For exactly quadratic, the coefficient of \(x^6\) must be zero and the coefficient of \(x^2\) is non-zero. From (a-5=0), (a=5).
Step 2
Why this answer is correct
The correct answer is A. (5). For exactly quadratic, the coefficient of \(x^6\) must be zero and the coefficient of \(x^2\) is non-zero. From (a-5=0), (a=5).
Step 3
Exam Tip
ठीक द्विघात के लिए \(x^6\) का गुणांक शून्य होना चाहिए और \(x^2\) का गुणांक अशून्य है। (a-5=0) से (a=5) मिलता है।
\(\frac{2}{x}=2x^{-1}\), where the variable has a negative power. One invalid term prevents the whole expression from being a polynomial.
Step 2
Why this answer is correct
The correct answer is B. \(x^4+\frac{2}{x}-1\). \(\frac{2}{x}=2x^{-1}\), where the variable has a negative power. One invalid term prevents the whole expression from being a polynomial.
Step 3
Exam Tip
\(\frac{2}{x}=2x^{-1}\) है, जिसमें चर की घात ऋणात्मक है। एक अमान्य पद पूरे व्यंजक को बहुपद नहीं रहने देता।
For degree (3), the \(x^4\) coefficient must be zero and the \(x^3\) coefficient must be non-zero. At (k=1), (k+2=3) is also non-zero.
Step 2
Why this answer is correct
The correct answer is A. (k=1). For degree (3), the \(x^4\) coefficient must be zero and the \(x^3\) coefficient must be non-zero. At (k=1), (k+2=3) is also non-zero.
Step 3
Exam Tip
घात (3) के लिए \(x^4\) का गुणांक शून्य और \(x^3\) का गुणांक अशून्य होना चाहिए। (k=1) पर (k+2=3) भी अशून्य है।
The degree is (4) only when the coefficient of \(x^4\) is not zero. Therefore, \(\lambda\neq0\) is required.
Step 2
Why this answer is correct
The correct answer is B. \(\lambda\neq0\). The degree is (4) only when the coefficient of \(x^4\) is not zero. Therefore, \(\lambda\neq0\) is required.
Step 3
Exam Tip
घात (4) तभी होगी जब \(x^4\) का गुणांक शून्य न हो। इसलिए \(\lambda\neq0\) आवश्यक है।
For a constant polynomial, both (r-3=0) and (r+1=0) are needed. One value of (r) cannot satisfy both conditions.
Step 2
Why this answer is correct
The correct answer is C. ऐसा कोई (r) नहीं है / No such (r) exists. For a constant polynomial, both (r-3=0) and (r+1=0) are needed. One value of (r) cannot satisfy both conditions.
Step 3
Exam Tip
स्थिर बहुपद के लिए (r-3=0) और (r+1=0) दोनों चाहिए। एक ही (r) दोनों शर्तें पूरी नहीं कर सकता।
The degree of the zero polynomial is not defined. Keep it separate from a non-zero constant polynomial.
Step 2
Why this answer is correct
The correct answer is C. घात परिभाषित नहीं है / Degree is not defined. The degree of the zero polynomial is not defined. Keep it separate from a non-zero constant polynomial.
Step 3
Exam Tip
शून्य बहुपद की घात परिभाषित नहीं होती। इसे शून्य से अलग स्थिर बहुपद से अलग समझें।
For degree (4), the \(x^5\) coefficient must be zero and the \(x^4\) coefficient must be non-zero. At (m=-4), (m-1=-5) is non-zero.
Step 2
Why this answer is correct
The correct answer is B. (-4). For degree (4), the \(x^5\) coefficient must be zero and the \(x^4\) coefficient must be non-zero. At (m=-4), (m-1=-5) is non-zero.
Step 3
Exam Tip
घात (4) के लिए \(x^5\) का गुणांक शून्य और \(x^4\) का गुणांक अशून्य चाहिए। (m=-4) पर (m-1=-5) अशून्य है।
B. यह बहुपद नहीं है क्योंकि चर मूल के अंदर है/It is not a polynomial because the variable is inside a root
Step 1
Concept
\(\sqrt{x^2+4}\) has the variable inside a root. In the class (9) definition, such a term is not treated as a polynomial term.
Step 2
Why this answer is correct
The correct answer is B. यह बहुपद नहीं है क्योंकि चर मूल के अंदर है / It is not a polynomial because the variable is inside a root. \(\sqrt{x^2+4}\) has the variable inside a root. In the class (9) definition, such a term is not treated as a polynomial term.
Step 3
Exam Tip
\(\sqrt{x^2+4}\) में चर मूल के अंदर है। कक्षा (9) की परिभाषा में ऐसा पद बहुपद पद नहीं माना जाता।
B. यह बहुपद नहीं है क्योंकि \(x^{-2}\) है/It is not a polynomial because it has \(x^{-2}\)
Step 1
Concept
\(x^0\) is valid, but \(x^{-2}\) is invalid. Negative powers are not allowed in a polynomial.
Step 2
Why this answer is correct
The correct answer is B. यह बहुपद नहीं है क्योंकि \(x^{-2}\) है / It is not a polynomial because it has \(x^{-2}\). \(x^0\) is valid, but \(x^{-2}\) is invalid. Negative powers are not allowed in a polynomial.
Step 3
Exam Tip
\(x^0\) मान्य है, लेकिन \(x^{-2}\) अमान्य है। बहुपद में ऋणात्मक घात स्वीकार्य नहीं होती।
The zero-coefficient terms disappear and (p(x)=9) remains. It is a non-zero constant polynomial.
Step 2
Why this answer is correct
The correct answer is C. स्थिर बहुपद / Constant polynomial. The zero-coefficient terms disappear and (p(x)=9) remains. It is a non-zero constant polynomial.
Step 3
Exam Tip
शून्य गुणांक वाले पद हट जाते हैं और (p(x)=9) बचता है। यह शून्य से अलग स्थिर बहुपद है।
For degree (2), the \(x^3\) coefficient must be zero and the \(x^2\) coefficient must be non-zero. At (k=0), (k-2=-2) is non-zero.
Step 2
Why this answer is correct
The correct answer is A. (0). For degree (2), the \(x^3\) coefficient must be zero and the \(x^2\) coefficient must be non-zero. At (k=0), (k-2=-2) is non-zero.
Step 3
Exam Tip
घात (2) के लिए \(x^3\) का गुणांक शून्य चाहिए और \(x^2\) का गुणांक अशून्य चाहिए। (k=0) पर (k-2=-2) अशून्य है।
\(\sqrt{7}\) is a real coefficient and is allowed. All powers of the variable are non-negative integers.
Step 2
Why this answer is correct
The correct answer is B. \(\sqrt{7}x^4-3x+2\). \(\sqrt{7}\) is a real coefficient and is allowed. All powers of the variable are non-negative integers.
Step 3
Exam Tip
\(\sqrt{7}\) एक वास्तविक गुणांक है और मान्य है। चर की सभी घातें शून्य या धनात्मक पूर्णांक हैं।
For a constant polynomial, both (a+1=0) and (a-2=0) are required. These cannot hold together.
Step 2
Why this answer is correct
The correct answer is C. ऐसा कोई (a) नहीं है / No such (a) exists. For a constant polynomial, both (a+1=0) and (a-2=0) are required. These cannot hold together.
Step 3
Exam Tip
स्थिर बहुपद के लिए (a+1=0) और (a-2=0) दोनों चाहिए। ये दोनों एक साथ संभव नहीं हैं।
A. \(x^n+2\), जहाँ (n) बदलता है/\(x^n+2\), where (n) varies
Step 1
Concept
In a polynomial, powers are fixed non-negative integers. A power depending on a varying variable is not valid in the definition.
Step 2
Why this answer is correct
The correct answer is A. \(x^n+2\), जहाँ (n) बदलता है / \(x^n+2\), where (n) varies. In a polynomial, powers are fixed non-negative integers. A power depending on a varying variable is not valid in the definition.
Step 3
Exam Tip
बहुपद में घातें निश्चित शून्य या धनात्मक पूर्णांक होती हैं। बदलते चर पर निर्भर घात परिभाषा में मान्य नहीं है।
The coefficient of \(x^3\) is always non-zero. Therefore, the degree remains (3) for every (h).
Step 2
Why this answer is correct
The correct answer is B. कोई भी वास्तविक (h) / Any real (h). The coefficient of \(x^3\) is always non-zero. Therefore, the degree remains (3) for every (h).
Step 3
Exam Tip
\(x^3\) का गुणांक (2) हमेशा अशून्य है। इसलिए घात (3) हर (h) के लिए रहेगी।
A. यह \(x^3+1\) बनता है, इसलिए बहुपद है/It becomes \(x^3+1\), so it is a polynomial
Step 1
Concept
\(\frac{x^5+x^2}{x^2}=x^3+1\). Its powers are (3) and (0), so it is a polynomial.
Step 2
Why this answer is correct
The correct answer is A. यह \(x^3+1\) बनता है, इसलिए बहुपद है / It becomes \(x^3+1\), so it is a polynomial. \(\frac{x^5+x^2}{x^2}=x^3+1\). Its powers are (3) and (0), so it is a polynomial.
Step 3
Exam Tip
\(\frac{x^5+x^2}{x^2}=x^3+1\) है। इसमें घातें (3) और (0) हैं, इसलिए यह बहुपद है।
B. यह बहुपद नहीं है क्योंकि \(\sin x\) बहुपद पद नहीं है/It is not a polynomial because \(\sin x\) is not a polynomial term
Step 1
Concept
In a polynomial, terms are powers of the variable with non-negative integer exponents. \(\sin x\) is not such a term.
Step 2
Why this answer is correct
The correct answer is B. यह बहुपद नहीं है क्योंकि \(\sin x\) बहुपद पद नहीं है / It is not a polynomial because \(\sin x\) is not a polynomial term. In a polynomial, terms are powers of the variable with non-negative integer exponents. \(\sin x\) is not such a term.
Step 3
Exam Tip
बहुपद में पद चर की शून्य या धनात्मक पूर्णांक घातों के रूप में होते हैं। \(\sin x\) ऐसा पद नहीं है।
The coefficient of \(x^2\) is (1), which is non-zero. Therefore, the degree is (2) for every (a).
Step 2
Why this answer is correct
The correct answer is C. सभी वास्तविक (a) के लिए / For all real (a). The coefficient of \(x^2\) is (1), which is non-zero. Therefore, the degree is (2) for every (a).
Step 3
Exam Tip
\(x^2\) का गुणांक (1) है, जो शून्य नहीं है। इसलिए घात हर (a) के लिए (2) है।
\(\log x\) is not a term with a non-negative integer power of the variable. Therefore, it is not a polynomial.
Step 2
Why this answer is correct
The correct answer is A. \(x^2+\log x\). \(\log x\) is not a term with a non-negative integer power of the variable. Therefore, it is not a polynomial.
Step 3
Exam Tip
\(\log x\) चर की शून्य या धनात्मक पूर्णांक घात वाला पद नहीं है। इसलिए यह बहुपद नहीं है।
C. यह शून्य बहुपद है जिसकी घात परिभाषित नहीं है/It is the zero polynomial whose degree is not defined
Step 1
Concept
All terms have zero coefficients, so it is the zero polynomial. The degree of the zero polynomial is not defined.
Step 2
Why this answer is correct
The correct answer is C. यह शून्य बहुपद है जिसकी घात परिभाषित नहीं है / It is the zero polynomial whose degree is not defined. All terms have zero coefficients, so it is the zero polynomial. The degree of the zero polynomial is not defined.
Step 3
Exam Tip
सभी पदों के गुणांक शून्य हैं, इसलिए यह शून्य बहुपद है। शून्य बहुपद की घात परिभाषित नहीं होती।
A. चर की घातें निश्चित शून्य या धनात्मक पूर्णांक हों/Variable powers are fixed zero or positive integers
Step 1
Concept
A polynomial is identified mainly by the powers of the variable. The powers must be fixed non-negative integers.
Step 2
Why this answer is correct
The correct answer is A. चर की घातें निश्चित शून्य या धनात्मक पूर्णांक हों / Variable powers are fixed zero or positive integers. A polynomial is identified mainly by the powers of the variable. The powers must be fixed non-negative integers.
Step 3
Exam Tip
बहुपद की पहचान मुख्य रूप से चर की घातों से होती है। घातें निश्चित शून्य या धनात्मक पूर्णांक होनी चाहिए।
For degree (4), the coefficient of \(x^5\) must be zero. From \(k^2-1=0\), \(k=\pm1\), and \(2x^4\) remains non-zero.
Step 2
Why this answer is correct
The correct answer is A. (k=1) या (k=-1) / (k=1) or (k=-1). For degree (4), the coefficient of \(x^5\) must be zero. From \(k^2-1=0\), \(k=\pm1\), and \(2x^4\) remains non-zero.
Step 3
Exam Tip
घात (4) के लिए \(x^5\) का गुणांक शून्य होना चाहिए। \(k^2-1=0\) से \(k=\pm1\) और \(2x^4\) अशून्य रहता है।
A. किसी भी (c), (d), (e) के लिए/For any (c), (d), (e)
Step 1
Concept
The coefficient of \(x^4\) is (1), which is never zero. Therefore, the degree is always (4).
Step 2
Why this answer is correct
The correct answer is A. किसी भी (c), (d), (e) के लिए / For any (c), (d), (e). The coefficient of \(x^4\) is (1), which is never zero. Therefore, the degree is always (4).
Step 3
Exam Tip
\(x^4\) का गुणांक (1) है, जो कभी शून्य नहीं होता। इसलिए घात हमेशा (4) रहेगी।
For degree (5), the \(x^6\) coefficient must be zero and the \(x^5\) coefficient must be non-zero. At (a=-2), (a-4=-6) remains non-zero.
Step 2
Why this answer is correct
The correct answer is B. (-2). For degree (5), the \(x^6\) coefficient must be zero and the \(x^5\) coefficient must be non-zero. At (a=-2), (a-4=-6) remains non-zero.
Step 3
Exam Tip
घात (5) के लिए \(x^6\) का गुणांक शून्य और \(x^5\) का गुणांक अशून्य होना चाहिए। (a=-2) पर (a-4=-6) अशून्य रहता है।
For degree (3), the \(x^4\) coefficient must be zero and the \(x^3\) coefficient must be non-zero. At (a=2), \(a^2-4=0\) and (a+2=4).
Step 2
Why this answer is correct
The correct answer is B. (2). For degree (3), the \(x^4\) coefficient must be zero and the \(x^3\) coefficient must be non-zero. At (a=2), \(a^2-4=0\) and (a+2=4).
Step 3
Exam Tip
घात (3) के लिए \(x^4\) का गुणांक शून्य और \(x^3\) का गुणांक अशून्य होना चाहिए। (a=2) पर \(a^2-4=0\) और (a+2=4) है।
