbers suc...

Let `a, b, c` be non-zero real numbers such that ; `int_0^1 (1 + cos^8 x)(ax^2 + bx + c)dx = int_0^2 (1+cos^8 x)(ax^2 + bx +c)dx` then the quadratic equation `ax^2 + bx + c = 0` has -

Text Solution

Verified by Experts

The correct Answer is:
B
Consider, `f(x)int_(0)^(1)(1+cos^(8)x)(ax^(2)+bx+c)dx`
Obviously, f(x) is cotinuous and differentiable in the interval [1,2].
Also, `f(1)=f(2)" "["given"]`
`therefore` By Rolle's theorem, there exist atleast one point `k in (1,2),` such that `f'(k)=0.`
Now, `f'(x)=(1+cos^(8)x)(ax^(2)+bx+x)f'(k)=0`
`implies(1+cos^(8)k)(ak^(2)+bk+c)=0`
`impliesak^(2)+bk+c=0," "[as (1+cos^(8)k)ne0]`
`thereforex=k"is root of"ax^(2)+bx+c=0, where kin (1,2)`
Promotional Banner

Similar Questions

Explore conceptually related problems

Let a, b and c be real numbers such that 4a + 2b + c = 0 and ab gt 0. Then the equation ax^(2) + bx + c = 0 has

Let a,b and c be real numbers such that 4a+2b+c=0 and ab lt 0 .Then the equation ax^2+bx+c=0 .

If a, b, c are positive real numbers such that the equations ax^(2) + bx + c = 0 and bx^(2) + cx + a = 0 , have a common root, then

Let a gt 0, b gt 0, c gt 0 . Then both the roots of the equation ax^2+ bx + c = 0 are

Show that the product of the roots of a quadratic equation ax^(2)+bx+c=0 (a ne 0) is (c )/(a) .

Show that the sum of roots of a quadratic equation ax^(2)+bx+c=0 (a ne 0) is (-b)/(a) .

Find the derivative of (1)/(ax^(2)+bx+c)

Let a, b, c be real numbers, a != 0. If alpha is a zero of a^2 x^2+bx+c=0, beta is the zero of a^2x^2-bx-c=0 and 0 , alpha < beta then prove that the equation a^2x^2+2bx+2c=0 has a root gamma that always satisfies alpha < gamma < beta.

If int_0^3 (3ax^2+2bx+c)dx=int_1^3 (3ax^2+2bx+c)dx where a,b,c are constants then a+b+c=

IF tan 18^@ and tan 27^@ are the roots of ax^2 - bx +c=0 then