If (p>0), the (x)-intercept \(\frac{6}{p}\) is finite. If \(p\leq 0\), the region is not bounded in the (x)-direction in the first quadrant.
Step 2
Why this answer is correct
The correct answer is C. (p>0). If (p>0), the (x)-intercept \(\frac{6}{p}\) is finite. If \(p\leq 0\), the region is not bounded in the (x)-direction in the first quadrant.
Step 3
Exam Tip
यदि (p>0), तो (x)-अवरोध \(\frac{6}{p}\) सीमित होता है। \(p\leq 0\) होने पर प्रथम चतुर्थांश में क्षेत्र (x) दिशा में सीमित नहीं रहता।
The \(\geq\) conditions give an upper-side region in the first quadrant. Boundaries are included and the region extends infinitely.
Step 2
Why this answer is correct
The correct answer is D. सीमा रहित और बंद / Unbounded and closed. The \(\geq\) conditions give an upper-side region in the first quadrant. Boundaries are included and the region extends infinitely.
Step 3
Exam Tip
\(\geq\) वाली शर्तें प्रथम चतुर्थांश में ऊपर की ओर क्षेत्र देती हैं। सीमाएं शामिल हैं और क्षेत्र अनंत तक जाता है।
The \(\geq\) inequalities give the upper-side region in the first quadrant. Boundaries are included and the region extends infinitely.
Step 2
Why this answer is correct
The correct answer is C. सीमा रहित और बंद / Unbounded and closed. The \(\geq\) inequalities give the upper-side region in the first quadrant. Boundaries are included and the region extends infinitely.
Step 3
Exam Tip
\(\geq\) वाली असमानताएं प्रथम चतुर्थांश में ऊपर की दिशा का क्षेत्र देती हैं। सीमाएं शामिल हैं और क्षेत्र अनंत तक जाता है।
Below or on the line means \(\leq\), and the first quadrant needs \(x\geq 0\), \(y\geq 0\). The boundary line is solid.
Step 2
Why this answer is correct
The correct answer is B. \(2x+3y\leq 18\), \(x\geq 0\), \(y\geq 0\). Below or on the line means \(\leq\), and the first quadrant needs \(x\geq 0\), \(y\geq 0\). The boundary line is solid.
Step 3
Exam Tip
नीचे या उसी पर होने से \(\leq\) लगेगा और प्रथम चतुर्थांश के लिए \(x\geq 0\), \(y\geq 0\) चाहिए। ठोस सीमा रेखा बनती है।
A. यह हल है और सीमा पर है/It is a solution and lies on a boundary
Step 1
Concept
At ((9,0)), (x+y=9) and (x+2y=9). It satisfies both conditions and lies on a boundary.
Step 2
Why this answer is correct
The correct answer is A. यह हल है और सीमा पर है / It is a solution and lies on a boundary. At ((9,0)), (x+y=9) and (x+2y=9). It satisfies both conditions and lies on a boundary.
Step 3
Exam Tip
((9,0)) पर (x+y=9) और (x+2y=9) मिलता है। यह दोनों शर्तें पूरी करता है और एक सीमा पर है।
Both inequalities select the upper sides of the lines, and in the first quadrant the region extends infinitely. Since \(\geq\) is used, boundaries are included.
Step 2
Why this answer is correct
The correct answer is A. सीमा रहित और बंद / Unbounded and closed. Both inequalities select the upper sides of the lines, and in the first quadrant the region extends infinitely. Since \(\geq\) is used, boundaries are included.
Step 3
Exam Tip
दोनों असमानताएं रेखाओं के ऊपर वाले भाग को चुनती हैं और प्रथम चतुर्थांश में क्षेत्र ऊपर की ओर अनंत है। \(\geq\) होने से सीमाएं शामिल हैं।
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।
The intercepts are ((15,0)) and ((0,10)), so the area is \(\frac{1}{2}\times 15\times 10=75\). Finding intercepts first is the fastest method.
Step 2
Why this answer is correct
The correct answer is B. (75). The intercepts are ((15,0)) and ((0,10)), so the area is \(\frac{1}{2}\times 15\times 10=75\). Finding intercepts first is the fastest method.
Step 3
Exam Tip
अवरोध ((15,0)) और ((0,10)) हैं, इसलिए क्षेत्रफल \(\frac{1}{2}\times 15\times 10=75\)। पहले अवरोध निकालना सबसे तेज तरीका है।
The first-quadrant region \(x+y\ge 5\) extends outward infinitely. A closed polygon needs bounds in every direction.
Step 2
Why this answer is correct
The correct answer is C. \(x+y\ge 5\), \(x\ge 0\), \(y\ge 0\). The first-quadrant region \(x+y\ge 5\) extends outward infinitely. A closed polygon needs bounds in every direction.
Step 3
Exam Tip
\(x+y\ge 5\) वाला प्रथम चतुर्थांश क्षेत्र बाहर की ओर अनंत फैलता है। बंद बहुभुज के लिए हर दिशा में सीमा चाहिए।
A. \(x+y\le 6\), \(x+y\ge 6\), \(x\ge 0\), \(y\ge 0\)
Step 1
Concept
Together \(x+y\le 6\) and \(x+y\ge 6\) force (x+y=6). Equality can be written using two opposite inequalities.
Step 2
Why this answer is correct
The correct answer is A. \(x+y\le 6\), \(x+y\ge 6\), \(x\ge 0\), \(y\ge 0\). Together \(x+y\le 6\) and \(x+y\ge 6\) force (x+y=6). Equality can be written using two opposite inequalities.
Step 3
Exam Tip
दोनों \(x+y\le 6\) और \(x+y\ge 6\) मिलकर (x+y=6) बनाते हैं। बराबरी को दो विपरीत असमानताओं से लिखा जा सकता है।
The upper bound \(2x+3y\le 18\) and the first quadrant make the region bounded. The lower line \(x+y\ge 4\) only cuts an inner part.
Step 2
Why this answer is correct
The correct answer is A. सीमित / Bounded. The upper bound \(2x+3y\le 18\) and the first quadrant make the region bounded. The lower line \(x+y\ge 4\) only cuts an inner part.
Step 3
Exam Tip
ऊपरी सीमा \(2x+3y\le 18\) और प्रथम चतुर्थांश क्षेत्र को सीमित करते हैं। निचली रेखा \(x+y\ge 4\) केवल अंदर का हिस्सा काटती है।
The intercepts are ((9,0)) and ((0,3)), so area is \(\frac{1}{2}\times 9\times 3=\frac{27}{2}\). For an axial triangle, use intercepts as base and height.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{27}{2}\). The intercepts are ((9,0)) and ((0,3)), so area is \(\frac{1}{2}\times 9\times 3=\frac{27}{2}\). For an axial triangle, use intercepts as base and height.
Step 3
Exam Tip
अवरोध ((9,0)) और ((0,3)) हैं, इसलिए क्षेत्रफल \(\frac{1}{2}\times 9\times 3=\frac{27}{2}\)। अक्षीय त्रिभुज में अवरोधों को आधार और ऊंचाई लें।
The region above the line in the first quadrant extends indefinitely. Conditions with \(\ge\) often create outward unbounded regions.
Step 2
Why this answer is correct
The correct answer is B. असीम / Unbounded. The region above the line in the first quadrant extends indefinitely. Conditions with \(\ge\) often create outward unbounded regions.
Step 3
Exam Tip
रेखा से ऊपर और प्रथम चतुर्थांश में क्षेत्र अनंत तक फैलता है। \(\ge\) वाली शर्तें अक्सर बाहर की ओर असीम क्षेत्र देती हैं।
Counting possible (y)-values for (x=0) to (8) gives a total of (33) points. Exam tip: For such questions, write the maximum (y) for each (x).
Step 2
Why this answer is correct
The correct answer is C. (33). Counting possible (y)-values for (x=0) to (8) gives a total of (33) points. Exam tip: For such questions, write the maximum (y) for each (x).
Step 3
Exam Tip
(x=0) से (8) तक (y) के संभव मान गिनने पर कुल (33) बिंदु मिलते हैं। परीक्षा सुझाव: ऐसे प्रश्नों में प्रत्येक (x) के लिए अधिकतम (y) लिखें।
At ((2,4)), \(4\cdot2+4=12\), so it lies on the boundary. Exam tip: For boundary checking, replace the inequality by its equation.
Step 2
Why this answer is correct
The correct answer is A. ((2,4)). At ((2,4)), \(4\cdot2+4=12\), so it lies on the boundary. Exam tip: For boundary checking, replace the inequality by its equation.
Step 3
Exam Tip
((2,4)) पर \(4\cdot2+4=12\) इसलिए यह सीमा पर है। परीक्षा सुझाव: सीमा के लिए असमानता की जगह समीकरण लगाएँ।
A. असीमित क्षेत्र जो मूल बिंदु को शामिल नहीं करता/Unbounded region not containing the origin
Step 1
Concept
The origin makes \(0\geq 10\) false, so the first-quadrant side away from the line is taken. Exam tip: With \(\geq\), the boundary line remains included.
Step 2
Why this answer is correct
The correct answer is A. असीमित क्षेत्र जो मूल बिंदु को शामिल नहीं करता / Unbounded region not containing the origin. The origin makes \(0\geq 10\) false, so the first-quadrant side away from the line is taken. Exam tip: With \(\geq\), the boundary line remains included.
Step 3
Exam Tip
मूल बिंदु \(0\geq 10\) असत्य करता है इसलिए रेखा से दूर वाला प्रथम चतुर्थांश भाग लिया जाता है। परीक्षा सुझाव: \(\geq\) वाली रेखा के लिए सीमा शामिल रहती है।
B. कटान ((9,0)), ((0,3)) हैं और मूल बिंदु अंदर है/Intercepts are ((9,0)), ((0,3)) and origin is inside
Step 1
Concept
Putting (y=0) gives (x=9), and putting (x=0) gives (y=3). Exam tip: Set one variable to zero to find intercepts.
Step 2
Why this answer is correct
The correct answer is B. कटान ((9,0)), ((0,3)) हैं और मूल बिंदु अंदर है / Intercepts are ((9,0)), ((0,3)) and origin is inside. Putting (y=0) gives (x=9), and putting (x=0) gives (y=3). Exam tip: Set one variable to zero to find intercepts.
Step 3
Exam Tip
(y=0) पर (x=9) और (x=0) पर (y=3) मिलता है। परीक्षा सुझाव: कटान निकालते समय एक चर को शून्य रखें।
The two lines intersect at ((2,2)), and the axes give the limits ((3,0)) and ((0,3)). Exam tip: Verify all possible vertices with the inequalities.
Step 2
Why this answer is correct
The correct answer is D. ((0,0)), ((3,0)), ((2,2)), ((0,3)). The two lines intersect at ((2,2)), and the axes give the limits ((3,0)) and ((0,3)). Exam tip: Verify all possible vertices with the inequalities.
Step 3
Exam Tip
दोनों रेखाओं का प्रतिच्छेद ((2,2)) है और अक्षों पर सीमाएँ ((3,0)), ((0,3)) देती हैं। परीक्षा सुझाव: सभी संभावित शीर्षों को असमानताओं से सत्यापित करें।
The axis directions give \(x\geq 0\), \(y\geq 0\), and being below the line gives \(2x+y\leq 10\). Exam tip: Convert words into axis inequalities first.
Step 2
Why this answer is correct
The correct answer is D. \(x\geq 0\), \(y\geq 0\), \(2x+y\leq 10\). The axis directions give \(x\geq 0\), \(y\geq 0\), and being below the line gives \(2x+y\leq 10\). Exam tip: Convert words into axis inequalities first.
Step 3
Exam Tip
अक्षों की दिशा से \(x\geq 0\), \(y\geq 0\) और रेखा के नीचे से \(2x+y\leq 10\) मिलता है। परीक्षा सुझाव: शब्दों को पहले अक्षीय असमानताओं में बदलें।
D. रेखा के ऊपर वाला असीमित भाग सीमा सहित/Unbounded part above the line with boundary
Step 1
Concept
The origin gives \(0\geq 12\), which is false, so the side away from the origin is chosen. Exam tip: Always combine the region with the axes in the first quadrant.
Step 2
Why this answer is correct
The correct answer is D. रेखा के ऊपर वाला असीमित भाग सीमा सहित / Unbounded part above the line with boundary. The origin gives \(0\geq 12\), which is false, so the side away from the origin is chosen. Exam tip: Always combine the region with the axes in the first quadrant.
Step 3
Exam Tip
मूल बिंदु से \(0\geq 12\) असत्य है इसलिए रेखा से दूर वाला भाग चुना जाता है। परीक्षा सुझाव: प्रथम चतुर्थांश क्षेत्र को हमेशा अक्षों से भी मिलाइए।
