A. क्योंकि इसमें दो चर हैं/Because it has two variables
Step 1
Concept
It contains both (x) and (y). A polynomial in one variable should contain only one variable.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि इसमें दो चर हैं / Because it has two variables. It contains both (x) and (y). A polynomial in one variable should contain only one variable.
Step 3
Exam Tip
इसमें (x) और (y) दोनों चर हैं। एक चर वाले बहुपद में केवल एक ही चर होना चाहिए।
Since \(\sqrt{x}=x^{\frac{1}{2}}\), the variable has a fractional power, so it is not a polynomial. In exams, powers must be whole numbers.
Step 2
Why this answer is correct
The correct answer is C. \(\sqrt{x}+5\). Since \(\sqrt{x}=x^{\frac{1}{2}}\), the variable has a fractional power, so it is not a polynomial. In exams, powers must be whole numbers.
Step 3
Exam Tip
\(\sqrt{x}=x^{\frac{1}{2}}\) में चर की घात भिन्न है इसलिए यह बहुपद नहीं है। परीक्षा में घात पूर्ण संख्या होनी चाहिए।
The degree of the zero polynomial is undefined. In exams, 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 undefined. The degree of the zero polynomial is undefined. In exams, keep it separate from a non-zero constant polynomial.
Step 3
Exam Tip
शून्य बहुपद की घात परिभाषित नहीं होती। परीक्षा में इसे स्थिर शून्य से भिन्न बहुपद से अलग रखें।
The term \(x^{-1}\) has a negative power of the variable, so it is not a polynomial. In exams, powers should be like \(0,1,2,\ldots\).
Step 2
Why this answer is correct
The correct answer is C. \(x^{-1}+4\). The term \(x^{-1}\) has a negative power of the variable, so it is not a polynomial. In exams, powers should be like \(0,1,2,\ldots\).
Step 3
Exam Tip
\(x^{-1}\) में चर की घात ऋणात्मक है इसलिए यह बहुपद नहीं है। परीक्षा में घातें \(0,1,2,\ldots\) जैसी होनी चाहिए।
B. संख्या और उसके (3) अधिक मान का गुणनफल (88) है/A number and (3) more than it have product (88)
Step 1
Concept
Option (B) forms (x(x+3)=88), which is quadratic. When a variable is multiplied by a variable expression, an \(x^2\) term appears.
Step 2
Why this answer is correct
The correct answer is B. संख्या और उसके (3) अधिक मान का गुणनफल (88) है / A number and (3) more than it have product (88). Option (B) forms (x(x+3)=88), which is quadratic. When a variable is multiplied by a variable expression, an \(x^2\) term appears.
Step 3
Exam Tip
विकल्प (B) में (x(x+3)=88) बनता है, जो द्विघात है। गुणनफल में चर के साथ चर हो तो \(x^2\) पद आता है।
When \(b^2=4ac\), \(D=b^2-4ac=0\), so the roots are equal and real. In exams, treat this as the discriminant zero condition.
Step 2
Why this answer is correct
The correct answer is A. दो बराबर वास्तविक मूल / Two equal real roots. When \(b^2=4ac\), \(D=b^2-4ac=0\), so the roots are equal and real. In exams, treat this as the discriminant zero condition.
Step 3
Exam Tip
\(b^2=4ac\) होने पर \(D=b^2-4ac=0\), इसलिए मूल बराबर वास्तविक होते हैं। परीक्षा में इसे discriminant zero condition मानें।
(D=-5) is negative, so there will be no real roots. In exams, (D<0) may also be linked with complex roots.
Step 2
Why this answer is correct
The correct answer is C. वास्तविक मूल नहीं / No real roots. (D=-5) is negative, so there will be no real roots. In exams, (D<0) may also be linked with complex roots.
Step 3
Exam Tip
(D=-5) ऋणात्मक है, इसलिए वास्तविक मूल नहीं होंगे। परीक्षा में (D<0) को complex roots से भी जोड़ा जा सकता है।
When (D=0), both roots are equal and real. In exams, these may also be called repeated roots.
Step 2
Why this answer is correct
The correct answer is B. दो बराबर वास्तविक मूल / Two equal real roots. When (D=0), both roots are equal and real. In exams, these may also be called repeated roots.
Step 3
Exam Tip
जब (D=0) होता है, तब दोनों मूल समान और वास्तविक होते हैं। परीक्षा में इसे repeated roots भी कहा जा सकता है।
For two distinct real roots, the discriminant \(D=b^2-4ac\) is positive. In exams, first calculate (D) to decide the nature.
Step 2
Why this answer is correct
The correct answer is A. \(b^2-4ac>0\). For two distinct real roots, the discriminant \(D=b^2-4ac\) is positive. In exams, first calculate (D) to decide the nature.
Step 3
Exam Tip
दो भिन्न वास्तविक मूलों के लिए विविक्तकर \(D=b^2-4ac\) धनात्मक होता है। परीक्षा में प्रकृति तय करने के लिए पहले (D) निकालें।
When (D=0), both roots are equal. This is a main rule to remember for nature of roots.
Step 2
Why this answer is correct
The correct answer is A. दोनों मूल बराबर हैं / Both roots are equal. When (D=0), both roots are equal. This is a main rule to remember for nature of roots.
Step 3
Exam Tip
(D=0) होने पर दोनों मूल बराबर होते हैं। यह प्रकृति याद करने का मुख्य नियम है।
A. मूल वास्तविक और भिन्न हैं/Roots are real and distinct
Step 1
Concept
Since (4>0), we have (D>0). Therefore the roots are real and distinct.
Step 2
Why this answer is correct
The correct answer is A. मूल वास्तविक और भिन्न हैं / Roots are real and distinct. Since (4>0), we have (D>0). Therefore the roots are real and distinct.
Step 3
Exam Tip
(4>0) है इसलिए (D>0) होगा। अतः मूल वास्तविक और भिन्न होंगे।
A. वे एक दूसरे के व्युत्क्रम हैं/They are reciprocals of each other
Step 1
Concept
\(6\cdot\frac{1}{6}=1\), so the roots are reciprocals. Reciprocal roots have product (1).
Step 2
Why this answer is correct
The correct answer is A. वे एक दूसरे के व्युत्क्रम हैं / They are reciprocals of each other. \(6\cdot\frac{1}{6}=1\), so the roots are reciprocals. Reciprocal roots have product (1).
Step 3
Exam Tip
\(6\cdot\frac{1}{6}=1\) है इसलिए दोनों व्युत्क्रम मूल हैं। व्युत्क्रम मूलों का गुणनफल (1) होता है।
A. वे एक दूसरे के व्युत्क्रम हैं/They are reciprocals of each other
Step 1
Concept
\(5\cdot\frac{1}{5}=1\), so the roots are reciprocals. Reciprocal roots have product (1).
Step 2
Why this answer is correct
The correct answer is A. वे एक दूसरे के व्युत्क्रम हैं / They are reciprocals of each other. \(5\cdot\frac{1}{5}=1\), so the roots are reciprocals. Reciprocal roots have product (1).
Step 3
Exam Tip
\(5\cdot\frac{1}{5}=1\) है इसलिए दोनों व्युत्क्रम मूल हैं। व्युत्क्रम मूलों का गुणनफल (1) होता है।
A. वे एक दूसरे के व्युत्क्रम हैं/They are reciprocals of each other
Step 1
Concept
\(4\cdot\frac{1}{4}=1\), so the roots are reciprocals. Reciprocal roots have product (1).
Step 2
Why this answer is correct
The correct answer is A. वे एक दूसरे के व्युत्क्रम हैं / They are reciprocals of each other. \(4\cdot\frac{1}{4}=1\), so the roots are reciprocals. Reciprocal roots have product (1).
Step 3
Exam Tip
\(4\cdot\frac{1}{4}=1\) है इसलिए दोनों व्युत्क्रम मूल हैं। व्युत्क्रम मूलों का गुणनफल (1) होता है।
