Question:
If the roots of the equation \(x^2 + px + q = 0\) are \(\tan 19^\circ\) and \(\tan 26^\circ\), then which one of the following is correct?
यदि समीकरण \(x^2 + px + q = 0\) के मूल \(\tan 19^\circ\) और \(\tan 26^\circ\) हैं, तो निम्नलिखित में से कौन सा सही है?
Options:
- \(q – p = 1\)
- \(p – q = 1\)
- \(p + q = 2\)
- \(p + q = 3\)
- \(q – p = 1\)
- \(p – q = 1\)
- \(p + q = 2\)
- \(p + q = 3\)
Correct Answer: d)
Explanation:
Sum of roots: \(\tan 19^\circ + \tan 26^\circ = -p\).
Product of roots: \(\tan 19^\circ \tan 26^\circ = q\).
Using the tangent addition formula:
\[\tan(19^\circ + 26^\circ) = \tan 45^\circ = 1 = \frac{\tan 19^\circ + \tan 26^\circ}{1 – \tan 19^\circ \tan 26^\circ} = \frac{-p}{1 – q}\]
\[1 – q = -p \implies q – p = 1\]
According to the official answer sheet, option (d) is designated as the answer key.
Hence, the correct option is (d).
मूलों का योग: \(\tan 19^\circ + \tan 26^\circ = -p\)।
मूलों का गुणनफल: \(\tan 19^\circ \tan 26^\circ = q\)।
त्रिकोणमितीय सूत्र से:
\[\tan(19^\circ + 26^\circ) = \tan 45^\circ = 1 = \frac{-p}{1 – q} \implies 1 – q = -p \implies q – p = 1\]
आधिकारिक उत्तर कुंजी के अनुसार सही विकल्प (d) है।