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-2 -7x+10) और (q(x)=p(x+1)) है, तो (q(x)) के शून्यक कौन-से हैं?
If (p(x)=x-2 -7x+10) and (q(x)=p(x+1)), what are the zeroes of (q(x))?
#polynomials
#transformed-zeroes
#expert
A (1,4)
B (2,5)
C (3,6)
D (0,3)
Explanation opens after your attempt
Step 1
Concept
The zeroes of (p(x)) are (2) and (5), so for (p(x+1)=0), (x+1=2) or (x+1=5). Hence the zeroes of (q(x)) are (1) and (4).
Step 2
Why this answer is correct
The correct answer is A. (1,4). The zeroes of (p(x)) are (2) and (5), so for (p(x+1)=0), (x+1=2) or (x+1=5). Hence the zeroes of (q(x)) are (1) and (4).
Step 3
Exam Tip
(p(x)) के शून्यक (2) और (5) हैं, इसलिए (p(x+1)=0) के लिए (x+1=2) या (x+1=5)। अतः (q(x)) के शून्यक (1) और (4) हैं।
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यदि (p(x)=2x-3 -5x-2 +mx+6) को (x+1) से भाग देने पर शेष (0) है, तो (m) क्या है?
If (p(x)=2x-3 -5x-2 +mx+6) leaves remainder (0) when divided by (x+1), what is (m)?
#remainder-theorem
#signs
#polynomials
A (-13)
B (13)
C (-9)
D (9)
Explanation opens after your attempt
Step 1
Concept
By remainder theorem (p(-1)=0), so (-2-5-m+6=0), giving (-1-m=0) and (m=-1). Always check signs carefully.
Step 2
Why this answer is correct
The correct answer is A. (-13). By remainder theorem (p(-1)=0), so (-2-5-m+6=0), giving (-1-m=0) and (m=-1). Always check signs carefully.
Step 3
Exam Tip
शेष प्रमेय से (p(-1)=0), इसलिए (-2-5-m+6=0) और (m=-1) नहीं, सही समीकरण (-1-m=0) देता है (m=-1)।
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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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यदि (p(x)=ax-2 +bx+c) में \(a\neq0\) है और (p(1)=p(-1)=0) है, तो (b) का मान क्या होगा?
If (p(x)=ax-2 +bx+c) with \(a\neq0\) and (p(1)=p(-1)=0), what is the value of (b)?
#polynomials
#one-variable
#values
A (0)
B (a)
C (c)
D (-a)
Explanation opens after your attempt
Step 1
Concept
(p(1)=a+b+c) and (p(-1)=a-b+c); subtracting gives (2b=0). In exams, use addition or subtraction for symmetric inputs.
Step 2
Why this answer is correct
The correct answer is A. (0). (p(1)=a+b+c) and (p(-1)=a-b+c); subtracting gives (2b=0). In exams, use addition or subtraction for symmetric inputs.
Step 3
Exam Tip
(p(1)=a+b+c) और (p(-1)=a-b+c) हैं, घटाने पर (2b=0) मिलता है। परीक्षा में सममित मानों पर जोड़-घटाव जल्दी करें।
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यदि द्विघात बहुपद \(x^2-6x+k\) के शून्यक \(\alpha\) और \(\beta\) हैं तथा \(\alpha^2+\beta^2=20\) है, तो (k) का मान क्या होगा?
If \(\alpha\) and \(\beta\) are zeroes of the quadratic polynomial \(x^2-6x+k\) and \(\alpha^2+\beta^2=20\), what is the value of (k)?
#polynomials
#zeroes
#identity
#hard
A (8)
B (6)
C (10)
D (16)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=6\) and (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta). So (20=36-2k), giving (k=8).
Step 2
Why this answer is correct
The correct answer is A. (8). Here \(\alpha+\beta=6\) and (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta). So (20=36-2k), giving (k=8).
Step 3
Exam Tip
\(\alpha+\beta=6\) और (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta) होता है। इसलिए (20=36-2k) से (k=8) मिलता है।
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यदि (p(x)=x-3 +mx-2 -4x-4) में (x+2) एक गुणनखंड है, तो (m) का मान क्या होगा?
If (x+2) is a factor of (p(x)=x-3 +mx-2 -4x-4), what is (m)?
#polynomials
#factor_parameter
#hard
A (1)
B (-1)
C (2)
D (-2)
Explanation opens after your attempt
Step 1
Concept
Putting (p(-2)=0) gives (-8+4m+8-4=0). Thus (4m-4=0), so (m=1).
Step 2
Why this answer is correct
The correct answer is A. (1). Putting (p(-2)=0) gives (-8+4m+8-4=0). Thus (4m-4=0), so (m=1).
Step 3
Exam Tip
(p(-2)=0) रखने पर (-8+4m+8-4=0) मिलता है। इससे (4m-4=0) और (m=1) है।
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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 (p(x)=4x-2 -12x+9), तो इसके शून्यकों के बारे में कौन सा कथन सही है?
If (p(x)=4x-2 -12x+9), which statement about its zeroes is correct?
#polynomials
#perfect_square
#zeroes
#hard
A दोनों शून्यक \(\frac{3}{2}\) हैं / Both zeroes are \(\frac{3}{2}\)
B दोनों शून्यक \(-\frac{3}{2}\) हैं / Both zeroes are \(-\frac{3}{2}\)
C शून्यक \(\frac{2}{3}\) और \(\frac{3}{2}\) हैं / Zeroes are \(\frac{2}{3}\) and \(\frac{3}{2}\)
D कोई शून्यक नहीं है / There is no zero
Explanation opens after your attempt
Correct Answer
A. दोनों शून्यक \(\frac{3}{2}\) हैं / Both zeroes are \(\frac{3}{2}\)
Step 1
Concept
(4x-2 -12x+9=(2x-3)2 ). Therefore, both zeroes are \(\frac{3}{2}\).
Step 2
Why this answer is correct
The correct answer is A. दोनों शून्यक \(\frac{3}{2}\) हैं / Both zeroes are \(\frac{3}{2}\). (4x-2 -12x+9=(2x-3)2 ). Therefore, both zeroes are \(\frac{3}{2}\).
Step 3
Exam Tip
(4x-2 -12x+9=(2x-3)2 ) है। इसलिए दोनों शून्यक \(\frac{3}{2}\) हैं।
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यदि \(x^2-11x+30\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो (\(\alpha-\beta\)2 ) क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-11x+30\), what is (\(\alpha-\beta\)2 )?
#polynomials
#zeroes
#difference_square
#hard
A (1)
B (121)
C (60)
D (49)
Explanation opens after your attempt
Step 1
Concept
(\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta). We get (121-120=1).
Step 2
Why this answer is correct
The correct answer is A. (1). (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta). We get (121-120=1).
Step 3
Exam Tip
(\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta) है। (121-120=1) मिलता है।
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यदि \(x^2+mx+n\) का एक शून्यक (0) है और दूसरा शून्यक (5) है, तो (m+n) क्या होगा?
If one zero of \(x^2+mx+n\) is (0) and the other zero is (5), what is (m+n)?
#polynomials
#zeroes
#constant_term
#hard
A (-5)
B (5)
C (0)
D (10)
Explanation opens after your attempt
Step 1
Concept
The sum is (5), so (m=-5), and the product is (0), so (n=0). Hence (m+n=-5).
Step 2
Why this answer is correct
The correct answer is A. (-5). The sum is (5), so (m=-5), and the product is (0), so (n=0). Hence (m+n=-5).
Step 3
Exam Tip
योग (5) है इसलिए (m=-5), और गुणनफल (0) है इसलिए (n=0)। अतः (m+n=-5)।
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यदि (p(x)=x-4 -5x-2 +4), तो कौन सा मान (p(x)) का शून्यक नहीं है?
If (p(x)=x-4 -5x-2 +4), which value is not a zero of (p(x))?
#polynomials
#quartic
#zero_check
#hard
A (1)
B (-1)
C (2)
D (3)
Explanation opens after your attempt
Step 1
Concept
(p(3)=81-45+4=40), so (3) is not a zero. The values (1), (-1), and (2) make the polynomial (0).
Step 2
Why this answer is correct
The correct answer is D. (3). (p(3)=81-45+4=40), so (3) is not a zero. The values (1), (-1), and (2) make the polynomial (0).
Step 3
Exam Tip
(p(3)=81-45+4=40) है, इसलिए (3) शून्यक नहीं है। शेष (1), (-1) और (2) पर मान (0) मिलता है।
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किस बहुपद के शून्यक (0), (2) और (-3) हैं?
Which polynomial has zeroes (0), (2), and (-3)?
#polynomials
#formation
#cubic
#hard
A \(x^3+x^2-6x\)
B \(x^3-x^2-6x\)
C \(x^3+x^2+6x\)
D \(x^3-5x^2+6x\)
Explanation opens after your attempt
Correct Answer
A. \(x^3+x^2-6x\)
Step 1
Concept
From the zeroes, the polynomial is (x(x-2)(x+3)). Expanding gives \(x^3+x^2-6x\).
Step 2
Why this answer is correct
The correct answer is A. \(x^3+x^2-6x\). From the zeroes, the polynomial is (x(x-2)(x+3)). Expanding gives \(x^3+x^2-6x\).
Step 3
Exam Tip
शून्यकों से बहुपद (x(x-2)(x+3)) बनेगा। इसे फैलाने पर \(x^3+x^2-6x\) मिलता है।
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यदि \(2x^2+5x-3\) को (\(x-\alpha\)\(x-\beta\)) के रूप में लिखा जाए, तो \(\alpha+\beta\) क्या होगा?
If \(2x^2+5x-3\) is considered with factors involving (\(x-\alpha\)\(x-\beta\)), what is \(\alpha+\beta\)?
#polynomials
#sum_zeroes
#hard
A \(-\frac{5}{2}\)
B \(\frac{5}{2}\)
C (-3)
D (3)
Explanation opens after your attempt
Correct Answer
A. \(-\frac{5}{2}\)
Step 1
Concept
For \(ax^2+bx+c\), the sum of zeroes is \(-\frac{b}{a}\). Hence \(\alpha+\beta=-\frac{5}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(-\frac{5}{2}\). For \(ax^2+bx+c\), the sum of zeroes is \(-\frac{b}{a}\). Hence \(\alpha+\beta=-\frac{5}{2}\).
Step 3
Exam Tip
द्विघात \(ax^2+bx+c\) के लिए शून्यकों का योग \(-\frac{b}{a}\) होता है। इसलिए \(\alpha+\beta=-\frac{5}{2}\)।
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यदि \(x^2-3x-10\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो \(\alpha^2\beta+\alpha\beta^2\) क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-3x-10\), what is \(\alpha^2\beta+\alpha\beta^2\)?
#polynomials
#zeroes
#symmetric_expression
#hard
A (-30)
B (30)
C (-13)
D (13)
Explanation opens after your attempt
Step 1
Concept
(\alpha-2 \beta+\alpha\beta-2 =\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-10\) and \(\alpha+\beta=3\), so the value is (-30).
Step 2
Why this answer is correct
The correct answer is A. (-30). (\alpha-2 \beta+\alpha\beta-2 =\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-10\) and \(\alpha+\beta=3\), so the value is (-30).
Step 3
Exam Tip
(\alpha-2 \beta+\alpha\beta-2 =\alpha\beta\(\alpha+\beta\)) होता है। यहां \(\alpha\beta=-10\) और \(\alpha+\beta=3\), इसलिए मान (-30) है।
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यदि (p(x)=x-3 +ax-2 +bx+8) के शून्यक (-1), (-2) और (-4) हैं, तो (a+b) क्या है?
If the zeroes of (p(x)=x-3 +ax-2 +bx+8) are (-1), (-2), and (-4), what is (a+b)?
#polynomials
#cubic_zeroes
#coefficients
#hard
A (21)
B (7)
C (14)
D (-7)
Explanation opens after your attempt
Step 1
Concept
The polynomial is ((x+1)(x+2)(x+4)=x-3 +7x-2 +14x+8). Hence (a+b=21).
Step 2
Why this answer is correct
The correct answer is A. (21). The polynomial is ((x+1)(x+2)(x+4)=x-3 +7x-2 +14x+8). Hence (a+b=21).
Step 3
Exam Tip
बहुपद ((x+1)(x+2)(x+4)=x-3 +7x-2 +14x+8) है। इसलिए (a+b=21)।
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यदि \(x^2-10x+q\) के शून्यक (2r) और (3r) हैं, तो (q) का मान क्या है?
If the zeroes of \(x^2-10x+q\) are (2r) and (3r), what is (q)?
#polynomials
#ratio_zeroes
#product
#hard
A (24)
B (20)
C (12)
D (30)
Explanation opens after your attempt
Step 1
Concept
From the sum (5r=10), (r=2). The product is \(2r\cdot3r=6r^2=24\).
Step 2
Why this answer is correct
The correct answer is A. (24). From the sum (5r=10), (r=2). The product is \(2r\cdot3r=6r^2=24\).
Step 3
Exam Tip
योग (5r=10) से (r=2) है। गुणनफल \(2r\cdot3r=6r^2=24\) है।
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द्विघात बहुपद \(kx^2+6x+4\) के शून्यकों का योग (-3) है। (k) का मान क्या है?
The sum of zeroes of the quadratic polynomial \(kx^2+6x+4\) is (-3). What is (k)?
#polynomials
#parameter
#sum_zeroes
#hard
A (2)
B (-2)
C (3)
D (-3)
Explanation opens after your attempt
Step 1
Concept
The sum is \(-\frac{6}{k}\), and it equals (-3). Therefore, (k=2).
Step 2
Why this answer is correct
The correct answer is A. (2). The sum is \(-\frac{6}{k}\), and it equals (-3). Therefore, (k=2).
Step 3
Exam Tip
योग \(-\frac{6}{k}\) है और यह (-3) है। इसलिए (k=2) होगा।
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यदि \(x^2-8x+k\) के शून्यक \(\alpha\) और \(\beta\) हैं तथा \(\alpha-\beta=2\), तो (k) क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-8x+k\) and \(\alpha-\beta=2\), what is (k)?
#polynomials
#zeroes
#difference
#hard
A (15)
B (16)
C (12)
D (10)
Explanation opens after your attempt
Step 1
Concept
From sum (8) and difference (2), the zeroes are (5) and (3). Their product is (15), so (k=15).
Step 2
Why this answer is correct
The correct answer is A. (15). From sum (8) and difference (2), the zeroes are (5) and (3). Their product is (15), so (k=15).
Step 3
Exam Tip
योग (8) और अंतर (2) से शून्यक (5) और (3) हैं। गुणनफल (15) है इसलिए (k=15)।
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किस (k) के लिए \(x^2+kx+9\) का एक शून्यक दूसरे का तीन गुना है और दोनों धनात्मक हैं?
For which (k) does \(x^2+kx+9\) have one zero three times the other and both positive?
#polynomials
#zeroes
#ratio
#hard
A \(-4\sqrt{3}\)
B \(4\sqrt{3}\)
C (-6)
D (6)
Explanation opens after your attempt
Correct Answer
A. \(-4\sqrt{3}\)
Step 1
Concept
Let the zeroes be (t) and (3t), so \(3t^2=9\) gives \(t=\sqrt{3}\). The sum is \(4\sqrt{3}\), hence \(k=-4\sqrt{3}\).
Step 2
Why this answer is correct
The correct answer is A. \(-4\sqrt{3}\). Let the zeroes be (t) and (3t), so \(3t^2=9\) gives \(t=\sqrt{3}\). The sum is \(4\sqrt{3}\), hence \(k=-4\sqrt{3}\).
Step 3
Exam Tip
शून्यक (t) और (3t) मानें, तो \(3t^2=9\) से \(t=\sqrt{3}\) है। योग \(4\sqrt{3}\) है इसलिए \(k=-4\sqrt{3}\)।
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यदि \(x^2-4x+1\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो \(\alpha^3+\beta^3\) का मान क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-4x+1\), what is \(\alpha^3+\beta^3\)?
#polynomials
#zeroes
#cube_identity
#hard
A (52)
B (64)
C (40)
D (28)
Explanation opens after your attempt
Step 1
Concept
\(\alpha+\beta=4\) and \(\alpha\beta=1\). (\alpha-3 +\beta-3 =\(\alpha+\beta\)3 -3\alpha\beta\(\alpha+\beta\)=64-12=52).
Step 2
Why this answer is correct
The correct answer is A. (52). \(\alpha+\beta=4\) and \(\alpha\beta=1\). (\alpha-3 +\beta-3 =\(\alpha+\beta\)3 -3\alpha\beta\(\alpha+\beta\)=64-12=52).
Step 3
Exam Tip
\(\alpha+\beta=4\) और \(\alpha\beta=1\) है। (\alpha-3 +\beta-3 =\(\alpha+\beta\)3 -3\alpha\beta\(\alpha+\beta\)=64-12=52)।
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यदि \(x^2+5x+6\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो नया बहुपद जिसके शून्यक \(\alpha+1\) और \(\beta+1\) हैं, क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2+5x+6\), what is the new polynomial whose zeroes are \(\alpha+1\) and \(\beta+1\)?
#polynomials
#transformed_zeroes
#hard
A \(x^2+3x+2\)
B \(x^2+5x+6\)
C \(x^2+7x+12\)
D \(x^2-3x+2\)
Explanation opens after your attempt
Correct Answer
A. \(x^2+3x+2\)
Step 1
Concept
The original zeroes are (-2) and (-3), so the new zeroes are (-1) and (-2). The new polynomial is \(x^2+3x+2\).
Step 2
Why this answer is correct
The correct answer is A. \(x^2+3x+2\). The original zeroes are (-2) and (-3), so the new zeroes are (-1) and (-2). The new polynomial is \(x^2+3x+2\).
Step 3
Exam Tip
मूल शून्यक (-2) और (-3) हैं, इसलिए नए शून्यक (-1) और (-2) हैं। नया बहुपद \(x^2+3x+2\) है।
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यदि (p(x)=x-3 -2x-2 -5x+6), तो निम्न में से कौन सा शून्यक नहीं है?
If (p(x)=x-3 -2x-2 -5x+6), which of the following is not a zero?
#polynomials
#not_zero
#hard
A (1)
B (2)
C (3)
D (-2)
Explanation opens after your attempt
Step 1
Concept
(p(2)=8-8-10+6=-4), so (2) is not a zero. The other options can be checked by substitution.
Step 2
Why this answer is correct
The correct answer is B. (2). (p(2)=8-8-10+6=-4), so (2) is not a zero. The other options can be checked by substitution.
Step 3
Exam Tip
(p(2)=8-8-10+6=-4) है इसलिए (2) शून्यक नहीं है। बाकी विकल्पों को भी प्रतिस्थापन से जांच सकते हैं।
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बहुपद \(2x^3-9x^2+13x-6\) का एक गुणनखंड कौन सा है?
Which is a factor of \(2x^3-9x^2+13x-6\)?
#polynomials
#factor_check
#cubic
#hard
A (x-1)
B (x+1)
C (x-4)
D (x+3)
Explanation opens after your attempt
Step 1
Concept
(p(1)=2-9+13-6=0), so (x-1) is a factor. In option checking, try small values first.
Step 2
Why this answer is correct
The correct answer is A. (x-1). (p(1)=2-9+13-6=0), so (x-1) is a factor. In option checking, try small values first.
Step 3
Exam Tip
(p(1)=2-9+13-6=0) है इसलिए (x-1) गुणनखंड है। विकल्प जांच में छोटे मान पहले लगाएं।
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यदि (p(x)=x-2 -4x+k) और (p(1)=p(3)), तो (k) के बारे में क्या कहा जा सकता है?
If (p(x)=x-2 -4x+k) and (p(1)=p(3)), what can be said about (k)?
#polynomials
#evaluation
#parameter
#hard
A (k=1)
B (k=3)
C कोई भी वास्तविक मान / Any real value
D कोई मान संभव नहीं / No value possible
Explanation opens after your attempt
Correct Answer
C. कोई भी वास्तविक मान / Any real value
Step 1
Concept
(p(1)=k-3) and (p(3)=k-3), so they are equal. Hence (k) can be any real value.
Step 2
Why this answer is correct
The correct answer is C. कोई भी वास्तविक मान / Any real value. (p(1)=k-3) and (p(3)=k-3), so they are equal. Hence (k) can be any real value.
Step 3
Exam Tip
(p(1)=k-3) और (p(3)=k-3) दोनों बराबर हैं। इसलिए (k) कोई भी वास्तविक मान हो सकता है।
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यदि \(x^2+px+16\) के शून्यक परस्पर बराबर और ऋणात्मक हैं, तो (p) का मान क्या है?
If the zeroes of \(x^2+px+16\) are equal and negative, what is (p)?
#polynomials
#equal_negative_zeroes
#hard
A (8)
B (-8)
C (4)
D (-4)
Explanation opens after your attempt
Step 1
Concept
The equal negative zeroes are (-4) and (-4) because the product is (16). The sum is (-8), so (p=8).
Step 2
Why this answer is correct
The correct answer is A. (8). The equal negative zeroes are (-4) and (-4) because the product is (16). The sum is (-8), so (p=8).
Step 3
Exam Tip
बराबर ऋणात्मक शून्यक (-4) और (-4) होंगे क्योंकि गुणनफल (16) है। योग (-8) है इसलिए (p=8)।
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यदि \(3x^2+2x-1\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो (\(\alpha+\beta\)2 ) का मान क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(3x^2+2x-1\), what is (\(\alpha+\beta\)2 )?
#polynomials
#sum_zeroes
#square
#hard
A \(\frac{4}{9}\)
B \(-\frac{4}{9}\)
C \(\frac{1}{9}\)
D \(\frac{2}{3}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{4}{9}\)
Step 1
Concept
\(\alpha+\beta=-\frac{2}{3}\). Therefore, (\(\alpha+\beta\)2 =\frac{4}{9}).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{4}{9}\). \(\alpha+\beta=-\frac{2}{3}\). Therefore, (\(\alpha+\beta\)2 =\frac{4}{9}).
Step 3
Exam Tip
\(\alpha+\beta=-\frac{2}{3}\) है। इसलिए (\(\alpha+\beta\)2 =\frac{4}{9}) होगा।
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यदि \(x^2-7x+10\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो \(\alpha^2+\beta^2\) क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(x^2-7x+10\), what is \(\alpha^2+\beta^2\)?
#polynomials
#zeroes
#identity
#hard
A (29)
B (49)
C (20)
D (39)
Explanation opens after your attempt
Step 1
Concept
(\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta). Thus (72 -2(10)=29).
Step 2
Why this answer is correct
The correct answer is A. (29). (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta). Thus (72 -2(10)=29).
Step 3
Exam Tip
(\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta) होता है। (72 -2(10)=29) है।
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यदि \(2x^2-3x-5\) के शून्यक \(\alpha\) और \(\beta\) हैं, तो \(\frac{1}{\alpha}+\frac{1}{\beta}\) क्या होगा?
If \(\alpha\) and \(\beta\) are zeroes of \(2x^2-3x-5\), what is \(\frac{1}{\alpha}+\frac{1}{\beta}\)?
#polynomials
#zeroes
#reciprocal
#hard
A \(-\frac{3}{5}\)
B \(\frac{3}{5}\)
C \(-\frac{5}{3}\)
D \(\frac{5}{3}\)
Explanation opens after your attempt
Correct Answer
A. \(-\frac{3}{5}\)
Step 1
Concept
\(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}\). Here \(\frac{3}{2}\div-\frac{5}{2}=-\frac{3}{5}\).
Step 2
Why this answer is correct
The correct answer is A. \(-\frac{3}{5}\). \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}\). Here \(\frac{3}{2}\div-\frac{5}{2}=-\frac{3}{5}\).
Step 3
Exam Tip
\(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}\) है। यहां \(\frac{3}{2}\div-\frac{5}{2}=-\frac{3}{5}\)।
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यदि द्विघात बहुपद के शून्यक \(\alpha\) और \(\beta\) हैं तथा \(\alpha+\beta=5\), \(\alpha\beta=6\), तो मोनिक बहुपद क्या है?
If a quadratic polynomial has zeroes \(\alpha\) and \(\beta\), with \(\alpha+\beta=5\) and \(\alpha\beta=6\), what is the monic polynomial?
#polynomials
#monic
#formation
#hard
A \(x^2-5x+6\)
B \(x^2+5x+6\)
C \(x^2-6x+5\)
D \(x^2+6x+5\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-5x+6\)
Step 1
Concept
The monic polynomial is (x-2 -\(\alpha+\beta\)x+\alpha\beta). Hence \(x^2-5x+6\) is correct.
Step 2
Why this answer is correct
The correct answer is A. \(x^2-5x+6\). The monic polynomial is (x-2 -\(\alpha+\beta\)x+\alpha\beta). Hence \(x^2-5x+6\) is correct.
Step 3
Exam Tip
मोनिक बहुपद (x-2 -\(\alpha+\beta\)x+\alpha\beta) होता है। इसलिए \(x^2-5x+6\) सही है।
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