A. (11) पर खाली वृत्त और दाईं ओर किरण/Open circle at (11) and ray to the right
Step 1
Concept
The sign (>) excludes the boundary (11) and greater numbers lie to the right. Check the circle type first and then the direction.
Step 2
Why this answer is correct
The correct answer is A. (11) पर खाली वृत्त और दाईं ओर किरण / Open circle at (11) and ray to the right. The sign (>) excludes the boundary (11) and greater numbers lie to the right. Check the circle type first and then the direction.
Step 3
Exam Tip
चिह्न (>) में सीमा (11) शामिल नहीं होती और बड़ी संख्याएँ दाईं ओर होती हैं। पहले वृत्त का प्रकार और फिर दिशा देखें।
B. (-9) पर भरा वृत्त और दाईं ओर छायांकन/Closed circle at (-9) and shading right
Step 1
Concept
\(\ge\) includes the boundary and greater numbers are on the right. Therefore a closed circle is drawn at (-9).
Step 2
Why this answer is correct
The correct answer is B. (-9) पर भरा वृत्त और दाईं ओर छायांकन / Closed circle at (-9) and shading right. \(\ge\) includes the boundary and greater numbers are on the right. Therefore a closed circle is drawn at (-9).
Step 3
Exam Tip
\(\ge\) में सीमा शामिल होती है और दाईं ओर बड़ी संख्याएँ मिलती हैं। इसलिए (-9) पर भरा वृत्त बनेगा।
The endpoint with a square bracket is included and the endpoint with a parenthesis is excluded. Therefore (-4) is closed.
Step 2
Why this answer is correct
The correct answer is B. केवल (-4) / Only (-4). The endpoint with a square bracket is included and the endpoint with a parenthesis is excluded. Therefore (-4) is closed.
Step 3
Exam Tip
वर्ग कोष्ठक वाला सिरा शामिल होता है और गोल कोष्ठक वाला सिरा शामिल नहीं होता। इसलिए (-4) बंद होगा।
C. (4) और (12) पर खाली वृत्त और बीच में छायांकन/Open circles at (4) and (12) with shading between
Step 1
Concept
Both inequalities are strict, so both endpoints are excluded. The part between them is the solution.
Step 2
Why this answer is correct
The correct answer is C. (4) और (12) पर खाली वृत्त और बीच में छायांकन / Open circles at (4) and (12) with shading between. Both inequalities are strict, so both endpoints are excluded. The part between them is the solution.
Step 3
Exam Tip
दोनों असमताएँ कठोर हैं इसलिए दोनों सीमाएँ शामिल नहीं हैं। बीच वाला भाग ही समाधान है।
The closed circle includes (7) and the right ray shows numbers greater than (7). Hence the interval is \([7,\infty\)).
Step 2
Why this answer is correct
The correct answer is D. \([7,\infty\)). The closed circle includes (7) and the right ray shows numbers greater than (7). Hence the interval is \([7,\infty\)).
Step 3
Exam Tip
भरा वृत्त (7) को शामिल करता है और दाईं किरण (7) से बड़ी संख्याएँ दिखाती है। इसलिए अंतराल \([7,\infty\)) है।
Parentheses exclude both boundary points from the solution. Therefore both (-8) and (0) have open circles.
Step 2
Why this answer is correct
The correct answer is C. (-8) और (0) दोनों / Both (-8) and (0). Parentheses exclude both boundary points from the solution. Therefore both (-8) and (0) have open circles.
Step 3
Exam Tip
गोल कोष्ठक दोनों सीमाओं को समाधान से बाहर रखता है। इसलिए (-8) और (0) दोनों पर खाली वृत्त होंगे।
Equality is included on both sides, so both endpoints are included. In this case, square brackets are used at both ends.
Step 2
Why this answer is correct
The correct answer is D. ([-7,-1]). Equality is included on both sides, so both endpoints are included. In this case, square brackets are used at both ends.
Step 3
Exam Tip
दोनों ओर बराबरी शामिल है इसलिए दोनों सीमाएँ शामिल होंगी। ऐसे में दोनों सिरों पर वर्ग कोष्ठक लगते हैं।
B. (6) पर खाली वृत्त और बाईं ओर किरण/Open circle at (6) and ray left
Step 1
Concept
In (\(-\infty,6\)), (6) is not included and smaller numbers are present. So an open circle at (6) and a left ray are drawn.
Step 2
Why this answer is correct
The correct answer is B. (6) पर खाली वृत्त और बाईं ओर किरण / Open circle at (6) and ray left. In (\(-\infty,6\)), (6) is not included and smaller numbers are present. So an open circle at (6) and a left ray are drawn.
Step 3
Exam Tip
(\(-\infty,6\)) में (6) शामिल नहीं है और उससे छोटी संख्याएँ हैं। इसलिए (6) पर खाली वृत्त और बाईं किरण बनेगी।
(5) has a parenthesis, so it is excluded, and (10) has a square bracket, so it is included. This decides the open and closed endpoints.
Step 2
Why this answer is correct
The correct answer is B. (5) खुला और (10) बंद / (5) open and (10) closed. (5) has a parenthesis, so it is excluded, and (10) has a square bracket, so it is included. This decides the open and closed endpoints.
Step 3
Exam Tip
(5) के साथ गोल कोष्ठक है इसलिए वह शामिल नहीं है और (10) के साथ वर्ग कोष्ठक है इसलिए वह शामिल है। यही खुले और बंद सिरों को तय करता है।
B. पूरी रेखा छायांकित और (-3) पर खाली वृत्त/Whole line shaded with an open circle at (-3)
Step 1
Concept
In \(x\ne -3\), all real numbers except (-3) are included. The excluded point is shown by an open circle.
Step 2
Why this answer is correct
The correct answer is B. पूरी रेखा छायांकित और (-3) पर खाली वृत्त / Whole line shaded with an open circle at (-3). In \(x\ne -3\), all real numbers except (-3) are included. The excluded point is shown by an open circle.
Step 3
Exam Tip
\(x\ne -3\) में (-3) को छोड़कर सभी वास्तविक संख्याएँ शामिल हैं। छोड़े गए बिंदु पर खाली वृत्त बनता है।
The symbol \(\cup\) is used to combine two separate parts. Here both (-2) and (5) are not included.
Step 2
Why this answer is correct
The correct answer is B. (\(-\infty,-2\)\cup\(5,\infty\)). The symbol \(\cup\) is used to combine two separate parts. Here both (-2) and (5) are not included.
Step 3
Exam Tip
दो अलग-अलग भागों को मिलाने के लिए \(\cup\) का प्रयोग होता है। यहाँ (-2) और (5) दोनों शामिल नहीं हैं।
No number can be greater than (3) and less than (3) at the same time. Therefore ((3,3)) has no member.
Step 2
Why this answer is correct
The correct answer is C. रिक्त समुच्चय / Empty set. No number can be greater than (3) and less than (3) at the same time. Therefore ((3,3)) has no member.
Step 3
Exam Tip
कोई संख्या (3) से बड़ी और (3) से छोटी एक साथ नहीं हो सकती। इसलिए ((3,3)) में कोई सदस्य नहीं है।
A strict inequality does not include \(\frac{3}{2}\). Therefore an open circle is drawn at the boundary.
Step 2
Why this answer is correct
The correct answer is B. खाली वृत्त / Open circle. A strict inequality does not include \(\frac{3}{2}\). Therefore an open circle is drawn at the boundary.
Step 3
Exam Tip
कठोर असमता में \(\frac{3}{2}\) शामिल नहीं है। इसलिए सीमा पर खाली वृत्त बनेगा।
C. दोनों सिरों पर भरे वृत्त/Closed circles at both endpoints
Step 1
Concept
Square brackets include both endpoints. Therefore both endpoints have closed circles.
Step 2
Why this answer is correct
The correct answer is C. दोनों सिरों पर भरे वृत्त / Closed circles at both endpoints. Square brackets include both endpoints. Therefore both endpoints have closed circles.
Step 3
Exam Tip
वर्ग कोष्ठक दोनों सीमाओं को शामिल करते हैं। इसलिए दोनों सिरों पर भरे वृत्त होंगे।
Both closed endpoints show that both boundaries are included. So inclusive inequalities are used on both sides.
Step 2
Why this answer is correct
The correct answer is D. \(-11\le x\le -2\). Both closed endpoints show that both boundaries are included. So inclusive inequalities are used on both sides.
Step 3
Exam Tip
दोनों बंद सिरे बताते हैं कि दोनों सीमाएँ शामिल हैं। इसलिए दोनों तरफ बराबरी सहित असमता होगी।
A. (18) पर भरा वृत्त और दाईं ओर तीर/Closed circle at (18) and arrow right
Step 1
Concept
In \(x\ge 18\), (18) is included and numbers greater than it are taken. So a closed circle with a right arrow is used.
Step 2
Why this answer is correct
The correct answer is A. (18) पर भरा वृत्त और दाईं ओर तीर / Closed circle at (18) and arrow right. In \(x\ge 18\), (18) is included and numbers greater than it are taken. So a closed circle with a right arrow is used.
Step 3
Exam Tip
\(x\ge 18\) में (18) शामिल है और उससे बड़ी संख्याएँ ली जाती हैं। इसलिए भरा वृत्त दाईं ओर तीर के साथ होगा।
B. \(x\le -1\) या \(x\ge 4\)/\(x\le -1\) or \(x\ge 4\)
Step 1
Concept
(\(-\infty,-1]\) means \(x\le -1\), and \([4,\infty\)) means \(x\ge 4\). The symbol \(\cup\) means the union of both parts.
Step 2
Why this answer is correct
The correct answer is B. \(x\le -1\) या \(x\ge 4\) / \(x\le -1\) or \(x\ge 4\). (\(-\infty,-1]\) means \(x\le -1\), and \([4,\infty\)) means \(x\ge 4\). The symbol \(\cup\) means the union of both parts.
Step 3
Exam Tip
(\(-\infty,-1]\) में \(x\le -1\) और \([4,\infty\)) में \(x\ge 4\) है। \(\cup\) का अर्थ दोनों भागों का मिलन है।
A. सिर्फ (12) पर भरा बिंदु/Only a closed point at (12)
Step 1
Concept
The interval ([12,12]) includes only (12). A closed interval with equal endpoints shows one closed point.
Step 2
Why this answer is correct
The correct answer is A. सिर्फ (12) पर भरा बिंदु / Only a closed point at (12). The interval ([12,12]) includes only (12). A closed interval with equal endpoints shows one closed point.
Step 3
Exam Tip
([12,12]) में केवल (12) शामिल है। समान सीमाओं वाला बंद अंतराल एक भरा बिंदु दिखाता है।
\(\ge\) includes the boundary \(-\frac{5}{2}\) and takes numbers to the right. Therefore it starts with a square bracket.
Step 2
Why this answer is correct
The correct answer is C. \([-\frac{5}{2},\infty\)). \(\ge\) includes the boundary \(-\frac{5}{2}\) and takes numbers to the right. Therefore it starts with a square bracket.
Step 3
Exam Tip
\(\ge\) सीमा \(-\frac{5}{2}\) को शामिल करता है और दाईं ओर की संख्याएँ लेता है। इसलिए वर्ग कोष्ठक से शुरू होगा।
(-4) is not included, so the first integer after it is (-3). On the number line, do not count an open endpoint as a member.
Step 2
Why this answer is correct
The correct answer is B. (-3). (-4) is not included, so the first integer after it is (-3). On the number line, do not count an open endpoint as a member.
Step 3
Exam Tip
(-4) शामिल नहीं है इसलिए उसके बाद पहला पूर्णांक (-3) होगा। संख्या रेखा पर खुले सिरे को सदस्य न मानें।
B. (x=0) समाधान में नहीं है/(x=0) is not in the solution
Step 1
Concept
An open circle keeps the boundary out of the solution. Therefore (x=0) is not included.
Step 2
Why this answer is correct
The correct answer is B. (x=0) समाधान में नहीं है / (x=0) is not in the solution. An open circle keeps the boundary out of the solution. Therefore (x=0) is not included.
Step 3
Exam Tip
खुला वृत्त सीमा को समाधान से बाहर रखता है। इसलिए (x=0) शामिल नहीं है।
A. पहले में दोनों सिरे खुले और दूसरे में दोनों सिरे बंद/The first has both endpoints open and the second has both endpoints closed
Step 1
Concept
Parentheses exclude endpoints and square brackets include endpoints. This changes the endpoints on the number line.
Step 2
Why this answer is correct
The correct answer is A. पहले में दोनों सिरे खुले और दूसरे में दोनों सिरे बंद / The first has both endpoints open and the second has both endpoints closed. Parentheses exclude endpoints and square brackets include endpoints. This changes the endpoints on the number line.
Step 3
Exam Tip
गोल कोष्ठक सीमा को बाहर रखता है और वर्ग कोष्ठक सीमा को शामिल करता है। यही संख्या रेखा पर सिरे बदलता है।
(\(-\infty,\infty\)) represents all real numbers. Therefore the entire number line will be shaded.
Step 2
Why this answer is correct
The correct answer is C. पूरी संख्या रेखा पर / On the entire number line. (\(-\infty,\infty\)) represents all real numbers. Therefore the entire number line will be shaded.
Step 3
Exam Tip
(\(-\infty,\infty\)) सभी वास्तविक संख्याओं को दर्शाता है। इसलिए पूरी संख्या रेखा छायांकित होगी।
Outward shading shows two separate parts and open circles exclude the boundaries. Hence (\(-\infty,-1\)\cup\(4,\infty\)) is correct.
Step 2
Why this answer is correct
The correct answer is B. (\(-\infty,-1\)\cup\(4,\infty\)). Outward shading shows two separate parts and open circles exclude the boundaries. Hence (\(-\infty,-1\)\cup\(4,\infty\)) is correct.
Step 3
Exam Tip
बाहर की छायांकन दो अलग भाग दिखाती है और खुले वृत्त सीमाओं को बाहर रखते हैं। इसलिए (\(-\infty,-1\)\cup\(4,\infty\)) सही है।
A closed interval with equal endpoints includes only that number. Therefore one closed point is drawn at (-6).
Step 2
Why this answer is correct
The correct answer is B. (-6) पर एक भरा बिंदु / One closed point at (-6). A closed interval with equal endpoints includes only that number. Therefore one closed point is drawn at (-6).
Step 3
Exam Tip
समान सीमाओं वाला बंद अंतराल केवल उसी संख्या को शामिल करता है। इसलिए (-6) पर एक भरा बिंदु बनेगा।
The open circle excludes (-8), and the closed circle includes (2). Therefore the correct inequality is \(-8<x\le 2\).
Step 2
Why this answer is correct
The correct answer is A. \(-8<x\le 2\). The open circle excludes (-8), and the closed circle includes (2). Therefore the correct inequality is \(-8<x\le 2\).
Step 3
Exam Tip
खाली वृत्त (-8) को बाहर रखता है और भरा वृत्त (2) को शामिल करता है। इसलिए सही असमता \(-8<x\le 2\) है।
The square bracket shows that \(-\frac{1}{2}\) is included, and the values extend toward \(-\infty\). Hence \(x\le -\frac{1}{2}\) is correct.
Step 2
Why this answer is correct
The correct answer is B. \(x\le -\frac{1}{2}\). The square bracket shows that \(-\frac{1}{2}\) is included, and the values extend toward \(-\infty\). Hence \(x\le -\frac{1}{2}\) is correct.
Step 3
Exam Tip
वर्ग कोष्ठक बताता है कि \(-\frac{1}{2}\) शामिल है और \(-\infty\) की ओर छोटी संख्याएँ हैं। इसलिए \(x\le -\frac{1}{2}\) सही है।
B. सीमा \(-\frac{3}{4}\), खाली वृत्त और दाईं ओर/Boundary \(-\frac{3}{4}\), open circle and right side
Step 1
Concept
The strict sign (>) does not include the boundary, and greater numbers lie to the right. So use an open circle with rightward direction.
Step 2
Why this answer is correct
The correct answer is B. सीमा \(-\frac{3}{4}\), खाली वृत्त और दाईं ओर / Boundary \(-\frac{3}{4}\), open circle and right side. The strict sign (>) does not include the boundary, and greater numbers lie to the right. So use an open circle with rightward direction.
Step 3
Exam Tip
कठोर (>) में सीमा शामिल नहीं होती और बड़ी संख्याएँ दाईं ओर होती हैं। इसलिए खाली वृत्त के साथ दाईं दिशा बनेगी।
