युग्म (kx+5y=10) और (6x+15y=18) का अद्वितीय हल कब होगा?
When will the pair (kx+5y=10) and (6x+15y=18) have a unique solution?
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#linear-equations
#solvability
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A (k=2)
B \(k\neq2\)
C (k=6)
D \(k\neq6\)
Explanation opens after your attempt
Correct Answer
B. \(k\neq2\)
Step 1
Concept
For a unique solution, \(\frac{k}{6}\neq\frac{5}{15}\) must hold. Therefore, \(k\neq2\) is correct.
Step 2
Why this answer is correct
The correct answer is B. \(k\neq2\). For a unique solution, \(\frac{k}{6}\neq\frac{5}{15}\) must hold. Therefore, \(k\neq2\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{k}{6}\neq\frac{5}{15}\) होना चाहिए। इसलिए \(k\neq2\) सही है।
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युग्म (4x+my=8) और (10x+15y=21) में कोई हल न होने के लिए (m) का मान क्या है?
What is the value of (m) for no solution in (4x+my=8) and (10x+15y=21)?
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A (m=4)
B (m=5)
C (m=6)
D (m=8)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios gives (m=6). The constant ratio is different, so there is no solution.
Step 2
Why this answer is correct
The correct answer is C. (m=6). Equating coefficient ratios gives (m=6). The constant ratio is different, so there is no solution.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर (m=6) मिलता है। स्थिर पद का अनुपात अलग है इसलिए कोई हल नहीं होगा।
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युग्म ((p-2)x+7y=14) और (9x+21y=42) के अनंत हलों के लिए (p) क्या होगा?
For infinitely many solutions of ((p-2)x+7y=14) and (9x+21y=42), what is (p)?
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A (p=3)
B (p=4)
C (p=5)
D (p=7)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, all three ratios must be equal. From \(\frac{p-2}{9}=\frac{7}{21}\), (p=5).
Step 2
Why this answer is correct
The correct answer is C. (p=5). For infinitely many solutions, all three ratios must be equal. From \(\frac{p-2}{9}=\frac{7}{21}\), (p=5).
Step 3
Exam Tip
अनंत हलों के लिए तीनों अनुपात समान होने चाहिए। \(\frac{p-2}{9}=\frac{7}{21}\) से (p=5) मिलता है।
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युग्म (8x+ay=12) और (20x+10y=30) के अनंत हलों के लिए (a) का मान बताइए।
Find the value of (a) for infinitely many solutions of (8x+ay=12) and (20x+10y=30).
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A (a=3)
B (a=4)
C (a=5)
D (a=6)
Explanation opens after your attempt
Step 1
Concept
Here \(\frac{8}{20}=\frac{12}{30}\). Thus \(\frac{a}{10}=\frac{2}{5}\) gives (a=4).
Step 2
Why this answer is correct
The correct answer is B. (a=4). Here \(\frac{8}{20}=\frac{12}{30}\). Thus \(\frac{a}{10}=\frac{2}{5}\) gives (a=4).
Step 3
Exam Tip
यहाँ \(\frac{8}{20}=\frac{12}{30}\) है। अतः \(\frac{a}{10}=\frac{2}{5}\) से (a=4) होगा।
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युग्म (3x-2y=5) और (12x-8y=c) में कोई हल न हो इसके लिए (c) पर कौन-सी शर्त होगी?
For (3x-2y=5) and (12x-8y=c) to have no solution, what condition is required on (c)?
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A (c=20)
B \(c\neq20\)
C (c=15)
D \(c\neq15\)
Explanation opens after your attempt
Correct Answer
B. \(c\neq20\)
Step 1
Concept
The coefficient ratio is \(\frac{1}{4}\). For no solution, \(\frac{5}{c}\neq\frac{1}{4}\), so \(c\neq20\).
Step 2
Why this answer is correct
The correct answer is B. \(c\neq20\). The coefficient ratio is \(\frac{1}{4}\). For no solution, \(\frac{5}{c}\neq\frac{1}{4}\), so \(c\neq20\).
Step 3
Exam Tip
गुणांक अनुपात \(\frac{1}{4}\) है। कोई हल नहीं के लिए \(\frac{5}{c}\neq\frac{1}{4}\) इसलिए \(c\neq20\)।
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युग्म (qx+4y=9) और (15x+10y=18) का अद्वितीय हल कब होगा?
When will (qx+4y=9) and (15x+10y=18) have a unique solution?
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#linear-equations
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A \(q\neq6\)
B (q=6)
C \(q\neq15\)
D (q=15)
Explanation opens after your attempt
Correct Answer
A. \(q\neq6\)
Step 1
Concept
For a unique solution, \(\frac{q}{15}\neq\frac{4}{10}\) must hold. Hence \(q\neq6\) is correct.
Step 2
Why this answer is correct
The correct answer is A. \(q\neq6\). For a unique solution, \(\frac{q}{15}\neq\frac{4}{10}\) must hold. Hence \(q\neq6\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{q}{15}\neq\frac{4}{10}\) होना चाहिए। इसलिए \(q\neq6\) सही शर्त है।
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युग्म (5x+(r+1)y=11) और (15x+18y=33) के अनंत हलों के लिए (r) क्या होगा?
For infinitely many solutions of (5x+(r+1)y=11) and (15x+18y=33), what is (r)?
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A (r=4)
B (r=5)
C (r=6)
D (r=7)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{5}{15}=\frac{r+1}{18}=\frac{11}{33}\) is needed. This gives (r=5).
Step 2
Why this answer is correct
The correct answer is B. (r=5). For infinitely many solutions, \(\frac{5}{15}=\frac{r+1}{18}=\frac{11}{33}\) is needed. This gives (r=5).
Step 3
Exam Tip
अनंत हलों के लिए \(\frac{5}{15}=\frac{r+1}{18}=\frac{11}{33}\) चाहिए। इससे (r=5) मिलता है।
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युग्म (2x+ny=7) और (6x+15y=19) में कोई हल न होने के लिए (n) का मान क्या है?
What is the value of (n) for no solution in (2x+ny=7) and (6x+15y=19)?
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A (n=3)
B (n=4)
C (n=5)
D (n=6)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios, \(\frac{2}{6}=\frac{n}{15}\) gives (n=5). The constant ratio is different, so the pair is inconsistent.
Step 2
Why this answer is correct
The correct answer is C. (n=5). Equating coefficient ratios, \(\frac{2}{6}=\frac{n}{15}\) gives (n=5). The constant ratio is different, so the pair is inconsistent.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\frac{2}{6}=\frac{n}{15}\) से (n=5) मिलता है। स्थिर अनुपात अलग है इसलिए युग्म असंगत है।
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युग्म (11x+3y=4) और (22x+sy=10) का अद्वितीय हल कब होगा?
When will (11x+3y=4) and (22x+sy=10) have a unique solution?
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A (s=6)
B \(s\neq6\)
C (s=3)
D \(s\neq3\)
Explanation opens after your attempt
Correct Answer
B. \(s\neq6\)
Step 1
Concept
For a unique solution, \(\frac{11}{22}\neq\frac{3}{s}\) must hold. Hence \(s\neq6\) is correct.
Step 2
Why this answer is correct
The correct answer is B. \(s\neq6\). For a unique solution, \(\frac{11}{22}\neq\frac{3}{s}\) must hold. Hence \(s\neq6\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{11}{22}\neq\frac{3}{s}\) होना चाहिए। अतः \(s\neq6\) सही है।
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युग्म (6x-ty=18) और (14x-21y=42) के अनंत हलों के लिए (t) का मान क्या होगा?
What is the value of (t) for infinitely many solutions of (6x-ty=18) and (14x-21y=42)?
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A (t=6)
B (t=8)
C (t=9)
D (t=12)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{6}{14}=\frac{-t}{-21}=\frac{18}{42}\) must hold. This gives (t=9).
Step 2
Why this answer is correct
The correct answer is C. (t=9). For infinitely many solutions, \(\frac{6}{14}=\frac{-t}{-21}=\frac{18}{42}\) must hold. This gives (t=9).
Step 3
Exam Tip
अनंत हलों के लिए \(\frac{6}{14}=\frac{-t}{-21}=\frac{18}{42}\) होना चाहिए। इससे (t=9) मिलता है।
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रेखाएँ (7x+2y=9) और (21x+6y=25) किस प्रकार का युग्म बनाती हैं?
What type of pair is formed by the lines (7x+2y=9) and (21x+6y=25)?
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A संगत और स्वतंत्र / Consistent and independent
B संगत और आश्रित / Consistent and dependent
C असंगत / Inconsistent
D सदैव अद्वितीय / Always unique
Explanation opens after your attempt
Correct Answer
C. असंगत / Inconsistent
Step 1
Concept
Coefficient ratios are equal but the constant ratio is different. Hence the lines are parallel and the pair is inconsistent.
Step 2
Why this answer is correct
The correct answer is C. असंगत / Inconsistent. Coefficient ratios are equal but the constant ratio is different. Hence the lines are parallel and the pair is inconsistent.
Step 3
Exam Tip
गुणांक अनुपात समान है पर स्थिर पद का अनुपात अलग है। इसलिए रेखाएँ समांतर हैं और युग्म असंगत है।
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रेखाएँ (9x-5y=13) और (18x-10y=26) के लिए सही निष्कर्ष क्या है?
What is the correct conclusion for the lines (9x-5y=13) and (18x-10y=26)?
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A कोई हल नहीं / No solution
B अद्वितीय हल / Unique solution
C अनंत हल / Infinitely many solutions
D ठीक दो हल / Exactly two solutions
Explanation opens after your attempt
Correct Answer
C. अनंत हल / Infinitely many solutions
Step 1
Concept
The second equation is (2) times the first. So both lines are coincident and give infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. The second equation is (2) times the first. So both lines are coincident and give infinitely many solutions.
Step 3
Exam Tip
दूसरा समीकरण पहले का (2) गुना है। इसलिए दोनों रेखाएँ संपाती हैं और अनंत हल देती हैं।
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युग्म (10x+3y=17) और (20x+9y=31) में कितने हल होंगे?
How many solutions will the pair (10x+3y=17) and (20x+9y=31) have?
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A कोई हल नहीं / No solution
B अनंत हल / Infinitely many solutions
C अद्वितीय हल / Unique solution
D निर्धारित नहीं / Cannot be decided
Explanation opens after your attempt
Correct Answer
C. अद्वितीय हल / Unique solution
Step 1
Concept
Here \(\frac{10}{20}\neq\frac{3}{9}\). Therefore, the lines meet at one point.
Step 2
Why this answer is correct
The correct answer is C. अद्वितीय हल / Unique solution. Here \(\frac{10}{20}\neq\frac{3}{9}\). Therefore, the lines meet at one point.
Step 3
Exam Tip
यहाँ \(\frac{10}{20}\neq\frac{3}{9}\) है। इसलिए रेखाएँ एक बिंदु पर मिलती हैं।
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युग्म (4x+9y=2) और (8x+18y=7) की हल-स्थिति क्या है?
What is the solution status of (4x+9y=2) and (8x+18y=7)?
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A अद्वितीय हल / Unique solution
B कोई हल नहीं / No solution
C अनंत हल / Infinitely many solutions
D सभी बिंदु हल / All points are solutions
Explanation opens after your attempt
Correct Answer
B. कोई हल नहीं / No solution
Step 1
Concept
The coefficient ratio is equal but \(\frac{2}{7}\) is different. Hence both are distinct parallel lines.
Step 2
Why this answer is correct
The correct answer is B. कोई हल नहीं / No solution. The coefficient ratio is equal but \(\frac{2}{7}\) is different. Hence both are distinct parallel lines.
Step 3
Exam Tip
गुणांक अनुपात समान है लेकिन \(\frac{2}{7}\) अलग है। इसलिए दोनों अलग समांतर रेखाएँ हैं।
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युग्म (13x+4y=6) और (26x+8y=12) के लिए सही विकल्प चुनिए।
Choose the correct option for (13x+4y=6) and (26x+8y=12).
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A अद्वितीय हल / Unique solution
B कोई हल नहीं / No solution
C अनंत हल / Infinitely many solutions
D केवल (x=0) हल / Only (x=0) solution
Explanation opens after your attempt
Correct Answer
C. अनंत हल / Infinitely many solutions
Step 1
Concept
All three ratios are equal. Thus the lines are coincident and the pair has infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. All three ratios are equal. Thus the lines are coincident and the pair has infinitely many solutions.
Step 3
Exam Tip
तीनों अनुपात समान हैं। अतः रेखाएँ संपाती हैं और युग्म के अनंत हल हैं।
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युग्म (12x-7y=5) और (18x-11y=8) की हल-संख्या क्या है?
What is the number of solutions of (12x-7y=5) and (18x-11y=8)?
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A कोई हल नहीं / No solution
B अनंत हल / Infinitely many solutions
C अद्वितीय हल / Unique solution
D शून्य या अनंत / Zero or infinite
Explanation opens after your attempt
Correct Answer
C. अद्वितीय हल / Unique solution
Step 1
Concept
Since \(\frac{12}{18}\neq\frac{-7}{-11}\), the lines intersect. Hence a unique solution is obtained.
Step 2
Why this answer is correct
The correct answer is C. अद्वितीय हल / Unique solution. Since \(\frac{12}{18}\neq\frac{-7}{-11}\), the lines intersect. Hence a unique solution is obtained.
Step 3
Exam Tip
\(\frac{12}{18}\neq\frac{-7}{-11}\) होने से रेखाएँ काटती हैं। इसलिए अद्वितीय हल मिलेगा।
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युग्म ((u+3)x+8y=16) और (10x+20y=40) के अनंत हलों के लिए (u) क्या होगा?
For infinitely many solutions of ((u+3)x+8y=16) and (10x+20y=40), what is (u)?
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A (u=1)
B (u=2)
C (u=3)
D (u=4)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{u+3}{10}=\frac{8}{20}=\frac{16}{40}\) is needed. This gives (u=1).
Step 2
Why this answer is correct
The correct answer is A. (u=1). For infinitely many solutions, \(\frac{u+3}{10}=\frac{8}{20}=\frac{16}{40}\) is needed. This gives (u=1).
Step 3
Exam Tip
अनंत हलों के लिए \(\frac{u+3}{10}=\frac{8}{20}=\frac{16}{40}\) चाहिए। इससे (u=1) मिलता है।
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युग्म (5x+(v-2)y=3) और (15x+9y=14) में कोई हल न होने के लिए (v) का मान क्या है?
What is the value of (v) for no solution in (5x+(v-2)y=3) and (15x+9y=14)?
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A (v=3)
B (v=4)
C (v=5)
D (v=6)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios gives (v-2=3). Thus (v=5), and the different constant ratio gives no solution.
Step 2
Why this answer is correct
The correct answer is C. (v=5). Equating coefficient ratios gives (v-2=3). Thus (v=5), and the different constant ratio gives no solution.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर (v-2=3) मिलता है। इसलिए (v=5) और स्थिर अनुपात अलग होने से कोई हल नहीं।
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युग्म (wx-6y=4) और (16x-24y=12) का अद्वितीय हल कब होगा?
When will (wx-6y=4) and (16x-24y=12) have a unique solution?
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#linear-equations
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A (w=4)
B \(w\neq4\)
C (w=16)
D \(w\neq16\)
Explanation opens after your attempt
Correct Answer
B. \(w\neq4\)
Step 1
Concept
For a unique solution, \(\frac{w}{16}\neq\frac{-6}{-24}\) must hold. Therefore, \(w\neq4\).
Step 2
Why this answer is correct
The correct answer is B. \(w\neq4\). For a unique solution, \(\frac{w}{16}\neq\frac{-6}{-24}\) must hold. Therefore, \(w\neq4\).
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{w}{16}\neq\frac{-6}{-24}\) होना चाहिए। इसलिए \(w\neq4\) होगा।
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युग्म \(3x+2y=\lambda\) और (18x+12y=48) के अनंत हलों के लिए \(\lambda\) क्या होगा?
For infinitely many solutions of \(3x+2y=\lambda\) and (18x+12y=48), what is \(\lambda\)?
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A \(\lambda=6\)
B \(\lambda=8\)
C \(\lambda=10\)
D \(\lambda=12\)
Explanation opens after your attempt
Correct Answer
B. \(\lambda=8\)
Step 1
Concept
The coefficient ratio is \(\frac{1}{6}\). For infinitely many solutions, \(\frac{\lambda}{48}=\frac{1}{6}\), so \(\lambda=8\).
Step 2
Why this answer is correct
The correct answer is B. \(\lambda=8\). The coefficient ratio is \(\frac{1}{6}\). For infinitely many solutions, \(\frac{\lambda}{48}=\frac{1}{6}\), so \(\lambda=8\).
Step 3
Exam Tip
गुणांक अनुपात \(\frac{1}{6}\) है। अनंत हलों के लिए \(\frac{\lambda}{48}=\frac{1}{6}\) इसलिए \(\lambda=8\)।
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युग्म \(14x+\mu y=9\) और (21x+6y=20) में कोई हल न होने के लिए \(\mu\) क्या होगा?
What is \(\mu\) for no solution in \(14x+\mu y=9\) and (21x+6y=20)?
#class10
#linear-equations
#solvability
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A \(\mu=3\)
B \(\mu=4\)
C \(\mu=5\)
D \(\mu=6\)
Explanation opens after your attempt
Correct Answer
B. \(\mu=4\)
Step 1
Concept
\(\frac{14}{21}=\frac{2}{3}\). Equating coefficient ratios, \(\frac{\mu}{6}=\frac{2}{3}\) gives \(\mu=4\).
Step 2
Why this answer is correct
The correct answer is B. \(\mu=4\). \(\frac{14}{21}=\frac{2}{3}\). Equating coefficient ratios, \(\frac{\mu}{6}=\frac{2}{3}\) gives \(\mu=4\).
Step 3
Exam Tip
\(\frac{14}{21}=\frac{2}{3}\) है। गुणांक अनुपात समान करने पर \(\frac{\mu}{6}=\frac{2}{3}\) से \(\mu=4\)।
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युग्म (\(\alpha-1\)x+5y=2) और (8x+10y=6) का अद्वितीय हल कब होगा?
When will (\(\alpha-1\)x+5y=2) and (8x+10y=6) have a unique solution?
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#solvability
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A \(\alpha=5\)
B \(\alpha\neq5\)
C \(\alpha=4\)
D \(\alpha\neq4\)
Explanation opens after your attempt
Correct Answer
B. \(\alpha\neq5\)
Step 1
Concept
For a unique solution, \(\frac{\alpha-1}{8}\neq\frac{5}{10}\) is required. Hence \(\alpha\neq5\) is correct.
Step 2
Why this answer is correct
The correct answer is B. \(\alpha\neq5\). For a unique solution, \(\frac{\alpha-1}{8}\neq\frac{5}{10}\) is required. Hence \(\alpha\neq5\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{\alpha-1}{8}\neq\frac{5}{10}\) चाहिए। इसलिए \(\alpha\neq5\) सही है।
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युग्म (9x+\(\beta+4\)y=15) और (27x+18y=45) के अनंत हलों के लिए \(\beta\) क्या होगा?
For infinitely many solutions of (9x+\(\beta+4\)y=15) and (27x+18y=45), what is \(\beta\)?
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A \(\beta=1\)
B \(\beta=2\)
C \(\beta=3\)
D \(\beta=4\)
Explanation opens after your attempt
Correct Answer
B. \(\beta=2\)
Step 1
Concept
For infinitely many solutions, \(\frac{9}{27}=\frac{\beta+4}{18}=\frac{15}{45}\) must hold. This gives \(\beta=2\).
Step 2
Why this answer is correct
The correct answer is B. \(\beta=2\). For infinitely many solutions, \(\frac{9}{27}=\frac{\beta+4}{18}=\frac{15}{45}\) must hold. This gives \(\beta=2\).
Step 3
Exam Tip
अनंत हलों में \(\frac{9}{27}=\frac{\beta+4}{18}=\frac{15}{45}\) होना चाहिए। इससे \(\beta=2\) मिलता है।
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युग्म \(5x+\gamma y=1\) और (25x+15y=11) में कोई हल न होने के लिए \(\gamma\) का मान क्या होगा?
What is the value of \(\gamma\) for no solution in \(5x+\gamma y=1\) and (25x+15y=11)?
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A \(\gamma=2\)
B \(\gamma=3\)
C \(\gamma=4\)
D \(\gamma=5\)
Explanation opens after your attempt
Correct Answer
B. \(\gamma=3\)
Step 1
Concept
Equating coefficient ratios, \(\frac{5}{25}=\frac{\gamma}{15}\) gives \(\gamma=3\). The constant ratio is different.
Step 2
Why this answer is correct
The correct answer is B. \(\gamma=3\). Equating coefficient ratios, \(\frac{5}{25}=\frac{\gamma}{15}\) gives \(\gamma=3\). The constant ratio is different.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\frac{5}{25}=\frac{\gamma}{15}\) से \(\gamma=3\) मिलता है। स्थिर अनुपात अलग है।
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सामान्य युग्म \(a_1x+b_1y+c_1=0\) और \(a_2x+b_2y+c_2=0\) में अद्वितीय हल की शर्त क्या है?
For the general pair \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\), what is the condition for a unique solution?
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A \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\)
B \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\)
C \(\frac{a_1}{a_2}=\frac{c_1}{c_2}\)
D \(c_1=c_2=0\)
Explanation opens after your attempt
Correct Answer
B. \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\)
Step 1
Concept
A unique solution occurs when the lines intersect at one point. Its ratio form is \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\).
Step 2
Why this answer is correct
The correct answer is B. \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\). A unique solution occurs when the lines intersect at one point. Its ratio form is \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\).
Step 3
Exam Tip
अद्वितीय हल तब मिलता है जब रेखाएँ एक बिंदु पर कटती हैं। इसका अनुपात रूप \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\) है।
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यदि \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) है, तो युग्म की हल-स्थिति क्या होगी?
If \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\), what is the solution status of the pair?
#class10
#linear-equations
#solvability
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A कोई हल नहीं / No solution
B अद्वितीय हल / Unique solution
C अनंत हल / Infinitely many solutions
D केवल दो हल / Only two solutions
Explanation opens after your attempt
Correct Answer
C. अनंत हल / Infinitely many solutions
Step 1
Concept
When all three ratios are equal, the lines are coincident. Therefore, infinitely many solutions occur.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. When all three ratios are equal, the lines are coincident. Therefore, infinitely many solutions occur.
Step 3
Exam Tip
तीनों अनुपात समान होने पर रेखाएँ संपाती होती हैं। इसलिए अनंत हल मिलते हैं।
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यदि दो रेखाओं की ढाल समान और अवरोध भी समान हो, तो उनके समीकरणों के युग्म में कितने हल होंगे?
If two lines have the same slope and the same intercept, how many solutions will their pair of equations have?
#class10
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#solvability
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A अद्वितीय हल / Unique solution
B कोई हल नहीं / No solution
C अनंत हल / Infinitely many solutions
D सिर्फ मूल बिंदु / Only origin
Explanation opens after your attempt
Correct Answer
C. अनंत हल / Infinitely many solutions
Step 1
Concept
Same slope and same intercept mean the same line. Therefore, such a pair has infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. Same slope and same intercept mean the same line. Therefore, such a pair has infinitely many solutions.
Step 3
Exam Tip
समान ढाल और समान अवरोध का अर्थ एक ही रेखा है। इसलिए ऐसे युग्म में अनंत हल होते हैं।
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यदि दो रेखाएँ अलग-अलग समांतर हैं, तो अनुपातों की सही स्थिति कौन-सी होगी?
If two lines are distinct and parallel, what is the correct ratio condition?
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A \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\)
B \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\)
C \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)
D \(a_1b_2-a_2b_1\neq0\)
Explanation opens after your attempt
Correct Answer
B. \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\)
Step 1
Concept
For distinct parallel lines, coefficient ratios are equal and the constant ratio is different. This is the condition for no solution.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\). For distinct parallel lines, coefficient ratios are equal and the constant ratio is different. This is the condition for no solution.
Step 3
Exam Tip
अलग समांतर रेखाओं में गुणांक अनुपात समान और स्थिर पद अनुपात अलग होता है। यही कोई हल नहीं की शर्त है।
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यदि \(a_1b_2-a_2b_1\neq0\) है, तो युग्म के बारे में सही कथन क्या है?
If \(a_1b_2-a_2b_1\neq0\), what is the correct statement about the pair?
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A युग्म असंगत है / The pair is inconsistent
B युग्म के अनंत हल हैं / The pair has infinitely many solutions
C युग्म का अद्वितीय हल है / The pair has a unique solution
D युग्म में कोई रेखा नहीं / The pair has no line
Explanation opens after your attempt
Correct Answer
C. युग्म का अद्वितीय हल है / The pair has a unique solution
Step 1
Concept
When the determinant is non-zero, the lines intersect at one point. Hence there is a unique solution.
Step 2
Why this answer is correct
The correct answer is C. युग्म का अद्वितीय हल है / The pair has a unique solution. When the determinant is non-zero, the lines intersect at one point. Hence there is a unique solution.
Step 3
Exam Tip
सारणिक शून्य नहीं होने पर रेखाएँ एक बिंदु पर कटती हैं। इसलिए अद्वितीय हल होता है।
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यदि (D=0), \(D_x=0\) और \(D_y=0\) हैं, तो दो रैखिक समीकरणों के युग्म में क्या होगा?
If (D=0), \(D_x=0\), and \(D_y=0\), what happens in a pair of two linear equations?
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A कोई हल नहीं / No solution
B अद्वितीय हल / Unique solution
C अनंत हल / Infinitely many solutions
D हमेशा गलत समीकरण / Always false equations
Explanation opens after your attempt
Correct Answer
C. अनंत हल / Infinitely many solutions
Step 1
Concept
When all three determinants are zero, the equations may be dependent. In Class (10), link this with infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. When all three determinants are zero, the equations may be dependent. In Class (10), link this with infinitely many solutions.
Step 3
Exam Tip
तीनों सारणिक शून्य होने पर समीकरण आश्रित हो सकते हैं। कक्षा (10) में इसे अनंत हल की स्थिति से जोड़कर देखें।
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युग्म ((z+2)x+3y=5) और (6x+(z-1)y=7) अद्वितीय न हो, इसके लिए (z) के संभावित मान कौन-से हैं?
For ((z+2)x+3y=5) and (6x+(z-1)y=7) to be non-unique, what are the possible values of (z)?
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A (z=4,-3)
B (z=3,-4)
C (z=2,-5)
D (z=5,-2)
Explanation opens after your attempt
Correct Answer
A. (z=4,-3)
Step 1
Concept
For non-unique solutions, ((z+2)(z-1)-18=0) must hold. This gives (z=4) or (z=-3).
Step 2
Why this answer is correct
The correct answer is A. (z=4,-3). For non-unique solutions, ((z+2)(z-1)-18=0) must hold. This gives (z=4) or (z=-3).
Step 3
Exam Tip
अद्वितीय न होने के लिए ((z+2)(z-1)-18=0) होना चाहिए। इससे (z=4) या (z=-3) मिलता है।
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युग्म (ax+2y=3) और (4x+ay=9) के अद्वितीय हल की शर्त क्या है?
What is the condition for a unique solution of (ax+2y=3) and (4x+ay=9)?
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A \(a^2=8\)
B \(a^2\neq8\)
C (a=4)
D \(a\neq2\)
Explanation opens after your attempt
Correct Answer
B. \(a^2\neq8\)
Step 1
Concept
The determinant is \(D=a^2-8\). For a unique solution, \(D\neq0\), so \(a^2\neq8\).
Step 2
Why this answer is correct
The correct answer is B. \(a^2\neq8\). The determinant is \(D=a^2-8\). For a unique solution, \(D\neq0\), so \(a^2\neq8\).
Step 3
Exam Tip
सारणिक \(D=a^2-8\) है। अद्वितीय हल के लिए \(D\neq0\) यानी \(a^2\neq8\)।
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युग्म (2x+by=6) और (bx+8y=24) के अनंत हलों के लिए (b) का मान क्या होगा?
What is the value of (b) for infinitely many solutions of (2x+by=6) and (bx+8y=24)?
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A (b=2)
B (b=3)
C (b=4)
D (b=6)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{2}{b}=\frac{b}{8}=\frac{6}{24}\) must hold. This gives (b=4).
Step 2
Why this answer is correct
The correct answer is C. (b=4). For infinitely many solutions, \(\frac{2}{b}=\frac{b}{8}=\frac{6}{24}\) must hold. This gives (b=4).
Step 3
Exam Tip
अनंत हलों के लिए \(\frac{2}{b}=\frac{b}{8}=\frac{6}{24}\) होना चाहिए। इससे (b=4) मिलता है।
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युग्म (cx+6y=5) और (9x+18y=10) में कोई हल न होने के लिए (c) क्या होगा?
What is (c) for no solution in (cx+6y=5) and (9x+18y=10)?
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A (c=2)
B (c=3)
C (c=4)
D (c=5)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios, \(\frac{c}{9}=\frac{6}{18}\) gives (c=3). The constant ratio \(\frac{5}{10}\) is different.
Step 2
Why this answer is correct
The correct answer is B. (c=3). Equating coefficient ratios, \(\frac{c}{9}=\frac{6}{18}\) gives (c=3). The constant ratio \(\frac{5}{10}\) is different.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\frac{c}{9}=\frac{6}{18}\) से (c=3) आता है। स्थिर अनुपात \(\frac{5}{10}\) अलग है।
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युग्म (4x+dy=7) और (12x+15y=21) के अनंत हलों के लिए (d) का मान क्या है?
What is the value of (d) for infinitely many solutions of (4x+dy=7) and (12x+15y=21)?
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A (d=3)
B (d=4)
C (d=5)
D (d=6)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{4}{12}=\frac{d}{15}=\frac{7}{21}\) must hold. So (d=5) is correct.
Step 2
Why this answer is correct
The correct answer is C. (d=5). For infinitely many solutions, \(\frac{4}{12}=\frac{d}{15}=\frac{7}{21}\) must hold. So (d=5) is correct.
Step 3
Exam Tip
अनंत हलों में \(\frac{4}{12}=\frac{d}{15}=\frac{7}{21}\) होना चाहिए। इसलिए (d=5) सही है।
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युग्म (ex-4y=9) और (18x-12y=27) का अद्वितीय हल कब होगा?
When will (ex-4y=9) and (18x-12y=27) have a unique solution?
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#linear-equations
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A (e=6)
B \(e\neq6\)
C (e=18)
D \(e\neq18\)
Explanation opens after your attempt
Correct Answer
B. \(e\neq6\)
Step 1
Concept
For a unique solution, \(\frac{e}{18}\neq\frac{-4}{-12}\) is needed. Hence \(e\neq6\).
Step 2
Why this answer is correct
The correct answer is B. \(e\neq6\). For a unique solution, \(\frac{e}{18}\neq\frac{-4}{-12}\) is needed. Hence \(e\neq6\).
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{e}{18}\neq\frac{-4}{-12}\) चाहिए। अतः \(e\neq6\)।
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युग्म (7x+fy=2) और (28x+20y=13) में कोई हल न होने के लिए (f) क्या होगा?
What is (f) for no solution in (7x+fy=2) and (28x+20y=13)?
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A (f=4)
B (f=5)
C (f=6)
D (f=7)
Explanation opens after your attempt
Step 1
Concept
To make the coefficient ratio \(\frac{1}{4}\), (f=5) is required. The different constant ratio gives no solution.
Step 2
Why this answer is correct
The correct answer is B. (f=5). To make the coefficient ratio \(\frac{1}{4}\), (f=5) is required. The different constant ratio gives no solution.
Step 3
Exam Tip
गुणांक अनुपात \(\frac{1}{4}\) बनाने के लिए (f=5) चाहिए। स्थिर पद अनुपात अलग होने से कोई हल नहीं होगा।
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युग्म ((g-1 )x+10y=15) और (8x+20y=30) के अनंत हलों के लिए (g) क्या होगा?
For infinitely many solutions of ((g-1 )x+10y=15) and (8x+20y=30), what is (g)?
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A (g=4)
B (g=5)
C (g=6)
D (g=7)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{g-1}{8}=\frac{10}{20}=\frac{15}{30}\) must hold. This gives (g=5).
Step 2
Why this answer is correct
The correct answer is B. (g=5). For infinitely many solutions, \(\frac{g-1}{8}=\frac{10}{20}=\frac{15}{30}\) must hold. This gives (g=5).
Step 3
Exam Tip
अनंत हलों के लिए \(\frac{g-1}{8}=\frac{10}{20}=\frac{15}{30}\) होना चाहिए। इससे (g=5) मिलता है।
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युग्म (hx+12y=6) और (10x+15y=20) का अद्वितीय हल कब होगा?
When will (hx+12y=6) and (10x+15y=20) have a unique solution?
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#linear-equations
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A (h=8)
B \(h\neq8\)
C (h=10)
D \(h\neq10\)
Explanation opens after your attempt
Correct Answer
B. \(h\neq8\)
Step 1
Concept
For a unique solution, \(\frac{h}{10}\neq\frac{12}{15}\) must hold. Hence \(h\neq8\) is correct.
Step 2
Why this answer is correct
The correct answer is B. \(h\neq8\). For a unique solution, \(\frac{h}{10}\neq\frac{12}{15}\) must hold. Hence \(h\neq8\) is correct.
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{h}{10}\neq\frac{12}{15}\) होना चाहिए। इसलिए \(h\neq8\) सही है।
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युग्म (6x+iy=12) और (18x+27y=50) में कोई हल न होने के लिए (i) का मान क्या है?
What is the value of (i) for no solution in (6x+iy=12) and (18x+27y=50)?
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#solvability
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A (i=6)
B (i=7)
C (i=8)
D (i=9)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios, \(\frac{6}{18}=\frac{i}{27}\) gives (i=9). The constant ratio is not equal.
Step 2
Why this answer is correct
The correct answer is D. (i=9). Equating coefficient ratios, \(\frac{6}{18}=\frac{i}{27}\) gives (i=9). The constant ratio is not equal.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\frac{6}{18}=\frac{i}{27}\) से (i=9) आता है। स्थिर पद अनुपात समान नहीं है।
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युग्म (jx+5y=10) और (14x+7y=14) के अनंत हलों के लिए (j) क्या होगा?
For infinitely many solutions of (jx+5y=10) and (14x+7y=14), what is (j)?
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A (j=8)
B (j=9)
C (j=10)
D (j=11)
Explanation opens after your attempt
Step 1
Concept
All three ratios must be \(\frac{5}{7}\). Therefore, \(\frac{j}{14}=\frac{5}{7}\) gives (j=10).
Step 2
Why this answer is correct
The correct answer is C. (j=10). All three ratios must be \(\frac{5}{7}\). Therefore, \(\frac{j}{14}=\frac{5}{7}\) gives (j=10).
Step 3
Exam Tip
तीनों अनुपात \(\frac{5}{7}\) होने चाहिए। इसलिए \(\frac{j}{14}=\frac{5}{7}\) से (j=10)।
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युग्म (2x+3y=4) और (kx+6y=11) में कोई हल न होने के लिए (k) का मान क्या है?
What is the value of (k) for no solution in (2x+3y=4) and (kx+6y=11)?
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A (k=2)
B (k=3)
C (k=4)
D (k=6)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios, \(\frac{2}{k}=\frac{3}{6}\) gives (k=4). The constant ratio is different.
Step 2
Why this answer is correct
The correct answer is C. (k=4). Equating coefficient ratios, \(\frac{2}{k}=\frac{3}{6}\) gives (k=4). The constant ratio is different.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\frac{2}{k}=\frac{3}{6}\) से (k=4) है। स्थिर पद अनुपात अलग है।
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युग्म (mx+7y=13) और (16x+14y=26) का अद्वितीय हल कब होगा?
When will (mx+7y=13) and (16x+14y=26) have a unique solution?
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#solvability
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A (m=8)
B \(m\neq8\)
C (m=16)
D \(m\neq16\)
Explanation opens after your attempt
Correct Answer
B. \(m\neq8\)
Step 1
Concept
For a unique solution, \(\frac{m}{16}\neq\frac{7}{14}\) must hold. Therefore, \(m\neq8\).
Step 2
Why this answer is correct
The correct answer is B. \(m\neq8\). For a unique solution, \(\frac{m}{16}\neq\frac{7}{14}\) must hold. Therefore, \(m\neq8\).
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{m}{16}\neq\frac{7}{14}\) होना चाहिए। इसलिए \(m\neq8\)।
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युग्म (3x+ny=18) और (5x+10y=30) के अनंत हलों के लिए (n) क्या होगा?
For infinitely many solutions of (3x+ny=18) and (5x+10y=30), what is (n)?
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A (n=4)
B (n=5)
C (n=6)
D (n=8)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{3}{5}=\frac{n}{10}=\frac{18}{30}\) is needed. This gives (n=6).
Step 2
Why this answer is correct
The correct answer is C. (n=6). For infinitely many solutions, \(\frac{3}{5}=\frac{n}{10}=\frac{18}{30}\) is needed. This gives (n=6).
Step 3
Exam Tip
अनंत हलों के लिए \(\frac{3}{5}=\frac{n}{10}=\frac{18}{30}\) चाहिए। इससे (n=6) मिलता है।
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युग्म (px-5y=7) और (12x-15y=19) में कोई हल न होने के लिए (p) क्या होगा?
What is (p) for no solution in (px-5y=7) and (12x-15y=19)?
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A (p=4)
B (p=5)
C (p=6)
D (p=7)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios, \(\frac{p}{12}=\frac{-5}{-15}\) gives (p=4). The constant ratio is different.
Step 2
Why this answer is correct
The correct answer is A. (p=4). Equating coefficient ratios, \(\frac{p}{12}=\frac{-5}{-15}\) gives (p=4). The constant ratio is different.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\frac{p}{12}=\frac{-5}{-15}\) से (p=4) है। स्थिर अनुपात अलग है।
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युग्म (qx+11y=22) और (18x+33y=66) के अनंत हलों के लिए (q) का मान क्या है?
What is the value of (q) for infinitely many solutions of (qx+11y=22) and (18x+33y=66)?
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A (q=4)
B (q=5)
C (q=6)
D (q=7)
Explanation opens after your attempt
Step 1
Concept
For infinitely many solutions, \(\frac{q}{18}=\frac{11}{33}=\frac{22}{66}\). Therefore, (q=6) is correct.
Step 2
Why this answer is correct
The correct answer is C. (q=6). For infinitely many solutions, \(\frac{q}{18}=\frac{11}{33}=\frac{22}{66}\). Therefore, (q=6) is correct.
Step 3
Exam Tip
अनंत हलों में \(\frac{q}{18}=\frac{11}{33}=\frac{22}{66}\) होता है। इसलिए (q=6) सही मान है।
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युग्म (rx+2y=5) और (9x+3y=8) का अद्वितीय हल कब होगा?
When will (rx+2y=5) and (9x+3y=8) have a unique solution?
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A (r=6)
B \(r\neq6\)
C (r=9)
D \(r\neq9\)
Explanation opens after your attempt
Correct Answer
B. \(r\neq6\)
Step 1
Concept
For a unique solution, \(\frac{r}{9}\neq\frac{2}{3}\) is required. Hence \(r\neq6\).
Step 2
Why this answer is correct
The correct answer is B. \(r\neq6\). For a unique solution, \(\frac{r}{9}\neq\frac{2}{3}\) is required. Hence \(r\neq6\).
Step 3
Exam Tip
अद्वितीय हल के लिए \(\frac{r}{9}\neq\frac{2}{3}\) चाहिए। इसलिए \(r\neq6\) होगा।
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युग्म (sx+4y=12) और (21x+12y=40) में कोई हल न होने के लिए (s) क्या होगा?
What is (s) for no solution in (sx+4y=12) and (21x+12y=40)?
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A (s=5)
B (s=6)
C (s=7)
D (s=8)
Explanation opens after your attempt
Step 1
Concept
Equating coefficient ratios, \(\frac{s}{21}=\frac{4}{12}\) gives (s=7). The constant ratio is not equal.
Step 2
Why this answer is correct
The correct answer is C. (s=7). Equating coefficient ratios, \(\frac{s}{21}=\frac{4}{12}\) gives (s=7). The constant ratio is not equal.
Step 3
Exam Tip
गुणांक अनुपात समान करने पर \(\frac{s}{21}=\frac{4}{12}\) से (s=7) है। स्थिर पद अनुपात समान नहीं है।
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युग्म (tx+9y=6) और (20x+15y=10) के अनंत हलों के लिए (t) का मान क्या होगा?
What is the value of (t) for infinitely many solutions of (tx+9y=6) and (20x+15y=10)?
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A (t=10)
B (t=11)
C (t=12)
D (t=13)
Explanation opens after your attempt
Step 1
Concept
All three ratios must be \(\frac{3}{5}\). Therefore, \(\frac{t}{20}=\frac{3}{5}\) gives (t=12).
Step 2
Why this answer is correct
The correct answer is C. (t=12). All three ratios must be \(\frac{3}{5}\). Therefore, \(\frac{t}{20}=\frac{3}{5}\) gives (t=12).
Step 3
Exam Tip
तीनों अनुपात \(\frac{3}{5}\) होने चाहिए। इसलिए \(\frac{t}{20}=\frac{3}{5}\) से (t=12)।
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युग्म ((y+1)x+2y=3) और (5x+(y-2)y=4) के अद्वितीय हल की सही शर्त कौन-सी है?
Which condition gives a unique solution for ((y+1)x+2y=3) and (5x+(y-2)y=4)?
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A \(y^2-y-12=0\)
B \(y^2-y-12\neq0\)
C \(y^2+y-12=0\)
D \(y^2+y-12\neq0\)
Explanation opens after your attempt
Correct Answer
B. \(y^2-y-12\neq0\)
Step 1
Concept
The determinant is (D=(y+1)(y-2)-10=y-2 -y-12). For a unique solution, \(D\neq0\) is required.
Step 2
Why this answer is correct
The correct answer is B. \(y^2-y-12\neq0\). The determinant is (D=(y+1)(y-2)-10=y-2 -y-12). For a unique solution, \(D\neq0\) is required.
Step 3
Exam Tip
सारणिक (D=(y+1)(y-2)-10=y-2 -y-12) है। अद्वितीय हल के लिए \(D\neq0\) चाहिए।
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