\(\frac{4}{x}=4x^{-1}\), where the power of (x) is negative. Having real coefficients alone does not guarantee a polynomial.
Step 2
Why this answer is correct
The correct answer is C. \(7x^2+\frac{4}{x}-9\). \(\frac{4}{x}=4x^{-1}\), where the power of (x) is negative. Having real coefficients alone does not guarantee a polynomial.
Step 3
Exam Tip
\(\frac{4}{x}=4x^{-1}\) है, जिसमें (x) की घात ऋणात्मक है। वास्तविक गुणांक होना अकेले बहुपद की गारंटी नहीं देता।
For degree (2), the coefficient of \(x^3\) must be zero and the coefficient of \(x^2\) must remain non-zero. From (k+2=0), (k=-2).
Step 2
Why this answer is correct
The correct answer is D. (k=-2). For degree (2), the coefficient of \(x^3\) must be zero and the coefficient of \(x^2\) must remain non-zero. From (k+2=0), (k=-2).
Step 3
Exam Tip
घात (2) के लिए \(x^3\) का गुणांक शून्य होना चाहिए और \(x^2\) का गुणांक अशून्य रहना चाहिए। (k+2=0) से (k=-2) मिलता है।
B. यह बहुपद नहीं है क्योंकि (\sqrt{x}=x^{1/2}) है / It is not a polynomial because \(\sqrt{x}=x^{1 / 2}\)
Step 1
Concept
In a polynomial, powers of the variable must be integers. \(\frac{1}{2}\) is not an integer.
Step 2
Why this answer is correct
The correct answer is B. यह बहुपद नहीं है क्योंकि \(\sqrt{x}=x^{1 / 2}\) है / It is not a polynomial because \(\sqrt{x}=x^{1 / 2}\). In a polynomial, powers of the variable must be integers. \(\frac{1}{2}\) is not an integer.
Step 3
Exam Tip
बहुपद में चर की घातें पूर्णांक होनी चाहिए। \(\frac{1}{2}\) पूर्णांक नहीं है।
For a constant polynomial, both (r-1=0) and (r+2=0) are required. These cannot hold together.
Step 2
Why this answer is correct
The correct answer is B. ऐसा कोई (r) नहीं है / No such (r) exists. For a constant polynomial, both (r-1=0) and (r+2=0) are required. These cannot hold together.
Step 3
Exam Tip
स्थिर बहुपद के लिए (r-1=0) और (r+2=0) दोनों चाहिए। ये दोनों एक साथ संभव नहीं हैं।
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 D. \(x^{5 / 2}+x+1\). 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।
A. \(5x^6-2x^3+1\), घात (6)/\(5x^6-2x^3+1\), degree (6)
Step 1
Concept
In the first option, all powers are non-negative integers and the highest power is (6). The other options are not polynomials.
Step 2
Why this answer is correct
The correct answer is A. \(5x^6-2x^3+1\), घात (6) / \(5x^6-2x^3+1\), degree (6). In the first option, all powers are non-negative integers and the highest power is (6). The other options are not polynomials.
Step 3
Exam Tip
पहले विकल्प में सभी घातें शून्य या धनात्मक पूर्णांक हैं और सबसे बड़ी घात (6) है। बाकी विकल्प बहुपद नहीं हैं।
For degree (2), the coefficient of \(x^3\) must be zero. From (c+5=0), (c=-5), and the \(x^2\) coefficient (-7) remains non-zero.
Step 2
Why this answer is correct
The correct answer is C. (c=-5). For degree (2), the coefficient of \(x^3\) must be zero. From (c+5=0), (c=-5), and the \(x^2\) coefficient (-7) remains non-zero.
Step 3
Exam Tip
घात (2) के लिए \(x^3\) का गुणांक शून्य होना चाहिए। (c+5=0) से (c=-5) और \(x^2\) का गुणांक (-7) अशून्य रहता है।
B. यह बहुपद है और घात (2) है/It is a polynomial and has degree (2)
Step 1
Concept
\(\frac{x}{2}\) means \(\frac{1}{2}x\), where the power is (1). All powers are valid.
Step 2
Why this answer is correct
The correct answer is B. यह बहुपद है और घात (2) है / It is a polynomial and has degree (2). \(\frac{x}{2}\) means \(\frac{1}{2}x\), where the power is (1). All powers are valid.
Step 3
Exam Tip
\(\frac{x}{2}\) का अर्थ \(\frac{1}{2}x\) है, जहाँ घात (1) है। सभी घातें मान्य हैं।
For it to be linear, the \(x^4\) coefficient must be zero and the (2x) term is present. From \(a^2-9=0\), \(a=\pm3\).
Step 2
Why this answer is correct
The correct answer is C. (a=3) या (a=-3) / (a=3) or (a=-3). For it to be linear, the \(x^4\) coefficient must be zero and the (2x) term is present. From \(a^2-9=0\), \(a=\pm3\).
Step 3
Exam Tip
रैखिक होने के लिए \(x^4\) का गुणांक शून्य होना चाहिए और (2x) पद मौजूद है। \(a^2-9=0\) से \(a=\pm3\)।
With respect to (u), \(\frac{1}{x}\) is like a constant, so it is a polynomial in (u). With respect to (x), \(\frac{1}{x}\) has power (-1).
Step 2
Why this answer is correct
The correct answer is B. \(u^2+\frac{1}{x}\). With respect to (u), \(\frac{1}{x}\) is like a constant, so it is a polynomial in (u). With respect to (x), \(\frac{1}{x}\) has power (-1).
Step 3
Exam Tip
(u) के संदर्भ में \(\frac{1}{x}\) स्थिर जैसा है, इसलिए यह (u) में बहुपद है। लेकिन (x) के संदर्भ में \(\frac{1}{x}\) की घात (-1) है।
\(0x^5\) is a zero term and does not raise the degree. The highest non-zero power is (3).
Step 2
Why this answer is correct
The correct answer is C. घात (3) का बहुपद / Polynomial of degree (3). \(0x^5\) is a zero term and does not raise the degree. The highest non-zero power is (3).
Step 3
Exam Tip
\(0x^5\) शून्य पद है और घात नहीं बढ़ाता। सबसे बड़ी अशून्य घात (3) है।
C. यह बहुपद नहीं है क्योंकि \(x^{-1}\) है/It is not a polynomial because it has \(x^{-1}\)
Step 1
Concept
\(x^0\) is valid but \(x^{-1}\) is not valid. One invalid term prevents the whole expression from being a polynomial.
Step 2
Why this answer is correct
The correct answer is C. यह बहुपद नहीं है क्योंकि \(x^{-1}\) है / It is not a polynomial because it has \(x^{-1}\). \(x^0\) is valid but \(x^{-1}\) is not valid. One invalid term prevents the whole expression from being a polynomial.
Step 3
Exam Tip
\(x^0\) मान्य है लेकिन \(x^{-1}\) मान्य नहीं है। एक अमान्य पद पूरे व्यंजक को बहुपद नहीं रहने देता।
D. यह घात (4) का बहुपद है/It is a polynomial of degree (4)
Step 1
Concept
Since \(a\neq0\), the \(x^4\) term remains present. Therefore, the highest power is definitely (4).
Step 2
Why this answer is correct
The correct answer is D. यह घात (4) का बहुपद है / It is a polynomial of degree (4). Since \(a\neq0\), the \(x^4\) term remains present. Therefore, the highest power is definitely (4).
Step 3
Exam Tip
\(a\neq0\) होने से \(x^4\) पद मौजूद रहता है। इसलिए सबसे बड़ी घात (4) निश्चित है।
A polynomial need not contain all middle powers. In \(x^5+2x^2-3\), the present powers (5), (2), (0) are valid.
Step 2
Why this answer is correct
The correct answer is A. \(x^5+2x^2-3\). A polynomial need not contain all middle powers. In \(x^5+2x^2-3\), the present powers (5), (2), (0) are valid.
Step 3
Exam Tip
बहुपद में बीच की घातों का होना जरूरी नहीं है। \(x^5+2x^2-3\) में उपस्थित घातें (5), (2), (0) हैं और मान्य हैं।
A. चर की घातें शून्य या धनात्मक पूर्णांक हों और गुणांक वास्तविक हो सकते हैं/Variable powers are zero or positive integers and coefficients may be real
Step 1
Concept
The main condition for a polynomial is about the powers of the variable. The powers must be zero or positive integers.
Step 2
Why this answer is correct
The correct answer is A. चर की घातें शून्य या धनात्मक पूर्णांक हों और गुणांक वास्तविक हो सकते हैं / Variable powers are zero or positive integers and coefficients may be real. The main condition for a polynomial is about the powers of the variable. The powers must be zero or positive integers.
Step 3
Exam Tip
बहुपद की मुख्य शर्त चर की घातों पर होती है। घातें शून्य या धनात्मक पूर्णांक होनी चाहिए।
For a constant polynomial, both (s-2=0) and (s+1=0) are required. One value of (s) cannot satisfy both conditions.
Step 2
Why this answer is correct
The correct answer is C. कोई ऐसा (s) नहीं है / No such (s) exists. For a constant polynomial, both (s-2=0) and (s+1=0) are required. One value of (s) cannot satisfy both conditions.
Step 3
Exam Tip
स्थिर बहुपद के लिए (s-2=0) और (s+1=0) दोनों चाहिए। एक ही (s) दोनों शर्तें पूरी नहीं कर सकता।
The degree will be (2) only if the coefficient of \(x^2\) is non-zero. Therefore, \(\lambda\neq0\) is required.
Step 2
Why this answer is correct
The correct answer is B. \(\lambda\neq0\). The degree will be (2) only if the coefficient of \(x^2\) is non-zero. Therefore, \(\lambda\neq0\) is required.
Step 3
Exam Tip
घात (2) तभी होगी जब \(x^2\) का गुणांक शून्य न हो। इसलिए \(\lambda\neq0\) चाहिए।
The coefficient of \(x^3\) is (2), which is never zero. So 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 (2), which is never zero. So the degree remains (3) for every (h).
Step 3
Exam Tip
\(x^3\) का गुणांक (2) है जो कभी शून्य नहीं होता। इसलिए घात (3) हर (h) के लिए रहेगी।
A. क्योंकि यह \(x^2-\frac{1}{x^2}\) बनता है/Because it becomes \(x^2-\frac{1}{x^2}\)
Step 1
Concept
The simplified form is \(x^2-\frac{1}{x^2}\). Because \(\frac{1}{x^2}=x^{-2}\), it is not a polynomial.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि यह \(x^2-\frac{1}{x^2}\) बनता है / Because it becomes \(x^2-\frac{1}{x^2}\). The simplified form is \(x^2-\frac{1}{x^2}\). Because \(\frac{1}{x^2}=x^{-2}\), it is not a polynomial.
Step 3
Exam Tip
सरल रूप \(x^2-\frac{1}{x^2}\) है। \(\frac{1}{x^2}=x^{-2}\) के कारण यह बहुपद 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।
A. \(x^n+1\), जहाँ (n) बदलता चर है/\(x^n+1\), where (n) is a changing variable
Step 1
Concept
In a polynomial, powers must be fixed non-negative integers. A power depending on a changing variable is not valid in the definition.
Step 2
Why this answer is correct
The correct answer is A. \(x^n+1\), जहाँ (n) बदलता चर है / \(x^n+1\), where (n) is a changing variable. In a polynomial, powers must be fixed non-negative integers. A power depending on a changing variable is not valid in the definition.
Step 3
Exam Tip
बहुपद में घातें निश्चित शून्य या धनात्मक पूर्णांक होनी चाहिए। बदलते चर वाली घात परिभाषा में मान्य नहीं है।
For degree (0), both (a-1=0) and (a+3=0) are required. These cannot be true together.
Step 2
Why this answer is correct
The correct answer is C. ऐसा कोई (a) नहीं है / No such (a) exists. For degree (0), both (a-1=0) and (a+3=0) are required. These cannot be true together.
Step 3
Exam Tip
घात (0) के लिए (a-1=0) और (a+3=0) दोनों चाहिए। ये दोनों साथ-साथ संभव नहीं हैं।
\(\sqrt{x^2+1}\) has the variable inside a root. In the class (9) definition, it is not treated as a polynomial term.
Step 2
Why this answer is correct
The correct answer is C. \(\sqrt{x^2+1}+x\). \(\sqrt{x^2+1}\) has the variable inside a root. In the class (9) definition, it is not treated as a polynomial term.
Step 3
Exam Tip
\(\sqrt{x^2+1}\) में चर मूल के अंदर है। कक्षा (9) की परिभाषा में यह बहुपद पद नहीं माना जाता।
\(\frac{x^3+x}{x}=x^2+1\), which is a polynomial. Check powers carefully before and after simplification.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{x^3+x}{x}\). \(\frac{x^3+x}{x}=x^2+1\), which is a polynomial. Check powers carefully before and after simplification.
Step 3
Exam Tip
\(\frac{x^3+x}{x}=x^2+1\) है, जो बहुपद है। सरल करने से पहले और बाद की घातों को ध्यान से देखें।
(p(x)=0) 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. घात परिभाषित नहीं है / Degree is not defined. (p(x)=0) is the zero polynomial. The degree of the zero polynomial is not defined.
Step 3
Exam Tip
(p(x)=0) शून्य बहुपद है। शून्य बहुपद की घात परिभाषित नहीं होती।
For exactly cubic, the coefficient of \(x^4\) must be zero and the coefficient of \(x^3\) must remain non-zero. From (a-2=0), (a=2).
Step 2
Why this answer is correct
The correct answer is A. (2). For exactly cubic, the coefficient of \(x^4\) must be zero and the coefficient of \(x^3\) must remain non-zero. From (a-2=0), (a=2).
Step 3
Exam Tip
ठीक घन के लिए \(x^4\) का गुणांक शून्य होना चाहिए और \(x^3\) का गुणांक अशून्य रहना चाहिए। (a-2=0) से (a=2) मिलता है।
