B. मूल बिंदु हल क्षेत्र में है/origin is in the solution region
Step 1
Concept
Substituting ((0,0)) gives (0<8), which is true. Hence the side containing the origin is the solution region.
Step 2
Why this answer is correct
The correct answer is B. मूल बिंदु हल क्षेत्र में है / origin is in the solution region. Substituting ((0,0)) gives (0<8), which is true. Hence the side containing the origin is the solution region.
Step 3
Exam Tip
((0,0)) रखने पर (0<8) सत्य है। इसलिए मूल बिंदु वाला भाग हल क्षेत्र है।
The point ((1,2)) is the direct intersection of (x=1) and (y=2). To identify vertices, intersect boundary lines pairwise.
Step 2
Why this answer is correct
The correct answer is D. (x=1) और (y=2) / (x=1) and (y=2). The point ((1,2)) is the direct intersection of (x=1) and (y=2). To identify vertices, intersect boundary lines pairwise.
Step 3
Exam Tip
((1,2)) सीधे (x=1) और (y=2) का प्रतिच्छेद है। शीर्ष पहचानने के लिए सीमाओं को जोड़ी में काटें।
The region is a rectangle of width (4) and height (3). Hence the area is \(4\times 3=12\).
Step 2
Why this answer is correct
The correct answer is C. (12) वर्ग इकाई / (12) square units. The region is a rectangle of width (4) and height (3). Hence the area is \(4\times 3=12\).
Step 3
Exam Tip
क्षेत्र (4) चौड़ाई और (3) ऊंचाई वाला आयत है। इसलिए क्षेत्रफल \(4\times 3=12\) है।
In the first quadrant, the region above (x+y=9) extends infinitely. Hence the solution region is unbounded.
Step 2
Why this answer is correct
The correct answer is D. असीमित क्षेत्र / unbounded region. In the first quadrant, the region above (x+y=9) extends infinitely. Hence the solution region is unbounded.
Step 3
Exam Tip
प्रथम चतुर्थांश में (x+y=9) के ऊपर का क्षेत्र अनंत तक फैलता है। इसलिए हल क्षेत्र असीमित है।
To the right, (x)-values are greater than (3), and a solid line includes equality. Therefore \(x\ge 3\) is correct.
Step 2
Why this answer is correct
The correct answer is A. \(x\ge 3\). To the right, (x)-values are greater than (3), and a solid line includes equality. Therefore \(x\ge 3\) is correct.
Step 3
Exam Tip
दाईं ओर (x) के मान (3) से बड़े होते हैं और ठोस रेखा बराबरी शामिल करती है। इसलिए \(x\ge 3\) सही है।
C. \(y\ge \frac{5x-10}{2}\) के ऊपर/above \(y\ge \frac{5x-10}{2}\)
Step 1
Concept
From \(-2y\le 10-5x\), reversing the sign gives \(y\ge \frac{5x-10}{2}\). The inequality sign changes when dividing by a negative.
Step 2
Why this answer is correct
The correct answer is C. \(y\ge \frac{5x-10}{2}\) के ऊपर / above \(y\ge \frac{5x-10}{2}\). From \(-2y\le 10-5x\), reversing the sign gives \(y\ge \frac{5x-10}{2}\). The inequality sign changes when dividing by a negative.
Step 3
Exam Tip
\(-2y\le 10-5x\) से चिन्ह पलटकर \(y\ge \frac{5x-10}{2}\) मिलता है। ऋणात्मक से भाग देने पर असमानता का चिन्ह बदलता है।
D. ((2,3)) नहीं बल्कि ((2,3))/not ((2,3)) but ((2,3))
Step 1
Concept
Solving (x+2y=8) and (3x-y=3) gives (x=2) and (y=3). Verify the intersection in both equations.
Step 2
Why this answer is correct
The correct answer is D. ((2,3)) नहीं बल्कि ((2,3)) / not ((2,3)) but ((2,3)). Solving (x+2y=8) and (3x-y=3) gives (x=2) and (y=3). Verify the intersection in both equations.
Step 3
Exam Tip
(x+2y=8) और (3x-y=3) हल करने पर (x=2) और (y=3) मिलता है। प्रतिच्छेद बिंदु को दोनों समीकरणों में जांचें।
The point ((3,2)) satisfies all conditions because \(3\ge 2\), \(2\ge 1\), and \(3+2\le 7\). Check each condition separately.
Step 2
Why this answer is correct
The correct answer is A. ((3,2)). The point ((3,2)) satisfies all conditions because \(3\ge 2\), \(2\ge 1\), and \(3+2\le 7\). Check each condition separately.
Step 3
Exam Tip
((3,2)) सभी शर्तों को संतुष्ट करता है क्योंकि \(3\ge 2\), \(2\ge 1\) और \(3+2\le 7\)। हर शर्त अलग से जांचें।
In \(y\le -3x+9\), (y) is less than or equal to the line value. Therefore the region below the line including the boundary is the solution.
Step 2
Why this answer is correct
The correct answer is D. रेखा के नीचे सहित / below the line including it. In \(y\le -3x+9\), (y) is less than or equal to the line value. Therefore the region below the line including the boundary is the solution.
Step 3
Exam Tip
\(y\le -3x+9\) में (y) रेखा के मान से कम या बराबर है। इसलिए रेखा के नीचे का भाग सीमा सहित हल है।
C. टूटी रेखा और मूल बिंदु के विपरीत ओर/dashed line and opposite to origin
Step 1
Concept
The sign (>) makes the boundary dashed. Substituting ((0,0)) gives (0>8), false, so shade the opposite side.
Step 2
Why this answer is correct
The correct answer is C. टूटी रेखा और मूल बिंदु के विपरीत ओर / dashed line and opposite to origin. The sign (>) makes the boundary dashed. Substituting ((0,0)) gives (0>8), false, so shade the opposite side.
Step 3
Exam Tip
चिन्ह (>) होने से रेखा टूटी होगी। ((0,0)) रखने पर (0>8) असत्य है इसलिए विपरीत ओर शेड होगा।
B. मूल बिंदु वाली ओर शेड होगी/side containing origin is shaded
Step 1
Concept
Substituting ((0,0)) gives (0<12), which is true. Hence the half-plane containing the origin is the solution.
Step 2
Why this answer is correct
The correct answer is B. मूल बिंदु वाली ओर शेड होगी / side containing origin is shaded. Substituting ((0,0)) gives (0<12), which is true. Hence the half-plane containing the origin is the solution.
Step 3
Exam Tip
((0,0)) रखने पर (0<12) सत्य है। इसलिए मूल बिंदु वाली ओर का अर्ध-समतल हल है।
In the first quadrant, the region above both lines extends infinitely. Therefore the solution region is unbounded.
Step 2
Why this answer is correct
The correct answer is B. असीमित क्षेत्र / unbounded region. In the first quadrant, the region above both lines extends infinitely. Therefore the solution region is unbounded.
Step 3
Exam Tip
प्रथम चतुर्थांश में दोनों रेखाओं के ऊपर का क्षेत्र अनंत तक फैलता है। इसलिए हल क्षेत्र असीमित है।
A. हल में शामिल और सीमा पर/included in solution and on boundary
Step 1
Concept
Substituting ((0,0)) gives \(0\le 0\), which is true with equality. So the origin is included on the boundary line.
Step 2
Why this answer is correct
The correct answer is A. हल में शामिल और सीमा पर / included in solution and on boundary. Substituting ((0,0)) gives \(0\le 0\), which is true with equality. So the origin is included on the boundary line.
Step 3
Exam Tip
((0,0)) रखने पर \(0\le 0\) सत्य है और बराबरी भी है। इसलिए मूल बिंदु सीमा रेखा पर शामिल है।
The same (y)-value cannot be both greater than and less than (1). Therefore there is no common solution.
Step 2
Why this answer is correct
The correct answer is C. रिक्त समुच्चय / empty set. The same (y)-value cannot be both greater than and less than (1). Therefore there is no common solution.
Step 3
Exam Tip
एक ही (y) मान (1) से बड़ा और छोटा दोनों नहीं हो सकता। इसलिए कोई संयुक्त हल नहीं है।
Both conditions together give (x=2). When \(\ge\) and \(\le\) occur at the same number, equality is obtained.
Step 2
Why this answer is correct
The correct answer is A. रेखा (x=2) / line (x=2). Both conditions together give (x=2). When \(\ge\) and \(\le\) occur at the same number, equality is obtained.
Step 3
Exam Tip
दोनों शर्तें मिलकर (x=2) देती हैं। जब \(\ge\) और \(\le\) एक ही संख्या पर हों तो बराबरी मिलती है।
B. एक-आयामी सीमित हल/one-dimensional bounded solution
Step 1
Concept
A line segment has infinitely many points but not a two-dimensional area. It is considered a bounded linear solution.
Step 2
Why this answer is correct
The correct answer is B. एक-आयामी सीमित हल / one-dimensional bounded solution. A line segment has infinitely many points but not a two-dimensional area. It is considered a bounded linear solution.
Step 3
Exam Tip
रेखाखंड में अनंत बिंदु होते हैं लेकिन वह क्षेत्रफल वाला द्वि-आयामी भाग नहीं होता। इसे सीमित रैखिक हल माना जाता है।
