A. (3) पर खाली वृत्त और दाईं ओर किरण/Open circle at (3) and ray to the right
Step 1
Concept
The sign (>) does not include the boundary so an open circle is used. In exams check the direction toward greater numbers.
Step 2
Why this answer is correct
The correct answer is A. (3) पर खाली वृत्त और दाईं ओर किरण / Open circle at (3) and ray to the right. The sign (>) does not include the boundary so an open circle is used. In exams check the direction toward greater numbers.
Step 3
Exam Tip
चिह्न (>) में सीमा संख्या शामिल नहीं होती इसलिए खाली वृत्त बनता है। परीक्षा में दिशा हमेशा बड़ी संख्याओं की ओर देखें।
B. (-2) पर भरा वृत्त और बाईं ओर छायांकन/Closed circle at (-2) and shading left
Step 1
Concept
The sign \(\le\) includes (-2) and smaller numbers lie to the left. Use a closed circle when the endpoint is included.
Step 2
Why this answer is correct
The correct answer is B. (-2) पर भरा वृत्त और बाईं ओर छायांकन / Closed circle at (-2) and shading left. The sign \(\le\) includes (-2) and smaller numbers lie to the left. Use a closed circle when the endpoint is included.
Step 3
Exam Tip
चिह्न \(\le\) में (-2) शामिल है और छोटी संख्याएँ बाईं ओर होती हैं। सीमा शामिल हो तो भरा वृत्त बनाइए।
Both closed circles show that both endpoints are included. The shaded middle part gives a compound inequality.
Step 2
Why this answer is correct
The correct answer is B. \(-1\le x\le 4\). Both closed circles show that both endpoints are included. The shaded middle part gives a compound inequality.
Step 3
Exam Tip
दोनों भरे वृत्त बताते हैं कि दोनों सीमाएँ शामिल हैं। बीच का भाग संयुक्त असमता देता है।
B. (2) पर खाली वृत्त और (7) पर भरा वृत्त/Open circle at (2) and closed circle at (7)
Step 1
Concept
A parenthesis endpoint is not included and a square bracket endpoint is included. Therefore (2) is open and (7) is closed.
Step 2
Why this answer is correct
The correct answer is B. (2) पर खाली वृत्त और (7) पर भरा वृत्त / Open circle at (2) and closed circle at (7). A parenthesis endpoint is not included and a square bracket endpoint is included. Therefore (2) is open and (7) is closed.
Step 3
Exam Tip
गोल कोष्ठक वाली सीमा शामिल नहीं होती और वर्ग कोष्ठक वाली सीमा शामिल होती है। इसलिए (2) खुला और (7) बंद होगा।
B. (0) पर भरा वृत्त और दाईं ओर किरण/Closed circle at (0) and ray right
Step 1
Concept
The sign \(\ge\) includes (0), and greater numbers are to the right. So it is a closed circle with a right ray.
Step 2
Why this answer is correct
The correct answer is B. (0) पर भरा वृत्त और दाईं ओर किरण / Closed circle at (0) and ray right. The sign \(\ge\) includes (0), and greater numbers are to the right. So it is a closed circle with a right ray.
Step 3
Exam Tip
चिह्न \(\ge\) में (0) शामिल है और उससे बड़ी संख्याएँ दाईं ओर होती हैं। इसलिए भरा वृत्त और दाईं किरण बनती है।
In \(x\le 4\), (4) is included so a square bracket is used at (4). Smaller numbers extend toward \(-\infty\).
Step 2
Why this answer is correct
The correct answer is B. (\(-\infty,4]\). In \(x\le 4\), (4) is included so a square bracket is used at (4). Smaller numbers extend toward \(-\infty\).
Step 3
Exam Tip
\(x\le 4\) में (4) शामिल है इसलिए (4) पर वर्ग कोष्ठक आएगा। छोटी संख्याएँ \(-\infty\) की ओर जाती हैं।
B. दोनों सिरों पर खाली वृत्त होंगे/Both endpoints have open circles
Step 1
Concept
Parentheses mean that boundary numbers are not included. Thus both (-4) and (3) have open circles.
Step 2
Why this answer is correct
The correct answer is B. दोनों सिरों पर खाली वृत्त होंगे / Both endpoints have open circles. Parentheses mean that boundary numbers are not included. Thus both (-4) and (3) have open circles.
Step 3
Exam Tip
गोल कोष्ठक का अर्थ है कि सीमा संख्या शामिल नहीं है। इसलिए (-4) और (3) दोनों पर खाली वृत्त होंगे।
The sign (<) shows smaller numbers, and smaller numbers are on the left of the number line. Identifying direction is the first step.
Step 2
Why this answer is correct
The correct answer is B. बाईं ओर / Left side. The sign (<) shows smaller numbers, and smaller numbers are on the left of the number line. Identifying direction is the first step.
Step 3
Exam Tip
चिह्न (<) छोटी संख्याओं को दिखाता है और छोटी संख्याएँ संख्या रेखा पर बाईं ओर होती हैं। दिशा पहचानना पहला कदम है।
B. (9) पर भरा वृत्त और दाईं ओर/Closed circle at (9) and right side
Step 1
Concept
The sign \(\ge\) includes the boundary and greater numbers lie to the right. So a closed circle with a right ray is drawn.
Step 2
Why this answer is correct
The correct answer is B. (9) पर भरा वृत्त और दाईं ओर / Closed circle at (9) and right side. The sign \(\ge\) includes the boundary and greater numbers lie to the right. So a closed circle with a right ray is drawn.
Step 3
Exam Tip
\(\ge\) में सीमा शामिल होती है और बड़ी संख्याएँ दाईं ओर होती हैं। इसलिए भरा वृत्त दाईं किरण के साथ बनेगा।
A. (4) पर भरा वृत्त और दाईं ओर किरण/Closed circle at (4) and ray right
Step 1
Concept
The interval \([4,\infty\)) includes (4) and all larger numbers. Hence a closed circle with a right ray is correct.
Step 2
Why this answer is correct
The correct answer is A. (4) पर भरा वृत्त और दाईं ओर किरण / Closed circle at (4) and ray right. The interval \([4,\infty\)) includes (4) and all larger numbers. Hence a closed circle with a right ray is correct.
Step 3
Exam Tip
\([4,\infty\)) में (4) शामिल है और आगे सभी बड़ी संख्याएँ हैं। इसलिए भरा वृत्त और दाईं किरण सही है।
(-6) is included so a square bracket is used, and (1) is not included so a parenthesis is used. Read brackets to decide endpoints.
Step 2
Why this answer is correct
The correct answer is C. ([-6,1)). (-6) is included so a square bracket is used, and (1) is not included so a parenthesis is used. Read brackets to decide endpoints.
Step 3
Exam Tip
(-6) शामिल है इसलिए वर्ग कोष्ठक और (1) शामिल नहीं है इसलिए गोल कोष्ठक आएगा। कोष्ठक देखकर सीमा तय करें।
The equation (x=3) represents only one number. For a single included value, draw a closed point.
Step 2
Why this answer is correct
The correct answer is A. (3) पर एक भरा बिंदु / A closed point at (3). The equation (x=3) represents only one number. For a single included value, draw a closed point.
Step 3
Exam Tip
समीकरण (x=3) केवल एक संख्या को दर्शाता है। एकमात्र मान के लिए भरा बिंदु बनाते हैं।
An open circle means the boundary value is not part of the solution. In such questions, check the type of circle first.
Step 2
Why this answer is correct
The correct answer is B. (a) शामिल नहीं है / (a) is not included. An open circle means the boundary value is not part of the solution. In such questions, check the type of circle first.
Step 3
Exam Tip
खाली वृत्त का अर्थ है सीमा मान समाधान में नहीं है। ऐसे प्रश्नों में वृत्त का प्रकार पहले देखें।
A. (b) समाधान में शामिल है/(b) is included in the solution
Step 1
Concept
A closed circle shows that the boundary number is included. Usually \(\le\) and \(\ge\) use closed circles.
Step 2
Why this answer is correct
The correct answer is A. (b) समाधान में शामिल है / (b) is included in the solution. A closed circle shows that the boundary number is included. Usually \(\le\) and \(\ge\) use closed circles.
Step 3
Exam Tip
भरा वृत्त बताता है कि सीमा संख्या शामिल है। \(\le\) और \(\ge\) में सामान्यतः भरा वृत्त आता है।
A. (3) पर खाली वृत्त और दाईं ओर किरण/Open circle at (3) and ray right
Step 1
Concept
The interval (\(3,\infty\)) does not include (3) and contains larger numbers. So it uses an open circle and a right ray.
Step 2
Why this answer is correct
The correct answer is A. (3) पर खाली वृत्त और दाईं ओर किरण / Open circle at (3) and ray right. The interval (\(3,\infty\)) does not include (3) and contains larger numbers. So it uses an open circle and a right ray.
Step 3
Exam Tip
(\(3,\infty\)) में (3) शामिल नहीं है और उससे बड़ी संख्याएँ आती हैं। इसलिए खाली वृत्त और दाईं किरण बनती है।
B. (-8) पर खाली वृत्त और (-3) पर भरा वृत्त/Open circle at (-8) and closed circle at (-3)
Step 1
Concept
(-8) is not included but (-3) is included. So the left endpoint is open and the right endpoint is closed.
Step 2
Why this answer is correct
The correct answer is B. (-8) पर खाली वृत्त और (-3) पर भरा वृत्त / Open circle at (-8) and closed circle at (-3). (-8) is not included but (-3) is included. So the left endpoint is open and the right endpoint is closed.
Step 3
Exam Tip
(-8) शामिल नहीं है लेकिन (-3) शामिल है। इसलिए बायाँ सिरा खुला और दायाँ सिरा बंद रहेगा।
A parenthesis gives a strict inequality and a square bracket gives an inclusive inequality. Therefore \(-2<x\le 4\) is correct.
Step 2
Why this answer is correct
The correct answer is A. \(-2<x\le 4\). A parenthesis gives a strict inequality and a square bracket gives an inclusive inequality. Therefore \(-2<x\le 4\) is correct.
Step 3
Exam Tip
गोल कोष्ठक पर कठोर असमता और वर्ग कोष्ठक पर बराबरी सहित असमता लगती है। इसलिए \(-2<x\le 4\) सही है।
A. सभी संख्या रेखा छायांकित और (2) पर खाली वृत्त/Whole number line shaded with an open circle at (2)
Step 1
Concept
In \(x\ne 2\), all real numbers except (2) are included. The excluded point is shown by an open circle.
Step 2
Why this answer is correct
The correct answer is A. सभी संख्या रेखा छायांकित और (2) पर खाली वृत्त / Whole number line shaded with an open circle at (2). In \(x\ne 2\), all real numbers except (2) are included. The excluded point is shown by an open circle.
Step 3
Exam Tip
\(x\ne 2\) में (2) को छोड़कर सभी वास्तविक संख्याएँ आती हैं। हटाए गए बिंदु के लिए खाली वृत्त बनता है।
B. सीमा (10), बाईं दिशा/Boundary (10), left direction
Step 1
Concept
In \(x\le 10\), the boundary is (10), and smaller numbers are on the left. Equality means the point is closed.
Step 2
Why this answer is correct
The correct answer is B. सीमा (10), बाईं दिशा / Boundary (10), left direction. In \(x\le 10\), the boundary is (10), and smaller numbers are on the left. Equality means the point is closed.
Step 3
Exam Tip
\(x\le 10\) में सीमा (10) है और उससे छोटी संख्याएँ बाईं ओर होती हैं। बराबरी होने से बिंदु भरा होगा।
Because \(3\le x\), (3) is included, and because (x<9), (9) is not included. So (3) is closed and (9) is open.
Step 2
Why this answer is correct
The correct answer is B. (3) बंद और (9) खुला / (3) closed and (9) open. Because \(3\le x\), (3) is included, and because (x<9), (9) is not included. So (3) is closed and (9) is open.
Step 3
Exam Tip
\(3\le x\) के कारण (3) शामिल है और (x<9) के कारण (9) शामिल नहीं है। इसलिए (3) बंद और (9) खुला होगा।
Since (5) is not included, a parenthesis is used, and smaller numbers extend toward \(-\infty\). Hence (\(-\infty,5\)) is correct.
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,5\)). Since (5) is not included, a parenthesis is used, and smaller numbers extend toward \(-\infty\). Hence (\(-\infty,5\)) is correct.
Step 3
Exam Tip
(5) शामिल नहीं है इसलिए गोल कोष्ठक लगेगा और छोटी संख्याएँ \(-\infty\) की ओर होंगी। इसलिए (\(-\infty,5\)) सही है।
\(x\le 2\) includes all numbers up to (2), and (x>2) includes all numbers after (2). Together they form the whole number line.
Step 2
Why this answer is correct
The correct answer is A. सभी वास्तविक संख्याएँ / All real numbers. \(x\le 2\) includes all numbers up to (2), and (x>2) includes all numbers after (2). Together they form the whole number line.
Step 3
Exam Tip
\(x\le 2\) में (2) तक सभी संख्याएँ और (x>2) में (2) से आगे सभी संख्याएँ आती हैं। दोनों मिलकर पूरी संख्या रेखा बनाते हैं।
A. पहले में सिरे बंद हैं और दूसरे में सिरे खुले हैं/The first has closed endpoints and the second has open endpoints
Step 1
Concept
Square brackets include endpoints and parentheses exclude endpoints. This creates the open or closed endpoint difference on the number line.
Step 2
Why this answer is correct
The correct answer is A. पहले में सिरे बंद हैं और दूसरे में सिरे खुले हैं / The first has closed endpoints and the second has open endpoints. Square brackets include endpoints and parentheses exclude endpoints. This creates the open or closed endpoint difference on the number line.
Step 3
Exam Tip
वर्ग कोष्ठक सीमा को शामिल करता है और गोल कोष्ठक सीमा को बाहर रखता है। यही संख्या रेखा पर खुले या बंद सिरों का अंतर है।
A. पूरी संख्या रेखा छायांकित/Entire number line shaded
Step 1
Concept
\(x\in \mathbb{R}\) means all real numbers. On the number line it represents the entire line.
Step 2
Why this answer is correct
The correct answer is A. पूरी संख्या रेखा छायांकित / Entire number line shaded. \(x\in \mathbb{R}\) means all real numbers. On the number line it represents the entire line.
Step 3
Exam Tip
\(x\in \mathbb{R}\) का अर्थ है सभी वास्तविक संख्याएँ। संख्या रेखा पर यह पूरी रेखा को दर्शाता है।
(\(-\infty,\infty\)) represents the whole real number line. Infinity is not an endpoint, so parentheses are used with it.
Step 2
Why this answer is correct
The correct answer is B. सभी वास्तविक संख्याएँ / All real numbers. (\(-\infty,\infty\)) represents the whole real number line. Infinity is not an endpoint, so parentheses are used with it.
Step 3
Exam Tip
(\(-\infty,\infty\)) पूरी वास्तविक संख्या रेखा को दर्शाता है। अनंत सीमा नहीं है इसलिए उसके साथ गोल कोष्ठक ही आता है।
A. (0) से (3) के बीच और दोनों सिरे खुले/Between (0) and (3) with both endpoints open
Step 1
Concept
Strict inequalities exclude the endpoints and the middle part is the solution. Therefore both endpoints remain open.
Step 2
Why this answer is correct
The correct answer is A. (0) से (3) के बीच और दोनों सिरे खुले / Between (0) and (3) with both endpoints open. Strict inequalities exclude the endpoints and the middle part is the solution. Therefore both endpoints remain open.
Step 3
Exam Tip
कठोर असमता में सीमाएँ शामिल नहीं होतीं और बीच का भाग समाधान होता है। इसलिए दोनों सिरे खुले रहेंगे।
B. सिर्फ (7) पर भरा बिंदु/Only a closed point at (7)
Step 1
Concept
In ([7,7]), the start and end are the same and that value is included. So only a closed point at (7) is drawn.
Step 2
Why this answer is correct
The correct answer is B. सिर्फ (7) पर भरा बिंदु / Only a closed point at (7). In ([7,7]), the start and end are the same and that value is included. So only a closed point at (7) is drawn.
Step 3
Exam Tip
([7,7]) में आरंभ और अंत समान हैं और वह मान शामिल है। इसलिए केवल (7) पर भरा बिंदु बनेगा।
In ((7,7)), no number can be greater than (7) and less than (7) at the same time. So it is an empty set.
Step 2
Why this answer is correct
The correct answer is C. रिक्त समुच्चय / Empty set. In ((7,7)), no number can be greater than (7) and less than (7) at the same time. So it is an empty set.
Step 3
Exam Tip
((7,7)) में कोई संख्या (7) से बड़ी और (7) से छोटी एक साथ नहीं हो सकती। इसलिए यह रिक्त समुच्चय है।
The sign \(\le\) includes equality, so the boundary (-6) is in the solution. An included boundary is shown by a closed circle.
Step 2
Why this answer is correct
The correct answer is B. भरा वृत्त / Closed circle. The sign \(\le\) includes equality, so the boundary (-6) is in the solution. An included boundary is shown by a closed circle.
Step 3
Exam Tip
\(\le\) में बराबरी शामिल होती है इसलिए सीमा (-6) समाधान में है। शामिल सीमा के लिए भरा वृत्त बनता है।
(-2) is closed so a square bracket is used, and (5) is open so a parenthesis is used. The correct interval is ([-2,5)).
Step 2
Why this answer is correct
The correct answer is C. ([-2,5)). (-2) is closed so a square bracket is used, and (5) is open so a parenthesis is used. The correct interval is ([-2,5)).
Step 3
Exam Tip
(-2) बंद है इसलिए वर्ग कोष्ठक और (5) खुला है इसलिए गोल कोष्ठक लगेगा। सही अंतराल ([-2,5)) है।
An open circle means (-6) is not included, and a closed circle means (2) is included. Therefore the correct inequality is \(-6<x\le 2\).
Step 2
Why this answer is correct
The correct answer is A. \(-6<x\le 2\). An open circle means (-6) is not included, and a closed circle means (2) is included. Therefore the correct inequality is \(-6<x\le 2\).
Step 3
Exam Tip
खाली वृत्त का अर्थ है (-6) शामिल नहीं है और भरा वृत्त बताता है कि (2) शामिल है। इसलिए सही असमता \(-6<x\le 2\) है।
