Both zeroes are \(\sqrt{3}\), so the sum is \(2\sqrt{3}\). In \(x^2+kx+3\), the sum is (-k), hence \(k=-2\sqrt{3}\).
Step 2
Why this answer is correct
The correct answer is A. \(-2\sqrt{3}\). Both zeroes are \(\sqrt{3}\), so the sum is \(2\sqrt{3}\). In \(x^2+kx+3\), the sum is (-k), hence \(k=-2\sqrt{3}\).
Step 3
Exam Tip
दोनों शून्यक \(\sqrt{3}\) हैं, इसलिए योग \(2\sqrt{3}\) है। \(x^2+kx+3\) में योग (-k) है, अतः \(k=-2\sqrt{3}\)।
Three distinct (x)-values give three distinct (x)-axis points. Tip: distinct zeroes make distinct intersection points.
Step 2
Why this answer is correct
The correct answer is C. तीन / Three. Three distinct (x)-values give three distinct (x)-axis points. Tip: distinct zeroes make distinct intersection points.
Step 3
Exam Tip
तीन अलग (x)-मान तीन अलग (x)-अक्ष बिंदु देते हैं। टिप: अलग शून्यक अलग कटान बिंदु बनाते हैं।
Both have the same (x)-value (3), so there is one distinct zero. Tip: count a repeated value once for distinct count.
Step 2
Why this answer is correct
The correct answer is A. एक / One. Both have the same (x)-value (3), so there is one distinct zero. Tip: count a repeated value once for distinct count.
Step 3
Exam Tip
दोनों में (x)-मान समान (3) है इसलिए अलग शून्यक एक है। टिप: दोहराए मान को अलग गिनती में एक बार लें।
The same (x)-value (4) is repeated, so there is one distinct zero. Tip: do not count repetition for distinct zeroes.
Step 2
Why this answer is correct
The correct answer is A. एक / One. The same (x)-value (4) is repeated, so there is one distinct zero. Tip: do not count repetition for distinct zeroes.
Step 3
Exam Tip
एक ही (x)-मान (4) दोहराया गया है इसलिए अलग शून्यक एक है। टिप: अलग शून्यक में दोहराव न गिनें।
Repeated points give the same (x)-values, so the distinct zeroes are (-11) and (4). Tip: count the same (x)-value once.
Step 2
Why this answer is correct
The correct answer is B. दो / Two. Repeated points give the same (x)-values, so the distinct zeroes are (-11) and (4). Tip: count the same (x)-value once.
Step 3
Exam Tip
दोहराए बिंदु समान (x)-मान देते हैं, इसलिए अलग शून्यक (-11) और (4) हैं। टिप: समान (x)-मान को एक बार गिनें।
Repeated points give the same (x)-values, so the distinct zeroes are (-7) and (2). Tip: count the same (x)-value once.
Step 2
Why this answer is correct
The correct answer is B. दो / Two. Repeated points give the same (x)-values, so the distinct zeroes are (-7) and (2). Tip: count the same (x)-value once.
Step 3
Exam Tip
दोहराए बिंदु समान (x)-मान देते हैं, इसलिए अलग शून्यक (-7) और (2) हैं। टिप: समान (x)-मान को एक बार गिनें।
The zeroes are (4) and (-7), but (-7) is repeated. Tip: count repetition once for distinct zeroes.
Step 2
Why this answer is correct
The correct answer is A. (4) और (-7) / (4) and (-7). The zeroes are (4) and (-7), but (-7) is repeated. Tip: count repetition once for distinct zeroes.
Step 3
Exam Tip
शून्यक (4) और (-7) हैं पर (-7) दोहराया गया है। टिप: अलग शून्यक में दोहराव को एक बार गिनें।
The zeroes are (2) and (-6), but (-6) is repeated. Tip: count repetition once for distinct zeroes.
Step 2
Why this answer is correct
The correct answer is A. (2) और (-6) / (2) and (-6). The zeroes are (2) and (-6), but (-6) is repeated. Tip: count repetition once for distinct zeroes.
Step 3
Exam Tip
शून्यक (2) और (-6) हैं, पर (-6) दोहराया गया है। टिप: अलग शून्यक में दोहराव एक बार गिनें।
The zeroes are (-1) and (4), but (4) is repeated. Tip: do not count repetition in distinct zeroes.
Step 2
Why this answer is correct
The correct answer is A. (-1) और (4) / (-1) and (4). The zeroes are (-1) and (4), but (4) is repeated. Tip: do not count repetition in distinct zeroes.
Step 3
Exam Tip
शून्यक (-1) और (4) हैं, पर (4) दोहराया गया है। टिप: अलग शून्यक में दोहराव न गिनें।
The zeroes are (-3) and (1), so there are two distinct meeting points. Tip: count the repeated zero (1) only once for distinct points.
Step 2
Why this answer is correct
The correct answer is B. दो / Two. The zeroes are (-3) and (1), so there are two distinct meeting points. Tip: count the repeated zero (1) only once for distinct points.
Step 3
Exam Tip
शून्यक (-3) और (1) हैं, इसलिए दो अलग बिंदु मिलेंगे। टिप: दोहराए हुए शून्यक (1) को अलग गिनती में एक बार गिनें।
The repeated (2) is counted once for distinct zeroes. Tip: do not rewrite the same value for distinct zeroes.
Step 2
Why this answer is correct
The correct answer is A. (2) और (-5) / (2) and (-5). The repeated (2) is counted once for distinct zeroes. Tip: do not rewrite the same value for distinct zeroes.
Step 3
Exam Tip
दोहराया (2) अलग शून्यक में एक बार गिना जाता है। टिप: अलग शून्यक में समान मान पुनः न लिखें।
Substituting (x=0) in \(4x^3-7x\) gives (0), and it is not the zero polynomial. For (x=0), the constant term must be (0).
Step 2
Why this answer is correct
The correct answer is B. \(4x^3-7x\). Substituting (x=0) in \(4x^3-7x\) gives (0), and it is not the zero polynomial. For (x=0), the constant term must be (0).
Step 3
Exam Tip
\(4x^3-7x\) में (x=0) रखने पर (0) मिलता है और यह शून्य बहुपद नहीं है। (x=0) के लिए अचर पद (0) होना चाहिए।
Distinct zeroes are counted from distinct meeting points with the (x)-axis. Tip: degree gives the maximum, but the actual count is read from the graph.
Step 2
Why this answer is correct
The correct answer is B. दो / Two. Distinct zeroes are counted from distinct meeting points with the (x)-axis. Tip: degree gives the maximum, but the actual count is read from the graph.
Step 3
Exam Tip
अलग शून्यक अलग (x)-अक्ष मिलने वाले बिंदुओं की संख्या से मिलते हैं। टिप: घात से अधिकतम संख्या मिलती है, वास्तविक गिनती ग्राफ से पढ़ें।
B. सतह का भ्रम बनाने के लिए/To create illusion of surface
Step 1
Concept
Repeated marks create surface feeling. Exam tip: understand texture effect through marks.
Step 2
Why this answer is correct
The correct answer is B. सतह का भ्रम बनाने के लिए / To create illusion of surface. Repeated marks create surface feeling. Exam tip: understand texture effect through marks.
Step 3
Exam Tip
दोहराए चिह्न सतह की अनुभूति बनाते हैं। परीक्षा में marks से texture effect समझें।
Repeated lines create pattern and rhythm in cloth. Exam tip: connect textile design with repetition.
Step 2
Why this answer is correct
The correct answer is C. पैटर्न और लय / Pattern and rhythm. Repeated lines create pattern and rhythm in cloth. Exam tip: connect textile design with repetition.
Step 3
Exam Tip
दोहराई रेखाएं कपड़े में पैटर्न और लय देती हैं। परीक्षा में वस्त्र डिजाइन को पुनरावृत्ति से जोड़ें।
The vertex lies on the (x)-axis, so the parabola touches at ((-14,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 2
Why this answer is correct
The correct answer is B. एक / One. The vertex lies on the (x)-axis, so the parabola touches at ((-14,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 3
Exam Tip
शीर्ष (x)-अक्ष पर है, इसलिए परवलय ((-14,0)) पर स्पर्श करेगा। टिप: शीर्ष का (y)-मान (0) हो तो एक अलग शून्यक होता है।
The vertex lies on the (x)-axis, so the parabola touches at ((12,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 2
Why this answer is correct
The correct answer is B. एक / One. The vertex lies on the (x)-axis, so the parabola touches at ((12,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 3
Exam Tip
शीर्ष (x)-अक्ष पर है, इसलिए परवलय ((12,0)) पर स्पर्श करेगा। टिप: शीर्ष का (y)-मान (0) हो तो एक अलग शून्यक होता है।
The vertex lies on the (x)-axis, so the parabola touches at ((-5,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 2
Why this answer is correct
The correct answer is B. एक / One. The vertex lies on the (x)-axis, so the parabola touches at ((-5,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 3
Exam Tip
शीर्ष (x)-अक्ष पर है इसलिए परवलय ((-5,0)) पर स्पर्श करेगा। टिप: शीर्ष का (y)-मान (0) हो तो एक अलग शून्यक होता है।
The vertex lies on the (x)-axis, so the parabola touches at ((4,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 2
Why this answer is correct
The correct answer is B. एक / One. The vertex lies on the (x)-axis, so the parabola touches at ((4,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 3
Exam Tip
शीर्ष (x)-अक्ष पर है, इसलिए परवलय ((4,0)) पर स्पर्श करेगा। टिप: शीर्ष का (y)-मान (0) हो तो एक अलग शून्यक होता है।
Both cutting and touching count as meeting the (x)-axis. If the two points are distinct, there are two distinct real zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो / Two. Both cutting and touching count as meeting the (x)-axis. If the two points are distinct, there are two distinct real zeroes.
Step 3
Exam Tip
कटना और छूना दोनों (x)-अक्ष से मिलना है। यदि दोनों बिंदु अलग हैं, तो दो अलग वास्तविक शून्यक होंगे।
The sum of zeroes is (2), so the other zero is (2-\(1+\sqrt{3}\)=1-\sqrt{3}). With rational coefficients, the conjugate also appears.
Step 2
Why this answer is correct
The correct answer is A. \(1-\sqrt{3}\). The sum of zeroes is (2), so the other zero is (2-\(1+\sqrt{3}\)=1-\sqrt{3}). With rational coefficients, the conjugate also appears.
Step 3
Exam Tip
शून्यकों का योग (2) है, इसलिए दूसरा शून्यक (2-\(1+\sqrt{3}\)=1-\sqrt{3}) है। परिमेय गुणांकों में संयुग्मी भी मिलता है।
A. दूसरा (7), कटान ((6,0)), ((7,0))/Other (7), intersections ((6,0)), ((7,0))
Step 1
Concept
In the quadratic, the sum of zeroes is (13), so the other zero is (7). Tip: convert a zero into ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. दूसरा (7), कटान ((6,0)), ((7,0)) / Other (7), intersections ((6,0)), ((7,0)). In the quadratic, the sum of zeroes is (13), so the other zero is (7). Tip: convert a zero into ((x,0)).
Step 3
Exam Tip
द्विघात में शून्यकों का योग (13) है, इसलिए दूसरा शून्यक (7) है। टिप: शून्यक को ((x,0)) में बदलें।
The average of the two zeroes is (5), so the other zero is (11). Tip: the axis of symmetry passes through the midpoint of zeroes.
Step 2
Why this answer is correct
The correct answer is A. (11). The average of the two zeroes is (5), so the other zero is (11). Tip: the axis of symmetry passes through the midpoint of zeroes.
Step 3
Exam Tip
दो शून्यकों का औसत (5) है इसलिए दूसरा शून्यक (11) होगा। टिप: सममिति अक्ष शून्यकों के मध्य से गुजरता है।
A. दूसरा (7), कटान ((4,0)), ((7,0))/Other (7), intersections ((4,0)), ((7,0))
Step 1
Concept
In the quadratic, the sum of zeroes is (11), so the other zero is (7). Tip: convert a zero into ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. दूसरा (7), कटान ((4,0)), ((7,0)) / Other (7), intersections ((4,0)), ((7,0)). In the quadratic, the sum of zeroes is (11), so the other zero is (7). Tip: convert a zero into ((x,0)).
Step 3
Exam Tip
द्विघात में शून्यकों का योग (11) है, इसलिए दूसरा शून्यक (7) है। टिप: शून्यक को ((x,0)) में बदलें।
The average of the two zeroes is (-2), so the other zero is (-9). Tip: connect the axis of symmetry with the midpoint of zeroes.
Step 2
Why this answer is correct
The correct answer is A. (-9). The average of the two zeroes is (-2), so the other zero is (-9). Tip: connect the axis of symmetry with the midpoint of zeroes.
Step 3
Exam Tip
दो शून्यकों का औसत (-2) है, इसलिए दूसरा शून्यक (-9) होगा। टिप: सममिति अक्ष को शून्यकों के मध्य से जोड़ें।
A. दूसरा (5), कटान ((4,0)), ((5,0))/Other (5), intersections ((4,0)), ((5,0))
Step 1
Concept
In the quadratic, the sum of zeroes is (9), so the other zero is (5). Tip: quickly convert a zero to ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. दूसरा (5), कटान ((4,0)), ((5,0)) / Other (5), intersections ((4,0)), ((5,0)). In the quadratic, the sum of zeroes is (9), so the other zero is (5). Tip: quickly convert a zero to ((x,0)).
Step 3
Exam Tip
द्विघात में शून्यकों का योग (9) है इसलिए दूसरा शून्यक (5) है। टिप: शून्यक को तुरंत ((x,0)) में बदलें।
The average of the two zeroes is (3), so the other zero is (-5). Tip: set \(\frac{a+b}{2}\) equal to the axis of symmetry.
Step 2
Why this answer is correct
The correct answer is A. (-5). The average of the two zeroes is (3), so the other zero is (-5). Tip: set \(\frac{a+b}{2}\) equal to the axis of symmetry.
Step 3
Exam Tip
दो शून्यकों का औसत (3) है इसलिए दूसरा शून्यक (-5) होगा। टिप: \(\frac{a+b}{2}\) को सममिति अक्ष के बराबर रखें।
The average of the two zeroes is (4), so the other zero is (10). Tip: connect the axis of symmetry with the midpoint of zeroes.
Step 2
Why this answer is correct
The correct answer is A. (10). The average of the two zeroes is (4), so the other zero is (10). Tip: connect the axis of symmetry with the midpoint of zeroes.
Step 3
Exam Tip
दोनों शून्यकों का औसत (4) है इसलिए दूसरा शून्यक (10) होगा। टिप: सममिति अक्ष को शून्यकों के मध्य मान से जोड़ें।
A. दूसरा (4), कटान ((3,0)), ((4,0))/Other (4), intersections ((3,0)), ((4,0))
Step 1
Concept
In the quadratic, the sum of zeroes is (7), so the other zero is (4). Tip: quickly convert a zero to ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. दूसरा (4), कटान ((3,0)), ((4,0)) / Other (4), intersections ((3,0)), ((4,0)). In the quadratic, the sum of zeroes is (7), so the other zero is (4). Tip: quickly convert a zero to ((x,0)).
Step 3
Exam Tip
द्विघात में शून्यकों का योग (7) है, इसलिए दूसरा शून्यक (4) है। टिप: शून्यक को तुरंत ((x,0)) में बदलें।
The average of the two zeroes is (2), so the other zero is (-5). Tip: set \( \frac{a+b}{2} \) equal to the axis of symmetry.
Step 2
Why this answer is correct
The correct answer is A. (-5). The average of the two zeroes is (2), so the other zero is (-5). Tip: set \( \frac{a+b}{2} \) equal to the axis of symmetry.
Step 3
Exam Tip
दो शून्यकों का औसत (2) है, इसलिए दूसरा शून्यक (-5) होगा। टिप: \( \frac{a+b}{2} \) को सममिति अक्ष के बराबर रखें।
The average of the two zeroes is (1), so the other zero is (7). Tip: the axis of symmetry passes through the midpoint of zeroes.
Step 2
Why this answer is correct
The correct answer is C. (7). The average of the two zeroes is (1), so the other zero is (7). Tip: the axis of symmetry passes through the midpoint of zeroes.
Step 3
Exam Tip
दो शून्यकों का औसत (1) होगा इसलिए दूसरा शून्यक (7) है। टिप: सममिति अक्ष शून्यकों के मध्य से गुजरता है।
A. दूसरा (3), कटान ((2,0)), ((3,0))/Other (3), intersections ((2,0)), ((3,0))
Step 1
Concept
In the quadratic, the sum of zeroes is (5), so the other zero is (3). Tip: immediately convert a zero to ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. दूसरा (3), कटान ((2,0)), ((3,0)) / Other (3), intersections ((2,0)), ((3,0)). In the quadratic, the sum of zeroes is (5), so the other zero is (3). Tip: immediately convert a zero to ((x,0)).
Step 3
Exam Tip
द्विघात में शून्यकों का योग (5) है, इसलिए दूसरा शून्यक (3) है। टिप: शून्यक को तुरंत ((x,0)) में बदलें।
(p(x)=\(x+\sqrt{5}\)2), so the zero is \(-\sqrt{5}\) twice. A perfect-square form gives a repeated zero.
Step 2
Why this answer is correct
The correct answer is A. \(-\sqrt{5}\) दो बार / \(-\sqrt{5}\) twice. (p(x)=\(x+\sqrt{5}\)2), so the zero is \(-\sqrt{5}\) twice. A perfect-square form gives a repeated zero.
Step 3
Exam Tip
(p(x)=\(x+\sqrt{5}\)2), इसलिए शून्यक \(-\sqrt{5}\) दो बार है। पूर्ण वर्ग रूप से दोहराया शून्यक मिलता है।
The coefficient ratio is \(\frac{1}{2}\), so coincidence needs (k=90). For (k=88), the lines are distinct and parallel.
Step 2
Why this answer is correct
The correct answer is B. (k=88). The coefficient ratio is \(\frac{1}{2}\), so coincidence needs (k=90). For (k=88), the lines are distinct and parallel.
Step 3
Exam Tip
गुणांक का अनुपात \(\frac{1}{2}\) है, इसलिए संपाती होने के लिए (k=90) चाहिए। (k=88) पर रेखाएं समांतर अलग-अलग हैं।
For parallel lines, \(\frac{3}{9}=\frac{a}{12}\), so (a=4). Since \(\frac{15}{47}\neq\frac{1}{3}\), they will not be coincident.
Step 2
Why this answer is correct
The correct answer is C. (a=4). For parallel lines, \(\frac{3}{9}=\frac{a}{12}\), so (a=4). Since \(\frac{15}{47}\neq\frac{1}{3}\), they will not be coincident.
Step 3
Exam Tip
समांतर के लिए \(\frac{3}{9}=\frac{a}{12}\), इसलिए (a=4)। चूंकि \(\frac{15}{47}\neq\frac{1}{3}\), वे संपाती नहीं होंगी।
The coefficient ratio is \(\frac{1}{2}\), so coincidence needs (k=42). For (k=40), the lines are distinct and parallel.
Step 2
Why this answer is correct
The correct answer is B. (k=40). The coefficient ratio is \(\frac{1}{2}\), so coincidence needs (k=42). For (k=40), the lines are distinct and parallel.
Step 3
Exam Tip
गुणांक का अनुपात \(\frac{1}{2}\) है, इसलिए संपाती होने के लिए (k=42) चाहिए। (k=40) पर रेखाएं समांतर अलग-अलग हैं।
For parallel lines, \(\frac{4}{8}=\frac{a}{10}\), so (a=5). Since \(\frac{16}{35}\neq\frac{1}{2}\), they will not be coincident.
Step 2
Why this answer is correct
The correct answer is B. (a=5). For parallel lines, \(\frac{4}{8}=\frac{a}{10}\), so (a=5). Since \(\frac{16}{35}\neq\frac{1}{2}\), they will not be coincident.
Step 3
Exam Tip
समांतर के लिए \(\frac{4}{8}=\frac{a}{10}\), इसलिए (a=5)। चूंकि \(\frac{16}{35}\neq\frac{1}{2}\), वे संपाती नहीं होंगी।
The coefficient ratio is \(\frac{1}{2}\), so coincidence needs (k=24). For (k=20), the lines are distinct and parallel.
Step 2
Why this answer is correct
The correct answer is B. (k=20). The coefficient ratio is \(\frac{1}{2}\), so coincidence needs (k=24). For (k=20), the lines are distinct and parallel.
Step 3
Exam Tip
गुणांक का अनुपात \(\frac{1}{2}\) है, इसलिए संपाती होने के लिए (k=24) चाहिए। (k=20) पर रेखाएं समांतर अलग-अलग हैं।
For parallel lines, \(\frac{2}{6}=\frac{a}{9}\), so (a=3). Since \(\frac{10}{31}\neq\frac{1}{3}\), the lines are not coincident.
Step 2
Why this answer is correct
The correct answer is B. (a=3). For parallel lines, \(\frac{2}{6}=\frac{a}{9}\), so (a=3). Since \(\frac{10}{31}\neq\frac{1}{3}\), the lines are not coincident.
Step 3
Exam Tip
समांतर के लिए \(\frac{2}{6}=\frac{a}{9}\), इसलिए (a=3)। चूंकि \(\frac{10}{31}\neq\frac{1}{3}\), रेखाएं संपाती नहीं होंगी।
The coefficient ratio is \(\frac{3}{6}=\frac{2}{4}=\frac{1}{2}\); for coincidence, (k=14) is needed. With (k=7), the lines are parallel and distinct.
Step 2
Why this answer is correct
The correct answer is B. (7). The coefficient ratio is \(\frac{3}{6}=\frac{2}{4}=\frac{1}{2}\); for coincidence, (k=14) is needed. With (k=7), the lines are parallel and distinct.
Step 3
Exam Tip
गुणांक अनुपात \(\frac{3}{6}=\frac{2}{4}=\frac{1}{2}\) है; संपाती होने के लिए (k=14) चाहिए। (k=7) होने पर रेखाएं समांतर अलग-अलग होंगी।
Dividing (6x-4y=20) by (2) gives (3x-2y=10). Same left side with different constants gives distinct parallel lines.
Step 2
Why this answer is correct
The correct answer is C. (6x-4y=20). Dividing (6x-4y=20) by (2) gives (3x-2y=10). Same left side with different constants gives distinct parallel lines.
Step 3
Exam Tip
(6x-4y=20) को (2) से भाग देने पर (3x-2y=10) मिलता है। समान बाएँ पक्ष और अलग नियतांक अलग समांतर रेखाएँ देते हैं।
Dividing (4x-6y=18) by (2) gives (2x-3y=9). Same left side with different constants gives distinct parallel lines.
Step 2
Why this answer is correct
The correct answer is C. (4x-6y=18). Dividing (4x-6y=18) by (2) gives (2x-3y=9). Same left side with different constants gives distinct parallel lines.
Step 3
Exam Tip
(4x-6y=18) को (2) से भाग देने पर (2x-3y=9) मिलता है। समान बाएँ पक्ष और अलग नियतांक अलग समांतर रेखाएँ देते हैं।
Dividing (2x+6y=30) by (2) gives (x+3y=15). Same left side with different constants gives distinct parallel lines.
Step 2
Why this answer is correct
The correct answer is C. (2x+6y=30). Dividing (2x+6y=30) by (2) gives (x+3y=15). Same left side with different constants gives distinct parallel lines.
Step 3
Exam Tip
(2x+6y=30) को (2) से भाग देने पर (x+3y=15) मिलता है। समान बाएँ पक्ष और अलग नियतांक समांतर अलग रेखाएँ देते हैं।
A. \(\theta<0\) या \(\theta>3\)/\(\theta<0\) or \(\theta>3\)
Step 1
Concept
Here (D=4\theta-2-12\theta=4\theta\(\theta-3\)). From (D>0), \(\theta<0\) or \(\theta>3\).
Step 2
Why this answer is correct
The correct answer is A. \(\theta<0\) या \(\theta>3\) / \(\theta<0\) or \(\theta>3\). Here (D=4\theta-2-12\theta=4\theta\(\theta-3\)). From (D>0), \(\theta<0\) or \(\theta>3\).
Step 3
Exam Tip
यहाँ (D=4\theta-2-12\theta=4\theta\(\theta-3\)) है। (D>0) से \(\theta<0\) या \(\theta>3\)।
A. \(\lambda<0\) या \(\lambda>1\)/\(\lambda<0\) or \(\lambda>1\)
Step 1
Concept
Here (D=4\lambda-2-4\lambda=4\lambda\(\lambda-1\)). For distinct real roots (D>0), so \(\lambda<0\) or \(\lambda>1\).
Step 2
Why this answer is correct
The correct answer is A. \(\lambda<0\) या \(\lambda>1\) / \(\lambda<0\) or \(\lambda>1\). Here (D=4\lambda-2-4\lambda=4\lambda\(\lambda-1\)). For distinct real roots (D>0), so \(\lambda<0\) or \(\lambda>1\).
Step 3
Exam Tip
यहाँ (D=4\lambda-2-4\lambda=4\lambda\(\lambda-1\)) है। असमान वास्तविक मूलों के लिए (D>0), इसलिए \(\lambda<0\) या \(\lambda>1\)।
A. \(k<2-4\sqrt{3}\) या \(k>2+4\sqrt{3}\)/\(k<2-4\sqrt{3}\) or \(k>2+4\sqrt{3}\)
Step 1
Concept
Here (D=(k-2)2-48). For distinct real roots (D>0), so ((k-2)2>48).
Step 2
Why this answer is correct
The correct answer is A. \(k<2-4\sqrt{3}\) या \(k>2+4\sqrt{3}\) / \(k<2-4\sqrt{3}\) or \(k>2+4\sqrt{3}\). Here (D=(k-2)2-48). For distinct real roots (D>0), so ((k-2)2>48).
Step 3
Exam Tip
यहाँ (D=(k-2)2-48) है। असमान वास्तविक मूलों के लिए (D>0), इसलिए ((k-2)2>48)।
A. \(k<-1-2\sqrt{6}\) या \(k>-1+2\sqrt{6}\)/\(k<-1-2\sqrt{6}\) or \(k>-1+2\sqrt{6}\)
Step 1
Concept
Here (D=(k+1)2-24). For distinct real roots (D>0), so ((k+1)2>24).
Step 2
Why this answer is correct
The correct answer is A. \(k<-1-2\sqrt{6}\) या \(k>-1+2\sqrt{6}\) / \(k<-1-2\sqrt{6}\) or \(k>-1+2\sqrt{6}\). Here (D=(k+1)2-24). For distinct real roots (D>0), so ((k+1)2>24).
Step 3
Exam Tip
यहाँ (D=(k+1)2-24) है। असमान वास्तविक मूलों के लिए (D>0), इसलिए ((k+1)2>24)।
In the first equation, (D=100-92=8>0), and (8) is not a perfect square. So the roots are real, irrational and distinct.
Step 2
Why this answer is correct
The correct answer is A. \(x^2-10x+23=0\). In the first equation, (D=100-92=8>0), and (8) is not a perfect square. So the roots are real, irrational and distinct.
Step 3
Exam Tip
पहले समीकरण में (D=100-92=8>0) है और (8) पूर्ण वर्ग नहीं है। इसलिए मूल वास्तविक, अपरिमेय और भिन्न हैं।
A. \(k^2<9\) और \(k\neq0\)/\(k^2<9\) and \(k\neq0\)
Step 1
Concept
Here \(D=36-4k^2\). For real and distinct roots (D>0) and \(k\neq0\), hence \(k^2<9\).
Step 2
Why this answer is correct
The correct answer is A. \(k^2<9\) और \(k\neq0\) / \(k^2<9\) and \(k\neq0\). Here \(D=36-4k^2\). For real and distinct roots (D>0) and \(k\neq0\), hence \(k^2<9\).
Step 3
Exam Tip
यहाँ \(D=36-4k^2\) है। वास्तविक और भिन्न मूलों के लिए (D>0) और \(k\neq0\), अतः \(k^2<9\)।
