यदि (p(x)=x-3 -5x-2 +ax+12) में (x-3) गुणनखंड है, तो (a) का मान क्या है?
If (x-3) is a factor of (p(x)=x-3 -5x-2 +ax+12), what is the value of (a)?
#factor-theorem
#polynomials
#parameter
A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
By the factor theorem (p(3)=0), so (27-45+3a+12=0) and (a=2). In exams, substitute the zero from the given factor directly.
Step 2
Why this answer is correct
The correct answer is A. (2). By the factor theorem (p(3)=0), so (27-45+3a+12=0) and (a=2). In exams, substitute the zero from the given factor directly.
Step 3
Exam Tip
गुणनखंड प्रमेय से (p(3)=0), इसलिए (27-45+3a+12=0) और (a=2)। परीक्षा में दिए गए गुणनखंड से मूल तुरंत रखिए।
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यदि (p(x)=x-3 +3x-2 -4x-12), तो निम्न में से कौन-सा गुणनखंड है?
If (p(x)=x-3 +3x-2 -4x-12), which of the following is a factor?
#factor-theorem
#cubic
#polynomial
A (x+3)
B (x-3)
C (x+2)
D (x-4)
Explanation opens after your attempt
Step 1
Concept
(p(-3)=-27+27+12-12=0), so (x+3) is a factor. For (x+a), test (x=-a).
Step 2
Why this answer is correct
The correct answer is A. (x+3). (p(-3)=-27+27+12-12=0), so (x+3) is a factor. For (x+a), test (x=-a).
Step 3
Exam Tip
(p(-3)=-27+27+12-12=0), इसलिए (x+3) गुणनखंड है। (x+a) के लिए (x=-a) रखकर जाँचें।
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यदि (x-2) बहुपद (p(x)=x-3 +kx-2 -4x-4) का गुणनखंड है, तो (k) का मान क्या है?
If (x-2) is a factor of (p(x)=x-3 +kx-2 -4x-4), what is the value of (k)?
#factor-theorem
#polynomials
#expert
A (1)
B (2)
C (-1)
D (-2)
Explanation opens after your attempt
Step 1
Concept
By factor theorem (p(2)=0), so (8+4k-8-4=0) and (k=1). In exams, substitute the given zero directly.
Step 2
Why this answer is correct
The correct answer is A. (1). By factor theorem (p(2)=0), so (8+4k-8-4=0) and (k=1). In exams, substitute the given zero directly.
Step 3
Exam Tip
गुणनखंड प्रमेय से (p(2)=0), इसलिए (8+4k-8-4=0) और (k=1)। परीक्षा में पहले दिए गए मूल को सीधे रखिए।
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\(x^3-7x+6\) के लिए कौन सा कथन सही है?
Which statement is correct for \(x^3-7x+6\)?
#polynomials
#factor_theorem
#two_factors
#hard
A (x-1) और (x-2) दोनों गुणनखंड हैं / Both (x-1) and (x-2) are factors
B केवल (x-1) गुणनखंड है / Only (x-1) is a factor
C केवल (x-2) गुणनखंड है / Only (x-2) is a factor
D दोनों गुणनखंड नहीं हैं / Neither is a factor
Explanation opens after your attempt
Correct Answer
A. (x-1) और (x-2) दोनों गुणनखंड हैं / Both (x-1) and (x-2) are factors
Step 1
Concept
Both (p(1)=0) and (p(2)=0). Hence (x-1) and (x-2) are both factors.
Step 2
Why this answer is correct
The correct answer is A. (x-1) और (x-2) दोनों गुणनखंड हैं / Both (x-1) and (x-2) are factors. Both (p(1)=0) and (p(2)=0). Hence (x-1) and (x-2) are both factors.
Step 3
Exam Tip
(p(1)=0) और (p(2)=0) दोनों हैं। इसलिए (x-1) और (x-2) दोनों गुणनखंड हैं।
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यदि (x+1) बहुपद \(2x^3+kx^2-5x+2\) का गुणनखंड है, तो (k) का सही मान क्या है?
If (x+1) is a factor of \(2x^3+kx^2-5x+2\), what is the correct value of (k)?
#polynomials
#factor_theorem
#parameter
#hard
A (-5)
B (5)
C (-1)
D (1)
Explanation opens after your attempt
Step 1
Concept
Putting (p(-1)=0) gives (-2+k+5+2=0). Therefore, (k=-5) is correct.
Step 2
Why this answer is correct
The correct answer is A. (-5). Putting (p(-1)=0) gives (-2+k+5+2=0). Therefore, (k=-5) is correct.
Step 3
Exam Tip
(p(-1)=0) रखने पर (-2+k+5+2=0) मिलता है। इसलिए (k=-5) सही है।
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यदि (x+1) बहुपद \(2x^3+kx^2-5x+2\) का गुणनखंड है, तो (k) का मान क्या है?
If (x+1) is a factor of \(2x^3+kx^2-5x+2\), what is (k)?
#polynomials
#factor_theorem
#parameter
#hard
A (5)
B (-5)
C (1)
D (-1)
Explanation opens after your attempt
Step 1
Concept
If (x+1) is a factor, then (p(-1)=0). From (-2+k+5+2=0), we get (k=-5).
Step 2
Why this answer is correct
The correct answer is A. (5). If (x+1) is a factor, then (p(-1)=0). From (-2+k+5+2=0), we get (k=-5).
Step 3
Exam Tip
(x+1) गुणनखंड होने पर (p(-1)=0) होगा। (-2+k+5+2=0) से (k=-5) नहीं बल्कि (k=-5) मिलता है।
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यदि (p(x)=x-3 -3x-2 -4x+12) है, तो कौन सा (p(x)) का गुणनखंड है?
If (p(x)=x-3 -3x-2 -4x+12), which is a factor of (p(x))?
#polynomials
#factor_theorem
#hard
A (x-2)
B (x+2)
C (x-3)
D (x+3)
Explanation opens after your attempt
Step 1
Concept
Since (p(2)=8-12-8+12=0), (x-2) is a factor. Use the factor theorem.
Step 2
Why this answer is correct
The correct answer is A. (x-2). Since (p(2)=8-12-8+12=0), (x-2) is a factor. Use the factor theorem.
Step 3
Exam Tip
(p(2)=8-12-8+12=0) है इसलिए (x-2) गुणनखंड है। गुणनखंड प्रमेय का प्रयोग करें।
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यदि (x-9) किसी द्विघात बहुपद का गुणनखंड है तो कौन सा मूल निश्चित होगा?
If (x-9) is a factor of a quadratic polynomial, which root is certain?
#roots
#factor_theorem
#concept
A (9)
B (-9)
C (0)
D \(\frac{1}{9}\)
Explanation opens after your attempt
Step 1
Concept
Solving (x-9=0) gives (x=9). Therefore the certain root is (9).
Step 2
Why this answer is correct
The correct answer is A. (9). Solving (x-9=0) gives (x=9). Therefore the certain root is (9).
Step 3
Exam Tip
(x-9=0) करने पर (x=9) मिलता है। इसलिए निश्चित मूल (9) है।
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यदि (x+5) किसी द्विघात बहुपद का गुणनखंड है तो कौन सा मूल निश्चित होगा?
If (x+5) is a factor of a quadratic polynomial, which root is certain?
#roots
#factor_theorem
#concept
A (5)
B (-5)
C (0)
D \(\frac{1}{5}\)
Explanation opens after your attempt
Step 1
Concept
Solving (x+5=0) gives (x=-5). Therefore the certain root is (-5).
Step 2
Why this answer is correct
The correct answer is B. (-5). Solving (x+5=0) gives (x=-5). Therefore the certain root is (-5).
Step 3
Exam Tip
(x+5=0) करने पर (x=-5) मिलता है। इसलिए निश्चित मूल (-5) है।
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यदि (x-a) किसी द्विघात बहुपद का गुणनखंड है तो (a) क्या होगा?
If (x-a) is a factor of a quadratic polynomial then what is (a)?
#roots
#factor_theorem
#concept
A गुणांक / Coefficient
B मूल / Root
C घात / Degree
D अचर पद / Constant term
Explanation opens after your attempt
Correct Answer
B. मूल / Root
Step 1
Concept
If (x-a) is a factor then substituting (x=a) makes the value (0). So (a) is a root.
Step 2
Why this answer is correct
The correct answer is B. मूल / Root. If (x-a) is a factor then substituting (x=a) makes the value (0). So (a) is a root.
Step 3
Exam Tip
गुणनखंड (x-a) होने पर (x=a) रखने से मान (0) होता है। इसलिए (a) मूल है।
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यदि \(2+\sqrt{3}\) किसी परिमेय गुणांक वाले बहुपद का शून्यक है, तो किस रैखिक गुणनखंड का साथ आना अपेक्षित है?
If \(2+\sqrt{3}\) is a zero of a polynomial with rational coefficients, which linear factor is expected to accompany it?
#factor-theorem
#conjugate-zeroes
#polynomials
A (x-\(2-\sqrt{3}\))
B (x-\(2+\sqrt{3}\)) ही केवल / (x-\(2+\sqrt{3}\)) only
C (x-\(\sqrt{3}-2\))
D (x+\(2+\sqrt{3}\))
Explanation opens after your attempt
Correct Answer
A. (x-\(2-\sqrt{3}\))
Step 1
Concept
The companion zero is \(2-\sqrt{3}\), so the factor is (x-\(2-\sqrt{3}\)). In exams remember the relation between a zero and factor as \(x-\alpha\).
Step 2
Why this answer is correct
The correct answer is A. (x-\(2-\sqrt{3}\)). The companion zero is \(2-\sqrt{3}\), so the factor is (x-\(2-\sqrt{3}\)). In exams remember the relation between a zero and factor as \(x-\alpha\).
Step 3
Exam Tip
साथी शून्यक \(2-\sqrt{3}\) होगा, इसलिए गुणनखंड (x-\(2-\sqrt{3}\)) है। परीक्षा में शून्यक और गुणनखंड का संबंध \(x-\alpha\) याद रखें।
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यदि (p(x)=x(x-5)), तो ग्राफ किन (x)-मानों पर (x)-अक्ष को काटेगा?
If (p(x)=x(x-5)), at which (x)-values will the graph cut the (x)-axis?
#factor theorem
#zeroes
#graph
A (0) और (5) / (0) and (5)
B (1) और (5) / (1) and (5)
C (0) और (-5) / (0) and (-5)
D केवल (5) / Only (5)
Explanation opens after your attempt
Correct Answer
A. (0) और (5) / (0) and (5)
Step 1
Concept
From (x=0) or (x-5=0), we get (x=0,5). Tip: if a product is zero, at least one factor is zero.
Step 2
Why this answer is correct
The correct answer is A. (0) और (5) / (0) and (5). From (x=0) or (x-5=0), we get (x=0,5). Tip: if a product is zero, at least one factor is zero.
Step 3
Exam Tip
(x=0) या (x-5=0) से (x=0,5) मिलते हैं। टिप: गुणनफल शून्य हो तो कम से कम एक कारक शून्य होता है।
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