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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।
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।
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 \(k^2<9\)।
For both roots to be negative, the sum (-12) and product \(\lambda>0\) are needed. For real distinct roots, \(144-4\lambda>0\), so \(0<\lambda<36\).
Step 2
Why this answer is correct
The correct answer is A. \(0<\lambda<36\). For both roots to be negative, the sum (-12) and product \(\lambda>0\) are needed. For real distinct roots, \(144-4\lambda>0\), so \(0<\lambda<36\).
Step 3
Exam Tip
दोनों ऋणात्मक जड़ों के लिए योग (-12) और गुणनफल \(\lambda>0\) चाहिए। वास्तविक भिन्न जड़ों के लिए \(144-4\lambda>0\), इसलिए \(0<\lambda<36\)।
For both roots to be negative, the sum (-10) and product \(\lambda>0\) are needed. For real distinct roots, \(100-4\lambda>0\), hence \(0<\lambda<25\).
Step 2
Why this answer is correct
The correct answer is B. \(0<\lambda<25\). For both roots to be negative, the sum (-10) and product \(\lambda>0\) are needed. For real distinct roots, \(100-4\lambda>0\), hence \(0<\lambda<25\).
Step 3
Exam Tip
दोनों ऋणात्मक जड़ों के लिए योग (-10) और गुणनफल \(\lambda>0\) चाहिए। वास्तविक भिन्न जड़ों के लिए \(100-4\lambda>0\), इसलिए \(0<\lambda<25\)।
For both roots to be negative, the sum (-2) and product \(\lambda>0\) are needed. For real distinct roots, \(4-4\lambda>0\), hence \(0<\lambda<1\).
Step 2
Why this answer is correct
The correct answer is A. \(0<\lambda<1\). For both roots to be negative, the sum (-2) and product \(\lambda>0\) are needed. For real distinct roots, \(4-4\lambda>0\), hence \(0<\lambda<1\).
Step 3
Exam Tip
दोनों ऋणात्मक जड़ों के लिए योग (-2) और गुणनफल \(\lambda>0\) चाहिए। वास्तविक भिन्न जड़ों के लिए \(4-4\lambda>0\), इसलिए \(0<\lambda<1\)।
A. दो भिन्न वास्तविक शून्यक/Two distinct real zeroes
Step 1
Concept
Two separate intersections give two distinct real zeroes. Different (x)-intercepts mean different zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो भिन्न वास्तविक शून्यक / Two distinct real zeroes. Two separate intersections give two distinct real zeroes. Different (x)-intercepts mean different zeroes.
Step 3
Exam Tip
दो अलग कटान दो अलग वास्तविक शून्यक देते हैं। ग्राफ में अलग (x)-प्रतिच्छेद अलग शून्यक होते हैं।
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।
A. दो बिंदु, (x=-5) पर स्पर्श/Two points, touching at (x=-5)
Step 1
Concept
The zeroes are (-5) and (14), and ((x+5)2) causes touching at (-5). Tip: the outside (11) does not change the zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो बिंदु, (x=-5) पर स्पर्श / Two points, touching at (x=-5). The zeroes are (-5) and (14), and ((x+5)2) causes touching at (-5). Tip: the outside (11) does not change the zeroes.
Step 3
Exam Tip
शून्यक (-5) और (14) हैं तथा ((x+5)2) के कारण (-5) पर स्पर्श है। टिप: बाहरी (11) शून्यक नहीं बदलता।
For eight distinct real zeroes, the degree must be at least (8). Tip: the number of distinct zeroes cannot exceed the degree.
Step 2
Why this answer is correct
The correct answer is C. (8). For eight distinct real zeroes, the degree must be at least (8). Tip: the number of distinct zeroes cannot exceed the degree.
Step 3
Exam Tip
आठ अलग वास्तविक शून्यकों के लिए घात कम से कम (8) होनी चाहिए। टिप: अलग शून्यकों की संख्या घात से अधिक नहीं हो सकती।
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।
From (x-c=0) we get (c), and from (x+d=0) we get (-d). Tip: do not count repetition among distinct zeroes.
Step 2
Why this answer is correct
The correct answer is A. (c) और (-d) / (c) and (-d). From (x-c=0) we get (c), and from (x+d=0) we get (-d). Tip: do not count repetition among distinct zeroes.
Step 3
Exam Tip
(x-c=0) से (c) और (x+d=0) से (-d) मिलता है। टिप: अलग शून्यकों में दोहराव न गिनें।
There are two distinct zeroes (-2) and (5), and both have even powers. Tip: at an even-power zero the graph usually touches.
Step 2
Why this answer is correct
The correct answer is A. दो, दोनों पर स्पर्श / Two, touches at both. There are two distinct zeroes (-2) and (5), and both have even powers. Tip: at an even-power zero the graph usually touches.
Step 3
Exam Tip
दो अलग शून्यक (-2) और (5) हैं तथा दोनों की घात सम है। टिप: सम घात वाले शून्यक पर ग्राफ सामान्यतः स्पर्श करता 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।
A. दो बिंदु, (x=-4) पर स्पर्श/Two points, touching at (x=-4)
Step 1
Concept
The zeroes are (-4) and (12), and ((x+4)2) causes touching at (-4). Tip: the outside (9) does not change the zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो बिंदु, (x=-4) पर स्पर्श / Two points, touching at (x=-4). The zeroes are (-4) and (12), and ((x+4)2) causes touching at (-4). Tip: the outside (9) does not change the zeroes.
Step 3
Exam Tip
शून्यक (-4) और (12) हैं तथा ((x+4)2) के कारण (-4) पर स्पर्श है। टिप: बाहरी (9) शून्यक नहीं बदलता।
For seven distinct real zeroes, the degree must be at least (7). Tip: the number of distinct zeroes cannot exceed the degree.
Step 2
Why this answer is correct
The correct answer is C. (7). For seven distinct real zeroes, the degree must be at least (7). Tip: the number of distinct zeroes cannot exceed the degree.
Step 3
Exam Tip
सात अलग वास्तविक शून्यकों के लिए घात कम से कम (7) होनी चाहिए। टिप: अलग शून्यकों की संख्या घात से अधिक नहीं हो सकती।
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।
From (x+a=0) we get (-a), and from (x-b=0) we get (b). Tip: do not count repetition among distinct zeroes.
Step 2
Why this answer is correct
The correct answer is A. (-a) और (b) / (-a) and (b). From (x+a=0) we get (-a), and from (x-b=0) we get (b). Tip: do not count repetition among distinct zeroes.
Step 3
Exam Tip
(x+a=0) से (-a) और (x-b=0) से (b) मिलता है। टिप: अलग शून्यकों में दोहराव न गिनें।
There are two distinct zeroes (1) and (-4), and both have even powers. Tip: at an even-power zero the graph usually touches.
Step 2
Why this answer is correct
The correct answer is A. दो, दोनों पर स्पर्श / Two, touches at both. There are two distinct zeroes (1) and (-4), and both have even powers. Tip: at an even-power zero the graph usually touches.
Step 3
Exam Tip
दो अलग शून्यक (1) और (-4) हैं तथा दोनों की घात सम है। टिप: सम घात वाले शून्यक पर ग्राफ सामान्यतः स्पर्श करता 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।
A. दो बिंदु, (x=-3) पर स्पर्श/Two points, touching at (x=-3)
Step 1
Concept
The zeroes are (-3) and (10), and ((x+3)2) causes touching at (-3). Tip: the outside (7) does not change the zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो बिंदु, (x=-3) पर स्पर्श / Two points, touching at (x=-3). The zeroes are (-3) and (10), and ((x+3)2) causes touching at (-3). Tip: the outside (7) does not change the zeroes.
Step 3
Exam Tip
शून्यक (-3) और (10) हैं तथा ((x+3)2) के कारण (-3) पर स्पर्श है। टिप: बाहरी (7) शून्यक नहीं बदलता।
For six distinct real zeroes, the degree must be at least (6). Tip: the number of distinct zeroes cannot exceed the degree.
Step 2
Why this answer is correct
The correct answer is C. (6). For six distinct real zeroes, the degree must be at least (6). Tip: the number of distinct zeroes cannot exceed the degree.
Step 3
Exam Tip
छह अलग वास्तविक शून्यकों के लिए घात कम से कम (6) होनी चाहिए। टिप: अलग शून्यकों की संख्या घात से अधिक नहीं हो सकती।
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 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।
A. दो बिंदु, (x=-2) पर स्पर्श/Two points, touching at (x=-2)
Step 1
Concept
The zeroes are (-2) and (7), and ((x+2)2) causes touching at (-2). Tip: the outside (5) does not change the zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो बिंदु, (x=-2) पर स्पर्श / Two points, touching at (x=-2). The zeroes are (-2) and (7), and ((x+2)2) causes touching at (-2). Tip: the outside (5) does not change the zeroes.
Step 3
Exam Tip
शून्यक (-2) और (7) हैं, तथा ((x+2)2) के कारण (-2) पर स्पर्श है। टिप: बाहरी (5) शून्यक नहीं बदलता।
For four distinct real zeroes, the degree must be at least (4). Tip: the number of distinct zeroes cannot exceed the degree.
Step 2
Why this answer is correct
The correct answer is C. (4). For four distinct real zeroes, the degree must be at least (4). Tip: the number of distinct zeroes cannot exceed the degree.
Step 3
Exam Tip
चार अलग वास्तविक शून्यकों के लिए घात कम से कम (4) होनी चाहिए। टिप: अलग शून्यकों की संख्या घात से अधिक नहीं हो सकती।
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।
A. दो बिंदु, (x=2) पर स्पर्श/Two points, touching at (x=2)
Step 1
Concept
The zeroes are (2) and (-1), and ((x-2)2) causes touching at (x=2). Tip: the outside (3) does not change the zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो बिंदु, (x=2) पर स्पर्श / Two points, touching at (x=2). The zeroes are (2) and (-1), and ((x-2)2) causes touching at (x=2). Tip: the outside (3) does not change the zeroes.
Step 3
Exam Tip
शून्यक (2) और (-1) हैं, तथा ((x-2)2) के कारण (x=2) पर स्पर्श है। टिप: बाहरी (3) शून्यक नहीं बदलता।
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।
(x-3-9x=x(x-3)(x+3)), so there are three distinct zeroes. Tip: take the common factor and use difference of squares.
Step 2
Why this answer is correct
The correct answer is C. तीन / Three. (x-3-9x=x(x-3)(x+3)), so there are three distinct zeroes. Tip: take the common factor and use difference of squares.
Step 3
Exam Tip
(x-3-9x=x(x-3)(x+3)), इसलिए तीन अलग शून्यक हैं। टिप: सामान्य गुणनखंड निकालकर वर्गों का अंतर देखें।
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।
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।
A. जो (x)-अक्ष को दो अलग बिंदुओं पर काटे/One that cuts the (x)-axis at two distinct points
Step 1
Concept
Two distinct (x)-axis intersections give two distinct real zeroes. Tip: do not count (y)-axis intersections as zeroes.
Step 2
Why this answer is correct
The correct answer is A. जो (x)-अक्ष को दो अलग बिंदुओं पर काटे / One that cuts the (x)-axis at two distinct points. Two distinct (x)-axis intersections give two distinct real zeroes. Tip: do not count (y)-axis intersections as zeroes.
Step 3
Exam Tip
दो अलग (x)-अक्ष कटान दो अलग वास्तविक शून्यक देते हैं। टिप: (y)-अक्ष कटान को शून्यक न गिनें।
The three different factors give three distinct zeroes (1), (4), (-2). Tip: each linear factor gives a possible intersection.
Step 2
Why this answer is correct
The correct answer is C. तीन / Three. The three different factors give three distinct zeroes (1), (4), (-2). Tip: each linear factor gives a possible intersection.
Step 3
Exam Tip
तीन अलग कारक तीन अलग शून्यक (1), (4), (-2) देते हैं। टिप: हर रैखिक कारक एक संभावित कटान देता है।
