यदि (p(x)=x-2 +px+q) और (p(1)=0,\ p(2)=0), तो (p+q) क्या है?
If (p(x)=x-2 +px+q) and (p(1)=0,\ p(2)=0), what is (p+q)?
#coefficient-finding
#zeroes
#quadratic
A -(1)
B (1)
C -(2)
D (2)
Explanation opens after your attempt
Step 1
Concept
The zeroes are (1) and (2), so the polynomial is \(x^2-3x+2\). Thus (p=-3,\ q=2), and (p+q=-1).
Step 2
Why this answer is correct
The correct answer is A. -(1). The zeroes are (1) and (2), so the polynomial is \(x^2-3x+2\). Thus (p=-3,\ q=2), and (p+q=-1).
Step 3
Exam Tip
शून्यक (1) और (2) हैं, इसलिए बहुपद \(x^2-3x+2\) है। अतः (p=-3,\ q=2) और (p+q=-1)।
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यदि (p(x)=x-2 -4x+3) है, तो कौन-सा मान (p(x)<0) बनाता है?
If (p(x)=x-2 -4x+3), which value makes (p(x)<0)?
#sign-of-polynomial
#quadratic
#zeroes
A (2)
B (0)
C (4)
D -(1)
Explanation opens after your attempt
Step 1
Concept
(p(x)=(x-1)(x-3)), and it is negative for (1<x<3). Therefore (x=2) is correct.
Step 2
Why this answer is correct
The correct answer is A. (2). (p(x)=(x-1)(x-3)), and it is negative for (1<x<3). Therefore (x=2) is correct.
Step 3
Exam Tip
(p(x)=(x-1)(x-3)) है और (1<x<3) में मान ऋणात्मक होता है। इसलिए (x=2) सही है।
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यदि (p(x)=x-2 -12x+35), तो \(\frac{1}{\alpha-1}+\frac{1}{\beta-1}\) क्या है, जहाँ \(\alpha,\beta\) शून्यक हैं?
If (p(x)=x-2 -12x+35), what is \(\frac{1}{\alpha-1}+\frac{1}{\beta-1}\), where \(\alpha,\beta\) are zeroes?
#zeroes
#reciprocal-expression
#expert
A \(\frac{5}{12}\)
B \(\frac{6}{11}\)
C \(\frac{12}{5}\)
D \(\frac{11}{24}\)
Explanation opens after your attempt
Correct Answer
D. \(\frac{11}{24}\)
Step 1
Concept
\(\alpha+\beta=12\) and \(\alpha\beta=35\). \(\frac{1}{\alpha-1}+\frac{1}{\beta-1}=\frac{\alpha+\beta-2}{\alpha\beta-\alpha-\beta+1}=\frac{10}{24}=\frac{5}{12}\).
Step 2
Why this answer is correct
The correct answer is D. \(\frac{11}{24}\). \(\alpha+\beta=12\) and \(\alpha\beta=35\). \(\frac{1}{\alpha-1}+\frac{1}{\beta-1}=\frac{\alpha+\beta-2}{\alpha\beta-\alpha-\beta+1}=\frac{10}{24}=\frac{5}{12}\).
Step 3
Exam Tip
\(\alpha+\beta=12\) और \(\alpha\beta=35\) हैं। \(\frac{1}{\alpha-1}+\frac{1}{\beta-1}=\frac{\alpha+\beta-2}{\alpha\beta-\alpha-\beta+1}=\frac{10}{24}=\frac{5}{12}\)।
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यदि (p(x)=2x-2 -7x+5), तो शून्यकों के योग और गुणनफल का अंतर क्या है?
If (p(x)=2x-2 -7x+5), what is the difference between the sum and product of its zeroes?
#sum-product
#quadratic
#zeroes
A (1)
B (2)
C \(\frac{1}{2}\)
D (3)
Explanation opens after your attempt
Step 1
Concept
The sum is \(\frac{7}{2}\) and the product is \(\frac{5}{2}\). Their difference is \(\frac{7}{2}-\frac{5}{2}=1\).
Step 2
Why this answer is correct
The correct answer is A. (1). The sum is \(\frac{7}{2}\) and the product is \(\frac{5}{2}\). Their difference is \(\frac{7}{2}-\frac{5}{2}=1\).
Step 3
Exam Tip
योग \(\frac{7}{2}\) और गुणनफल \(\frac{5}{2}\) है। उनका अंतर \(\frac{7}{2}-\frac{5}{2}=1\) है।
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यदि (p(x)=x-2 -5x+6), तो (p(2)+p(3)) का मान क्या है?
If (p(x)=x-2 -5x+6), what is the value of (p(2)+p(3))?
#evaluation
#zeroes
#quadratic
A (0)
B (1)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
(2) and (3) are zeroes of (p(x)), so (p(2)=0) and (p(3)=0). Hence the sum is (0).
Step 2
Why this answer is correct
The correct answer is A. (0). (2) and (3) are zeroes of (p(x)), so (p(2)=0) and (p(3)=0). Hence the sum is (0).
Step 3
Exam Tip
(2) और (3), (p(x)) के शून्यक हैं, इसलिए (p(2)=0) और (p(3)=0)। इसलिए योग (0) है।
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यदि (p(x)=x-4 -5x-2 +4), तो इसके वास्तविक शून्यक कौन-से हैं?
If (p(x)=x-4 -5x-2 +4), what are its real zeroes?
#quartic
#biquadratic
#zeroes
A -(2,-1,1,2)
B -(4,-1,1,4)
C -(2,0,1,2)
D -(1,1)
Explanation opens after your attempt
Correct Answer
A. -(2,-1,1,2)
Step 1
Concept
(x-4 -5x-2 +4=\(x^2-1\)\(x^2-4\)). Therefore the real zeroes are (-2,-1,1,2).
Step 2
Why this answer is correct
The correct answer is A. -(2,-1,1,2). (x-4 -5x-2 +4=\(x^2-1\)\(x^2-4\)). Therefore the real zeroes are (-2,-1,1,2).
Step 3
Exam Tip
(x-4 -5x-2 +4=\(x^2-1\)\(x^2-4\)) है। इसलिए वास्तविक शून्यक (-2,-1,1,2) हैं।
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यदि (p(x)=x-2 -3x-28), तो (p(x)) के शून्यकों के मध्य दूरी क्या है?
If (p(x)=x-2 -3x-28), what is the distance between its zeroes?
#zeroes
#distance
#factorisation
A (11)
B (7)
C (4)
D (3)
Explanation opens after your attempt
Step 1
Concept
(p(x)=(x-7)(x+4)), so the zeroes are (7) and (-4). The distance is (|7-(-4)|=11).
Step 2
Why this answer is correct
The correct answer is A. (11). (p(x)=(x-7)(x+4)), so the zeroes are (7) and (-4). The distance is (|7-(-4)|=11).
Step 3
Exam Tip
(p(x)=(x-7)(x+4)), इसलिए शून्यक (7) और (-4) हैं। दूरी (|7-(-4)|=11) है।
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यदि \(\alpha,\beta\), \(x^2-9x+20\) के शून्यक हैं, तो (\(\alpha+2\)\(\beta+2\)) का मान क्या है?
If \(\alpha,\beta\) are zeroes of \(x^2-9x+20\), what is the value of (\(\alpha+2\)\(\beta+2\))?
#zeroes
#algebraic-expression
#quadratic
A (42)
B (38)
C (31)
D (24)
Explanation opens after your attempt
Step 1
Concept
\(\alpha+\beta=9\) and \(\alpha\beta=20\). (\(\alpha+2\)\(\beta+2\)=\alpha\beta+2\(\alpha+\beta\)+4=20+18+4=42).
Step 2
Why this answer is correct
The correct answer is A. (42). \(\alpha+\beta=9\) and \(\alpha\beta=20\). (\(\alpha+2\)\(\beta+2\)=\alpha\beta+2\(\alpha+\beta\)+4=20+18+4=42).
Step 3
Exam Tip
\(\alpha+\beta=9\) और \(\alpha\beta=20\) हैं। (\(\alpha+2\)\(\beta+2\)=\alpha\beta+2\(\alpha+\beta\)+4=20+18+4=42)।
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यदि (p(x)=2x-2 -5x-3), तो \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\) क्या है, जहाँ \(\alpha,\beta\) शून्यक हैं?
If (p(x)=2x-2 -5x-3), what is \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\), where \(\alpha,\beta\) are zeroes?
#zeroes
#ratio-expression
#expert
A -\(\frac{37}{6}\)
B \(\frac{37}{6}\)
C -\(\frac{25}{6}\)
D \(\frac{25}{6}\)
Explanation opens after your attempt
Correct Answer
A. -\(\frac{37}{6}\)
Step 1
Concept
\(\alpha+\beta=\frac{5}{2}\) and \(\alpha\beta=-\frac{3}{2}\). (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\(\alpha+\beta\)2 -2\alpha\beta}{\alpha\beta}=-\frac{37}{6}).
Step 2
Why this answer is correct
The correct answer is A. -\(\frac{37}{6}\). \(\alpha+\beta=\frac{5}{2}\) and \(\alpha\beta=-\frac{3}{2}\). (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\(\alpha+\beta\)2 -2\alpha\beta}{\alpha\beta}=-\frac{37}{6}).
Step 3
Exam Tip
\(\alpha+\beta=\frac{5}{2}\) और \(\alpha\beta=-\frac{3}{2}\) हैं। (\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\(\alpha+\beta\)2 -2\alpha\beta}{\alpha\beta}=-\frac{37}{6})।
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यदि (p(x)=x-2 -6x+s) का एक शून्यक दूसरे से (2) अधिक है, तो (s) का मान क्या है?
If one zero of (p(x)=x-2 -6x+s) is (2) more than the other, what is (s)?
#zeroes
#difference
#parameter
A (8)
B (9)
C (10)
D (12)
Explanation opens after your attempt
Step 1
Concept
Let the zeroes be (t) and (t+2), then (2t+2=6) gives (t=2). The product is \(2\cdot4=8\), so (s=8).
Step 2
Why this answer is correct
The correct answer is A. (8). Let the zeroes be (t) and (t+2), then (2t+2=6) gives (t=2). The product is \(2\cdot4=8\), so (s=8).
Step 3
Exam Tip
शून्यक (t) और (t+2) मानें, तो (2t+2=6) से (t=2)। गुणनफल \(2\cdot4=8\), इसलिए (s=8)।
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यदि (x=2) और (x=-3), (p(x)=ax-2 +bx+c) के शून्यक हैं, तो \(\frac{c}{a}\) क्या है?
If (x=2) and (x=-3) are zeroes of (p(x)=ax-2 +bx+c), what is \(\frac{c}{a}\)?
#zeroes
#coefficient-relation
#quadratic
A -(6)
B (6)
C -(1)
D (1)
Explanation opens after your attempt
Step 1
Concept
The product of zeroes is (2\cdot(-3)=-6), and it equals \(\frac{c}{a}\). Pay special attention to the sign in products.
Step 2
Why this answer is correct
The correct answer is A. -(6). The product of zeroes is (2\cdot(-3)=-6), and it equals \(\frac{c}{a}\). Pay special attention to the sign in products.
Step 3
Exam Tip
शून्यकों का गुणनफल (2\cdot(-3)=-6) है और यह \(\frac{c}{a}\) के बराबर होता है। गुणनफल में संकेत पर विशेष ध्यान दें।
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यदि \(\alpha\) और \(\beta\), \(3x^2-10x+7\) के शून्यक हैं, तो \(\alpha^2+\beta^2\) का मान क्या है?
If \(\alpha\) and \(\beta\) are zeroes of \(3x^2-10x+7\), what is the value of \(\alpha^2+\beta^2\)?
#zeroes
#identity
#quadratic
A \(\frac{58}{9}\)
B \(\frac{100}{9}\)
C \(\frac{14}{3}\)
D \(\frac{49}{9}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{58}{9}\)
Step 1
Concept
Here \(\alpha+\beta=\frac{10}{3}\) and \(\alpha\beta=\frac{7}{3}\). Hence (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=\frac{100}{9}-\frac{14}{3}=\frac{58}{9}).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{58}{9}\). Here \(\alpha+\beta=\frac{10}{3}\) and \(\alpha\beta=\frac{7}{3}\). Hence (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=\frac{100}{9}-\frac{14}{3}=\frac{58}{9}).
Step 3
Exam Tip
यहाँ \(\alpha+\beta=\frac{10}{3}\) और \(\alpha\beta=\frac{7}{3}\) हैं। इसलिए (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=\frac{100}{9}-\frac{14}{3}=\frac{58}{9})।
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यदि (p(x)=x-2 -(m+4)x+4m) के शून्यक (4) और (m) हैं, तो कौन-सा कथन सही है?
If the zeroes of (p(x)=x-2 -(m+4)x+4m) are (4) and (m), which statement is correct?
#zeroes
#coefficients
#concept
A यह हर वास्तविक (m) के लिए सही है / It is true for every real (m)
B यह केवल (m=4) के लिए सही है / It is true only for (m=4)
C यह केवल (m=0) के लिए सही है / It is true only for (m=0)
D यह कभी सही नहीं है / It is never true
Explanation opens after your attempt
Correct Answer
A. यह हर वास्तविक (m) के लिए सही है / It is true for every real (m)
Step 1
Concept
The sum (4+m) and product (4m) match the given polynomial. Therefore the statement is true for every real (m).
Step 2
Why this answer is correct
The correct answer is A. यह हर वास्तविक (m) के लिए सही है / It is true for every real (m). The sum (4+m) and product (4m) match the given polynomial. Therefore the statement is true for every real (m).
Step 3
Exam Tip
योग (4+m) और गुणनफल (4m) हैं, जो दिए बहुपद से मिलते हैं। इसलिए कथन हर वास्तविक (m) के लिए सही है।
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किस बहुपद के शून्यक (-1) और (4) हैं?
Which polynomial has zeroes (-1) and (4)?
#construct-polynomial
#zeroes
#signs
A \(x^2-3x-4\)
B \(x^2+3x-4\)
C \(x^2-4x+1\)
D \(x^2+x-4\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-3x-4\)
Step 1
Concept
The polynomial is ((x+1)(x-4)=x-2 -3x-4). Be careful with signs while forming factors.
Step 2
Why this answer is correct
The correct answer is A. \(x^2-3x-4\). The polynomial is ((x+1)(x-4)=x-2 -3x-4). Be careful with signs while forming factors.
Step 3
Exam Tip
शून्यकों से बहुपद ((x+1)(x-4)=x-2 -3x-4) बनता है। चिन्ह बदलते समय सावधानी रखें।
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यदि (p(x)=x-2 +px+q) के शून्यक (4) और (-5) हैं, तो (p+q) क्या है?
If the zeroes of (p(x)=x-2 +px+q) are (4) and (-5), what is (p+q)?
#coefficient-finding
#zeroes
#quadratic
A (-19)
B (-21)
C (19)
D (21)
Explanation opens after your attempt
Step 1
Concept
The sum is (-1), so (p=1), and the product is (-20), so (q=-20). Hence (p+q=-19).
Step 2
Why this answer is correct
The correct answer is A. (-19). The sum is (-1), so (p=1), and the product is (-20), so (q=-20). Hence (p+q=-19).
Step 3
Exam Tip
योग (-1) है, इसलिए (p=1), और गुणनफल (-20) है, इसलिए (q=-20)। अतः (p+q=-19) है।
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यदि (p(x)=x-3 -6x-2 +11x-6), तो (p(x)) के शून्यक कौन-से हैं?
If (p(x)=x-3 -6x-2 +11x-6), what are the zeroes of (p(x))?
#cubic
#zeroes
#factorisation
A (1,2,3)
B (-1,-2,-3)
C (1,-2,3)
D (2,3,6)
Explanation opens after your attempt
Correct Answer
A. (1,2,3)
Step 1
Concept
(x-3 -6x-2 +11x-6=(x-1)(x-2)(x-3)). Zeroes are read directly from the factors.
Step 2
Why this answer is correct
The correct answer is A. (1,2,3). (x-3 -6x-2 +11x-6=(x-1)(x-2)(x-3)). Zeroes are read directly from the factors.
Step 3
Exam Tip
(x-3 -6x-2 +11x-6=(x-1)(x-2)(x-3)) है। गुणनखंडों से शून्यक सीधे मिलते हैं।
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यदि (p(x)=x-2 -(k+2)x+3k) के शून्यक (3) और (k) हैं, तो कौन-सा कथन सही है?
If the zeroes of (p(x)=x-2 -(k+2)x+3k) are (3) and (k), which statement is correct?
#zeroes
#consistency
#expert-trap
A (k=1)
B (k=2)
C (k=3)
D (k=0)
Explanation opens after your attempt
Step 1
Concept
The sum of zeroes is (3+k), while the polynomial gives sum (k+2), so (3+k=k+2) is impossible. This is a conceptual trap.
Step 2
Why this answer is correct
The correct answer is A. (k=1). The sum of zeroes is (3+k), while the polynomial gives sum (k+2), so (3+k=k+2) is impossible. This is a conceptual trap.
Step 3
Exam Tip
शून्यकों का योग (3+k) है और बहुपद से योग (k+2) है, अतः (3+k=k+2) असंभव है। इसलिए दिए गए विकल्पों में कोई नहीं होना चाहिए, यह अवधारणा जाँचने वाला जाल है।
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यदि किसी द्विघात बहुपद के शून्यक (2) और (5) हैं, तो एक मोनिक बहुपद क्या होगा?
If the zeroes of a quadratic polynomial are (2) and (5), what is one monic polynomial?
#construct-polynomial
#zeroes
#monic
A \(x^2-7x+10\)
B \(x^2+7x+10\)
C \(x^2-10x+7\)
D \(x^2+10x-7\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-7x+10\)
Step 1
Concept
With zeroes (2) and (5), the polynomial is ((x-2)(x-5)=x-2 -7x+10). A monic polynomial has leading coefficient (1).
Step 2
Why this answer is correct
The correct answer is A. \(x^2-7x+10\). With zeroes (2) and (5), the polynomial is ((x-2)(x-5)=x-2 -7x+10). A monic polynomial has leading coefficient (1).
Step 3
Exam Tip
शून्यक (2) और (5) होने पर बहुपद ((x-2)(x-5)=x-2 -7x+10) है। मोनिक बहुपद में \(x^2\) का गुणांक (1) होता है।
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यदि (p(x)=x-2 -9), तो (p(x)) के शून्यक कौन-से हैं?
If (p(x)=x-2 -9), what are the zeroes of (p(x))?
#zeroes
#factorisation
#polynomial
A (3,-3)
B (0,9)
C (1,-9)
D (9,-9)
Explanation opens after your attempt
Step 1
Concept
(x-2 -9=(x-3)(x+3)), so the zeroes are (3) and (-3). Identify difference of squares quickly.
Step 2
Why this answer is correct
The correct answer is A. (3,-3). (x-2 -9=(x-3)(x+3)), so the zeroes are (3) and (-3). Identify difference of squares quickly.
Step 3
Exam Tip
(x-2 -9=(x-3)(x+3)), इसलिए शून्यक (3) और (-3) हैं। वर्गों के अंतर को तुरंत पहचानें।
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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)=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 \(x^2-3x-10\) 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 \(x^2-8x+k\) 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 \(x^2-7x+10\) 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 (p(x)=x-2 -5x+6) है, तो (p(3)-p(2)) का मान क्या है?
If (p(x)=x-2 -5x+6), what is the value of (p(3)-p(2))?
#polynomials
#evaluation
#zeroes
#hard
A (0)
B (1)
C (-1)
D (6)
Explanation opens after your attempt
Step 1
Concept
Both (2) and (3) are zeroes, so (p(3)=0) and (p(2)=0). Therefore, the difference is (0).
Step 2
Why this answer is correct
The correct answer is A. (0). Both (2) and (3) are zeroes, so (p(3)=0) and (p(2)=0). Therefore, the difference is (0).
Step 3
Exam Tip
(2) और (3) दोनों शून्यक हैं, इसलिए (p(3)=0) और (p(2)=0)। अतः अंतर (0) है।
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