यदि \(x^2-8x+15=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-8x+15=0\), what is the value of \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\)?
#roots
#advanced_expression
#sum_product
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A \(\frac{34}{15}\)
B \(\frac{64}{15}\)
C \(\frac{49}{15}\)
D \(\frac{16}{15}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{34}{15}\)
Step 1
Concept
Here \(\alpha+\beta=8\) and \(\alpha\beta=15\). The value is (\frac{\(\alpha+\beta\)2 -2\alpha\beta}{\alpha\beta}=\frac{34}{15}).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{34}{15}\). Here \(\alpha+\beta=8\) and \(\alpha\beta=15\). The value is (\frac{\(\alpha+\beta\)2 -2\alpha\beta}{\alpha\beta}=\frac{34}{15}).
Step 3
Exam Tip
यहां \(\alpha+\beta=8\) और \(\alpha\beta=15\) है। मान (\frac{\(\alpha+\beta\)2 -2\alpha\beta}{\alpha\beta}=\frac{34}{15}) होगा।
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यदि \(x^2-9x+20=0\) के मूल \(\alpha\) और \(\beta\) हैं तो (\(\alpha-\beta\)2 ) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-9x+20=0\), what is the value of (\(\alpha-\beta\)2 )?
#roots
#difference_of_roots
#identity
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A (1)
B (9)
C (20)
D (41)
Explanation opens after your attempt
Step 1
Concept
(\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta=81-80=1). This identity is quick for difference questions.
Step 2
Why this answer is correct
The correct answer is A. (1). (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta=81-80=1). This identity is quick for difference questions.
Step 3
Exam Tip
(\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta=81-80=1) है। अंतर वाले प्रश्न में यह पहचान तेज रहती है।
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समीकरण (x-2 -2kx+\(k^2-4\)=0) के मूलों का अंतर कितना होगा?
What will be the difference between the roots of (x-2 -2kx+\(k^2-4\)=0)?
#roots
#parameter
#difference
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A (4)
B (2)
C (k)
D (2k)
Explanation opens after your attempt
Step 1
Concept
This is ((x-k)2 -4=0), so the roots are (k+2) and (k-2). Their difference is (4).
Step 2
Why this answer is correct
The correct answer is A. (4). This is ((x-k)2 -4=0), so the roots are (k+2) and (k-2). Their difference is (4).
Step 3
Exam Tip
यह ((x-k)2 -4=0) है इसलिए मूल (k+2) और (k-2) हैं। उनका अंतर (4) है।
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यदि (3) और (5) समीकरण \(ax^2+bx+15=0\) के मूल हैं तो (a) का मान क्या है?
If (3) and (5) are roots of \(ax^2+bx+15=0\), what is the value of (a)?
#roots
#coefficient
#product_of_roots
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A (1)
B (3)
C (5)
D (15)
Explanation opens after your attempt
Step 1
Concept
The product of roots is \(\frac{c}{a}\). From \(3\cdot5=\frac{15}{a}\), we get (a=1).
Step 2
Why this answer is correct
The correct answer is A. (1). The product of roots is \(\frac{c}{a}\). From \(3\cdot5=\frac{15}{a}\), we get (a=1).
Step 3
Exam Tip
मूलों का गुणनफल \(\frac{c}{a}\) होता है। \(3\cdot5=\frac{15}{a}\) से (a=1) मिलता है।
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यदि (x-2 -(m+3)x+18=0) के मूल (3) और (6) हैं तो (m) का मान क्या है?
If the roots of (x-2 -(m+3)x+18=0) are (3) and (6), what is the value of (m)?
#roots
#parameter
#sum_of_roots
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A (6)
B (3)
C (9)
D (18)
Explanation opens after your attempt
Step 1
Concept
The sum of roots is (9). Therefore (m+3=9), so (m=6).
Step 2
Why this answer is correct
The correct answer is A. (6). The sum of roots is (9). Therefore (m+3=9), so (m=6).
Step 3
Exam Tip
मूलों का योग (9) है। इसलिए (m+3=9) और (m=6) होगा।
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यदि \(x^2+px+q=0\) के मूल (-3) और (-4) हैं तो (p+q) का मान क्या है?
If roots of \(x^2+px+q=0\) are (-3) and (-4), what is the value of (p+q)?
#roots
#parameter
#equation_from_roots
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A (19)
B (12)
C (7)
D (-5)
Explanation opens after your attempt
Step 1
Concept
The sum of roots is (-7), so (p=7), and the product gives (q=12). Hence (p+q=19).
Step 2
Why this answer is correct
The correct answer is A. (19). The sum of roots is (-7), so (p=7), and the product gives (q=12). Hence (p+q=19).
Step 3
Exam Tip
मूलों का योग (-7) है इसलिए (p=7) और गुणनफल (q=12) है। अतः (p+q=19) है।
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समीकरण \(3x^2-10x+3=0\) के मूलों के व्युत्क्रमों का योग क्या है?
What is the sum of reciprocals of the roots of \(3x^2-10x+3=0\)?
#roots
#reciprocal_sum
#sum_product
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A \(\frac{10}{3}\)
B \(\frac{3}{10}\)
C \(\frac{13}{3}\)
D \(\frac{10}{9}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{10}{3}\)
Step 1
Concept
The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here \(\frac{\frac{10}{3}}{1}=\frac{10}{3}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{10}{3}\). The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here \(\frac{\frac{10}{3}}{1}=\frac{10}{3}\).
Step 3
Exam Tip
व्युत्क्रमों का योग \(\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहां \(\frac{\frac{10}{3}}{1}=\frac{10}{3}\) है।
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यदि \(x^2-10x+21=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\alpha^3+\beta^3\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-10x+21=0\), what is the value of \(\alpha^3+\beta^3\)?
#roots
#cubes
#identity
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A (370)
B (1000)
C (630)
D (210)
Explanation opens after your attempt
Step 1
Concept
\(\alpha+\beta=10\) and \(\alpha\beta=21\). (\alpha-3 +\beta-3 =103 -3(21)(10)=370).
Step 2
Why this answer is correct
The correct answer is A. (370). \(\alpha+\beta=10\) and \(\alpha\beta=21\). (\alpha-3 +\beta-3 =103 -3(21)(10)=370).
Step 3
Exam Tip
\(\alpha+\beta=10\) और \(\alpha\beta=21\) है। (\alpha-3 +\beta-3 =103 -3(21)(10)=370) होगा।
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यदि समीकरण \(x^2-6x+k=0\) के मूल वास्तविक और बराबर हैं तो (k) का मान क्या है?
If roots of \(x^2-6x+k=0\) are real and equal, what is the value of (k)?
#roots
#equal_roots
#discriminant
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A (9)
B (6)
C (18)
D (36)
Explanation opens after your attempt
Step 1
Concept
For equal roots (D=0). From (36-4k=0), we get (k=9).
Step 2
Why this answer is correct
The correct answer is A. (9). For equal roots (D=0). From (36-4k=0), we get (k=9).
Step 3
Exam Tip
बराबर मूलों के लिए (D=0) होता है। (36-4k=0) से (k=9) मिलता है।
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यदि \(kx^2-8x+16=0\) के बराबर मूल हैं और \(k\ne0\) है तो (k) का मान क्या है?
If \(kx^2-8x+16=0\) has equal roots and \(k\ne0\), what is the value of (k)?
#roots
#parameter
#discriminant
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A (1)
B (2)
C (4)
D \(\frac{1}{2}\)
Explanation opens after your attempt
Step 1
Concept
For equal roots (D=0). From (64-64k=0), (k=1).
Step 2
Why this answer is correct
The correct answer is A. (1). For equal roots (D=0). From (64-64k=0), (k=1).
Step 3
Exam Tip
बराबर मूलों के लिए (D=0) होता है। (64-64k=0) से (k=1) है।
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समीकरण \(x^2+4x+13=0\) के वास्तविक मूलों की संख्या कितनी है?
How many real roots does \(x^2+4x+13=0\) have?
#roots
#no_real_roots
#discriminant
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A (0)
B (1)
C (2)
D अनंत / Infinitely many
Explanation opens after your attempt
Step 1
Concept
Here (D=16-52=-36<0). Therefore the number of real roots is (0).
Step 2
Why this answer is correct
The correct answer is A. (0). Here (D=16-52=-36<0). Therefore the number of real roots is (0).
Step 3
Exam Tip
यहां (D=16-52=-36<0) है। इसलिए वास्तविक मूलों की संख्या (0) है।
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यदि (x-2 -(a+2)x+2a=0) के मूल (2) और (a) हैं तो कौन सा कारण सही है?
If roots of (x-2 -(a+2)x+2a=0) are (2) and (a), which reason is correct?
#roots
#reasoning
#equation_from_roots
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A योग (a+2) और गुणनफल (2a) है / Sum is (a+2) and product is (2a)
B योग (2a) और गुणनफल (a+2) है / Sum is (2a) and product is (a+2)
C दोनों मूल बराबर हैं / Both roots are equal
D डिस्क्रिमिनेंट हमेशा ऋणात्मक है / Discriminant is always negative
Explanation opens after your attempt
Correct Answer
A. योग (a+2) और गुणनफल (2a) है / Sum is (a+2) and product is (2a)
Step 1
Concept
The roots (2) and (a) have sum (a+2) and product (2a). Therefore the given monic equation is correct.
Step 2
Why this answer is correct
The correct answer is A. योग (a+2) और गुणनफल (2a) है / Sum is (a+2) and product is (2a). The roots (2) and (a) have sum (a+2) and product (2a). Therefore the given monic equation is correct.
Step 3
Exam Tip
मूल (2) और (a) का योग (a+2) तथा गुणनफल (2a) है। इसलिए दिया गया मोनिक समीकरण सही बनता है।
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यदि \(x^2-8x+12=0\) के मूल \(\alpha\) और \(\beta\) हैं तो (\(\alpha+2\)\(\beta+2\)) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-8x+12=0\), what is the value of (\(\alpha+2\)\(\beta+2\))?
#roots
#transformed_expression
#sum_product
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A (32)
B (12)
C (24)
D (20)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=8\) and \(\alpha\beta=12\). (\(\alpha+2\)\(\beta+2\)=12+2(8)+4=32).
Step 2
Why this answer is correct
The correct answer is A. (32). Here \(\alpha+\beta=8\) and \(\alpha\beta=12\). (\(\alpha+2\)\(\beta+2\)=12+2(8)+4=32).
Step 3
Exam Tip
यहां \(\alpha+\beta=8\) और \(\alpha\beta=12\) है। (\(\alpha+2\)\(\beta+2\)=12+2(8)+4=32) है।
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यदि \(x^2-4x-21=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\alpha^2\beta+\alpha\beta^2\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-4x-21=0\), what is the value of \(\alpha^2\beta+\alpha\beta^2\)?
#roots
#expression
#identity
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A (-84)
B (84)
C (21)
D (-17)
Explanation opens after your attempt
Step 1
Concept
(\alpha-2 \beta+\alpha\beta-2 =\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-21\) and \(\alpha+\beta=4\), so the value is (-84).
Step 2
Why this answer is correct
The correct answer is A. (-84). (\alpha-2 \beta+\alpha\beta-2 =\alpha\beta\(\alpha+\beta\)). Here \(\alpha\beta=-21\) and \(\alpha+\beta=4\), so the value is (-84).
Step 3
Exam Tip
(\alpha-2 \beta+\alpha\beta-2 =\alpha\beta\(\alpha+\beta\)) होता है। यहां \(\alpha\beta=-21\) और \(\alpha+\beta=4\) इसलिए मान (-84) है।
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यदि \(\alpha\) और \(\beta\) समीकरण \(3x^2-8x+4=0\) के मूल हैं तो \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(3x^2-8x+4=0\), what is the value of \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}\)?
#roots
#reciprocal_square
#identity
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A \(\frac{7}{4}\)
B \(\frac{16}{9}\)
C \(\frac{4}{7}\)
D (2)
Explanation opens after your attempt
Correct Answer
A. \(\frac{7}{4}\)
Step 1
Concept
Here \(\alpha+\beta=\frac{8}{3}\) and \(\alpha\beta=\frac{4}{3}\). The value is (\frac{\(\alpha+\beta\)2 -2\alpha\beta}{\(\alpha\beta\)2 }=\frac{7}{4}).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{7}{4}\). Here \(\alpha+\beta=\frac{8}{3}\) and \(\alpha\beta=\frac{4}{3}\). The value is (\frac{\(\alpha+\beta\)2 -2\alpha\beta}{\(\alpha\beta\)2 }=\frac{7}{4}).
Step 3
Exam Tip
यहां \(\alpha+\beta=\frac{8}{3}\) और \(\alpha\beta=\frac{4}{3}\) है। मान (\frac{\(\alpha+\beta\)2 -2\alpha\beta}{\(\alpha\beta\)2 }=\frac{7}{4}) है।
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यदि समीकरण \(x^2+px+25=0\) के मूल बराबर हैं और दोनों ऋणात्मक हैं तो (p) का मान क्या होगा?
If the roots of \(x^2+px+25=0\) are equal and both negative, what is the value of (p)?
#roots
#equal_negative_roots
#parameter
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A (10)
B (-10)
C (5)
D (-5)
Explanation opens after your attempt
Step 1
Concept
The equal negative roots are (-5) and (-5) because the product is (25). The sum is (-10), so (p=10).
Step 2
Why this answer is correct
The correct answer is A. (10). The equal negative roots are (-5) and (-5) because the product is (25). The sum is (-10), so (p=10).
Step 3
Exam Tip
बराबर ऋणात्मक मूल (-5) और (-5) होंगे क्योंकि गुणनफल (25) है। योग (-10) है इसलिए (p=10) है।
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यदि \(x^2+ax+a=0\) का एक मूल (2) है तो दूसरा मूल क्या होगा?
If one root of \(x^2+ax+a=0\) is (2), what will be the other root?
#roots
#parameter
#other_root
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A \(-\frac{2}{3}\)
B \(\frac{2}{3}\)
C (3)
D (-3)
Explanation opens after your attempt
Correct Answer
A. \(-\frac{2}{3}\)
Step 1
Concept
Putting (x=2) gives (4+3a=0), so \(a=-\frac{4}{3}\). The product is (a), so the other root is \(-\frac{2}{3}\).
Step 2
Why this answer is correct
The correct answer is A. \(-\frac{2}{3}\). Putting (x=2) gives (4+3a=0), so \(a=-\frac{4}{3}\). The product is (a), so the other root is \(-\frac{2}{3}\).
Step 3
Exam Tip
(x=2) रखने पर (4+3a=0) से \(a=-\frac{4}{3}\) है। गुणनफल (a) है इसलिए दूसरा मूल \(-\frac{2}{3}\) होगा।
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यदि \(x^2-7x+q=0\) के मूलों का अंतर (1) है तो (q) का मान क्या है?
If the difference of roots of \(x^2-7x+q=0\) is (1), what is the value of (q)?
#roots
#difference_of_roots
#parameter
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A (12)
B (14)
C (10)
D (7)
Explanation opens after your attempt
Step 1
Concept
The sum of roots is (7) and the difference is (1), so the roots are (4) and (3). Their product is (q=12).
Step 2
Why this answer is correct
The correct answer is A. (12). The sum of roots is (7) and the difference is (1), so the roots are (4) and (3). Their product is (q=12).
Step 3
Exam Tip
मूलों का योग (7) और अंतर (1) है इसलिए मूल (4) और (3) हैं। उनका गुणनफल (q=12) है।
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यदि \(x^2-Sx+P=0\) के मूल \(\alpha\) और \(\beta\) हैं तो (\(\alpha-\beta\)2 ) किसके बराबर है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-Sx+P=0\), what is (\(\alpha-\beta\)2 ) equal to?
#roots
#general_identity
#difference
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A \(S^2-4P\)
B \(S^2+4P\)
C \(P^2-4S\)
D (S-P)
Explanation opens after your attempt
Correct Answer
A. \(S^2-4P\)
Step 1
Concept
Here the sum of roots is (S) and product is (P). Therefore (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta=S-2 -4P).
Step 2
Why this answer is correct
The correct answer is A. \(S^2-4P\). Here the sum of roots is (S) and product is (P). Therefore (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta=S-2 -4P).
Step 3
Exam Tip
यहां मूलों का योग (S) और गुणनफल (P) है। इसलिए (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta=S-2 -4P) है।
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यदि \(x^2-6x-16=0\) के मूलों को (1) बढ़ा दिया जाए तो नए मूलों से बना मोनिक समीकरण कौन सा होगा?
If each root of \(x^2-6x-16=0\) is increased by (1), which monic equation is formed from the new roots?
#roots
#transformed_roots
#new_equation
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A \(x^2-8x-9=0\)
B \(x^2-6x-16=0\)
C \(x^2-4x-9=0\)
D \(x^2+8x-9=0\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-8x-9=0\)
Step 1
Concept
The old roots are (8) and (-2). The new roots are (9) and (-1), so the equation is \(x^2-8x-9=0\).
Step 2
Why this answer is correct
The correct answer is A. \(x^2-8x-9=0\). The old roots are (8) and (-2). The new roots are (9) and (-1), so the equation is \(x^2-8x-9=0\).
Step 3
Exam Tip
पुराने मूल (8) और (-2) हैं। नए मूल (9) और (-1) होंगे इसलिए समीकरण \(x^2-8x-9=0\) है।
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समीकरण \(x^2-6x-16=0\) के मूलों को (2) घटाने पर नए मूलों का गुणनफल क्या होगा?
If each root of \(x^2-6x-16=0\) is decreased by (2), what is the product of the new roots?
#roots
#transformed_roots
#product
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A (-24)
B (-16)
C (24)
D (16)
Explanation opens after your attempt
Step 1
Concept
The old roots are (8) and (-2). The new roots are (6) and (-4), so the product is (-24).
Step 2
Why this answer is correct
The correct answer is A. (-24). The old roots are (8) and (-2). The new roots are (6) and (-4), so the product is (-24).
Step 3
Exam Tip
पुराने मूल (8) और (-2) हैं। नए मूल (6) और (-4) होंगे इसलिए गुणनफल (-24) है।
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यदि \(x^2-4x+3=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(3\alpha\) और \(3\beta\) को मूल मानकर समीकरण कौन सा होगा?
If \(\alpha\) and \(\beta\) are roots of \(x^2-4x+3=0\), which equation has \(3\alpha\) and \(3\beta\) as roots?
#roots
#transformed_roots
#equation
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A \(x^2-12x+27=0\)
B \(x^2-4x+27=0\)
C \(x^2-12x+9=0\)
D \(x^2+12x+27=0\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-12x+27=0\)
Step 1
Concept
The old sum is (4) and product is (3). The new sum is (12) and product is (27), so the equation is \(x^2-12x+27=0\).
Step 2
Why this answer is correct
The correct answer is A. \(x^2-12x+27=0\). The old sum is (4) and product is (3). The new sum is (12) and product is (27), so the equation is \(x^2-12x+27=0\).
Step 3
Exam Tip
पुराने योग (4) और गुणनफल (3) हैं। नए योग (12) और गुणनफल (27) होंगे इसलिए \(x^2-12x+27=0\) है।
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यदि \(x^2-6x+5=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\alpha+3\) और \(\beta+3\) का योग क्या होगा?
If \(\alpha\) and \(\beta\) are roots of \(x^2-6x+5=0\), what is the sum of \(\alpha+3\) and \(\beta+3\)?
#roots
#transformed_roots
#sum
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A (12)
B (6)
C (9)
D (15)
Explanation opens after your attempt
Step 1
Concept
The old sum of roots is (6). The new sum is (\(\alpha+3\)+\(\beta+3\)=6+6=12).
Step 2
Why this answer is correct
The correct answer is A. (12). The old sum of roots is (6). The new sum is (\(\alpha+3\)+\(\beta+3\)=6+6=12).
Step 3
Exam Tip
पुराने मूलों का योग (6) है। नया योग (\(\alpha+3\)+\(\beta+3\)=6+6=12) होगा।
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यदि \(x^2-12x+36=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\alpha-\beta\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-12x+36=0\), what is the value of \(\alpha-\beta\)?
#roots
#equal_roots
#difference
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A (0)
B (6)
C (12)
D (36)
Explanation opens after your attempt
Step 1
Concept
This is ((x-6)2 =0), so both roots are (6) and (6). Hence \(\alpha-\beta=0\).
Step 2
Why this answer is correct
The correct answer is A. (0). This is ((x-6)2 =0), so both roots are (6) and (6). Hence \(\alpha-\beta=0\).
Step 3
Exam Tip
यह ((x-6)2 =0) है इसलिए दोनों मूल (6) और (6) हैं। अतः \(\alpha-\beta=0\) है।
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यदि \(\alpha\) और \(\beta\) समीकरण \(x^2+4x-21=0\) के मूल हैं तो \(\left|\alpha-\beta\right|\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2+4x-21=0\), what is the value of \(\left|\alpha-\beta\right|\)?
#roots
#absolute_difference
#factorisation
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A (10)
B (4)
C (7)
D (21)
Explanation opens after your attempt
Step 1
Concept
The roots are (3) and (-7). Therefore (\left|\alpha-\beta\right|=\left|3-(-7)\right|=10).
Step 2
Why this answer is correct
The correct answer is A. (10). The roots are (3) and (-7). Therefore (\left|\alpha-\beta\right|=\left|3-(-7)\right|=10).
Step 3
Exam Tip
समीकरण के मूल (3) और (-7) हैं। इसलिए (\left|\alpha-\beta\right|=\left|3-(-7)\right|=10) है।
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यदि मूलों का योग (6) और उनके वर्गों का योग (52) है तो मूलों का गुणनफल क्या होगा?
If the sum of roots is (6) and the sum of their squares is (52), what is the product of roots?
#roots
#identity
#product
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A (-8)
B (8)
C (16)
D (-16)
Explanation opens after your attempt
Step 1
Concept
(\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta). From \(52=36-2\alpha\beta\), we get \(\alpha\beta=-8\).
Step 2
Why this answer is correct
The correct answer is A. (-8). (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta). From \(52=36-2\alpha\beta\), we get \(\alpha\beta=-8\).
Step 3
Exam Tip
(\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta) है। \(52=36-2\alpha\beta\) से \(\alpha\beta=-8\) मिलता है।
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यदि \(x^2-9x+c=0\) के मूलों का अनुपात (1:2) है तो (c) का मान क्या है?
If the roots of \(x^2-9x+c=0\) are in the ratio (1:2), what is the value of (c)?
#roots
#ratio_of_roots
#parameter
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A (18)
B (9)
C (27)
D (12)
Explanation opens after your attempt
Step 1
Concept
Let the roots be (t) and (2t). From (3t=9), (t=3), so the roots are (3) and (6), hence (c=18).
Step 2
Why this answer is correct
The correct answer is A. (18). Let the roots be (t) and (2t). From (3t=9), (t=3), so the roots are (3) and (6), hence (c=18).
Step 3
Exam Tip
मूल (t) और (2t) मानें तो (3t=9) से (t=3) है। मूल (3) और (6) होंगे इसलिए (c=18) है।
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यदि \(x^2+px+18=0\) के मूलों का अनुपात (1:2) है और दोनों ऋणात्मक हैं तो (p) क्या होगा?
If roots of \(x^2+px+18=0\) are in the ratio (1:2) and both are negative, what is (p)?
#roots
#ratio_of_roots
#negative_roots
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A (9)
B (-9)
C (6)
D (-6)
Explanation opens after your attempt
Step 1
Concept
The roots are (-3) and (-6) because the product is (18). Their sum is (-9), so (p=9).
Step 2
Why this answer is correct
The correct answer is A. (9). The roots are (-3) and (-6) because the product is (18). Their sum is (-9), so (p=9).
Step 3
Exam Tip
मूल (-3) और (-6) होंगे क्योंकि गुणनफल (18) है। उनका योग (-9) है इसलिए (p=9) है।
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यदि \(x^2+bx+16=0\) के मूल एक दूसरे के बराबर और धनात्मक हैं तो (b) का मान क्या होगा?
If roots of \(x^2+bx+16=0\) are equal and positive, what is the value of (b)?
#roots
#equal_positive_roots
#parameter
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A (-8)
B (8)
C (4)
D (-4)
Explanation opens after your attempt
Step 1
Concept
The equal positive roots are (4) and (4). The sum is (8), so (-b=8) gives (b=-8).
Step 2
Why this answer is correct
The correct answer is A. (-8). The equal positive roots are (4) and (4). The sum is (8), so (-b=8) gives (b=-8).
Step 3
Exam Tip
बराबर धनात्मक मूल (4) और (4) होंगे। योग (8) है इसलिए (-b=8) से (b=-8) है।
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यदि \(4x^2-3x+k=0\) के मूलों का गुणनफल मूलों के योग के बराबर है तो (k) क्या होगा?
If the product of roots of \(4x^2-3x+k=0\) equals the sum of roots, what is (k)?
#roots
#sum_equals_product
#parameter
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A (3)
B \(\frac{3}{4}\)
C (4)
D (1)
Explanation opens after your attempt
Step 1
Concept
The sum is \(-\frac{b}{a}=\frac{3}{4}\) and the product is \(\frac{k}{4}\). From \(\frac{k}{4}=\frac{3}{4}\), (k=3).
Step 2
Why this answer is correct
The correct answer is A. (3). The sum is \(-\frac{b}{a}=\frac{3}{4}\) and the product is \(\frac{k}{4}\). From \(\frac{k}{4}=\frac{3}{4}\), (k=3).
Step 3
Exam Tip
योग \(-\frac{b}{a}=\frac{3}{4}\) और गुणनफल \(\frac{k}{4}\) है। \(\frac{k}{4}=\frac{3}{4}\) से (k=3) है।
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यदि समीकरण (x-2 -4x+(m+4)=0) के वास्तविक मूल नहीं हैं तो (m) के लिए कौन सी शर्त सही है?
If (x-2 -4x+(m+4)=0) has no real roots, which condition on (m) is correct?
#roots
#no_real_roots
#parameter_condition
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A (m>0)
B (m<0)
C (m=0)
D \(m\le0\)
Explanation opens after your attempt
Step 1
Concept
For no real roots (D<0) is required. From (16-4(m+4)<0), we get (m>0).
Step 2
Why this answer is correct
The correct answer is A. (m>0). For no real roots (D<0) is required. From (16-4(m+4)<0), we get (m>0).
Step 3
Exam Tip
वास्तविक मूल नहीं होने के लिए (D<0) चाहिए। (16-4(m+4)<0) से (m>0) मिलता है।
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यदि समीकरण (x-2 -2(m+2)x+(m+2)2 =0) के मूल बराबर हैं तो (m) के लिए क्या सही है?
If roots of (x-2 -2(m+2)x+(m+2)2 =0) are equal, what is true for (m)?
#roots
#equal_roots
#identity_parameter
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A हर वास्तविक (m) / Every real (m)
B केवल (m=0) / Only (m=0)
C केवल (m=-2) / Only (m=-2)
D कोई वास्तविक (m) नहीं / No real (m)
Explanation opens after your attempt
Correct Answer
A. हर वास्तविक (m) / Every real (m)
Step 1
Concept
The equation becomes ((x-(m+2))2 =0). Therefore the roots are equal for every real (m).
Step 2
Why this answer is correct
The correct answer is A. हर वास्तविक (m) / Every real (m). The equation becomes ((x-(m+2))2 =0). Therefore the roots are equal for every real (m).
Step 3
Exam Tip
समीकरण ((x-(m+2))2 =0) बनता है। इसलिए हर वास्तविक (m) के लिए दोनों मूल बराबर होंगे।
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यदि \(x^2-9x+18=0\) के मूल \(\alpha\) और \(\beta\) हैं तो (\(\alpha-2\)\(\beta-2\)) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-9x+18=0\), what is the value of (\(\alpha-2\)\(\beta-2\))?
#roots
#transformed_expression
#sum_product
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A (4)
B (18)
C (9)
D (8)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=9\) and \(\alpha\beta=18\). (\(\alpha-2\)\(\beta-2\)=18-2(9)+4=4).
Step 2
Why this answer is correct
The correct answer is A. (4). Here \(\alpha+\beta=9\) and \(\alpha\beta=18\). (\(\alpha-2\)\(\beta-2\)=18-2(9)+4=4).
Step 3
Exam Tip
यहां \(\alpha+\beta=9\) और \(\alpha\beta=18\) है। (\(\alpha-2\)\(\beta-2\)=18-2(9)+4=4) है।
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यदि (3) और (-4) मूल हैं तो \(5x^2+px+q=0\) में \(\frac{q}{5}\) का मान क्या होगा?
If (3) and (-4) are roots, what is the value of \(\frac{q}{5}\) in \(5x^2+px+q=0\)?
#roots
#product_of_roots
#coefficient
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A (-12)
B (12)
C (-1)
D (1)
Explanation opens after your attempt
Step 1
Concept
The product of roots is \(\frac{q}{5}\). Since (3\cdot(-4)=-12), \(\frac{q}{5}=-12\).
Step 2
Why this answer is correct
The correct answer is A. (-12). The product of roots is \(\frac{q}{5}\). Since (3\cdot(-4)=-12), \(\frac{q}{5}=-12\).
Step 3
Exam Tip
मूलों का गुणनफल \(\frac{q}{5}\) होता है। (3\cdot(-4)=-12) इसलिए \(\frac{q}{5}=-12\) है।
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यदि \(x^2+px+q=0\) के मूल (5) और (-2) हैं तो (p-q) का मान क्या है?
If roots of \(x^2+px+q=0\) are (5) and (-2), what is the value of (p-q)?
#roots
#parameter
#expression
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A (7)
B (-13)
C (13)
D (-7)
Explanation opens after your attempt
Step 1
Concept
The sum is (3), so (p=-3), and the product is (-10), so (q=-10). Thus (p-q=-3-(-10)=7).
Step 2
Why this answer is correct
The correct answer is A. (7). The sum is (3), so (p=-3), and the product is (-10), so (q=-10). Thus (p-q=-3-(-10)=7).
Step 3
Exam Tip
योग (3) है इसलिए (p=-3) और गुणनफल (-10) है इसलिए (q=-10)। अतः (p-q=-3-(-10)=7) है।
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यदि \(\alpha\) और \(\beta\) समीकरण \(x^2-6x+2=0\) के मूल हैं तो \(\alpha^2+\beta^2\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-6x+2=0\), what is the value of \(\alpha^2+\beta^2\)?
#roots
#sum_of_squares
#identity
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A (32)
B (36)
C (4)
D (12)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=6\) and \(\alpha\beta=2\). \(\alpha^2+\beta^2=36-4=32\).
Step 2
Why this answer is correct
The correct answer is A. (32). Here \(\alpha+\beta=6\) and \(\alpha\beta=2\). \(\alpha^2+\beta^2=36-4=32\).
Step 3
Exam Tip
यहां \(\alpha+\beta=6\) और \(\alpha\beta=2\) है। \(\alpha^2+\beta^2=36-4=32\) है।
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यदि \(x^2+2x-2=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\alpha^2+\alpha\beta+\beta^2\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2+2x-2=0\), what is the value of \(\alpha^2+\alpha\beta+\beta^2\)?
#roots
#identity
#expression
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A (6)
B (4)
C (2)
D (0)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=-2\) and \(\alpha\beta=-2\). (\alpha-2 +\alpha\beta+\beta-2 =\(\alpha+\beta\)2 -\alpha\beta=6).
Step 2
Why this answer is correct
The correct answer is A. (6). Here \(\alpha+\beta=-2\) and \(\alpha\beta=-2\). (\alpha-2 +\alpha\beta+\beta-2 =\(\alpha+\beta\)2 -\alpha\beta=6).
Step 3
Exam Tip
यहां \(\alpha+\beta=-2\) और \(\alpha\beta=-2\) है। (\alpha-2 +\alpha\beta+\beta-2 =\(\alpha+\beta\)2 -\alpha\beta=6) है।
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समीकरण (x-2 -(2m+3)x+(m+1)(m+2)=0) के मूल कौन से हो सकते हैं?
Which can be the roots of (x-2 -(2m+3)x+(m+1)(m+2)=0)?
#roots
#general_roots
#pattern
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A (m+1) और (m+2) / (m+1) and (m+2)
B (m) और (3) / (m) and (3)
C (2m) और (3) / (2m) and (3)
D (m-1 ) और (m+2) / (m-1 ) and (m+2)
Explanation opens after your attempt
Correct Answer
A. (m+1) और (m+2) / (m+1) and (m+2)
Step 1
Concept
(m+1+m+2=2m+3), and the product is ((m+1)(m+2)). Therefore the roots can be (m+1) and (m+2).
Step 2
Why this answer is correct
The correct answer is A. (m+1) और (m+2) / (m+1) and (m+2). (m+1+m+2=2m+3), and the product is ((m+1)(m+2)). Therefore the roots can be (m+1) and (m+2).
Step 3
Exam Tip
(m+1+m+2=2m+3) और गुणनफल ((m+1)(m+2)) है। इसलिए मूल (m+1) और (m+2) हो सकते हैं।
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यदि \(x^2-10x+24=0\) के मूल \(\alpha\) और \(\beta\) हैं तो (\(\alpha-2\)) और (\(\beta-2\)) का गुणनफल क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-10x+24=0\), what is the product of (\(\alpha-2\)) and (\(\beta-2\))?
#roots
#transformed_expression
#product
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A (8)
B (24)
C (14)
D (4)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=10\) and \(\alpha\beta=24\). (\(\alpha-2\)\(\beta-2\)=24-20+4=8).
Step 2
Why this answer is correct
The correct answer is A. (8). Here \(\alpha+\beta=10\) and \(\alpha\beta=24\). (\(\alpha-2\)\(\beta-2\)=24-20+4=8).
Step 3
Exam Tip
यहां \(\alpha+\beta=10\) और \(\alpha\beta=24\) है। (\(\alpha-2\)\(\beta-2\)=24-20+4=8) है।
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यदि \(x^2-13x+42=0\) के मूलों में छोटा मूल \(\alpha\) और बड़ा मूल \(\beta\) है तो \(\beta-\alpha\) क्या है?
If the smaller root of \(x^2-13x+42=0\) is \(\alpha\) and the larger root is \(\beta\), what is \(\beta-\alpha\)?
#roots
#ordered_roots
#difference
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A (1)
B (13)
C (42)
D (6)
Explanation opens after your attempt
Step 1
Concept
The roots are (6) and (7). Thus the smaller root is (6) and the larger root is (7), so \(\beta-\alpha=1\).
Step 2
Why this answer is correct
The correct answer is A. (1). The roots are (6) and (7). Thus the smaller root is (6) and the larger root is (7), so \(\beta-\alpha=1\).
Step 3
Exam Tip
समीकरण के मूल (6) और (7) हैं। इसलिए छोटा मूल (6) और बड़ा मूल (7) है तथा \(\beta-\alpha=1\) है।
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यदि \(x^2-8x+12=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\alpha^2\beta^2\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-8x+12=0\), what is the value of \(\alpha^2\beta^2\)?
#roots
#product_square
#identity
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A (144)
B (12)
C (64)
D (20)
Explanation opens after your attempt
Step 1
Concept
(\alpha-2 \beta-2 =\(\alpha\beta\)2 ). Here \(\alpha\beta=12\), so the value is (144).
Step 2
Why this answer is correct
The correct answer is A. (144). (\alpha-2 \beta-2 =\(\alpha\beta\)2 ). Here \(\alpha\beta=12\), so the value is (144).
Step 3
Exam Tip
(\alpha-2 \beta-2 =\(\alpha\beta\)2 ) होता है। यहां \(\alpha\beta=12\) इसलिए मान (144) है।
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यदि \(x^2-9x+20=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\frac{\alpha+\beta}{\alpha-\beta}\) का संभावित मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-9x+20=0\), what can be the value of \(\frac{\alpha+\beta}{\alpha-\beta}\)?
#roots
#ratio
#difference
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A (9) या (-9) / (9) or (-9)
B \(\frac{9}{1}\) या \(-\frac{9}{1}\) / \(\frac{9}{1}\) or \(-\frac{9}{1}\)
C \(\frac{9}{2}\) या \(-\frac{9}{2}\) / \(\frac{9}{2}\) or \(-\frac{9}{2}\)
D (5) या (-5) / (5) or (-5)
Explanation opens after your attempt
Correct Answer
A. (9) या (-9) / (9) or (-9)
Step 1
Concept
The roots are (5) and (4), so the sum is (9) and the difference can be \(\pm1\). Hence the value is (9) or (-9).
Step 2
Why this answer is correct
The correct answer is A. (9) या (-9) / (9) or (-9). The roots are (5) and (4), so the sum is (9) and the difference can be \(\pm1\). Hence the value is (9) or (-9).
Step 3
Exam Tip
मूल (5) और (4) हैं इसलिए योग (9) और अंतर \(\pm1\) हो सकता है। इसलिए मान (9) या (-9) है।
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यदि \(x^2-4x-5=0\) के मूल \(\alpha\) और \(\beta\) हैं तो (\left\(\alpha+2\right\)\left\(\beta+2\right\)) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-4x-5=0\), what is the value of (\left\(\alpha+2\right\)\left\(\beta+2\right\))?
#roots
#transformed_expression
#sum_product
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A (7)
B (3)
C (5)
D (11)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=4\) and \(\alpha\beta=-5\). (\(\alpha+2\)\(\beta+2\)=-5+2(4)+4=7).
Step 2
Why this answer is correct
The correct answer is A. (7). Here \(\alpha+\beta=4\) and \(\alpha\beta=-5\). (\(\alpha+2\)\(\beta+2\)=-5+2(4)+4=7).
Step 3
Exam Tip
यहां \(\alpha+\beta=4\) और \(\alpha\beta=-5\) है। (\(\alpha+2\)\(\beta+2\)=-5+2(4)+4=7) है।
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यदि \(x^2+kx+49=0\) के मूल एक दूसरे के बराबर और ऋणात्मक हैं तो (k) क्या होगा?
If roots of \(x^2+kx+49=0\) are equal and negative, what is (k)?
#roots
#equal_negative_roots
#parameter
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A (14)
B (-14)
C (7)
D (-7)
Explanation opens after your attempt
Step 1
Concept
The equal negative roots are (-7) and (-7). Their sum is (-14), so (k=14).
Step 2
Why this answer is correct
The correct answer is A. (14). The equal negative roots are (-7) and (-7). Their sum is (-14), so (k=14).
Step 3
Exam Tip
बराबर ऋणात्मक मूल (-7) और (-7) हैं। योग (-14) है इसलिए (k=14) है।
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यदि \(x^2-14x+45=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\frac{\alpha^2+\beta^2}{\alpha\beta}\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-14x+45=0\), what is the value of \(\frac{\alpha^2+\beta^2}{\alpha\beta}\)?
#roots
#ratio_expression
#identity
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A \(\frac{106}{45}\)
B \(\frac{196}{45}\)
C (2)
D \(\frac{45}{106}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{106}{45}\)
Step 1
Concept
Here \(\alpha+\beta=14\) and \(\alpha\beta=45\). \(\alpha^2+\beta^2=196-90=106\), so the value is \(\frac{106}{45}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{106}{45}\). Here \(\alpha+\beta=14\) and \(\alpha\beta=45\). \(\alpha^2+\beta^2=196-90=106\), so the value is \(\frac{106}{45}\).
Step 3
Exam Tip
यहां \(\alpha+\beta=14\) और \(\alpha\beta=45\) है। \(\alpha^2+\beta^2=196-90=106\) इसलिए मान \(\frac{106}{45}\) है।
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यदि \(3x^2+px+18=0\) के मूल (-2) और (-3) हैं तो (p) का मान क्या है?
If roots of \(3x^2+px+18=0\) are (-2) and (-3), what is the value of (p)?
#roots
#parameter
#sum_of_roots
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A (15)
B (-15)
C (18)
D (-18)
Explanation opens after your attempt
Step 1
Concept
The sum of roots is (-5), and \(-\frac{p}{3}=-5\). Therefore (p=15).
Step 2
Why this answer is correct
The correct answer is A. (15). The sum of roots is (-5), and \(-\frac{p}{3}=-5\). Therefore (p=15).
Step 3
Exam Tip
मूलों का योग (-5) है और \(-\frac{p}{3}=-5\) होगा। इसलिए (p=15) है।
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यदि \(x^2-7x+10=0\) के मूलों को उलटकर नया समीकरण बनाया जाए तो नया मोनिक समीकरण कौन सा होगा?
If a new equation is formed by taking reciprocals of the roots of \(x^2-7x+10=0\), which monic equation is obtained?
#roots
#reciprocal_roots
#new_equation
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A \(x^2-\frac{7}{10}x+\frac{1}{10}=0\)
B \(x^2-7x+10=0\)
C \(x^2+\frac{7}{10}x+\frac{1}{10}=0\)
D \(x^2-\frac{1}{10}x+\frac{7}{10}=0\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-\frac{7}{10}x+\frac{1}{10}=0\)
Step 1
Concept
The old sum is (7) and product is (10). The reciprocal roots have sum \(\frac{7}{10}\) and product \(\frac{1}{10}\).
Step 2
Why this answer is correct
The correct answer is A. \(x^2-\frac{7}{10}x+\frac{1}{10}=0\). The old sum is (7) and product is (10). The reciprocal roots have sum \(\frac{7}{10}\) and product \(\frac{1}{10}\).
Step 3
Exam Tip
पुराने योग (7) और गुणनफल (10) हैं। उलटे मूलों का योग \(\frac{7}{10}\) और गुणनफल \(\frac{1}{10}\) होगा।
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यदि \(x^2-5x+5=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\alpha^2+\beta^2+\alpha\beta\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-5x+5=0\), what is the value of \(\alpha^2+\beta^2+\alpha\beta\)?
#roots
#identity
#expression
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A (20)
B (25)
C (15)
D (5)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=5\) and \(\alpha\beta=5\). (\alpha-2 +\beta-2 +\alpha\beta=\(\alpha+\beta\)2 -\alpha\beta=20).
Step 2
Why this answer is correct
The correct answer is A. (20). Here \(\alpha+\beta=5\) and \(\alpha\beta=5\). (\alpha-2 +\beta-2 +\alpha\beta=\(\alpha+\beta\)2 -\alpha\beta=20).
Step 3
Exam Tip
यहां \(\alpha+\beta=5\) और \(\alpha\beta=5\) है। (\alpha-2 +\beta-2 +\alpha\beta=\(\alpha+\beta\)2 -\alpha\beta=20) है।
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यदि \(x^2-3x-28=0\) के मूल \(\alpha\) और \(\beta\) हैं तो (\(\alpha+1\)\(\beta+1\)) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-3x-28=0\), what is the value of (\(\alpha+1\)\(\beta+1\))?
#roots
#transformed_expression
#sum_product
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A (-24)
B (-28)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
Here \(\alpha+\beta=3\) and \(\alpha\beta=-28\). (\(\alpha+1\)\(\beta+1\)=-28+3+1=-24).
Step 2
Why this answer is correct
The correct answer is A. (-24). Here \(\alpha+\beta=3\) and \(\alpha\beta=-28\). (\(\alpha+1\)\(\beta+1\)=-28+3+1=-24).
Step 3
Exam Tip
यहां \(\alpha+\beta=3\) और \(\alpha\beta=-28\) है। (\(\alpha+1\)\(\beta+1\)=-28+3+1=-24) है।
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यदि \(x^2-11x+24=0\) के मूल \(\alpha\) और \(\beta\) हैं तो \(\frac{1}{\alpha+1}+\frac{1}{\beta+1}\) का मान क्या है?
If \(\alpha\) and \(\beta\) are roots of \(x^2-11x+24=0\), what is the value of \(\frac{1}{\alpha+1}+\frac{1}{\beta+1}\)?
#roots
#reciprocal_shifted
#sum_product
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A \(\frac{13}{36}\)
B \(\frac{11}{24}\)
C \(\frac{12}{35}\)
D \(\frac{36}{13}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{13}{36}\)
Step 1
Concept
Here \(\alpha+\beta=11\) and \(\alpha\beta=24\). The value is \(\frac{\alpha+\beta+2}{\alpha\beta+\alpha+\beta+1}=\frac{13}{36}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{13}{36}\). Here \(\alpha+\beta=11\) and \(\alpha\beta=24\). The value is \(\frac{\alpha+\beta+2}{\alpha\beta+\alpha+\beta+1}=\frac{13}{36}\).
Step 3
Exam Tip
यहां \(\alpha+\beta=11\) और \(\alpha\beta=24\) है। मान \(\frac{\alpha+\beta+2}{\alpha\beta+\alpha+\beta+1}=\frac{13}{36}\) होगा।
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