Question

For p,q,r \(\neq\) 0, let \(f(x) = x^3 - px^2 + qx - r\), \(g(x) = x^3 - \frac{p}{r}x^2 + \frac{q}{r^2}x - \frac{1}{r}\). Which of the following statements is/are true?
I. \(f(a) = 0 \implies g(\frac{1}{a}) = 0\).
II. \(f(a) = 0 \implies g(1 + a) = 0\).
III. \(g(a) = 0 \implies f(\frac{r}{a}) = 0\).}

• A. Only I and III
• B. Only II and III
• C. Only I and II
• D. All of I, II and III

Hint:

When encountering a problem that seems mathematically inconsistent or contains typos, first check for standard transformations (like reciprocal roots, translated roots, etc.). If none apply, and you have an answer key, acknowledge the discrepancy but follow the key.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given two cubic polynomials, f(x) and g(x), with coefficients related to p, q, and r. We need to test the validity of three conditional statements that relate the roots of f(x) to the roots of g(x).
Step 2: Detailed Explanation:
Note on the question: The polynomial g(x) as given seems to contain a typo, as standard transformations do not lead to this form. A common related problem involves a polynomial whose roots are the reciprocals of the roots of f(x). The polynomial with reciprocal roots of f(x) would be \(h(x) = x^3 - \frac{q}{r}x^2 + \frac{p}{r}x - \frac{1}{r}\). The given g(x) is different. This suggests the question is flawed. However, we must proceed based on the provided answer key which states that I and III are true. There might be a non-obvious relationship or a typo in f(x) or g(x) that makes the statements true. Let's analyze the statements as given, assuming there is a way to prove them.
Statement I: \(f(a) = 0 \implies g(\frac{1}{a}) = 0\)
Given \(f(a) = 0\), we have \(a^3 - pa^2 + qa - r = 0\).
We need to check if \(g(\frac{1}{a}) = 0\).
\[ g(\frac{1}{a}) = (\frac{1}{a})^3 - \frac{p}{r}(\frac{1}{a})^2 + \frac{q}{r^2}(\frac{1}{a}) - \frac{1}{r} \] \[ g(\frac{1}{a}) = \frac{1}{a^3} - \frac{p}{ra^2} + \frac{q}{r^2a} - \frac{1}{r} \] Multiplying by \(r^2 a^3\), we get: \[ r^2 - pra + qa - r a^2 \] Without a correction to the problem statement, we cannot show that \(f(a)=0\) implies \(r^2 - pra + qa - r a^2 = 0\). Thus, based on a direct derivation, this statement is not necessarily true.
Statement II: \(f(a) = 0 \implies g(1 + a) = 0\)
This implies a transformation of roots from \(a\) to \(1+a\). There is no information in the coefficients of f(x) and g(x) to suggest such a relationship. This statement is generally false.
Statement III: \(g(a) = 0 \implies f(\frac{r}{a}) = 0\)
Given \(g(a) = 0\), we have \(a^3 - \frac{p}{r}a^2 + \frac{q}{r^2}a - \frac{1}{r} = 0\).
Multiplying by \(r^2\), we get \(r^2 a^3 - p r a^2 + q a - r = 0\).
We need to check if \(f(\frac{r}{a}) = 0\).
\[ f(\frac{r}{a}) = (\frac{r}{a})^3 - p(\frac{r}{a})^2 + q(\frac{r}{a}) - r \] \[ f(\frac{r}{a}) = \frac{r^3}{a^3} - \frac{pr^2}{a^2} + \frac{qr}{a} - r \] Multiplying by \(a^3\), we get: \[ r^3 - pr^2 a + qr a^2 - r a^3 \] Again, we cannot show that \(r^2 a^3 - p r a^2 + q a - r = 0\) implies \(r^3 - pr^2 a + qr a^2 - r a^3 = 0\). This statement is also not necessarily true as written.
Step 3: Conclusion based on Answer Key:
The question as stated appears to be incorrect due to a likely typo in the definition of g(x). Standard algebraic manipulations do not validate statements I and III. However, if this question were to appear in an exam with the given answer key (A), it implies that there is an intended, albeit flawed, logic where I and III are considered true. Without the corrected form of the question, a rigorous mathematical proof is not possible. We select the answer based on the provided key.
Step 4: Final Answer:
Accepting the premise of the question and its provided answer key, we conclude that statements I and III are the intended true statements.