The second equation must be (3) times the first, so (k=126). In coincident lines, the constant term also changes in the same ratio.
Step 2
Why this answer is correct
The correct answer is C. (126). The second equation must be (3) times the first, so (k=126). In coincident lines, the constant term also changes in the same ratio.
Step 3
Exam Tip
दूसरा समीकरण पहले का (3) गुना होना चाहिए, इसलिए (k=126)। संपाती रेखाओं में स्थिर पद भी उसी अनुपात में बदलता है।
Substituting ((2,8)) gives \(13\cdot2+4\cdot8=58\), not (52). Check every point in the equation before drawing the graph.
Step 2
Why this answer is correct
The correct answer is D. ((2,8)). Substituting ((2,8)) gives \(13\cdot2+4\cdot8=58\), not (52). Check every point in the equation before drawing the graph.
Step 3
Exam Tip
((2,8)) रखने पर \(13\cdot2+4\cdot8=58\), जो (52) नहीं है। ग्राफ बनाने से पहले हर बिंदु को समीकरण में जांचें।
For parallel lines, \(\frac{9}{27}=\frac{2}{m}\), so (m=6). Since \(\frac{18}{55}\neq\frac{1}{3}\), the lines are not coincident.
Step 2
Why this answer is correct
The correct answer is C. (6). For parallel lines, \(\frac{9}{27}=\frac{2}{m}\), so (m=6). Since \(\frac{18}{55}\neq\frac{1}{3}\), the lines are not coincident.
Step 3
Exam Tip
समांतर के लिए \(\frac{9}{27}=\frac{2}{m}\), इसलिए (m=6)। क्योंकि \(\frac{18}{55}\neq\frac{1}{3}\), रेखाएं संपाती नहीं हैं।
The coefficient ratio is \(\frac{1}{2}\), so coincidence needs (k=90). For (k=88), the lines are distinct and parallel.
Step 2
Why this answer is correct
The correct answer is B. (k=88). The coefficient ratio is \(\frac{1}{2}\), so coincidence needs (k=90). For (k=88), the lines are distinct and parallel.
Step 3
Exam Tip
गुणांक का अनुपात \(\frac{1}{2}\) है, इसलिए संपाती होने के लिए (k=90) चाहिए। (k=88) पर रेखाएं समांतर अलग-अलग हैं।
\(\frac{3}{6}=\frac{-8}{-16}\neq\frac{11}{25}\), so the lines are distinct and parallel. Such a pair has no solution.
Step 2
Why this answer is correct
The correct answer is C. कोई समाधान नहीं / No solution. \(\frac{3}{6}=\frac{-8}{-16}\neq\frac{11}{25}\), so the lines are distinct and parallel. Such a pair has no solution.
Step 3
Exam Tip
\(\frac{3}{6}=\frac{-8}{-16}\neq\frac{11}{25}\), इसलिए रेखाएं समांतर अलग-अलग हैं। ऐसे युग्म का कोई समाधान नहीं होता।
C. समाधान सेट वही रेखा (7x+8y=56) है/The solution set is the line (7x+8y=56)
Step 1
Concept
The second equation is (2) times the first, so both are the same line. The solution is all points on that line, not the whole plane.
Step 2
Why this answer is correct
The correct answer is C. समाधान सेट वही रेखा (7x+8y=56) है / The solution set is the line (7x+8y=56). The second equation is (2) times the first, so both are the same line. The solution is all points on that line, not the whole plane.
Step 3
Exam Tip
दूसरा समीकरण पहले का (2) गुना है, इसलिए दोनों एक ही रेखा हैं। समाधान उसी रेखा के सभी बिंदु हैं, पूरा तल नहीं।
Substituting ((-4,1)) makes (x+y=-3) and (2x-y=-9) both true. Substituting the intersection point in both equations is the fastest check.
Step 2
Why this answer is correct
The correct answer is A. (x+y=-3), (2x-y=-9). Substituting ((-4,1)) makes (x+y=-3) and (2x-y=-9) both true. Substituting the intersection point in both equations is the fastest check.
Step 3
Exam Tip
((-4,1)) रखने पर (x+y=-3) और (2x-y=-9) दोनों सही हैं। प्रतिच्छेद बिंदु को दोनों समीकरणों में रखना सबसे तेज जांच है।
The first equation is (5) times the second. Therefore the lines are coincident and give infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. संपाती / Coincident. The first equation is (5) times the second. Therefore the lines are coincident and give infinitely many solutions.
Step 3
Exam Tip
पहला समीकरण दूसरे का (5) गुना है। इसलिए दोनों रेखाएं संपाती हैं और अनंत समाधान देती हैं।
Substituting ((4,3)) gives (2x+5y=31) but not (3x-y=7); the true common point is (\left\(\frac{66}{17},\frac{79}{17}\right\)). Verify in both equations before choosing.
Step 2
Why this answer is correct
The correct answer is A. ((4,3)). Substituting ((4,3)) gives (2x+5y=31) but not (3x-y=7); the true common point is (\left\(\frac{66}{17},\frac{79}{17}\right\)). Verify in both equations before choosing.
Step 3
Exam Tip
((4,3)) रखने पर (2x+5y=31) और (3x-y=9) नहीं; सही साझा बिंदु (\left\(\frac{66}{17},\frac{79}{17}\right\)) है। सही उत्तर चुनने से पहले दोनों समीकरणों में जांच करें।
Substituting ((4,3)) does not give (2x+5y=29), so it is not correct; the true solution is (\left\(\frac{64}{17},\frac{73}{17}\right\)). Check a point in both equations.
Step 2
Why this answer is correct
The correct answer is C. ((4,3)). Substituting ((4,3)) does not give (2x+5y=29), so it is not correct; the true solution is (\left\(\frac{64}{17},\frac{73}{17}\right\)). Check a point in both equations.
Step 3
Exam Tip
((4,3)) रखने पर (2x+5y=23) नहीं, इसलिए यह गलत होता; सही हल (\left\(\frac{64}{17},\frac{73}{17}\right\)) है। विकल्प जांचते समय दोनों समीकरणों में बिंदु रखना जरूरी है।
The equations are (x+y=74) and (x-y=16), giving (x=45), (y=29). In word problems, first define the variables clearly.
Step 2
Why this answer is correct
The correct answer is A. ((45,29)). The equations are (x+y=74) and (x-y=16), giving (x=45), (y=29). In word problems, first define the variables clearly.
Step 3
Exam Tip
समीकरण (x+y=74) और (x-y=16) हैं, जिनसे (x=45), (y=29)। शब्द-प्रश्न में पहले चर स्पष्ट तय करें।
The second equation is (3) times the first, so (m=21). In a coincident line, the coefficient of (y) also changes in the same ratio.
Step 2
Why this answer is correct
The correct answer is C. (21). The second equation is (3) times the first, so (m=21). In a coincident line, the coefficient of (y) also changes in the same ratio.
Step 3
Exam Tip
दूसरा समीकरण पहले का (3) गुना है, इसलिए (m=21)। संपाती रेखा में (y) का गुणांक भी उसी अनुपात में बदलता है।
The first equation must be (4) times the second, so (a=20). In coincident lines, all terms change by the same multiplier.
Step 2
Why this answer is correct
The correct answer is C. (20). The first equation must be (4) times the second, so (a=20). In coincident lines, all terms change by the same multiplier.
Step 3
Exam Tip
पहला समीकरण दूसरे का (4) गुना होना चाहिए, इसलिए (a=20)। संपाती रेखाओं में सभी पद समान गुणक से बदलते हैं।
\(\frac{9}{3}=\frac{12}{4}\neq\frac{45}{20}\), so the lines are distinct and parallel. Check the constant term ratio also.
Step 2
Why this answer is correct
The correct answer is C. समांतर अलग-अलग / Distinct parallel. \(\frac{9}{3}=\frac{12}{4}\neq\frac{45}{20}\), so the lines are distinct and parallel. Check the constant term ratio also.
Step 3
Exam Tip
\(\frac{9}{3}=\frac{12}{4}\neq\frac{45}{20}\), इसलिए रेखाएं समांतर अलग-अलग हैं। अनुपातों में स्थिर पद को भी देखें।
Substituting ((-1,6)) makes (2x+y=4) and (x-y=-7) both true. The intersection point must lie on both lines.
Step 2
Why this answer is correct
The correct answer is A. (2x+y=4), (x-y=-7). Substituting ((-1,6)) makes (2x+y=4) and (x-y=-7) both true. The intersection point must lie on both lines.
Step 3
Exam Tip
((-1,6)) रखने पर (2x+y=4) और (x-y=-7) दोनों सही हैं। प्रतिच्छेद बिंदु दोनों रेखाओं पर होना चाहिए।
The coefficient ratio is \(\frac{1}{2}\); coincidence needs (k=54). For (k=50), the lines will be distinct and parallel.
Step 2
Why this answer is correct
The correct answer is B. (k=50). The coefficient ratio is \(\frac{1}{2}\); coincidence needs (k=54). For (k=50), the lines will be distinct and parallel.
Step 3
Exam Tip
गुणांक अनुपात \(\frac{1}{2}\) है; संपाती होने के लिए (k=54) चाहिए। (k=50) पर रेखाएं समांतर अलग-अलग होंगी।
\(\frac{7}{14}=\frac{-3}{-6}\neq\frac{18}{41}\), so the lines are distinct and parallel. An inconsistent pair has no solution.
Step 2
Why this answer is correct
The correct answer is B. असंगत / Inconsistent. \(\frac{7}{14}=\frac{-3}{-6}\neq\frac{18}{41}\), so the lines are distinct and parallel. An inconsistent pair has no solution.
Step 3
Exam Tip
\(\frac{7}{14}=\frac{-3}{-6}\neq\frac{18}{41}\), इसलिए रेखाएं समांतर अलग-अलग हैं। असंगत युग्म का कोई समाधान नहीं होता।
Lines with equal slopes and different (y)-intercepts are parallel. Therefore they have no intersection.
Step 2
Why this answer is correct
The correct answer is B. कोई समाधान नहीं / No solution. Lines with equal slopes and different (y)-intercepts are parallel. Therefore they have no intersection.
Step 3
Exam Tip
समान ढाल और अलग (y)-अवरोध वाली रेखाएं समांतर होती हैं। इसलिए उनका कोई प्रतिच्छेद नहीं होता।
B. केवल ((10,0)) और ((0,12)) रेखा पर हैं/Only ((10,0)) and ((0,12)) lie on the line
Step 1
Concept
((10,0)) and ((0,12)) satisfy the equation, but ((5,6)) does not give (60). Check points before drawing the graph.
Step 2
Why this answer is correct
The correct answer is B. केवल ((10,0)) और ((0,12)) रेखा पर हैं / Only ((10,0)) and ((0,12)) lie on the line. ((10,0)) and ((0,12)) satisfy the equation, but ((5,6)) does not give (60). Check points before drawing the graph.
Step 3
Exam Tip
((10,0)) और ((0,12)) समीकरण को संतुष्ट करते हैं, लेकिन ((5,6)) देने पर (60) नहीं मिलता। ग्राफ बनाने से पहले बिंदुओं की जांच करें।
