For exactly quadratic, the coefficient of \(x^4\) must be zero and the coefficient of \(x^2\) must remain non-zero. From (k-3=0), we get (k=3).
Step 2
Why this answer is correct
The correct answer is A. (3). For exactly quadratic, the coefficient of \(x^4\) must be zero and the coefficient of \(x^2\) must remain non-zero. From (k-3=0), we get (k=3).
Step 3
Exam Tip
ठीक द्विघात के लिए \(x^4\) का गुणांक शून्य होना चाहिए और \(x^2\) का गुणांक अशून्य रहना चाहिए। (k-3=0) से (k=3) मिलता है।
B. क्योंकि चर मूल के अंदर है/Because the variable is under a root
Step 1
Concept
In class (9), when the variable is under a root, the power is not treated as a valid integer power. So it is not considered a polynomial.
Step 2
Why this answer is correct
The correct answer is B. क्योंकि चर मूल के अंदर है / Because the variable is under a root. In class (9), when the variable is under a root, the power is not treated as a valid integer power. So it is not considered a polynomial.
Step 3
Exam Tip
कक्षा (9) में चर मूल के अंदर हो तो घात पूर्णांक रूप में मान्य नहीं मानी जाती। इसलिए इसे बहुपद नहीं माना जाता।
C. इसकी घात परिभाषित नहीं होती/Its degree is not defined
Step 1
Concept
The zero polynomial is (0), but its degree is not defined. Keep zero and non-zero constant polynomials separate.
Step 2
Why this answer is correct
The correct answer is C. इसकी घात परिभाषित नहीं होती / Its degree is not defined. The zero polynomial is (0), but its degree is not defined. Keep zero and non-zero constant polynomials separate.
Step 3
Exam Tip
शून्य बहुपद (0) होता है, लेकिन उसकी घात परिभाषित नहीं होती। शून्य और अशून्य स्थिर बहुपद में अंतर रखें।
C. \(3x^5-2x^2+7\), बहुपद, घात (5)/\(3x^5-2x^2+7\), polynomial, degree (5)
Step 1
Concept
In \(3x^5-2x^2+7\), all powers are valid and the highest power is (5). The other options have invalid powers.
Step 2
Why this answer is correct
The correct answer is C. \(3x^5-2x^2+7\), बहुपद, घात (5) / \(3x^5-2x^2+7\), polynomial, degree (5). In \(3x^5-2x^2+7\), all powers are valid and the highest power is (5). The other options have invalid powers.
Step 3
Exam Tip
\(3x^5-2x^2+7\) में सभी घातें मान्य हैं और सबसे बड़ी घात (5) है। बाकी विकल्पों में अमान्य घातें हैं।
\(\frac{x^2+4}{x}=x+\frac{4}{x}\) contains \(\frac{4}{x}\), whose power is (-1). Hence it is not a polynomial.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{x^2+4}{x}\). \(\frac{x^2+4}{x}=x+\frac{4}{x}\) contains \(\frac{4}{x}\), whose power is (-1). Hence it is not a polynomial.
Step 3
Exam Tip
\(\frac{x^2+4}{x}=x+\frac{4}{x}\) में \(\frac{4}{x}\) की घात (-1) है। इसलिए यह बहुपद नहीं है।
B. घात (6) का चार-पदी बहुपद/Four-term polynomial of degree (6)
Step 1
Concept
It has four terms and the highest power is (6). Therefore, it is a four-term polynomial of degree (6).
Step 2
Why this answer is correct
The correct answer is B. घात (6) का चार-पदी बहुपद / Four-term polynomial of degree (6). It has four terms and the highest power is (6). Therefore, it is a four-term polynomial of degree (6).
Step 3
Exam Tip
इसमें चार पद हैं और सबसे बड़ी घात (6) है। इसलिए यह घात (6) का चार-पदी बहुपद है।
In \(x^{5/2}\), the power is \(\frac{5}{2}\), which is not an integer. Such a power is not valid in a polynomial.
Step 2
Why this answer is correct
The correct answer is C. \(x^{5 / 2}+3x\). In \(x^{5/2}\), the power is \(\frac{5}{2}\), which is not an integer. Such a power is not valid in a polynomial.
Step 3
Exam Tip
\(x^{5/2}\) में घात \(\frac{5}{2}\) है जो पूर्णांक नहीं है। बहुपद में ऐसी घात मान्य नहीं होती।
In \(3^x\), the variable is in the exponent. In a polynomial, variable powers must be fixed non-negative integers.
Step 2
Why this answer is correct
The correct answer is B. \(3^x+x^2\). In \(3^x\), the variable is in the exponent. In a polynomial, variable powers must be fixed non-negative integers.
Step 3
Exam Tip
\(3^x\) में चर घात में है। बहुपद में चर की घात निश्चित शून्य या धनात्मक पूर्णांक होनी चाहिए।
B. चर की घातें शून्य या धनात्मक पूर्णांक हों तो व्यंजक बहुपद हो सकता है/An expression can be a polynomial if variable powers are non-negative integers
Step 1
Concept
The main condition of a polynomial is about powers of the variable. Coefficients may be real, but powers must also be valid.
Step 2
Why this answer is correct
The correct answer is B. चर की घातें शून्य या धनात्मक पूर्णांक हों तो व्यंजक बहुपद हो सकता है / An expression can be a polynomial if variable powers are non-negative integers. The main condition of a polynomial is about powers of the variable. Coefficients may be real, but powers must also be valid.
Step 3
Exam Tip
बहुपद की मुख्य शर्त चर की घातों पर है। गुणांक वास्तविक हो सकते हैं लेकिन घातें भी मान्य होनी चाहिए।
\(\frac{1}{x+1}\) has the variable in the denominator and is not a polynomial term. A polynomial must not have the variable in the denominator.
Step 2
Why this answer is correct
The correct answer is C. \(x+\frac{1}{x+1}\). \(\frac{1}{x+1}\) has the variable in the denominator and is not a polynomial term. A polynomial must not have the variable in the denominator.
Step 3
Exam Tip
\(\frac{1}{x+1}\) में चर हर में है और यह बहुपद पद नहीं है। बहुपद में चर हर में नहीं होना चाहिए।
For degree (2), the coefficient of \(x^3\) must be (0) and the coefficient of \(x^2\) non-zero. This happens at (m=-1).
Step 2
Why this answer is correct
The correct answer is A. (m=-1). For degree (2), the coefficient of \(x^3\) must be (0) and the coefficient of \(x^2\) non-zero. This happens at (m=-1).
Step 3
Exam Tip
घात (2) के लिए \(x^3\) का गुणांक (0) और \(x^2\) का गुणांक अशून्य होना चाहिए। (m=-1) पर ऐसा होता है।
C. यह बहुपद है और घात (2) है/It is a polynomial and degree (2)
Step 1
Concept
Fractional and irrational real coefficients are allowed in a polynomial. The highest (x)-power is (2).
Step 2
Why this answer is correct
The correct answer is C. यह बहुपद है और घात (2) है / It is a polynomial and degree (2). Fractional and irrational real coefficients are allowed in a polynomial. The highest (x)-power is (2).
Step 3
Exam Tip
भिन्न और अपरिमेय वास्तविक गुणांक बहुपद में मान्य हैं। सबसे बड़ी (x)-घात (2) है।
All terms have coefficient (0), so the whole polynomial is (0). It is called the zero polynomial.
Step 2
Why this answer is correct
The correct answer is C. शून्य बहुपद / Zero polynomial. All terms have coefficient (0), so the whole polynomial is (0). It is called the zero polynomial.
Step 3
Exam Tip
सभी पदों के गुणांक (0) हैं इसलिए पूरा बहुपद (0) है। इसे शून्य बहुपद कहते हैं।
In \(x^3-5x+6\), the highest power is (3) and the \(x^2\) term is absent. A missing term does not make a polynomial invalid.
Step 2
Why this answer is correct
The correct answer is A. \(x^3-5x+6\). In \(x^3-5x+6\), the highest power is (3) and the \(x^2\) term is absent. A missing term does not make a polynomial invalid.
Step 3
Exam Tip
\(x^3-5x+6\) में सबसे बड़ी घात (3) है और \(x^2\) पद अनुपस्थित है। अनुपस्थित पद बहुपद को अमान्य नहीं बनाता।
B. हर बहुपद एक बीजीय व्यंजक है/Every polynomial is an algebraic expression
Step 1
Concept
A polynomial is a special type of algebraic expression. But every algebraic expression is not a polynomial.
Step 2
Why this answer is correct
The correct answer is B. हर बहुपद एक बीजीय व्यंजक है / Every polynomial is an algebraic expression. A polynomial is a special type of algebraic expression. But every algebraic expression is not a polynomial.
Step 3
Exam Tip
बहुपद बीजीय व्यंजक का विशेष प्रकार है। लेकिन हर बीजीय व्यंजक बहुपद नहीं होता।
C. क्योंकि \(\frac{1}{x^2}=x^{-2}\) है/Because \(\frac{1}{x^2}=x^{-2}\)
Step 1
Concept
In \(\frac{1}{x^2}\), the power of (x) is (-2). Due to the negative power, it is not a polynomial.
Step 2
Why this answer is correct
The correct answer is C. क्योंकि \(\frac{1}{x^2}=x^{-2}\) है / Because \(\frac{1}{x^2}=x^{-2}\). In \(\frac{1}{x^2}\), the power of (x) is (-2). Due to the negative power, it is not a polynomial.
Step 3
Exam Tip
\(\frac{1}{x^2}\) में (x) की घात (-2) है। ऋणात्मक घात के कारण यह बहुपद नहीं है।
For degree (3), the coefficient of \(x^5\) must be (0). From (k+2=0), (k=-2), and the \(x^3\) coefficient remains non-zero.
Step 2
Why this answer is correct
The correct answer is A. (k=-2). For degree (3), the coefficient of \(x^5\) must be (0). From (k+2=0), (k=-2), and the \(x^3\) coefficient remains non-zero.
Step 3
Exam Tip
घात (3) के लिए \(x^5\) का गुणांक (0) होना चाहिए। (k+2=0) से (k=-2), और \(x^3\) का गुणांक अशून्य रहता है।
C. (4x) पद को शून्य करना/Making the (4x) term zero
Step 1
Concept
The coefficient of (4x) is fixed and cannot become zero. Therefore, this expression cannot become a constant polynomial.
Step 2
Why this answer is correct
The correct answer is C. (4x) पद को शून्य करना / Making the (4x) term zero. The coefficient of (4x) is fixed and cannot become zero. Therefore, this expression cannot become a constant polynomial.
Step 3
Exam Tip
(4x) का गुणांक (4) तय है और शून्य नहीं हो सकता। इसलिए यह व्यंजक स्थिर बहुपद नहीं बन सकता।
It has four terms and the highest power is (3). Check the number of terms and degree separately.
Step 2
Why this answer is correct
The correct answer is B. (4) पद और घात (3) / (4) terms and degree (3). It has four terms and the highest power is (3). Check the number of terms and degree separately.
Step 3
Exam Tip
इसमें चार पद हैं और सबसे बड़ी घात (3) है। पदों की संख्या और घात अलग-अलग जाँचें।
In \(x^x\), the exponent is not a fixed integer. In a polynomial, the variable power must be a fixed non-negative integer.
Step 2
Why this answer is correct
The correct answer is B. \(x^x+1\). In \(x^x\), the exponent is not a fixed integer. In a polynomial, the variable power must be a fixed non-negative integer.
Step 3
Exam Tip
\(x^x\) में घात स्थिर पूर्णांक नहीं है। बहुपद में चर की घात निश्चित शून्य या धनात्मक पूर्णांक होनी चाहिए।
At (a=5), the \(x^2\) term disappears, but \(2x^4\) remains. Therefore, the degree remains (4).
Step 2
Why this answer is correct
The correct answer is B. घात (4) रहेगी / The degree remains (4). At (a=5), the \(x^2\) term disappears, but \(2x^4\) remains. Therefore, the degree remains (4).
Step 3
Exam Tip
(a=5) पर \(x^2\) पद हटता है, लेकिन \(2x^4\) मौजूद है। इसलिए घात (4) रहेगी।
B. यह बहुपद है और घात (2) है/It is a polynomial and degree (2)
Step 1
Concept
\(\sqrt{3}\) is a constant real number and is allowed. The highest (x)-power is (2).
Step 2
Why this answer is correct
The correct answer is B. यह बहुपद है और घात (2) है / It is a polynomial and degree (2). \(\sqrt{3}\) is a constant real number and is allowed. The highest (x)-power is (2).
Step 3
Exam Tip
\(\sqrt{3}\) स्थिर वास्तविक संख्या है और मान्य है। सबसे बड़ी (x)-घात (2) है।
B. क्योंकि (0) की घात परिभाषित नहीं होती/Because the degree of (0) is not defined
Step 1
Concept
(0) is the zero polynomial and its degree is not defined. Degree (0) belongs to non-zero constant polynomials only.
Step 2
Why this answer is correct
The correct answer is B. क्योंकि (0) की घात परिभाषित नहीं होती / Because the degree of (0) is not defined. (0) is the zero polynomial and its degree is not defined. Degree (0) belongs to non-zero constant polynomials only.
Step 3
Exam Tip
(0) शून्य बहुपद है और उसकी घात परिभाषित नहीं होती। घात (0) केवल अशून्य स्थिर बहुपद की होती है।
Having two terms does not guarantee being a polynomial. Due to the negative power \(x^{-1}\), it is not a polynomial.
Step 2
Why this answer is correct
The correct answer is C. \(x^{-1}+8\). Having two terms does not guarantee being a polynomial. Due to the negative power \(x^{-1}\), it is not a polynomial.
Step 3
Exam Tip
दो पद होना बहुपद होने की गारंटी नहीं है। \(x^{-1}\) की ऋणात्मक घात के कारण यह बहुपद नहीं है।
C. ऐसा कोई (k) संभव नहीं है/No such (k) is possible
Step 1
Concept
In the zero polynomial, all coefficients must be (0). The coefficients of (3x) and (2) are not zero, so no such (k) is possible.
Step 2
Why this answer is correct
The correct answer is C. ऐसा कोई (k) संभव नहीं है / No such (k) is possible. In the zero polynomial, all coefficients must be (0). The coefficients of (3x) and (2) are not zero, so no such (k) is possible.
Step 3
Exam Tip
शून्य बहुपद में सभी गुणांक (0) होने चाहिए। (3x) और (2) के गुणांक शून्य नहीं हैं, इसलिए ऐसा (k) संभव नहीं।
