Level 58 • 50/50 questions • 25 seconds per question.
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When will the pair (kx+5y=10) and (6x+15y=18) have a unique solution?
B. \(k\neq2\)
अद्वितीय हल के लिए \(\frac{k}{6}\neq\frac{5}{15}\) होना चाहिए। इसलिए \(k\neq2\) सही है। / For a unique solution, \(\frac{k}{6}\neq\frac{5}{15}\) must hold. Therefore, \(k\neq2\) is correct.
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What is the value of (m) for no solution in (4x+my=8) and (10x+15y=21)?
C. (m=6)
गुणांक अनुपात समान करने पर (m=6) मिलता है। स्थिर पद का अनुपात अलग है इसलिए कोई हल नहीं होगा। / Equating coefficient ratios gives (m=6). The constant ratio is different, so there is no solution.
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For infinitely many solutions of ((p-2)x+7y=14) and (9x+21y=42), what is (p)?
C. (p=5)
अनंत हलों के लिए तीनों अनुपात समान होने चाहिए। \(\frac{p-2}{9}=\frac{7}{21}\) से (p=5) मिलता है। / For infinitely many solutions, all three ratios must be equal. From \(\frac{p-2}{9}=\frac{7}{21}\), (p=5).
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Find the value of (a) for infinitely many solutions of (8x+ay=12) and (20x+10y=30).
B. (a=4)
यहाँ \(\frac{8}{20}=\frac{12}{30}\) है। अतः \(\frac{a}{10}=\frac{2}{5}\) से (a=4) होगा। / Here \(\frac{8}{20}=\frac{12}{30}\). Thus \(\frac{a}{10}=\frac{2}{5}\) gives (a=4).
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For (3x-2y=5) and (12x-8y=c) to have no solution, what condition is required on (c)?
B. \(c\neq20\)
गुणांक अनुपात \(\frac{1}{4}\) है। कोई हल नहीं के लिए \(\frac{5}{c}\neq\frac{1}{4}\) इसलिए \(c\neq20\)। / The coefficient ratio is \(\frac{1}{4}\). For no solution, \(\frac{5}{c}\neq\frac{1}{4}\), so \(c\neq20\).
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When will (qx+4y=9) and (15x+10y=18) have a unique solution?
A. \(q\neq6\)
अद्वितीय हल के लिए \(\frac{q}{15}\neq\frac{4}{10}\) होना चाहिए। इसलिए \(q\neq6\) सही शर्त है। / For a unique solution, \(\frac{q}{15}\neq\frac{4}{10}\) must hold. Hence \(q\neq6\) is correct.
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For infinitely many solutions of (5x+(r+1)y=11) and (15x+18y=33), what is (r)?
B. (r=5)
अनंत हलों के लिए \(\frac{5}{15}=\frac{r+1}{18}=\frac{11}{33}\) चाहिए। इससे (r=5) मिलता है। / For infinitely many solutions, \(\frac{5}{15}=\frac{r+1}{18}=\frac{11}{33}\) is needed. This gives (r=5).
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What is the value of (n) for no solution in (2x+ny=7) and (6x+15y=19)?
C. (n=5)
गुणांक अनुपात समान करने पर \(\frac{2}{6}=\frac{n}{15}\) से (n=5) मिलता है। स्थिर अनुपात अलग है इसलिए युग्म असंगत है। / Equating coefficient ratios, \(\frac{2}{6}=\frac{n}{15}\) gives (n=5). The constant ratio is different, so the pair is inconsistent.
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When will (11x+3y=4) and (22x+sy=10) have a unique solution?
B. \(s\neq6\)
अद्वितीय हल के लिए \(\frac{11}{22}\neq\frac{3}{s}\) होना चाहिए। अतः \(s\neq6\) सही है। / For a unique solution, \(\frac{11}{22}\neq\frac{3}{s}\) must hold. Hence \(s\neq6\) is correct.
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What is the value of (t) for infinitely many solutions of (6x-ty=18) and (14x-21y=42)?
C. (t=9)
अनंत हलों के लिए \(\frac{6}{14}=\frac{-t}{-21}=\frac{18}{42}\) होना चाहिए। इससे (t=9) मिलता है। / For infinitely many solutions, \(\frac{6}{14}=\frac{-t}{-21}=\frac{18}{42}\) must hold. This gives (t=9).
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What type of pair is formed by the lines (7x+2y=9) and (21x+6y=25)?
C. असंगतInconsistent
गुणांक अनुपात समान है पर स्थिर पद का अनुपात अलग है। इसलिए रेखाएँ समांतर हैं और युग्म असंगत है। / Coefficient ratios are equal but the constant ratio is different. Hence the lines are parallel and the pair is inconsistent.
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What is the correct conclusion for the lines (9x-5y=13) and (18x-10y=26)?
C. अनंत हलInfinitely many solutions
दूसरा समीकरण पहले का (2) गुना है। इसलिए दोनों रेखाएँ संपाती हैं और अनंत हल देती हैं। / The second equation is (2) times the first. So both lines are coincident and give infinitely many solutions.
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How many solutions will the pair (10x+3y=17) and (20x+9y=31) have?
C. अद्वितीय हलUnique solution
यहाँ \(\frac{10}{20}\neq\frac{3}{9}\) है। इसलिए रेखाएँ एक बिंदु पर मिलती हैं। / Here \(\frac{10}{20}\neq\frac{3}{9}\). Therefore, the lines meet at one point.
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What is the solution status of (4x+9y=2) and (8x+18y=7)?
B. कोई हल नहींNo solution
गुणांक अनुपात समान है लेकिन \(\frac{2}{7}\) अलग है। इसलिए दोनों अलग समांतर रेखाएँ हैं। / The coefficient ratio is equal but \(\frac{2}{7}\) is different. Hence both are distinct parallel lines.
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Choose the correct option for (13x+4y=6) and (26x+8y=12).
C. अनंत हलInfinitely many solutions
तीनों अनुपात समान हैं। अतः रेखाएँ संपाती हैं और युग्म के अनंत हल हैं। / All three ratios are equal. Thus the lines are coincident and the pair has infinitely many solutions.
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What is the number of solutions of (12x-7y=5) and (18x-11y=8)?
C. अद्वितीय हलUnique solution
\(\frac{12}{18}\neq\frac{-7}{-11}\) होने से रेखाएँ काटती हैं। इसलिए अद्वितीय हल मिलेगा। / Since \(\frac{12}{18}\neq\frac{-7}{-11}\), the lines intersect. Hence a unique solution is obtained.
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For infinitely many solutions of ((u+3)x+8y=16) and (10x+20y=40), what is (u)?
A. (u=1)
अनंत हलों के लिए \(\frac{u+3}{10}=\frac{8}{20}=\frac{16}{40}\) चाहिए। इससे (u=1) मिलता है। / For infinitely many solutions, \(\frac{u+3}{10}=\frac{8}{20}=\frac{16}{40}\) is needed. This gives (u=1).
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What is the value of (v) for no solution in (5x+(v-2)y=3) and (15x+9y=14)?
C. (v=5)
गुणांक अनुपात समान करने पर (v-2=3) मिलता है। इसलिए (v=5) और स्थिर अनुपात अलग होने से कोई हल नहीं। / Equating coefficient ratios gives (v-2=3). Thus (v=5), and the different constant ratio gives no solution.
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When will (wx-6y=4) and (16x-24y=12) have a unique solution?
B. \(w\neq4\)
अद्वितीय हल के लिए \(\frac{w}{16}\neq\frac{-6}{-24}\) होना चाहिए। इसलिए \(w\neq4\) होगा। / For a unique solution, \(\frac{w}{16}\neq\frac{-6}{-24}\) must hold. Therefore, \(w\neq4\).
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For infinitely many solutions of \(3x+2y=\lambda\) and (18x+12y=48), what is \(\lambda\)?
B. \(\lambda=8\)
गुणांक अनुपात \(\frac{1}{6}\) है। अनंत हलों के लिए \(\frac{\lambda}{48}=\frac{1}{6}\) इसलिए \(\lambda=8\)। / The coefficient ratio is \(\frac{1}{6}\). For infinitely many solutions, \(\frac{\lambda}{48}=\frac{1}{6}\), so \(\lambda=8\).
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What is \(\mu\) for no solution in \(14x+\mu y=9\) and (21x+6y=20)?
B. \(\mu=4\)
\(\frac{14}{21}=\frac{2}{3}\) है। गुणांक अनुपात समान करने पर \(\frac{\mu}{6}=\frac{2}{3}\) से \(\mu=4\)। / \(\frac{14}{21}=\frac{2}{3}\). Equating coefficient ratios, \(\frac{\mu}{6}=\frac{2}{3}\) gives \(\mu=4\).
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When will (\(\alpha-1\)x+5y=2) and (8x+10y=6) have a unique solution?
B. \(\alpha\neq5\)
अद्वितीय हल के लिए \(\frac{\alpha-1}{8}\neq\frac{5}{10}\) चाहिए। इसलिए \(\alpha\neq5\) सही है। / For a unique solution, \(\frac{\alpha-1}{8}\neq\frac{5}{10}\) is required. Hence \(\alpha\neq5\) is correct.
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For infinitely many solutions of (9x+\(\beta+4\)y=15) and (27x+18y=45), what is \(\beta\)?
B. \(\beta=2\)
अनंत हलों में \(\frac{9}{27}=\frac{\beta+4}{18}=\frac{15}{45}\) होना चाहिए। इससे \(\beta=2\) मिलता है। / For infinitely many solutions, \(\frac{9}{27}=\frac{\beta+4}{18}=\frac{15}{45}\) must hold. This gives \(\beta=2\).
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What is the value of \(\gamma\) for no solution in \(5x+\gamma y=1\) and (25x+15y=11)?
B. \(\gamma=3\)
गुणांक अनुपात समान करने पर \(\frac{5}{25}=\frac{\gamma}{15}\) से \(\gamma=3\) मिलता है। स्थिर अनुपात अलग है। / Equating coefficient ratios, \(\frac{5}{25}=\frac{\gamma}{15}\) gives \(\gamma=3\). The constant ratio is different.
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B. \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\)
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। / A unique solution occurs when the lines intersect at one point. Its ratio form is \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\).
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If \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\), what is the solution status of the pair?
C. अनंत हलInfinitely many solutions
तीनों अनुपात समान होने पर रेखाएँ संपाती होती हैं। इसलिए अनंत हल मिलते हैं। / When all three ratios are equal, the lines are coincident. Therefore, infinitely many solutions occur.
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C. अनंत हलInfinitely many solutions
समान ढाल और समान अवरोध का अर्थ एक ही रेखा है। इसलिए ऐसे युग्म में अनंत हल होते हैं। / Same slope and same intercept mean the same line. Therefore, such a pair has infinitely many solutions.
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If two lines are distinct and parallel, what is the correct ratio condition?
B. \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\)
अलग समांतर रेखाओं में गुणांक अनुपात समान और स्थिर पद अनुपात अलग होता है। यही कोई हल नहीं की शर्त है। / For distinct parallel lines, coefficient ratios are equal and the constant ratio is different. This is the condition for no solution.
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If \(a_1b_2-a_2b_1\neq0\), what is the correct statement about the pair?
C. युग्म का अद्वितीय हल हैThe pair has a unique solution
सारणिक शून्य नहीं होने पर रेखाएँ एक बिंदु पर कटती हैं। इसलिए अद्वितीय हल होता है। / When the determinant is non-zero, the lines intersect at one point. Hence there is a unique solution.
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If (D=0), \(D_x=0\), and \(D_y=0\), what happens in a pair of two linear equations?
C. अनंत हलInfinitely many solutions
तीनों सारणिक शून्य होने पर समीकरण आश्रित हो सकते हैं। कक्षा (10) में इसे अनंत हल की स्थिति से जोड़कर देखें। / When all three determinants are zero, the equations may be dependent. In Class (10), link this with infinitely many solutions.
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For ((z+2)x+3y=5) and (6x+(z-1)y=7) to be non-unique, what are the possible values of (z)?
A. (z=4,-3)
अद्वितीय न होने के लिए ((z+2)(z-1)-18=0) होना चाहिए। इससे (z=4) या (z=-3) मिलता है। / For non-unique solutions, ((z+2)(z-1)-18=0) must hold. This gives (z=4) or (z=-3).
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What is the condition for a unique solution of (ax+2y=3) and (4x+ay=9)?
B. \(a^2\neq8\)
सारणिक \(D=a^2-8\) है। अद्वितीय हल के लिए \(D\neq0\) यानी \(a^2\neq8\)। / The determinant is \(D=a^2-8\). For a unique solution, \(D\neq0\), so \(a^2\neq8\).
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What is the value of (b) for infinitely many solutions of (2x+by=6) and (bx+8y=24)?
C. (b=4)
अनंत हलों के लिए \(\frac{2}{b}=\frac{b}{8}=\frac{6}{24}\) होना चाहिए। इससे (b=4) मिलता है। / For infinitely many solutions, \(\frac{2}{b}=\frac{b}{8}=\frac{6}{24}\) must hold. This gives (b=4).
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What is (c) for no solution in (cx+6y=5) and (9x+18y=10)?
B. (c=3)
गुणांक अनुपात समान करने पर \(\frac{c}{9}=\frac{6}{18}\) से (c=3) आता है। स्थिर अनुपात \(\frac{5}{10}\) अलग है। / Equating coefficient ratios, \(\frac{c}{9}=\frac{6}{18}\) gives (c=3). The constant ratio \(\frac{5}{10}\) is different.
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What is the value of (d) for infinitely many solutions of (4x+dy=7) and (12x+15y=21)?
C. (d=5)
अनंत हलों में \(\frac{4}{12}=\frac{d}{15}=\frac{7}{21}\) होना चाहिए। इसलिए (d=5) सही है। / For infinitely many solutions, \(\frac{4}{12}=\frac{d}{15}=\frac{7}{21}\) must hold. So (d=5) is correct.
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When will (ex-4y=9) and (18x-12y=27) have a unique solution?
B. \(e\neq6\)
अद्वितीय हल के लिए \(\frac{e}{18}\neq\frac{-4}{-12}\) चाहिए। अतः \(e\neq6\)। / For a unique solution, \(\frac{e}{18}\neq\frac{-4}{-12}\) is needed. Hence \(e\neq6\).
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What is (f) for no solution in (7x+fy=2) and (28x+20y=13)?
B. (f=5)
गुणांक अनुपात \(\frac{1}{4}\) बनाने के लिए (f=5) चाहिए। स्थिर पद अनुपात अलग होने से कोई हल नहीं होगा। / To make the coefficient ratio \(\frac{1}{4}\), (f=5) is required. The different constant ratio gives no solution.
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For infinitely many solutions of ((g-1)x+10y=15) and (8x+20y=30), what is (g)?
B. (g=5)
अनंत हलों के लिए \(\frac{g-1}{8}=\frac{10}{20}=\frac{15}{30}\) होना चाहिए। इससे (g=5) मिलता है। / For infinitely many solutions, \(\frac{g-1}{8}=\frac{10}{20}=\frac{15}{30}\) must hold. This gives (g=5).
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When will (hx+12y=6) and (10x+15y=20) have a unique solution?
B. \(h\neq8\)
अद्वितीय हल के लिए \(\frac{h}{10}\neq\frac{12}{15}\) होना चाहिए। इसलिए \(h\neq8\) सही है। / For a unique solution, \(\frac{h}{10}\neq\frac{12}{15}\) must hold. Hence \(h\neq8\) is correct.
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What is the value of (i) for no solution in (6x+iy=12) and (18x+27y=50)?
D. (i=9)
गुणांक अनुपात समान करने पर \(\frac{6}{18}=\frac{i}{27}\) से (i=9) आता है। स्थिर पद अनुपात समान नहीं है। / Equating coefficient ratios, \(\frac{6}{18}=\frac{i}{27}\) gives (i=9). The constant ratio is not equal.
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For infinitely many solutions of (jx+5y=10) and (14x+7y=14), what is (j)?
C. (j=10)
तीनों अनुपात \(\frac{5}{7}\) होने चाहिए। इसलिए \(\frac{j}{14}=\frac{5}{7}\) से (j=10)। / All three ratios must be \(\frac{5}{7}\). Therefore, \(\frac{j}{14}=\frac{5}{7}\) gives (j=10).
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What is the value of (k) for no solution in (2x+3y=4) and (kx+6y=11)?
C. (k=4)
गुणांक अनुपात समान करने पर \(\frac{2}{k}=\frac{3}{6}\) से (k=4) है। स्थिर पद अनुपात अलग है। / Equating coefficient ratios, \(\frac{2}{k}=\frac{3}{6}\) gives (k=4). The constant ratio is different.
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When will (mx+7y=13) and (16x+14y=26) have a unique solution?
B. \(m\neq8\)
अद्वितीय हल के लिए \(\frac{m}{16}\neq\frac{7}{14}\) होना चाहिए। इसलिए \(m\neq8\)। / For a unique solution, \(\frac{m}{16}\neq\frac{7}{14}\) must hold. Therefore, \(m\neq8\).
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For infinitely many solutions of (3x+ny=18) and (5x+10y=30), what is (n)?
C. (n=6)
अनंत हलों के लिए \(\frac{3}{5}=\frac{n}{10}=\frac{18}{30}\) चाहिए। इससे (n=6) मिलता है। / For infinitely many solutions, \(\frac{3}{5}=\frac{n}{10}=\frac{18}{30}\) is needed. This gives (n=6).
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What is (p) for no solution in (px-5y=7) and (12x-15y=19)?
A. (p=4)
गुणांक अनुपात समान करने पर \(\frac{p}{12}=\frac{-5}{-15}\) से (p=4) है। स्थिर अनुपात अलग है। / Equating coefficient ratios, \(\frac{p}{12}=\frac{-5}{-15}\) gives (p=4). The constant ratio is different.
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What is the value of (q) for infinitely many solutions of (qx+11y=22) and (18x+33y=66)?
C. (q=6)
अनंत हलों में \(\frac{q}{18}=\frac{11}{33}=\frac{22}{66}\) होता है। इसलिए (q=6) सही मान है। / For infinitely many solutions, \(\frac{q}{18}=\frac{11}{33}=\frac{22}{66}\). Therefore, (q=6) is correct.
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When will (rx+2y=5) and (9x+3y=8) have a unique solution?
B. \(r\neq6\)
अद्वितीय हल के लिए \(\frac{r}{9}\neq\frac{2}{3}\) चाहिए। इसलिए \(r\neq6\) होगा। / For a unique solution, \(\frac{r}{9}\neq\frac{2}{3}\) is required. Hence \(r\neq6\).
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What is (s) for no solution in (sx+4y=12) and (21x+12y=40)?
C. (s=7)
गुणांक अनुपात समान करने पर \(\frac{s}{21}=\frac{4}{12}\) से (s=7) है। स्थिर पद अनुपात समान नहीं है। / Equating coefficient ratios, \(\frac{s}{21}=\frac{4}{12}\) gives (s=7). The constant ratio is not equal.
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What is the value of (t) for infinitely many solutions of (tx+9y=6) and (20x+15y=10)?
C. (t=12)
तीनों अनुपात \(\frac{3}{5}\) होने चाहिए। इसलिए \(\frac{t}{20}=\frac{3}{5}\) से (t=12)। / All three ratios must be \(\frac{3}{5}\). Therefore, \(\frac{t}{20}=\frac{3}{5}\) gives (t=12).
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Which condition gives a unique solution for ((y+1)x+2y=3) and (5x+(y-2)y=4)?
B. \(y^2-y-12\neq0\)
सारणिक (D=(y+1)(y-2)-10=y-2-y-12) है। अद्वितीय हल के लिए \(D\neq0\) चाहिए। / The determinant is (D=(y+1)(y-2)-10=y-2-y-12). For a unique solution, \(D\neq0\) is required.
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