The vertices are ((0,0)), ((6,0)), ((4,4)), and ((0,6)). The shoelace method gives area (24) square units.
Step 2
Why this answer is correct
The correct answer is C. (24) वर्ग इकाई / (24) square units. The vertices are ((0,0)), ((6,0)), ((4,4)), and ((0,6)). The shoelace method gives area (24) square units.
Step 3
Exam Tip
शीर्ष ((0,0)), ((6,0)), ((4,4)), ((0,6)) हैं। शूलेस विधि से क्षेत्रफल (24) वर्ग इकाई मिलता है।
The square has area (25), and the triangle below (x+y=4) has area (8). Therefore the remaining area is (17).
Step 2
Why this answer is correct
The correct answer is C. (17) वर्ग इकाई / (17) square units. The square has area (25), and the triangle below (x+y=4) has area (8). Therefore the remaining area is (17).
Step 3
Exam Tip
पूरे वर्ग का क्षेत्रफल (25) है और रेखा (x+y=4) के नीचे का त्रिभुज क्षेत्रफल (8) है। इसलिए बचा क्षेत्रफल (17) है।
A. ((0,0)), ((4,0)), (\left\(\frac{14}{5},\frac{18}{5}\right\)), ((0,5))
Step 1
Concept
The slant boundaries intersect at (\left\(\frac{14}{5},\frac{18}{5}\right\)). Use valid intercepts on the axes to list all corners.
Step 2
Why this answer is correct
The correct answer is A. ((0,0)), ((4,0)), (\left\(\frac{14}{5},\frac{18}{5}\right\)), ((0,5)). The slant boundaries intersect at (\left\(\frac{14}{5},\frac{18}{5}\right\)). Use valid intercepts on the axes to list all corners.
Step 3
Exam Tip
दोनों तिरछी सीमाओं का प्रतिच्छेद (\left\(\frac{14}{5},\frac{18}{5}\right\)) है। अक्षों पर वैध अवरोध लेकर सभी कोने चुनें।
The vertices are ((2,0)), ((6,0)), ((3,6)), and ((2,7)). Using the shoelace method or splitting into parts gives area (20).
Step 2
Why this answer is correct
The correct answer is A. (20) वर्ग इकाई / (20) square units. The vertices are ((2,0)), ((6,0)), ((3,6)), and ((2,7)). Using the shoelace method or splitting into parts gives area (20).
Step 3
Exam Tip
शीर्ष ((2,0)), ((6,0)), ((3,6)), ((2,7)) हैं। शूलेस विधि या भागों में बांटकर क्षेत्रफल (20) मिलता है।
A. ((0,0)), ((6,0)), (\left\(\frac{21}{5},\frac{18}{5}\right\)), ((0,5))
Step 1
Concept
The intersection of the two slant lines is (\left\(\frac{21}{5},\frac{18}{5}\right\)). Use valid intercepts on the axes to choose all polygon corners.
Step 2
Why this answer is correct
The correct answer is A. ((0,0)), ((6,0)), (\left\(\frac{21}{5},\frac{18}{5}\right\)), ((0,5)). The intersection of the two slant lines is (\left\(\frac{21}{5},\frac{18}{5}\right\)). Use valid intercepts on the axes to choose all polygon corners.
Step 3
Exam Tip
दोनों तिरछी रेखाओं का प्रतिच्छेद (\left\(\frac{21}{5},\frac{18}{5}\right\)) है। अक्षों पर वैध अवरोध लेकर पूरे बहुभुज के कोने चुनें।
Solving the two boundary equations gives \(x=\frac{16}{5}\) and \(y=\frac{27}{5}\). It is a good method to test the intersection in all inequalities.
Step 2
Why this answer is correct
The correct answer is A. (\left\(\frac{16}{5},\frac{27}{5}\right\)). Solving the two boundary equations gives \(x=\frac{16}{5}\) and \(y=\frac{27}{5}\). It is a good method to test the intersection in all inequalities.
Step 3
Exam Tip
दोनों सीमा समीकरण हल करने पर \(x=\frac{16}{5}\) और \(y=\frac{27}{5}\) मिलता है। प्रतिच्छेद को सभी असमानताओं से जांचना अच्छा तरीका है।
A. यह दोनों तिरछी सीमाओं के प्रतिच्छेद पर स्थित हल है/It is a solution at the intersection of both slant boundaries
Step 1
Concept
At ((6,4)), both (x+y=10) and (x-y=2) hold as equalities. So it is the valid intersection of both boundary lines.
Step 2
Why this answer is correct
The correct answer is A. यह दोनों तिरछी सीमाओं के प्रतिच्छेद पर स्थित हल है / It is a solution at the intersection of both slant boundaries. At ((6,4)), both (x+y=10) and (x-y=2) hold as equalities. So it is the valid intersection of both boundary lines.
Step 3
Exam Tip
((6,4)) पर (x+y=10) और (x-y=2) दोनों बराबरी देते हैं। इसलिए यह दोनों सीमा-रेखाओं का वैध प्रतिच्छेद है।
A. त्रिभुज जिसके शीर्ष ((2,5)), ((5,2)), ((5,5)) हैं/Triangle with vertices ((2,5)), ((5,2)), ((5,5))
Step 1
Concept
Inside the square \(0\leq x\leq 5\), \(0\leq y\leq 5\), the part above (x+y=7) remains. Its vertices are ((2,5)), ((5,2)), and ((5,5)).
Step 2
Why this answer is correct
The correct answer is A. त्रिभुज जिसके शीर्ष ((2,5)), ((5,2)), ((5,5)) हैं / Triangle with vertices ((2,5)), ((5,2)), ((5,5)). Inside the square \(0\leq x\leq 5\), \(0\leq y\leq 5\), the part above (x+y=7) remains. Its vertices are ((2,5)), ((5,2)), and ((5,5)).
Step 3
Exam Tip
वर्ग \(0\leq x\leq 5\), \(0\leq y\leq 5\) में रेखा (x+y=7) के ऊपर का कोना बचता है। उसके शीर्ष ((2,5)), ((5,2)), ((5,5)) हैं।
The vertices are ((2,1)), (\left\(\frac{15}{2},1\right\)), and (\(2,\frac{14}{3}\)). Check parallel distances carefully before using triangle area.
Step 2
Why this answer is correct
The correct answer is B. (15) वर्ग इकाई / (15) square units. The vertices are ((2,1)), (\left\(\frac{15}{2},1\right\)), and (\(2,\frac{14}{3}\)). Check parallel distances carefully before using triangle area.
Step 3
Exam Tip
शीर्ष ((2,1)), (\left\(\frac{15}{2},1\right\)), (\(2,\frac{14}{3}\)) हैं। आधार \(\frac{11}{2}\) और ऊंचाई \(\frac{11}{3}\) से क्षेत्रफल \(\frac{121}{12}\) नहीं बल्कि सही गणना के लिए अक्षों के समांतर दूरी जांचें।
Solving the two equations gives \(x=\frac{18}{5}\) and \(y=\frac{14}{5}\). This is the inner corner on the graph.
Step 2
Why this answer is correct
The correct answer is A. (\left\(\frac{18}{5},\frac{14}{5}\right\)). Solving the two equations gives \(x=\frac{18}{5}\) and \(y=\frac{14}{5}\). This is the inner corner on the graph.
Step 3
Exam Tip
दोनों समीकरण हल करने पर \(x=\frac{18}{5}\) और \(y=\frac{14}{5}\) मिलता है। ग्राफ में यही आंतरिक कोना है।
A. शीर्ष ((2,0)), ((6,0)), ((4,2)) वाला बंद त्रिभुज/Closed triangle with vertices ((2,0)), ((6,0)), ((4,2))
Step 1
Concept
The three half-planes form a closed triangle. In exams, first find the intersection points of boundary lines.
Step 2
Why this answer is correct
The correct answer is A. शीर्ष ((2,0)), ((6,0)), ((4,2)) वाला बंद त्रिभुज / Closed triangle with vertices ((2,0)), ((6,0)), ((4,2)). The three half-planes form a closed triangle. In exams, first find the intersection points of boundary lines.
Step 3
Exam Tip
तीनों अर्द्ध-तल मिलकर बंद त्रिभुज बनाते हैं। परीक्षा में पहले रेखाओं के प्रतिच्छेद बिंदु निकालें।
Solving (x+2y=16) and (2x+y=14) together gives (\left\(\frac{22}{3},\frac{13}{3}\right\)). In a graph, always check key vertices from boundary intersections.
Step 2
Why this answer is correct
The correct answer is D. (\left\(\frac{22}{3},\frac{13}{3}\right\)). Solving (x+2y=16) and (2x+y=14) together gives (\left\(\frac{22}{3},\frac{13}{3}\right\)). In a graph, always check key vertices from boundary intersections.
Step 3
Exam Tip
रेखाओं (x+2y=16) और (2x+y=14) को साथ हल करने पर (\left\(\frac{22}{3},\frac{13}{3}\right\)) मिलता है। ग्राफ में मुख्य शीर्ष हमेशा सीमा रेखाओं के प्रतिच्छेद से जांचें।
At (x=0), the conditions give \(y\le 10\) and \(3y\le 18\), so the maximum is (y=6). To maximize a variable, look for the tightest bound in that direction.
Step 2
Why this answer is correct
The correct answer is B. (6). At (x=0), the conditions give \(y\le 10\) and \(3y\le 18\), so the maximum is (y=6). To maximize a variable, look for the tightest bound in that direction.
Step 3
Exam Tip
(x=0) पर शर्तें \(y\le 10\) और \(3y\le 18\) देती हैं, इसलिए अधिकतम (y=6)। किसी चर का अधिकतम पाने के लिए उस दिशा की कठोर सीमा देखें।
At ((2,2)), (4<5) and (6>4), so it is inside and not on a boundary. An interior point does not satisfy any boundary as equality.
Step 2
Why this answer is correct
The correct answer is B. ((2,2)). At ((2,2)), (4<5) and (6>4), so it is inside and not on a boundary. An interior point does not satisfy any boundary as equality.
Step 3
Exam Tip
((2,2)) पर (4<5) और (6>4), इसलिए यह अंदर है और सीमा पर नहीं। अंदरूनी बिंदु में कोई भी सीमा बराबरी नहीं देती।
The vertices are ((2,2)), ((8,2)), and ((2,5)); the area is \(\frac{1}{2}\times 6\times 3=9\). So the correct value should be (9).
Step 2
Why this answer is correct
The correct answer is A. (12). The vertices are ((2,2)), ((8,2)), and ((2,5)); the area is \(\frac{1}{2}\times 6\times 3=9\). So the correct value should be (9).
Step 3
Exam Tip
शीर्ष ((2,2)), ((8,2)), ((2,5)) हैं; क्षेत्रफल \(\frac{1}{2}\times 6\times 3=9\) है। इसलिए सही मान (9) होना चाहिए।
None of the listed points fully satisfies the system, because each fails at least one inequality. Always verify all inequalities before choosing an axial point.
Step 2
Why this answer is correct
The correct answer is A. ((0,4)). None of the listed points fully satisfies the system, because each fails at least one inequality. Always verify all inequalities before choosing an axial point.
Step 3
Exam Tip
((0,4)) पर \(4\ge 6\) गलत है; सही अक्षीय जांच में ((3,0)) होता, पर विकल्पों में ((0,4)) नहीं चलता। इसलिए दिए विकल्पों में कोई पूर्ण सही नहीं है।
At ((6,1)), (x+2y=8) and \(x+y=7\le 10\), so it lies on the second boundary. To identify a boundary point, check equality.
Step 2
Why this answer is correct
The correct answer is C. दूसरी सीमा पर / On the second boundary. At ((6,1)), (x+2y=8) and \(x+y=7\le 10\), so it lies on the second boundary. To identify a boundary point, check equality.
Step 3
Exam Tip
((6,1)) पर (x+2y=8) और \(x+y=7\le 10\), इसलिए यह दूसरी सीमा पर है। सीमा पहचानने के लिए बराबरी वाली शर्त देखें।
The lines (x=2) and (y=1) meet at ((2,1)), and it also satisfies \(x+y\le 8\). Check every vertex in all inequalities.
Step 2
Why this answer is correct
The correct answer is A. ((2,1)). The lines (x=2) and (y=1) meet at ((2,1)), and it also satisfies \(x+y\le 8\). Check every vertex in all inequalities.
Step 3
Exam Tip
रेखाएं (x=2) और (y=1) मिलकर ((2,1)) देती हैं और यह \(x+y\le 8\) को भी संतुष्ट करता है। हर शीर्ष को सभी असमानताओं में जांचें।
A. समानांतर रेखाओं के बीच पट्टी/Strip between parallel lines
Step 1
Concept
Both boundaries are parallel, and the region lies between them. For inequalities with the same left side, compare the constants.
Step 2
Why this answer is correct
The correct answer is A. समानांतर रेखाओं के बीच पट्टी / Strip between parallel lines. Both boundaries are parallel, and the region lies between them. For inequalities with the same left side, compare the constants.
Step 3
Exam Tip
दोनों सीमाएं समानांतर हैं और क्षेत्र उनके बीच है। समान बाईं ओर वाली असमानताओं में स्थिर पदों की तुलना करें।
The vertices are ((0,0)), ((4,0)), (\left\(\frac{8}{3},\frac{8}{3}\right\)), and ((0,4)). List all valid corners in order.
Step 2
Why this answer is correct
The correct answer is B. (4). The vertices are ((0,0)), ((4,0)), (\left\(\frac{8}{3},\frac{8}{3}\right\)), and ((0,4)). List all valid corners in order.
Step 3
Exam Tip
शीर्ष ((0,0)), ((4,0)), (\left\(\frac{8}{3},\frac{8}{3}\right\)), और ((0,4)) हैं। सभी वैध कोनों को क्रम से सूचीबद्ध करें।
The slant lines intersect at (\left\(\frac{10}{3},\frac{10}{3}\right\)), where \(x+y=\frac{20}{3}\). Check linear expressions at vertices for maxima.
Step 2
Why this answer is correct
The correct answer is C. (\left\(\frac{10}{3},\frac{10}{3}\right\)). The slant lines intersect at (\left\(\frac{10}{3},\frac{10}{3}\right\)), where \(x+y=\frac{20}{3}\). Check linear expressions at vertices for maxima.
Step 3
Exam Tip
दो तिरछी रेखाओं का प्रतिच्छेद (\left\(\frac{10}{3},\frac{10}{3}\right\)) है और वहां \(x+y=\frac{20}{3}\) मिलता है। रैखिक अभिव्यक्ति का अधिकतम कोनों पर जांचें।
