Question

Consider a quadratic equation ax²+2bx+c=0 where a,b,c are positive real numbers. If the equation has no real root, then which of the following is true?

• A. a,b,c cannot be in A.P. or H.P. but can be in G.P.
• B. a,b,c cannot be in G.P. or H.P. but can be in A.P.
• C. a,b,c cannot be in A.P. or G.P. but can be in H.P.
• D. a,b,c cannot be in A.P.,G.P or H.P.

Answer 1

The Correct Option is C

The correct answer is option (C): a,b,c cannot be in A.P. or G.P. but can be in H.P.

In  \(ax^2+bx+c\), d should be less than 0.

So, \(4b^2-4ac=0\)

\(\Rightarrow\)\(b^2-ac=0\)

Putting  the values of \(b=a+\frac{c}{2}\) for A.P,(ac)\(^{\frac{1}{2}}\) for G.P and \(\frac{2ac}{a+c}\) for H.P, we get the above condition only satisfies H.P.

Answer 2

The correct answer is option (C): a, b, c cannot be in A.P. or G.P. but can be in H.P.
For the quadratic expression ax² + bx + c, the discriminant d should be less than 0.
So, 4b² - 4ac = 0
This simplifies to: b² = ac
Now, let's put the values of b for different progressions:
For Arithmetic Progression (A.P): b = a + c/2
For Geometric Progression (G.P): b = (ac)^1/2
For Harmonic Progression (H.P): b = 2ac / (a + c)