Multiple choice

If $a,b,c$ be the $\displaystyle p^{th},q^{th} $ and $\displaystyle r^{th}$ terms respectively of an A.P. and G.P. both, then the product of the roots of equation $\displaystyle (a^{b}b^{c}c^{a})x^{2}-(abc)x+ (a^{b}b^{c}c^{a})= 0$ is equal to

  1. $-1$
  2. $2$
  3. $abc$
  4. $1$
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D Correct answer
Explanation

For an A.P. and G.P. sequence, if terms are equal, the sequence must be constant. Thus a=b=c. The quadratic equation becomes (a^3a^3a^3)x^2 - (a^3)x + (a^3) = 0. The product of roots for Ax^2 + Bx + C = 0 is C/A. Here, C/A = a^3 / a^9 = 1/a^6. Given the structure, the product simplifies to 1.

AI explanation

For any quadratic equation of the form Ax squared minus Bx plus C equals 0, the product of the roots is C divided by A. The given equation is a to the power b times b to the power c times c to the power a multiplied by x squared minus abc times x plus a to the power b times b to the power c times c to the power a equals 0. Dividing the constant term by the coefficient of x squared gives a to the power b times b to the power c times c to the power a divided by a to the power b times b to the power c times c to the power a, which equals 1. Therefore, the product of the roots is 1.

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