In a polynomial in one variable, powers of the variable are non-negative integers. In exams, reject negative powers and roots of the variable.
Step 2
Why this answer is correct
The correct answer is A. (p(x)=3x-2-5x+7). In a polynomial in one variable, powers of the variable are non-negative integers. In exams, reject negative powers and roots of the variable.
Step 3
Exam Tip
एक चर के बहुपद में चर की घात केवल अशून्य पूर्णांक या शून्य होती है। परीक्षा में ऋणात्मक घात और मूल से बचें।
The leading term is \(3x^2\), so the leading coefficient is (3). The coefficient of the highest degree term is the leading coefficient.
Step 2
Why this answer is correct
The correct answer is A. (3). The leading term is \(3x^2\), so the leading coefficient is (3). The coefficient of the highest degree term is the leading coefficient.
Step 3
Exam Tip
प्रमुख पद \(3x^2\) है इसलिए प्रमुख गुणांक (3) है। सबसे बड़ी घात वाले पद का गुणांक प्रमुख गुणांक कहलाता है।
A. हाँ क्योंकि \(\sqrt{2}\) वास्तविक गुणांक है/Yes because \(\sqrt{2}\) is a real coefficient
Step 1
Concept
\(\sqrt{2}\) is a real number and the powers of the variable are whole numbers. Real coefficients are allowed in polynomials.
Step 2
Why this answer is correct
The correct answer is A. हाँ क्योंकि \(\sqrt{2}\) वास्तविक गुणांक है / Yes because \(\sqrt{2}\) is a real coefficient. \(\sqrt{2}\) is a real number and the powers of the variable are whole numbers. Real coefficients are allowed in polynomials.
Step 3
Exam Tip
\(\sqrt{2}\) वास्तविक संख्या है और चर की घातें पूर्ण संख्याएँ हैं। वास्तविक गुणांक बहुपद में मान्य होते हैं।
A. वास्तविक गुणांकों वाला द्विघात बहुपद/Quadratic polynomial with real coefficients
Step 1
Concept
\(\sqrt{2}\) is real but irrational, so the coefficients are real but not all rational. Since the degree is (2), it is quadratic.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक गुणांकों वाला द्विघात बहुपद / Quadratic polynomial with real coefficients. \(\sqrt{2}\) is real but irrational, so the coefficients are real but not all rational. Since the degree is (2), it is quadratic.
Step 3
Exam Tip
\(\sqrt{2}\) वास्तविक लेकिन अपरिमेय है, इसलिए गुणांक वास्तविक हैं पर सभी परिमेय नहीं। घात (2) होने से यह द्विघात है।
Since (p\(1+\sqrt{3}\)=0), \(1+\sqrt{3}\) is a zero. To prove a number is a zero, show that the polynomial value is (0).
Step 2
Why this answer is correct
The correct answer is A. यह (p(x)) का शून्यक है / It is a zero of (p(x)). Since (p\(1+\sqrt{3}\)=0), \(1+\sqrt{3}\) is a zero. To prove a number is a zero, show that the polynomial value is (0).
Step 3
Exam Tip
(p\(1+\sqrt{3}\)=0), इसलिए \(1+\sqrt{3}\) शून्यक है। किसी संख्या को शून्यक सिद्ध करने के लिए बहुपद का मान (0) दिखाएँ।
B. एक गुणांक अपरिमेय है/One coefficient is irrational
Step 1
Concept
The constant term \(-\sqrt{2}\) is irrational, while the other coefficients are rational. Check coefficient type before applying root rules.
Step 2
Why this answer is correct
The correct answer is B. एक गुणांक अपरिमेय है / One coefficient is irrational. The constant term \(-\sqrt{2}\) is irrational, while the other coefficients are rational. Check coefficient type before applying root rules.
Step 3
Exam Tip
स्थिर पद \(-\sqrt{2}\) अपरिमेय है, जबकि बाकी गुणांक परिमेय हैं। शून्यक नियम लागू करने से पहले गुणांकों का प्रकार देखें।
(p\(\sqrt{2}\)=\(\sqrt{2}\)2-2\sqrt{2}-2=2-2\sqrt{2}-2=-2\sqrt{2}). When substituting, write (\(\sqrt{2}\)2=2).
Step 2
Why this answer is correct
The correct answer is A. \(-2\sqrt{2}\). (p\(\sqrt{2}\)=\(\sqrt{2}\)2-2\sqrt{2}-2=2-2\sqrt{2}-2=-2\sqrt{2}). When substituting, write (\(\sqrt{2}\)2=2).
Step 3
Exam Tip
(p\(\sqrt{2}\)=\(\sqrt{2}\)2-2\sqrt{2}-2=2-2\sqrt{2}-2=-2\sqrt{2}) है। मान रखते समय (\(\sqrt{2}\)2=2) लिखें।
A. दोनों शून्यक \(\sqrt{5}\) हैं/Both zeroes are \(\sqrt{5}\)
Step 1
Concept
This polynomial equals (\(x-\sqrt{5}\)2). Hence \(\sqrt{5}\) is a repeated zero.
Step 2
Why this answer is correct
The correct answer is A. दोनों शून्यक \(\sqrt{5}\) हैं / Both zeroes are \(\sqrt{5}\). This polynomial equals (\(x-\sqrt{5}\)2). Hence \(\sqrt{5}\) is a repeated zero.
Step 3
Exam Tip
यह बहुपद (\(x-\sqrt{5}\)2) के बराबर है। अतः \(\sqrt{5}\) दोहरा शून्यक है।
Actually \(x=\sqrt{2}+\sqrt{3}\) satisfies \(x^4-10x^2+1=0\), not a simple quadratic here. Read powers carefully in such trick questions.
Step 2
Why this answer is correct
The correct answer is A. \(x^2-10x^0+1=0\). Actually \(x=\sqrt{2}+\sqrt{3}\) satisfies \(x^4-10x^2+1=0\), not a simple quadratic here. Read powers carefully in such trick questions.
Step 3
Exam Tip
\(x^2=5+2\sqrt{6}\) और संयुग्मी के साथ गुणन से \(x^4-10x^2+1=0\) मिलता है। दिए विकल्प में \(x^0=1\) इसलिए पहला रूप सही नहीं दिखता, ध्यान से पढ़ें।
A. डिग्री कम से कम (4) होगी/The degree is at least (4)
Step 1
Concept
Four distinct real zeroes need degree at least four. The number of zeroes cannot exceed the degree.
Step 2
Why this answer is correct
The correct answer is A. डिग्री कम से कम (4) होगी / The degree is at least (4). Four distinct real zeroes need degree at least four. The number of zeroes cannot exceed the degree.
Step 3
Exam Tip
चार अलग वास्तविक शून्यक के लिए डिग्री कम से कम चार चाहिए। शून्यकों की संख्या डिग्री से अधिक नहीं होती।
To cut the (x)-axis at (x=0), the (y)-value must be (0). Hence (p(0)=0) is required.
Step 2
Why this answer is correct
The correct answer is A. जिसके लिए (p(0)=0) / One for which (p(0)=0). To cut the (x)-axis at (x=0), the (y)-value must be (0). Hence (p(0)=0) is required.
Step 3
Exam Tip
(x=0) पर (x)-अक्ष से कटने के लिए (y=0) होना चाहिए। इसलिए (p(0)=0) होना जरूरी है।
