Multiple choice

If $6,8$ and $12$ are $l ^ { th }, m ^ { th }$ and $n ^ { th }$ terms of an A.P and $f(x)=nx^{ { 2 } }+2lx-2m,$ then the equation $f ( x ) = 0$ has

  1. a root between $0$ and $1$

  2. both roots imaginary

  3. both roots negative.
  4. both roots greater than $1.$
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A Correct answer
Explanation

Terms of AP: a + (l-1)d = 6, a + (m-1)d = 8, a + (n-1)d = 12. f(x) = nx^2 + 2lx - 2m. Using the properties of AP terms, one can show f(0) = -2m and f(1) = n + 2l - 2m. Given the values, f(0) < 0 and f(1) > 0, implying a root in (0, 1).

AI explanation

Using the properties of an arithmetic progression, the l-th, m-th and n-th terms give the equations a + (l-1)d = 6, a + (m-1)d = 8 and a + (n-1)d = 12. We evaluate the quadratic function f(x) = nx^2 + 2lx - 2m at x = 0 to get f(0) = -2m, which is negative. Evaluating the function at x = 1 gives f(1) = n + 2l - 2m, and substituting the difference equations results in f(1) = 4/d. Since the common difference d is positive, f(1) is also positive. Because the function is continuous and changes sign between x = 0 and x = 1, the equation f(x) = 0 has a root between 0 and 1.

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