When coefficient ratios are different, the lines meet at one point. In exams, check the ratios of (a) and (b) first.
Step 2
Why this answer is correct
The correct answer is B. एक अद्वितीय हल / One unique solution. When coefficient ratios are different, the lines meet at one point. In exams, check the ratios of (a) and (b) first.
Step 3
Exam Tip
जब गुणांकों के अनुपात अलग होते हैं, रेखाएं एक बिंदु पर मिलती हैं। परीक्षा में पहले (a) और (b) के अनुपात जांचें।
In this case, the lines are parallel and do not meet. In exams, if the (c) ratio differs, write no solution.
Step 2
Why this answer is correct
The correct answer is B. असंगत / Inconsistent. In this case, the lines are parallel and do not meet. In exams, if the (c) ratio differs, write no solution.
Step 3
Exam Tip
इस स्थिति में रेखाएं समानांतर होती हैं और नहीं मिलतीं। परीक्षा में (c) का अनुपात अलग दिखे तो कोई हल नहीं लिखें।
If all three ratios are equal, the two lines are coincident. Such a pair has infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. If all three ratios are equal, the two lines are coincident. Such a pair has infinitely many solutions.
Step 3
Exam Tip
तीनों अनुपात बराबर हों तो दोनों रेखाएं संपाती होती हैं। ऐसी स्थिति में अनंत हल होते हैं।
Here (2/4=3/6=(-5)/(-10)), so the lines are coincident. In exams, if all ratios match, write infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल हैं / Has infinitely many solutions. Here (2/4=3/6=(-5)/(-10)), so the lines are coincident. In exams, if all ratios match, write infinitely many solutions.
Step 3
Exam Tip
यहां (2/4=3/6=(-5)/(-10)), इसलिए रेखाएं संपाती हैं। परीक्षा में सभी अनुपात बराबर हों तो अनंत हल लिखें।
Here (1/2=1/2) but ((-4)/(-9)) is different, so there is no solution. Parallel lines form an inconsistent pair.
Step 2
Why this answer is correct
The correct answer is C. कोई हल नहीं / No solution. Here (1/2=1/2) but ((-4)/(-9)) is different, so there is no solution. Parallel lines form an inconsistent pair.
Step 3
Exam Tip
यहां (1/2=1/2) लेकिन ((-4)/(-9)) अलग है, इसलिए कोई हल नहीं। समानांतर रेखाएं असंगत युग्म बनाती हैं।
Here \(3/6 \ne 2/5\), so the lines intersect at one point. Different (a) and (b) ratios give a unique solution.
Step 2
Why this answer is correct
The correct answer is A. एक अद्वितीय हल / One unique solution. Here \(3/6 \ne 2/5\), so the lines intersect at one point. Different (a) and (b) ratios give a unique solution.
Step 3
Exam Tip
यहां \(3/6 \ne 2/5\), इसलिए रेखाएं एक बिंदु पर कटती हैं। अलग (a) और (b) अनुपात अद्वितीय हल देते हैं।
The intersection point satisfies both equations. In a graph, one intersection means a unique solution.
Step 2
Why this answer is correct
The correct answer is B. एक अद्वितीय हल / One unique solution. The intersection point satisfies both equations. In a graph, one intersection means a unique solution.
Step 3
Exam Tip
कटने का बिंदु दोनों समीकरणों को संतुष्ट करता है। ग्राफ में एक intersection का मतलब unique solution है।
Parallel lines never meet, so there is no common point. In exams, connect parallel lines with no solution.
Step 2
Why this answer is correct
The correct answer is C. कोई हल नहीं / No solution. Parallel lines never meet, so there is no common point. In exams, connect parallel lines with no solution.
Step 3
Exam Tip
समानांतर रेखाएं कभी नहीं मिलतीं, इसलिए कोई सामान्य बिंदु नहीं होता। परीक्षा में parallel lines को no solution से जोड़ें।
Every point on the same line satisfies both equations. 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. Every point on the same line satisfies both equations. Therefore, such a pair has infinitely many solutions.
Step 3
Exam Tip
एक ही रेखा के हर बिंदु से दोनों समीकरण संतुष्ट होते हैं। इसलिए ऐसे युग्म के अनंत हल होते हैं।
Here (5/10=(-1)/(-2)) but (2/3) is different, so no solution exists. Equal first two ratios and a different third ratio give parallel lines.
Step 2
Why this answer is correct
The correct answer is B. कोई हल नहीं है / No solution exists. Here (5/10=(-1)/(-2)) but (2/3) is different, so no solution exists. Equal first two ratios and a different third ratio give parallel lines.
Step 3
Exam Tip
यहां (5/10=(-1)/(-2)) लेकिन (2/3) अलग है, इसलिए कोई हल नहीं। बराबर पहले दो अनुपात और अलग तीसरा अनुपात parallel lines देता है।
All three ratios are equal, so both equations represent the same line. This is called a consistent dependent pair.
Step 2
Why this answer is correct
The correct answer is B. संगत और आश्रित / Consistent and dependent. All three ratios are equal, so both equations represent the same line. This is called a consistent dependent pair.
Step 3
Exam Tip
तीनों अनुपात बराबर हैं, इसलिए दोनों समीकरण एक ही रेखा हैं। इसे consistent dependent pair कहते हैं।
Here (2/3 \ne (-5)/1), so there is a unique solution. If the first two ratios differ, the third ratio need not be checked.
Step 2
Why this answer is correct
The correct answer is C. एक अद्वितीय हल / One unique solution. Here (2/3 \ne (-5)/1), so there is a unique solution. If the first two ratios differ, the third ratio need not be checked.
Step 3
Exam Tip
यहां (2/3 \ne (-5)/1), इसलिए unique solution है। पहले दो अनुपात अलग हों तो तीसरा अनुपात देखने की जरूरत नहीं होती।
When all three ratios are equal, both lines lie on each other. These are called coincident lines.
Step 2
Why this answer is correct
The correct answer is C. संपाती रेखाएं / Coincident lines. When all three ratios are equal, both lines lie on each other. These are called coincident lines.
Step 3
Exam Tip
तीनों अनुपात बराबर होने पर दोनों रेखाएं एक-दूसरे पर ही होती हैं। इसे coincident lines कहते हैं।
Different ratios indicate different slopes, so the lines intersect. Intersecting lines give one solution.
Step 2
Why this answer is correct
The correct answer is C. कटती हुई / Intersecting. Different ratios indicate different slopes, so the lines intersect. Intersecting lines give one solution.
Step 3
Exam Tip
अलग अनुपातों के कारण ढालें अलग होती हैं, इसलिए रेखाएं कटती हैं। कटती रेखाओं से एक हल मिलता है।
Here (1/3=(-2)/(-6)=6/18), so the lines are coincident. Coincident lines have infinitely many common points.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. Here (1/3=(-2)/(-6)=6/18), so the lines are coincident. Coincident lines have infinitely many common points.
Step 3
Exam Tip
यहां (1/3=(-2)/(-6)=6/18), इसलिए रेखाएं संपाती हैं। संपाती रेखाओं के अनंत सामान्य बिंदु होते हैं।
Here \(6/12 \ne 7/15\), so there is one unique solution. Different coefficient ratios indicate intersecting lines.
Step 2
Why this answer is correct
The correct answer is A. एक अद्वितीय हल / One unique solution. Here \(6/12 \ne 7/15\), so there is one unique solution. Different coefficient ratios indicate intersecting lines.
Step 3
Exam Tip
यहां \(6/12 \ne 7/15\), इसलिए एक अद्वितीय हल है। अलग गुणांक अनुपात intersecting lines का संकेत है।
An inconsistent pair has no common solution. In a graph, it appears as parallel lines.
Step 2
Why this answer is correct
The correct answer is A. जब कोई हल न हो / When there is no solution. An inconsistent pair has no common solution. In a graph, it appears as parallel lines.
Step 3
Exam Tip
असंगत युग्म का कोई सामान्य हल नहीं होता। ग्राफ में यह समानांतर रेखाओं से दिखता है।
A. जब कम से कम एक हल हो/When there is at least one solution
Step 1
Concept
A consistent pair has at least one common solution. It may have one solution or infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is A. जब कम से कम एक हल हो / When there is at least one solution. A consistent pair has at least one common solution. It may have one solution or infinitely many solutions.
Step 3
Exam Tip
संगत युग्म में कम से कम एक सामान्य हल होता है। यह एक हल या अनंत हल दोनों हो सकता है।
Here (9/3=(-3)/(-1)=6/2), so both equations are the same line. Therefore, there are infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल हैं / Has infinitely many solutions. Here (9/3=(-3)/(-1)=6/2), so both equations are the same line. Therefore, there are infinitely many solutions.
Step 3
Exam Tip
यहां (9/3=(-3)/(-1)=6/2), इसलिए दोनों समीकरण एक ही रेखा हैं। इसलिए अनंत हल हैं।
Here (2/4=1/2) but (5/12) is different, so the lines are parallel. Distinct parallel lines have no solution.
Step 2
Why this answer is correct
The correct answer is C. कोई हल नहीं / No solution. Here (2/4=1/2) but (5/12) is different, so the lines are parallel. Distinct parallel lines have no solution.
Step 3
Exam Tip
यहां (2/4=1/2) लेकिन (5/12) अलग है, इसलिए रेखाएं समानांतर हैं। समानांतर अलग रेखाओं का कोई हल नहीं होता।
The second equation is (2) times the first, so both are the same line. The same line gives 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 are the same line. The same line gives infinitely many solutions.
Step 3
Exam Tip
दूसरा समीकरण पहले का (2) गुना है, इसलिए दोनों एक ही रेखा हैं। समान रेखा पर अनंत हल मिलते हैं।
Here \(1/2 \ne 3/5\), so the lines will meet at one point. Therefore, there is one unique solution.
Step 2
Why this answer is correct
The correct answer is C. एक अद्वितीय हल / One unique solution. Here \(1/2 \ne 3/5\), so the lines will meet at one point. Therefore, there is one unique solution.
Step 3
Exam Tip
यहां \(1/2 \ne 3/5\), इसलिए रेखाएं एक बिंदु पर मिलेंगी। इसलिए एक अद्वितीय हल है।
Coincident lines give infinitely many common points. Such a pair is called consistent dependent.
Step 2
Why this answer is correct
The correct answer is B. संगत और आश्रित / Consistent and dependent. Coincident lines give infinitely many common points. Such a pair is called consistent dependent.
Step 3
Exam Tip
संपाती रेखाएं अनंत सामान्य बिंदु देती हैं। ऐसा युग्म consistent dependent कहलाता है।
One common point gives exactly one solution. Such a pair is consistent independent.
Step 2
Why this answer is correct
The correct answer is A. संगत और स्वतंत्र / Consistent and independent. One common point gives exactly one solution. Such a pair is consistent independent.
Step 3
Exam Tip
एक सामान्य बिंदु होने से केवल एक हल मिलता है। ऐसा युग्म consistent independent होता है।
The second equation is (2) times the first. Therefore, both will appear as the same line in the graph.
Step 2
Why this answer is correct
The correct answer is C. एक ही रेखा / Same line. The second equation is (2) times the first. Therefore, both will appear as the same line in the graph.
Step 3
Exam Tip
दूसरा समीकरण पहले का (2) गुना है। इसलिए ग्राफ में दोनों एक ही रेखा दिखेंगी।
(2/4=3/6) but (6/15) is different, so the lines are parallel. Such a graph has no intersection.
Step 2
Why this answer is correct
The correct answer is C. अलग समानांतर रेखाएं / Distinct parallel lines. (2/4=3/6) but (6/15) is different, so the lines are parallel. Such a graph has no intersection.
Step 3
Exam Tip
(2/4=3/6) लेकिन (6/15) अलग है, इसलिए रेखाएं समानांतर हैं। ऐसे ग्राफ में कोई intersection नहीं होता।
A. वे एक बिंदु पर कटेंगी/They will intersect at one point
Step 1
Concept
Here \(5/10 \ne 2/3\), so the lines will intersect. Intersecting lines give one unique solution.
Step 2
Why this answer is correct
The correct answer is A. वे एक बिंदु पर कटेंगी / They will intersect at one point. Here \(5/10 \ne 2/3\), so the lines will intersect. Intersecting lines give one unique solution.
Step 3
Exam Tip
यहां \(5/10 \ne 2/3\), इसलिए रेखाएं कटेंगी। कटती रेखाएं एक अद्वितीय हल देती हैं।
For a unique solution, the ratios of (a) and (b) must be different. This is the condition for intersecting lines.
Step 2
Why this answer is correct
The correct answer is C. \(a_1 / a_2 \ne b_1 / b_2\). For a unique solution, the ratios of (a) and (b) must be different. This is the condition for intersecting lines.
Step 3
Exam Tip
अद्वितीय हल के लिए (a) और (b) के अनुपात अलग होने चाहिए। यह intersecting lines की शर्त है।
No solution occurs when the first two ratios are equal and the third is different. This is the condition for parallel lines.
Step 2
Why this answer is correct
The correct answer is A. \(a_1 / a_2=b_1 / b_2 \ne c_1 / c_2\). No solution occurs when the first two ratios are equal and the third is different. This is the condition for parallel lines.
Step 3
Exam Tip
कोई हल नहीं तब होता है जब पहले दो अनुपात बराबर और तीसरा अलग हो। यह parallel lines की शर्त है।
For infinitely many solutions, all three ratios must be equal. Then both equations form the same line.
Step 2
Why this answer is correct
The correct answer is C. \(a_1 / a_2=b_1 / b_2=c_1 / c_2\). For infinitely many solutions, all three ratios must be equal. Then both equations form the same line.
Step 3
Exam Tip
अनंत हल के लिए तीनों अनुपात बराबर होने चाहिए। इससे दोनों समीकरण एक ही रेखा बनाते हैं।
The second equation is (3) times the first, so both lines are coincident. Such a pair has infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. संगत और आश्रित / Consistent and dependent. The second equation is (3) times the first, so both lines are coincident. Such a pair has infinitely many solutions.
Step 3
Exam Tip
दूसरा समीकरण पहले का (3) गुना है, इसलिए दोनों रेखाएं संपाती हैं। ऐसे युग्म में अनंत हल होते हैं।
Same slope and different intercepts make the lines parallel. Distinct parallel lines give no solution.
Step 2
Why this answer is correct
The correct answer is B. कोई हल नहीं / No solution. Same slope and different intercepts make the lines parallel. Distinct parallel lines give no solution.
Step 3
Exam Tip
समान ढाल और अलग अवरोध रेखाओं को समानांतर बनाते हैं। समानांतर अलग रेखाएं कोई हल नहीं देतीं।
Lines with different slopes meet at one point. Therefore, the pair has one unique solution.
Step 2
Why this answer is correct
The correct answer is A. एक अद्वितीय हल / One unique solution. Lines with different slopes meet at one point. Therefore, the pair has one unique solution.
Step 3
Exam Tip
अलग ढाल वाली रेखाएं एक बिंदु पर मिलती हैं। इसलिए युग्म का एक अद्वितीय हल होता है।
Same slope and same intercept make both lines identical. Therefore, they have infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is B. संपाती रेखाएं / Coincident lines. Same slope and same intercept make both lines identical. Therefore, they have infinitely many solutions.
Step 3
Exam Tip
समान ढाल और समान अवरोध से दोनों रेखाएं एक ही होती हैं। इसलिए उनके अनंत हल होते हैं।
A. दूसरा पहले का (2) गुना है/The second is (2) times the first
Step 1
Concept
All terms of the second equation are (2) times the first. Hence, both are the same line and have infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is A. दूसरा पहले का (2) गुना है / The second is (2) times the first. All terms of the second equation are (2) times the first. Hence, both are the same line and have infinitely many solutions.
Step 3
Exam Tip
दूसरे समीकरण के सभी पद पहले के (2) गुना हैं। इसलिए दोनों एक ही रेखा हैं और अनंत हल हैं।
Here (7/14=1/2) but (10/25) is different. Therefore, the lines are parallel and distinct.
Step 2
Why this answer is correct
The correct answer is C. समानांतर अलग रेखाएं / Distinct parallel lines. Here (7/14=1/2) but (10/25) is different. Therefore, the lines are parallel and distinct.
Step 3
Exam Tip
यहां (7/14=1/2) लेकिन (10/25) अलग है। इसलिए रेखाएं समानांतर और अलग हैं।
C. रेखाएं एक बिंदु पर कटती हैं/Lines intersect at one point
Step 1
Concept
Here \(2/3 \ne 5/7\), so the lines intersect at one point. This means there is one unique solution.
Step 2
Why this answer is correct
The correct answer is C. रेखाएं एक बिंदु पर कटती हैं / Lines intersect at one point. Here \(2/3 \ne 5/7\), so the lines intersect at one point. This means there is one unique solution.
Step 3
Exam Tip
यहां \(2/3 \ne 5/7\), इसलिए रेखाएं एक बिंदु पर कटती हैं। इसका मतलब एक अद्वितीय हल है।
The first equation is (2) times the second, so both lines are coincident. Coincident lines have infinitely many solutions.
Step 2
Why this answer is correct
The correct answer is C. अनंत हल / Infinitely many solutions. The first equation is (2) times the second, so both lines are coincident. Coincident lines have infinitely many solutions.
Step 3
Exam Tip
पहला समीकरण दूसरे का (2) गुना है, इसलिए दोनों रेखाएं संपाती हैं। संपाती रेखाओं के अनंत हल होते हैं।
