The average of the two zeroes is (2), so the other zero is (7). Tip: in a parabola the axis of symmetry passes through the midpoint of the zeroes.
Step 2
Why this answer is correct
The correct answer is A. (7). The average of the two zeroes is (2), so the other zero is (7). Tip: in a parabola the axis of symmetry passes through the midpoint of the zeroes.
Step 3
Exam Tip
दो शून्यकों का औसत (2) होगा, इसलिए दूसरा शून्यक (7) है। टिप: परवलय में सममिति अक्ष शून्यकों के मध्य से गुजरता है।
A. (x=-4) पर छुएगा और (x=3) पर काटेगा/It touches at (x=-4) and crosses at (x=3)
Step 1
Concept
The squared factor ((x+4)2) gives touching, and the single factor (x-3) gives crossing. Tip: an even-power factor usually shows touching.
Step 2
Why this answer is correct
The correct answer is A. (x=-4) पर छुएगा और (x=3) पर काटेगा / It touches at (x=-4) and crosses at (x=3). The squared factor ((x+4)2) gives touching, and the single factor (x-3) gives crossing. Tip: an even-power factor usually shows touching.
Step 3
Exam Tip
वर्ग कारक ((x+4)2) स्पर्श देता है और एकल कारक (x-3) कटान देता है। टिप: सम घात वाला कारक सामान्यतः स्पर्श दिखाता है।
For an upward-opening parabola, the graph lies below the (x)-axis between the two zeroes. Tip: identify the sign region between zeroes.
Step 2
Why this answer is correct
The correct answer is C. (x)-अक्ष के नीचे / Below the (x)-axis. For an upward-opening parabola, the graph lies below the (x)-axis between the two zeroes. Tip: identify the sign region between zeroes.
Step 3
Exam Tip
ऊपर खुलने वाले परवलय में दो शून्यकों के बीच ग्राफ (x)-अक्ष के नीचे होता है। टिप: शून्यकों के बीच के क्षेत्र का संकेत पहचानें।
Between them ((x-1)) is positive and ((x-5)) is negative, and the outside negative makes the value positive. Tip: graph position is decided by the sign of (p(x)).
Step 2
Why this answer is correct
The correct answer is A. ऊपर / Above. Between them ((x-1)) is positive and ((x-5)) is negative, and the outside negative makes the value positive. Tip: graph position is decided by the sign of (p(x)).
Step 3
Exam Tip
बीच में ((x-1)) धनात्मक और ((x-5)) ऋणात्मक है, बाहर का ऋण चिन्ह मान को धनात्मक बनाता है। टिप: ग्राफ की स्थिति (p(x)) के चिह्न से तय करें।
The vertex is below the (x)-axis and the graph opens upward, so it cuts the (x)-axis twice. Tip: check vertex height and opening direction together.
Step 2
Why this answer is correct
The correct answer is C. दो / Two. The vertex is below the (x)-axis and the graph opens upward, so it cuts the (x)-axis twice. Tip: check vertex height and opening direction together.
Step 3
Exam Tip
शीर्ष (x)-अक्ष के नीचे है और ग्राफ ऊपर खुलता है, इसलिए वह (x)-अक्ष को दो बार काटेगा। टिप: शीर्ष की ऊँचाई और खुलने की दिशा साथ देखें।
The vertex lies on the (x)-axis, so the graph touches there. Tip: the opening direction does not change the touching point.
Step 2
Why this answer is correct
The correct answer is C. (x=-1) पर स्पर्श करेगा / It will touch at (x=-1). The vertex lies on the (x)-axis, so the graph touches there. Tip: the opening direction does not change the touching point.
Step 3
Exam Tip
शीर्ष (x)-अक्ष पर है, इसलिए ग्राफ वहीं स्पर्श करता है। टिप: खुलने की दिशा स्पर्श बिंदु नहीं बदलती।
The polynomial equals ((x-a)(x-b)), so the zeroes are (a) and (b). Tip: connect factor form with graph intersections.
Step 2
Why this answer is correct
The correct answer is A. ((a,0)) और ((b,0)) / ((a,0)) and ((b,0)). The polynomial equals ((x-a)(x-b)), so the zeroes are (a) and (b). Tip: connect factor form with graph intersections.
Step 3
Exam Tip
बहुपद ((x-a)(x-b)) के बराबर है, इसलिए शून्यक (a) और (b) हैं। टिप: गुणनखंड रूप को ग्राफ कटान से जोड़ें।
In both crossing and touching, (p(x)=0). Tip: crossing is not necessary for a zero.
Step 2
Why this answer is correct
The correct answer is A. दोनों वास्तविक शून्यक हैं / Both are real zeroes. In both crossing and touching, (p(x)=0). Tip: crossing is not necessary for a zero.
Step 3
Exam Tip
कटान और स्पर्श दोनों स्थितियों में (p(x)=0) होता है। टिप: शून्यक के लिए पार करना जरूरी नहीं है।
(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।
B. दोनों शून्यकों पर स्पर्श करेगा/It will touch at both zeroes
Step 1
Concept
Both factors have even powers, so the graph touches at both points. Tip: at an even power the graph usually turns back.
Step 2
Why this answer is correct
The correct answer is B. दोनों शून्यकों पर स्पर्श करेगा / It will touch at both zeroes. Both factors have even powers, so the graph touches at both points. Tip: at an even power the graph usually turns back.
Step 3
Exam Tip
दोनों कारक सम घात में हैं, इसलिए दोनों बिंदुओं पर स्पर्श होगा। टिप: सम घात पर ग्राफ सामान्यतः दिशा बदलता है।
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-2+2x-15=(x+5)(x-3)), so the zeroes are (-5) and (3). Tip: form points ((x,0)) from factors.
Step 2
Why this answer is correct
The correct answer is A. ((3,0)) और ((-5,0)) / ((3,0)) and ((-5,0)). (x-2+2x-15=(x+5)(x-3)), so the zeroes are (-5) and (3). Tip: form points ((x,0)) from factors.
Step 3
Exam Tip
(x-2+2x-15=(x+5)(x-3)), इसलिए शून्यक (-5) और (3) हैं। टिप: गुणनखंडों से बिंदु ((x,0)) बनाएं।
B. कथन गलत है क्योंकि ((0,0)) (x)-अक्ष पर भी है/The statement is wrong because ((0,0)) is also on the (x)-axis
Step 1
Concept
The origin lies on both axes, so (p(0)=0). Tip: treat ((0,0)) as a special point.
Step 2
Why this answer is correct
The correct answer is B. कथन गलत है क्योंकि ((0,0)) (x)-अक्ष पर भी है / The statement is wrong because ((0,0)) is also on the (x)-axis. The origin lies on both axes, so (p(0)=0). Tip: treat ((0,0)) as a special point.
Step 3
Exam Tip
मूल बिंदु दोनों अक्षों पर होता है, इसलिए (p(0)=0) है। टिप: ((0,0)) को विशेष बिंदु समझें।
A. क्योंकि यह ((x-3)2+4) है/Because it is ((x-3)2+4)
Step 1
Concept
((x-3)2+4) is always positive, so (p(x)=0) will not occur. Tip: a positive number added to a square can prevent intersection.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि यह ((x-3)2+4) है / Because it is ((x-3)2+4). ((x-3)2+4) is always positive, so (p(x)=0) will not occur. Tip: a positive number added to a square can prevent intersection.
Step 3
Exam Tip
((x-3)2+4) हमेशा धनात्मक है, इसलिए (p(x)=0) नहीं होगा। टिप: पूर्ण वर्ग में जोड़ी गई धनात्मक संख्या कटान रोक सकती है।
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. ग्राफ (x)-अक्ष को स्पर्श करेगा/The graph will touch the (x)-axis
Step 1
Concept
((x-2)2) is an even-power factor, so the graph touches at (x=2). Tip: power (2) shows a repeated zero.
Step 2
Why this answer is correct
The correct answer is B. ग्राफ (x)-अक्ष को स्पर्श करेगा / The graph will touch the (x)-axis. ((x-2)2) is an even-power factor, so the graph touches at (x=2). Tip: power (2) shows a repeated zero.
Step 3
Exam Tip
((x-2)2) सम घात का कारक है, इसलिए (x=2) पर स्पर्श होगा। टिप: घात (2) दोहराया शून्यक दिखाती है।
The axis of symmetry is at the average of zeroes, (\frac{(a-2)+(a+4)}{2}=a+1). Tip: take the average even for symbolic zeroes.
Step 2
Why this answer is correct
The correct answer is A. (x=a+1). The axis of symmetry is at the average of zeroes, (\frac{(a-2)+(a+4)}{2}=a+1). Tip: take the average even for symbolic zeroes.
Step 3
Exam Tip
सममिति अक्ष शून्यकों के औसत पर है, (\frac{(a-2)+(a+4)}{2}=a+1)। टिप: प्रतीकात्मक शून्यकों में भी औसत लें।
In this interval the first two factors are positive and the third is negative, so the product is negative. Tip: check the sign of each factor separately.
Step 2
Why this answer is correct
The correct answer is B. ऋणात्मक / Negative. In this interval the first two factors are positive and the third is negative, so the product is negative. Tip: check the sign of each factor separately.
Step 3
Exam Tip
इस अंतराल में पहले दो कारक धनात्मक और तीसरा ऋणात्मक है, इसलिए गुणनफल ऋणात्मक है। टिप: प्रत्येक कारक का चिह्न अलग जांचें।
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।
B. वे परस्पर विपरीत हैं और योग (0) है/They are opposites and their sum is (0)
Step 1
Concept
The zeroes are (-5) and (5), which are opposite numbers. Tip: such zeroes are equally distant from the (y)-axis.
Step 2
Why this answer is correct
The correct answer is B. वे परस्पर विपरीत हैं और योग (0) है / They are opposites and their sum is (0). The zeroes are (-5) and (5), which are opposite numbers. Tip: such zeroes are equally distant from the (y)-axis.
Step 3
Exam Tip
शून्यक (-5) और (5) हैं, जो विपरीत संख्याएँ हैं। टिप: ऐसे शून्यक (y)-अक्ष से समान दूरी पर होते हैं।
A. (\left\(\frac{5}{2},0\right\)) और (\left\(-\frac{5}{2},0\right\))/(\left\(\frac{5}{2},0\right\)) and (\left\(-\frac{5}{2},0\right\))
Step 1
Concept
From \(4x^2-25=0\), \(x=\pm\frac{5}{2}\). Tip: treat \(4x^2\) as ((2x)2).
Step 2
Why this answer is correct
The correct answer is A. (\left\(\frac{5}{2},0\right\)) और (\left\(-\frac{5}{2},0\right\)) / (\left\(\frac{5}{2},0\right\)) and (\left\(-\frac{5}{2},0\right\)). From \(4x^2-25=0\), \(x=\pm\frac{5}{2}\). Tip: treat \(4x^2\) as ((2x)2).
Step 3
Exam Tip
\(4x^2-25=0\) से \(x=\pm\frac{5}{2}\) मिलता है। टिप: \(4x^2\) को ((2x)2) समझें।
The zeroes are (-2) and (5), so the polynomial is ((x+2)(x-5)=x-2-3x-10). Tip: form factors from graph intersections.
Step 2
Why this answer is correct
The correct answer is A. (p=-3, q=-10). The zeroes are (-2) and (5), so the polynomial is ((x+2)(x-5)=x-2-3x-10). Tip: form factors from graph intersections.
Step 3
Exam Tip
शून्यक (-2) और (5) हैं, इसलिए बहुपद ((x+2)(x-5)=x-2-3x-10) है। टिप: ग्राफ कटान से गुणनखंड बनाएं।
(x-4-1=\(x^2-1\)\(x^2+1\)), and the real zeroes are only \(\pm1\). Tip: \(x^2+1\) gives no real zero.
Step 2
Why this answer is correct
The correct answer is A. ((-1,0)) और ((1,0)) / ((-1,0)) and ((1,0)). (x-4-1=\(x^2-1\)\(x^2+1\)), and the real zeroes are only \(\pm1\). Tip: \(x^2+1\) gives no real zero.
Step 3
Exam Tip
(x-4-1=\(x^2-1\)\(x^2+1\)) है और वास्तविक शून्यक केवल \(\pm1\) हैं। टिप: \(x^2+1\) वास्तविक शून्यक नहीं देता।
For three distinct real zeroes, the degree must be at least (3). Tip: the number of distinct zeroes cannot exceed the degree.
Step 2
Why this answer is correct
The correct answer is C. (3). For three distinct real zeroes, the degree must be at least (3). Tip: the number of distinct zeroes cannot exceed the degree.
Step 3
Exam Tip
तीन अलग वास्तविक शून्यकों के लिए घात कम से कम (3) होनी चाहिए। टिप: अलग शून्यकों की संख्या घात से अधिक नहीं हो सकती।
C. ग्राफ (x)-अक्ष को नहीं काटता/The graph does not cut the (x)-axis
Step 1
Concept
(x-2-2x+5=(x-1)2+4), so it cannot be zero for real (x). Tip: an always positive form gives no real intersection.
Step 2
Why this answer is correct
The correct answer is C. ग्राफ (x)-अक्ष को नहीं काटता / The graph does not cut the (x)-axis. (x-2-2x+5=(x-1)2+4), so it cannot be zero for real (x). Tip: an always positive form gives no real intersection.
Step 3
Exam Tip
(x-2-2x+5=(x-1)2+4) है, इसलिए वास्तविक (x) पर शून्य नहीं बनेगा। टिप: हमेशा धनात्मक रूप कोई वास्तविक कटान नहीं देता।
A. यह दोनों शून्यकों का मध्य है/It is the midpoint of the two zeroes
Step 1
Concept
\(5=\frac{2+8}{2}\), so it is the midpoint of the two zeroes. Tip: a midpoint need not be a zero.
Step 2
Why this answer is correct
The correct answer is A. यह दोनों शून्यकों का मध्य है / It is the midpoint of the two zeroes. \(5=\frac{2+8}{2}\), so it is the midpoint of the two zeroes. Tip: a midpoint need not be a zero.
Step 3
Exam Tip
\(5=\frac{2+8}{2}\), इसलिए यह दोनों शून्यकों का मध्य है। टिप: मध्य का शून्यक होना जरूरी नहीं।
(2x-2+12x+18=2(x+3)2), so it touches at (x=-3). Tip: the outside (2) does not change the zero.
Step 2
Why this answer is correct
The correct answer is A. (x=-3) पर स्पर्श करेगा / It will touch at (x=-3). (2x-2+12x+18=2(x+3)2), so it touches at (x=-3). Tip: the outside (2) does not change the zero.
Step 3
Exam Tip
(2x-2+12x+18=2(x+3)2), इसलिए (x=-3) पर स्पर्श होगा। टिप: बाहरी (2) शून्यक नहीं बदलता।
For (x<-6), both factors are negative and the outside negative makes the value negative. Tip: first check factor signs and then apply the outside sign.
Step 2
Why this answer is correct
The correct answer is B. नीचे / Below. For (x<-6), both factors are negative and the outside negative makes the value negative. Tip: first check factor signs and then apply the outside sign.
Step 3
Exam Tip
(x<-6) पर दोनों कारक ऋणात्मक हैं और बाहर का ऋण चिन्ह मान को ऋणात्मक बनाता है। टिप: पहले कारकों का चिह्न देखें फिर बाहरी चिन्ह लगाएं।
For a downward-opening parabola, values outside the zeroes are negative. Tip: when the direction changes, sign regions also change.
Step 2
Why this answer is correct
The correct answer is B. (x)-अक्ष के नीचे / Below the (x)-axis. For a downward-opening parabola, values outside the zeroes are negative. Tip: when the direction changes, sign regions also change.
Step 3
Exam Tip
नीचे खुलने वाले परवलय में शून्यकों के बाहर मान ऋणात्मक होते हैं। टिप: दिशा बदलने पर संकेत क्षेत्र भी बदलता है।
A. (9) को (3) करना होगा/(9) must be changed to (3)
Step 1
Concept
For equal distance from the (y)-axis, zeroes should be opposites, so (3) is needed with (-3). Tip: symmetric zeroes are (a) and (-a).
Step 2
Why this answer is correct
The correct answer is A. (9) को (3) करना होगा / (9) must be changed to (3). For equal distance from the (y)-axis, zeroes should be opposites, so (3) is needed with (-3). Tip: symmetric zeroes are (a) and (-a).
Step 3
Exam Tip
(y)-अक्ष से समान दूरी के लिए शून्यक विपरीत होने चाहिए, इसलिए (-3) के साथ (3) चाहिए। टिप: सममित शून्यक (a) और (-a) होते हैं।
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 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. (x=3) पर स्पर्श और (x=-4) पर कटान/Touches at (x=3) and crosses at (x=-4)
Step 1
Concept
The even-power factor ((x-3)2) gives touching and the single factor (x+4) gives crossing. Tip: identify behavior from factor power.
Step 2
Why this answer is correct
The correct answer is A. (x=3) पर स्पर्श और (x=-4) पर कटान / Touches at (x=3) and crosses at (x=-4). The even-power factor ((x-3)2) gives touching and the single factor (x+4) gives crossing. Tip: identify behavior from factor power.
Step 3
Exam Tip
सम घात वाला कारक ((x-3)2) स्पर्श देता है और एकल कारक (x+4) कटान देता है। टिप: गुणनखंड की घात से व्यवहार पहचानें।
(x=-3) lies between the two zeroes, and an upward parabola is below the axis there. Tip: check the sign between zeroes.
Step 2
Why this answer is correct
The correct answer is B. (x)-अक्ष के नीचे / Below the (x)-axis. (x=-3) lies between the two zeroes, and an upward parabola is below the axis there. Tip: check the sign between zeroes.
Step 3
Exam Tip
(x=-3) दोनों शून्यकों के बीच है और ऊपर खुलने वाले परवलय में बीच का भाग नीचे होता है। टिप: शून्यकों के बीच संकेत देखें।
In this interval the first factor is positive and the second is negative, and the outside negative makes the value positive. Tip: check each factor's sign separately.
Step 2
Why this answer is correct
The correct answer is A. ऊपर / Above. In this interval the first factor is positive and the second is negative, and the outside negative makes the value positive. Tip: check each factor's sign separately.
Step 3
Exam Tip
इस अंतराल में पहला कारक धनात्मक और दूसरा ऋणात्मक है, बाहर का ऋण चिन्ह मान को धनात्मक करता है। टिप: हर कारक का चिह्न अलग देखें।
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।
B. दोनों शून्यकों पर स्पर्श करेगा/It will touch at both zeroes
Step 1
Concept
Both factors have even powers, so the graph touches at both places. Tip: an even-power zero usually gives touching, not crossing.
Step 2
Why this answer is correct
The correct answer is B. दोनों शून्यकों पर स्पर्श करेगा / It will touch at both zeroes. Both factors have even powers, so the graph touches at both places. Tip: an even-power zero usually gives touching, not crossing.
Step 3
Exam Tip
दोनों कारक सम घात में हैं, इसलिए दोनों स्थानों पर स्पर्श होगा। टिप: सम घात वाला शून्यक आमतौर पर कटान नहीं बल्कि स्पर्श देता है।
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।
(x-2-4x-21=(x-7)(x+3)), so the zeroes are (7) and (-3). Tip: form ((x,0)) points from factors.
Step 2
Why this answer is correct
The correct answer is A. ((7,0)) और ((-3,0)) / ((7,0)) and ((-3,0)). (x-2-4x-21=(x-7)(x+3)), so the zeroes are (7) and (-3). Tip: form ((x,0)) points from factors.
Step 3
Exam Tip
(x-2-4x-21=(x-7)(x+3)) है, इसलिए शून्यक (7) और (-3) हैं। टिप: गुणनखंडों से बिंदु ((x,0)) बनाएं।
A. (0) शून्यक है क्योंकि बिंदु (x)-अक्ष पर भी है/(0) is a zero because the point is also on the (x)-axis
Step 1
Concept
The origin lies on both axes, so (p(0)=0). Tip: treat ((0,0)) as a special point.
Step 2
Why this answer is correct
The correct answer is A. (0) शून्यक है क्योंकि बिंदु (x)-अक्ष पर भी है / (0) is a zero because the point is also on the (x)-axis. The origin lies on both axes, so (p(0)=0). Tip: treat ((0,0)) as a special point.
Step 3
Exam Tip
मूल बिंदु दोनों अक्षों पर होता है, इसलिए (p(0)=0) है। टिप: ((0,0)) को विशेष बिंदु मानें।
The zeroes are (a), (b), (c), so their product is (abc). Tip: even in symbolic points, the first coordinate is the zero.
Step 2
Why this answer is correct
The correct answer is B. (abc). The zeroes are (a), (b), (c), so their product is (abc). Tip: even in symbolic points, the first coordinate is the zero.
Step 3
Exam Tip
शून्यक (a), (b), (c) हैं, इसलिए गुणनफल (abc) है। टिप: प्रतीकात्मक बिंदु में भी पहला निर्देशांक शून्यक है।
A. क्योंकि यह ((x+2)2+4) है/Because it is ((x+2)2+4)
Step 1
Concept
((x+2)2+4) is always positive, so (p(x)=0) will not occur. Tip: adding a positive number to a square can prevent intersection.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि यह ((x+2)2+4) है / Because it is ((x+2)2+4). ((x+2)2+4) is always positive, so (p(x)=0) will not occur. Tip: adding a positive number to a square can prevent intersection.
Step 3
Exam Tip
((x+2)2+4) हमेशा धनात्मक है, इसलिए (p(x)=0) नहीं होगा। टिप: वर्ग में धनात्मक संख्या जुड़ने पर कटान रुक सकता है।
B. ग्राफ (x)-अक्ष को स्पर्श करेगा/The graph will touch the (x)-axis
Step 1
Concept
((x-3)2) is an even-power factor, so the graph touches at (x=3). Tip: power (2) shows a repeated zero.
Step 2
Why this answer is correct
The correct answer is B. ग्राफ (x)-अक्ष को स्पर्श करेगा / The graph will touch the (x)-axis. ((x-3)2) is an even-power factor, so the graph touches at (x=3). Tip: power (2) shows a repeated zero.
Step 3
Exam Tip
((x-3)2) सम घात का कारक है, इसलिए (x=3) पर स्पर्श होगा। टिप: घात (2) दोहराया शून्यक दिखाती है।
The axis of symmetry is at the average of the zeroes, (\frac{(b-5)+(b+1)}{2}=b-2). Tip: the average rule also works with symbols.
Step 2
Why this answer is correct
The correct answer is A. (x=b-2). The axis of symmetry is at the average of the zeroes, (\frac{(b-5)+(b+1)}{2}=b-2). Tip: the average rule also works with symbols.
Step 3
Exam Tip
सममिति अक्ष शून्यकों के औसत पर है, (\frac{(b-5)+(b+1)}{2}=b-2)। टिप: प्रतीकों में भी औसत का नियम लागू होता है।
In this interval the first two factors are positive and the third is negative, so the product is negative. Tip: check factor signs separately.
Step 2
Why this answer is correct
The correct answer is B. ऋणात्मक / Negative. In this interval the first two factors are positive and the third is negative, so the product is negative. Tip: check factor signs separately.
Step 3
Exam Tip
इस अंतराल में पहले दो कारक धनात्मक और तीसरा ऋणात्मक है, इसलिए गुणनफल ऋणात्मक है। टिप: कारकों के चिह्न अलग-अलग जाँचें।
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।
B. शून्यक परस्पर विपरीत हैं और योग (0) है/The zeroes are opposites and their sum is (0)
Step 1
Concept
The zeroes (-6) and (6) are opposite numbers. Tip: opposite zeroes are equally distant from the (y)-axis.
Step 2
Why this answer is correct
The correct answer is B. शून्यक परस्पर विपरीत हैं और योग (0) है / The zeroes are opposites and their sum is (0). The zeroes (-6) and (6) are opposite numbers. Tip: opposite zeroes are equally distant from the (y)-axis.
Step 3
Exam Tip
शून्यक (-6) और (6) विपरीत संख्याएँ हैं। टिप: विपरीत शून्यक (y)-अक्ष से समान दूरी पर होते हैं।
A. (\left\(\frac{4}{3},0\right\)) और (\left\(-\frac{4}{3},0\right\))/(\left\(\frac{4}{3},0\right\)) and (\left\(-\frac{4}{3},0\right\))
Step 1
Concept
From \(9x^2-16=0\), \(x=\pm\frac{4}{3}\). Tip: treat \(9x^2\) as ((3x)2).
Step 2
Why this answer is correct
The correct answer is A. (\left\(\frac{4}{3},0\right\)) और (\left\(-\frac{4}{3},0\right\)) / (\left\(\frac{4}{3},0\right\)) and (\left\(-\frac{4}{3},0\right\)). From \(9x^2-16=0\), \(x=\pm\frac{4}{3}\). Tip: treat \(9x^2\) as ((3x)2).
Step 3
Exam Tip
\(9x^2-16=0\) से \(x=\pm\frac{4}{3}\) मिलता है। टिप: \(9x^2\) को ((3x)2) समझें।
Real zeroes are counted from (x)-axis intersections, not from the (y)-axis intercept. Tip: ((0,-15)) does not show a zero.
Step 2
Why this answer is correct
The correct answer is B. दो / Two. Real zeroes are counted from (x)-axis intersections, not from the (y)-axis intercept. Tip: ((0,-15)) does not show a zero.
Step 3
Exam Tip
वास्तविक शून्यक (x)-अक्ष कटानों से गिने जाते हैं, (y)-अक्ष कटान से नहीं। टिप: ((0,-15)) शून्यक नहीं बताता।
(x-4-16=\(x^2-4\)\(x^2+4\)), and the real zeroes are only \(\pm2\). Tip: \(x^2+4\) gives no real zero.
Step 2
Why this answer is correct
The correct answer is A. ((-2,0)) और ((2,0)) / ((-2,0)) and ((2,0)). (x-4-16=\(x^2-4\)\(x^2+4\)), and the real zeroes are only \(\pm2\). Tip: \(x^2+4\) gives no real zero.
Step 3
Exam Tip
(x-4-16=\(x^2-4\)\(x^2+4\)) है और वास्तविक शून्यक केवल \(\pm2\) हैं। टिप: \(x^2+4\) वास्तविक शून्यक नहीं देता।
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।
C. ग्राफ (x)-अक्ष को नहीं काटता/The graph does not cut the (x)-axis
Step 1
Concept
(x-2+10x+29=(x+5)2+4), so there is no real zero. Tip: an always positive form gives no intersection.
Step 2
Why this answer is correct
The correct answer is C. ग्राफ (x)-अक्ष को नहीं काटता / The graph does not cut the (x)-axis. (x-2+10x+29=(x+5)2+4), so there is no real zero. Tip: an always positive form gives no intersection.
Step 3
Exam Tip
(x-2+10x+29=(x+5)2+4) है, इसलिए वास्तविक शून्यक नहीं है। टिप: हमेशा धनात्मक रूप कटान नहीं देता।
A. यह दोनों शून्यकों का मध्य है/It is the midpoint of the two zeroes
Step 1
Concept
\(8=\frac{4+12}{2}\), so it is the midpoint of the two zeroes. Tip: the midpoint need not be a zero.
Step 2
Why this answer is correct
The correct answer is A. यह दोनों शून्यकों का मध्य है / It is the midpoint of the two zeroes. \(8=\frac{4+12}{2}\), so it is the midpoint of the two zeroes. Tip: the midpoint need not be a zero.
Step 3
Exam Tip
\(8=\frac{4+12}{2}\), इसलिए यह दोनों शून्यकों का मध्य है। टिप: मध्य बिंदु शून्यक हो यह जरूरी नहीं।
(3x-2-18x+27=3(x-3)2), so it touches at (x=3). Tip: the outside (3) does not change the zero.
Step 2
Why this answer is correct
The correct answer is A. (x=3) पर स्पर्श करेगा / It will touch at (x=3). (3x-2-18x+27=3(x-3)2), so it touches at (x=3). Tip: the outside (3) does not change the zero.
Step 3
Exam Tip
(3x-2-18x+27=3(x-3)2) है, इसलिए (x=3) पर स्पर्श होगा। टिप: बाहरी (3) शून्यक नहीं बदलता।
For (x<-8), both factors are negative and the outside negative makes the value negative. Tip: check factor signs first.
Step 2
Why this answer is correct
The correct answer is B. नीचे / Below. For (x<-8), both factors are negative and the outside negative makes the value negative. Tip: check factor signs first.
Step 3
Exam Tip
(x<-8) पर दोनों कारक ऋणात्मक हैं और बाहर का ऋण चिन्ह मान को ऋणात्मक बनाता है। टिप: पहले कारकों का चिह्न देखें।
For a downward-opening parabola, values outside the zeroes are negative. Tip: opening direction changes sign regions.
Step 2
Why this answer is correct
The correct answer is B. (x)-अक्ष के नीचे / Below the (x)-axis. For a downward-opening parabola, values outside the zeroes are negative. Tip: opening direction changes sign regions.
Step 3
Exam Tip
नीचे खुलने वाले परवलय में शून्यकों के बाहर मान ऋणात्मक होते हैं। टिप: खुलने की दिशा संकेत क्षेत्र बदलती है।
A. (10) को (4) करना होगा/(10) must be changed to (4)
Step 1
Concept
For equal distance from the (y)-axis, zeroes should be opposites, so (4) is needed with (-4). Tip: symmetric zeroes are (a) and (-a).
Step 2
Why this answer is correct
The correct answer is A. (10) को (4) करना होगा / (10) must be changed to (4). For equal distance from the (y)-axis, zeroes should be opposites, so (4) is needed with (-4). Tip: symmetric zeroes are (a) and (-a).
Step 3
Exam Tip
(y)-अक्ष से समान दूरी के लिए शून्यक विपरीत होने चाहिए, इसलिए (-4) के साथ (4) चाहिए। टिप: सममित शून्यक (a) और (-a) होते हैं।
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।
In this interval the signs are (+), (-), (-), so the product is positive. Tip: an odd number of negative factors gives a negative value.
Step 2
Why this answer is correct
The correct answer is A. (x)-अक्ष के ऊपर / Above the (x)-axis. In this interval the signs are (+), (-), (-), so the product is positive. Tip: an odd number of negative factors gives a negative value.
Step 3
Exam Tip
इस अंतराल में चिह्न (+), (-), (-) हैं, इसलिए गुणनफल धनात्मक है। टिप: विषम संख्या ऋणात्मक कारकों से मान ऋणात्मक होता है।
A. क्योंकि इसका विविक्तकर ऋणात्मक है/Because its discriminant is negative
Step 1
Concept
The discriminant is \(c^2-4c^2=-3c^2<0\), so there are no real zeroes. Tip: a negative discriminant means no (x)-axis intersection.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि इसका विविक्तकर ऋणात्मक है / Because its discriminant is negative. The discriminant is \(c^2-4c^2=-3c^2<0\), so there are no real zeroes. Tip: a negative discriminant means no (x)-axis intersection.
Step 3
Exam Tip
विविक्तकर \(c^2-4c^2=-3c^2<0\) है, इसलिए वास्तविक शून्यक नहीं हैं। टिप: ऋणात्मक विविक्तकर का अर्थ (x)-अक्ष कटान नहीं 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।
The discriminant is (64-88=-24), so there are no real zeroes. Tip: with negative discriminant a parabola does not meet the (x)-axis.
Step 2
Why this answer is correct
The correct answer is C. नहीं काटेगा / It will not cut. The discriminant is (64-88=-24), so there are no real zeroes. Tip: with negative discriminant a parabola does not meet the (x)-axis.
Step 3
Exam Tip
विविक्तकर (64-88=-24) है, इसलिए वास्तविक शून्यक नहीं हैं। टिप: ऋणात्मक विविक्तकर पर परवलय (x)-अक्ष से नहीं मिलता।
It is ((x-a)2-9), so \(x-a=\pm3\) and the zeroes are (a-3), (a+3). Tip: use difference of squares.
Step 2
Why this answer is correct
The correct answer is A. (a-3) और (a+3) / (a-3) and (a+3). It is ((x-a)2-9), so \(x-a=\pm3\) and the zeroes are (a-3), (a+3). Tip: use difference of squares.
Step 3
Exam Tip
यह ((x-a)2-9) है, इसलिए \(x-a=\pm3\) और शून्यक (a-3), (a+3) हैं। टिप: वर्गों के अंतर का उपयोग करें।
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।
B. (x=-6) पर स्पर्श और (x=2) पर कटान/Touches at (x=-6) and crosses at (x=2)
Step 1
Concept
The even-power factor ((x+6)2) gives touching and the single factor (x-2) gives crossing. Tip: identify graph behavior from the power of the factor.
Step 2
Why this answer is correct
The correct answer is B. (x=-6) पर स्पर्श और (x=2) पर कटान / Touches at (x=-6) and crosses at (x=2). The even-power factor ((x+6)2) gives touching and the single factor (x-2) gives crossing. Tip: identify graph behavior from the power of the factor.
Step 3
Exam Tip
सम घात वाला कारक ((x+6)2) स्पर्श देता है और एकल कारक (x-2) कटान देता है। टिप: कारक की घात से ग्राफ का व्यवहार पहचानें।
(x=-1) lies between the zeroes and an upward-opening parabola is below the axis there. Tip: check the sign region between zeroes.
Step 2
Why this answer is correct
The correct answer is B. (x)-अक्ष के नीचे / Below the (x)-axis. (x=-1) lies between the zeroes and an upward-opening parabola is below the axis there. Tip: check the sign region between zeroes.
Step 3
Exam Tip
(x=-1) दोनों शून्यकों के बीच है और ऊपर खुलने वाले परवलय में बीच का भाग नीचे होता है। टिप: शून्यकों के बीच संकेत क्षेत्र देखें।
In this interval the first factor is positive and the second is negative, so the outside negative makes the value positive. Tip: check each factor's sign separately.
Step 2
Why this answer is correct
The correct answer is A. ऊपर / Above. In this interval the first factor is positive and the second is negative, so the outside negative makes the value positive. Tip: check each factor's sign separately.
Step 3
Exam Tip
इस अंतराल में पहला कारक धनात्मक और दूसरा ऋणात्मक है इसलिए बाहर का ऋण चिन्ह मान को धनात्मक बनाता है। टिप: हर कारक का चिह्न अलग जांचें।
It is ((x-u)(x-v)), so the zeroes are (u) and (v). Tip: write each zero as the point ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. ((u,0)) और ((v,0)) / ((u,0)) and ((v,0)). It is ((x-u)(x-v)), so the zeroes are (u) and (v). Tip: write each zero as the point ((x,0)).
Step 3
Exam Tip
यह ((x-u)(x-v)) है इसलिए शून्यक (u) और (v) हैं। टिप: शून्यक को ((x,0)) बिंदु के रूप में लिखें।
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।
B. दोनों शून्यकों पर स्पर्श करेगा/It will touch at both zeroes
Step 1
Concept
Both factors have even powers, so the graph touches at both places. Tip: an even-power zero usually gives touching, not crossing.
Step 2
Why this answer is correct
The correct answer is B. दोनों शून्यकों पर स्पर्श करेगा / It will touch at both zeroes. Both factors have even powers, so the graph touches at both places. Tip: an even-power zero usually gives touching, not crossing.
Step 3
Exam Tip
दोनों कारक सम घात में हैं इसलिए दोनों स्थानों पर स्पर्श होगा। टिप: सम घात वाला शून्यक सामान्यतः कटान नहीं बल्कि स्पर्श देता है।
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।
(x-2-8x-33=(x-11)(x+3)), so the zeroes are (11) and (-3). Tip: form ((x,0)) points from factors.
Step 2
Why this answer is correct
The correct answer is A. ((11,0)) और ((-3,0)) / ((11,0)) and ((-3,0)). (x-2-8x-33=(x-11)(x+3)), so the zeroes are (11) and (-3). Tip: form ((x,0)) points from factors.
Step 3
Exam Tip
(x-2-8x-33=(x-11)(x+3)) है इसलिए शून्यक (11) और (-3) हैं। टिप: गुणनखंडों से बिंदु ((x,0)) बनाएं।
A. (0) शून्यक है क्योंकि बिंदु (x)-अक्ष पर भी है/(0) is a zero because the point is also on the (x)-axis
Step 1
Concept
The origin lies on both axes, so (p(0)=0). Tip: treat ((0,0)) as a special point.
Step 2
Why this answer is correct
The correct answer is A. (0) शून्यक है क्योंकि बिंदु (x)-अक्ष पर भी है / (0) is a zero because the point is also on the (x)-axis. The origin lies on both axes, so (p(0)=0). Tip: treat ((0,0)) as a special point.
Step 3
Exam Tip
मूल बिंदु दोनों अक्षों पर होता है इसलिए (p(0)=0) है। टिप: ((0,0)) को विशेष बिंदु मानें।
The zeroes are (m), (n), (r), so the mean is \(\frac{m+n+r}{3}\). Tip: take the first coordinate even in symbolic points.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{m+n+r}{3}\). The zeroes are (m), (n), (r), so the mean is \(\frac{m+n+r}{3}\). Tip: take the first coordinate even in symbolic points.
Step 3
Exam Tip
शून्यक (m), (n), (r) हैं इसलिए माध्य \(\frac{m+n+r}{3}\) है। टिप: प्रतीकात्मक बिंदुओं में भी पहला निर्देशांक लें।
A. क्योंकि यह ((x-4)2+4) है/Because it is ((x-4)2+4)
Step 1
Concept
((x-4)2+4) is always positive, so (p(x)=0) will not occur. Tip: adding a positive number to a square can prevent intersection.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि यह ((x-4)2+4) है / Because it is ((x-4)2+4). ((x-4)2+4) is always positive, so (p(x)=0) will not occur. Tip: adding a positive number to a square can prevent intersection.
Step 3
Exam Tip
((x-4)2+4) हमेशा धनात्मक है इसलिए (p(x)=0) नहीं होगा। टिप: वर्ग में धनात्मक संख्या जुड़ने पर कटान रुक सकता है।
B. ग्राफ (x)-अक्ष को स्पर्श करेगा/The graph will touch the (x)-axis
Step 1
Concept
((x+4)2) is an even-power factor, so the graph touches at (x=-4). Tip: power (2) shows a repeated zero.
Step 2
Why this answer is correct
The correct answer is B. ग्राफ (x)-अक्ष को स्पर्श करेगा / The graph will touch the (x)-axis. ((x+4)2) is an even-power factor, so the graph touches at (x=-4). Tip: power (2) shows a repeated zero.
Step 3
Exam Tip
((x+4)2) सम घात का कारक है इसलिए (x=-4) पर स्पर्श होगा। टिप: घात (2) दोहराया शून्यक दिखाती है।
The axis of symmetry is at the average of the zeroes, (\frac{(c-7)+(c+3)}{2}=c-2). Tip: the average rule also works with symbols.
Step 2
Why this answer is correct
The correct answer is A. (x=c-2). The axis of symmetry is at the average of the zeroes, (\frac{(c-7)+(c+3)}{2}=c-2). Tip: the average rule also works with symbols.
Step 3
Exam Tip
सममिति अक्ष शून्यकों के औसत पर है (\frac{(c-7)+(c+3)}{2}=c-2)। टिप: प्रतीकों में भी औसत का नियम लागू होता है।
In this interval the first two factors are positive and the third is negative, so the product is negative. Tip: check factor signs separately.
Step 2
Why this answer is correct
The correct answer is B. ऋणात्मक / Negative. In this interval the first two factors are positive and the third is negative, so the product is negative. Tip: check factor signs separately.
Step 3
Exam Tip
इस अंतराल में पहले दो कारक धनात्मक और तीसरा ऋणात्मक है इसलिए गुणनफल ऋणात्मक है। टिप: कारकों के चिह्न अलग-अलग जांचें।
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।
B. शून्यक परस्पर विपरीत हैं और योग (0) है/The zeroes are opposites and their sum is (0)
Step 1
Concept
The zeroes (-8) and (8) are opposite numbers. Tip: opposite zeroes are equally distant from the (y)-axis.
Step 2
Why this answer is correct
The correct answer is B. शून्यक परस्पर विपरीत हैं और योग (0) है / The zeroes are opposites and their sum is (0). The zeroes (-8) and (8) are opposite numbers. Tip: opposite zeroes are equally distant from the (y)-axis.
Step 3
Exam Tip
शून्यक (-8) और (8) विपरीत संख्याएँ हैं। टिप: विपरीत शून्यक (y)-अक्ष से समान दूरी पर होते हैं।
A. (\left\(\frac{3}{4},0\right\)) और (\left\(-\frac{3}{4},0\right\))/(\left\(\frac{3}{4},0\right\)) and (\left\(-\frac{3}{4},0\right\))
Step 1
Concept
From \(16x^2-9=0\), \(x=\pm\frac{3}{4}\). Tip: treat \(16x^2\) as ((4x)2).
Step 2
Why this answer is correct
The correct answer is A. (\left\(\frac{3}{4},0\right\)) और (\left\(-\frac{3}{4},0\right\)) / (\left\(\frac{3}{4},0\right\)) and (\left\(-\frac{3}{4},0\right\)). From \(16x^2-9=0\), \(x=\pm\frac{3}{4}\). Tip: treat \(16x^2\) as ((4x)2).
Step 3
Exam Tip
\(16x^2-9=0\) से \(x=\pm\frac{3}{4}\) मिलता है। टिप: \(16x^2\) को ((4x)2) समझें।
Real zeroes are counted from (x)-axis intersections, not from the (y)-axis intercept. Tip: ((0,18)) does not show a zero.
Step 2
Why this answer is correct
The correct answer is B. दो / Two. Real zeroes are counted from (x)-axis intersections, not from the (y)-axis intercept. Tip: ((0,18)) does not show a zero.
Step 3
Exam Tip
वास्तविक शून्यक (x)-अक्ष कटानों से गिने जाते हैं (y)-अक्ष कटान से नहीं। टिप: ((0,18)) शून्यक नहीं बताता।
(x-4-81=\(x^2-9\)\(x^2+9\)), and the real zeroes are only \(\pm3\). Tip: \(x^2+9\) gives no real zero.
Step 2
Why this answer is correct
The correct answer is A. ((-3,0)) और ((3,0)) / ((-3,0)) and ((3,0)). (x-4-81=\(x^2-9\)\(x^2+9\)), and the real zeroes are only \(\pm3\). Tip: \(x^2+9\) gives no real zero.
Step 3
Exam Tip
(x-4-81=\(x^2-9\)\(x^2+9\)) है और वास्तविक शून्यक केवल \(\pm3\) हैं। टिप: \(x^2+9\) वास्तविक शून्यक नहीं देता।
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।
C. ग्राफ (x)-अक्ष को नहीं काटता/The graph does not cut the (x)-axis
Step 1
Concept
(x-2-2x+10=(x-1)2+9), so there is no real zero. Tip: an always positive form gives no intersection.
Step 2
Why this answer is correct
The correct answer is C. ग्राफ (x)-अक्ष को नहीं काटता / The graph does not cut the (x)-axis. (x-2-2x+10=(x-1)2+9), so there is no real zero. Tip: an always positive form gives no intersection.
Step 3
Exam Tip
(x-2-2x+10=(x-1)2+9) है इसलिए वास्तविक शून्यक नहीं है। टिप: हमेशा धनात्मक रूप कटान नहीं देता।
A. यह दोनों शून्यकों का मध्य है/It is the midpoint of the two zeroes
Step 1
Concept
\(12=\frac{6+18}{2}\), so it is the midpoint of the two zeroes. Tip: the midpoint need not be a zero.
Step 2
Why this answer is correct
The correct answer is A. यह दोनों शून्यकों का मध्य है / It is the midpoint of the two zeroes. \(12=\frac{6+18}{2}\), so it is the midpoint of the two zeroes. Tip: the midpoint need not be a zero.
Step 3
Exam Tip
\(12=\frac{6+18}{2}\) है इसलिए यह दोनों शून्यकों का मध्य है। टिप: मध्य बिंदु शून्यक हो यह जरूरी नहीं।
(4x-2-24x+36=4(x-3)2), so it touches at (x=3). Tip: the outside (4) does not change the zero.
Step 2
Why this answer is correct
The correct answer is A. (x=3) पर स्पर्श करेगा / It will touch at (x=3). (4x-2-24x+36=4(x-3)2), so it touches at (x=3). Tip: the outside (4) does not change the zero.
Step 3
Exam Tip
(4x-2-24x+36=4(x-3)2) है इसलिए (x=3) पर स्पर्श होगा। टिप: बाहरी (4) शून्यक नहीं बदलता।
For (x<-9), both factors are negative and the outside negative makes the value negative. Tip: check factor signs first.
Step 2
Why this answer is correct
The correct answer is B. नीचे / Below. For (x<-9), both factors are negative and the outside negative makes the value negative. Tip: check factor signs first.
Step 3
Exam Tip
(x<-9) पर दोनों कारक ऋणात्मक हैं और बाहर का ऋण चिन्ह मान को ऋणात्मक बनाता है। टिप: पहले कारकों का चिह्न देखें।
For a downward-opening parabola, values outside the zeroes are negative. Tip: opening direction changes sign regions.
Step 2
Why this answer is correct
The correct answer is B. (x)-अक्ष के नीचे / Below the (x)-axis. For a downward-opening parabola, values outside the zeroes are negative. Tip: opening direction changes sign regions.
Step 3
Exam Tip
नीचे खुलने वाले परवलय में शून्यकों के बाहर मान ऋणात्मक होते हैं। टिप: खुलने की दिशा संकेत क्षेत्र बदलती है।
A. (14) को (6) करना होगा/(14) must be changed to (6)
Step 1
Concept
For equal distance from the (y)-axis, zeroes should be opposites, so (6) is needed with (-6). Tip: symmetric zeroes are (a) and (-a).
Step 2
Why this answer is correct
The correct answer is A. (14) को (6) करना होगा / (14) must be changed to (6). For equal distance from the (y)-axis, zeroes should be opposites, so (6) is needed with (-6). Tip: symmetric zeroes are (a) and (-a).
Step 3
Exam Tip
(y)-अक्ष से समान दूरी के लिए शून्यक विपरीत होने चाहिए, इसलिए (-6) के साथ (6) चाहिए। टिप: सममित शून्यक (a) और (-a) होते हैं।
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।
In this interval the signs are (+), (-), (-), so the product is positive. Tip: the product of two negative factors is positive.
Step 2
Why this answer is correct
The correct answer is A. (x)-अक्ष के ऊपर / Above the (x)-axis. In this interval the signs are (+), (-), (-), so the product is positive. Tip: the product of two negative factors is positive.
Step 3
Exam Tip
इस अंतराल में चिह्न (+), (-), (-) हैं इसलिए गुणनफल धनात्मक है। टिप: दो ऋणात्मक कारकों का गुणन धनात्मक होता है।
A. क्योंकि इसका विविक्तकर ऋणात्मक है/Because its discriminant is negative
Step 1
Concept
The discriminant is \(d^2-4d^2=-3d^2<0\), so there are no real zeroes. Tip: a negative discriminant means no (x)-axis intersection.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि इसका विविक्तकर ऋणात्मक है / Because its discriminant is negative. The discriminant is \(d^2-4d^2=-3d^2<0\), so there are no real zeroes. Tip: a negative discriminant means no (x)-axis intersection.
Step 3
Exam Tip
विविक्तकर \(d^2-4d^2=-3d^2<0\) है इसलिए वास्तविक शून्यक नहीं हैं। टिप: ऋणात्मक विविक्तकर का अर्थ (x)-अक्ष कटान नहीं 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 discriminant is (144-204=-60), so there are no real zeroes. Tip: with negative discriminant a parabola does not meet the (x)-axis.
Step 2
Why this answer is correct
The correct answer is C. नहीं काटेगा / It will not cut. The discriminant is (144-204=-60), so there are no real zeroes. Tip: with negative discriminant a parabola does not meet the (x)-axis.
Step 3
Exam Tip
विविक्तकर (144-204=-60) है इसलिए वास्तविक शून्यक नहीं हैं। टिप: ऋणात्मक विविक्तकर पर परवलय (x)-अक्ष से नहीं मिलता।
It is ((x-b)2-16), so \(x-b=\pm4\) and the zeroes are (b-4), (b+4). Tip: use difference of squares.
Step 2
Why this answer is correct
The correct answer is A. (b-4) और (b+4) / (b-4) and (b+4). It is ((x-b)2-16), so \(x-b=\pm4\) and the zeroes are (b-4), (b+4). Tip: use difference of squares.
Step 3
Exam Tip
यह ((x-b)2-16) है इसलिए \(x-b=\pm4\) और शून्यक (b-4), (b+4) हैं। टिप: वर्गों के अंतर का उपयोग करें।
Here \(x^2=7+2\sqrt{10}\), so subtracting gives (7). In such questions expand the square first.
Step 2
Why this answer is correct
The correct answer is A. परिमेय संख्या / Rational number. Here \(x^2=7+2\sqrt{10}\), so subtracting gives (7). In such questions expand the square first.
Step 3
Exam Tip
\(x^2=7+2\sqrt{10}\) इसलिए घटाने पर (7) मिलता है। ऐसे प्रश्नों में पहले वर्ग विस्तार करें।
By identity the difference is (4ab), where \(a=\sqrt{11}\) and \(b=\sqrt{5}\). So the answer is \(4\sqrt{55}\).
Step 2
Why this answer is correct
The correct answer is A. \(4\sqrt{55}\). By identity the difference is (4ab), where \(a=\sqrt{11}\) and \(b=\sqrt{5}\). So the answer is \(4\sqrt{55}\).
Step 3
Exam Tip
सूत्र से अंतर (4ab) होता है जहाँ \(a=\sqrt{11}\) और \(b=\sqrt{5}\) हैं। इसलिए उत्तर \(4\sqrt{55}\) है।
The conjugate of the denominator is \(3-\sqrt{5}\) and the denominator becomes (9-5=4). Multiply by the conjugate to rationalize.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{3-\sqrt{5}}{4}\). The conjugate of the denominator is \(3-\sqrt{5}\) and the denominator becomes (9-5=4). Multiply by the conjugate to rationalize.
Step 3
Exam Tip
हर का संयुग्मी \(3-\sqrt{5}\) है और हर (9-5=4) बनता है। परिमेयकरण में संयुग्मी से गुणा करें।
Multiplying by the conjugate makes the denominator (7-3=4). Hence we get (\frac{2\(\sqrt{7}+\sqrt{3}\)}{4}).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{\sqrt{7}+\sqrt{3}}{2}\). Multiplying by the conjugate makes the denominator (7-3=4). Hence we get (\frac{2\(\sqrt{7}+\sqrt{3}\)}{4}).
Step 3
Exam Tip
संयुग्मी से गुणा करने पर हर (7-3=4) बनता है। इसलिए (\frac{2\(\sqrt{7}+\sqrt{3}\)}{4}) मिलता है।
Since (\(2+\sqrt{3}\)\(2-\sqrt{3}\)=1), the reciprocal is \(2-\sqrt{3}\). Recognizing conjugates is a fast method.
Step 2
Why this answer is correct
The correct answer is A. \(2-\sqrt{3}\). Since (\(2+\sqrt{3}\)\(2-\sqrt{3}\)=1), the reciprocal is \(2-\sqrt{3}\). Recognizing conjugates is a fast method.
Step 3
Exam Tip
क्योंकि (\(2+\sqrt{3}\)\(2-\sqrt{3}\)=1), इसलिए व्युत्क्रम \(2-\sqrt{3}\) है। संयुग्मी को पहचानना तेज तरीका है।
The first product is (25-6=19) and \(\sqrt{24}=2\sqrt{6}\) is irrational. A rational plus an irrational is irrational.
Step 2
Why this answer is correct
The correct answer is A. अपरिमेय संख्या / Irrational number. The first product is (25-6=19) and \(\sqrt{24}=2\sqrt{6}\) is irrational. A rational plus an irrational is irrational.
Step 3
Exam Tip
पहला गुणनफल (25-6=19) है और \(\sqrt{24}=2\sqrt{6}\) अपरिमेय है। परिमेय और अपरिमेय का योग अपरिमेय होता है।
\(\sqrt{8}=2\sqrt{2}\) and \(\sqrt{18}=3\sqrt{2}\), so \(a=5\sqrt{2}\). Its square is (50), a rational number.
Step 2
Why this answer is correct
The correct answer is A. परिमेय संख्या / Rational number. \(\sqrt{8}=2\sqrt{2}\) and \(\sqrt{18}=3\sqrt{2}\), so \(a=5\sqrt{2}\). Its square is (50), a rational number.
Step 3
Exam Tip
\(\sqrt{8}=2\sqrt{2}\) और \(\sqrt{18}=3\sqrt{2}\), इसलिए \(a=5\sqrt{2}\)। इसका वर्ग (50) परिमेय है।
The expression becomes \(3\sqrt{5}+4\sqrt{5}-5\sqrt{5}=2\sqrt{5}\). \(2\sqrt{5}\) is irrational.
Step 2
Why this answer is correct
The correct answer is A. अपरिमेय संख्या / Irrational number. The expression becomes \(3\sqrt{5}+4\sqrt{5}-5\sqrt{5}=2\sqrt{5}\). \(2\sqrt{5}\) is irrational.
Step 3
Exam Tip
अभिव्यक्ति \(3\sqrt{5}+4\sqrt{5}-5\sqrt{5}=2\sqrt{5}\) बनती है। \(2\sqrt{5}\) अपरिमेय है।
If it were rational, its square \(3+\sqrt{5}\) would be rational, which it is not. Hence it is real irrational.
Step 2
Why this answer is correct
The correct answer is A. अपरिमेय वास्तविक संख्या / Irrational real number. If it were rational, its square \(3+\sqrt{5}\) would be rational, which it is not. Hence it is real irrational.
Step 3
Exam Tip
यदि यह परिमेय होता तो उसका वर्ग \(3+\sqrt{5}\) परिमेय होता, जो नहीं है। इसलिए यह वास्तविक अपरिमेय है।
Multiplying by the conjugate makes the denominator (5-2=3). So the rationalized form is \(\frac{\sqrt{5}-\sqrt{2}}{3}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{\sqrt{5}-\sqrt{2}}{3}\). Multiplying by the conjugate makes the denominator (5-2=3). So the rationalized form is \(\frac{\sqrt{5}-\sqrt{2}}{3}\).
Step 3
Exam Tip
संयुग्मी से गुणा करने पर हर (5-2=3) हो जाता है। इसलिए परिमेय हर वाला रूप \(\frac{\sqrt{5}-\sqrt{2}}{3}\) है।
Actually \(x=\sqrt{2}+\sqrt{3}\) satisfies \(x^4-10x^2+1=0\), not a simple quadratic here. Read powers carefully in such trick questions.
Step 2
Why this answer is correct
The correct answer is A. \(x^2-10x^0+1=0\). Actually \(x=\sqrt{2}+\sqrt{3}\) satisfies \(x^4-10x^2+1=0\), not a simple quadratic here. Read powers carefully in such trick questions.
Step 3
Exam Tip
\(x^2=5+2\sqrt{6}\) और संयुग्मी के साथ गुणन से \(x^4-10x^2+1=0\) मिलता है। दिए विकल्प में \(x^0=1\) इसलिए पहला रूप सही नहीं दिखता, ध्यान से पढ़ें।
The average of two numbers lies between them. Hence \(\frac{\sqrt{2}+\sqrt{3}}{2}\) lies between them.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{\sqrt{2}+\sqrt{3}}{2}\). The average of two numbers lies between them. Hence \(\frac{\sqrt{2}+\sqrt{3}}{2}\) lies between them.
Step 3
Exam Tip
दो संख्याओं का औसत उनके बीच होता है। इसलिए \(\frac{\sqrt{2}+\sqrt{3}}{2}\) इनके बीच है।
For positive numbers a larger value inside the root gives a larger square root. Here (48<75<108).
Step 2
Why this answer is correct
The correct answer is A. \(\sqrt{48}<\sqrt{75}<\sqrt{108}\). For positive numbers a larger value inside the root gives a larger square root. Here (48<75<108).
Step 3
Exam Tip
धनात्मक संख्याओं में जड़ के अंदर बड़ी संख्या हो तो वर्गमूल भी बड़ा होता है। (48<75<108) है।
\(\sqrt{8}=2\sqrt{2}\), so the sum is \(3\sqrt{2}\). A non zero rational multiple of \(\sqrt{2}\) remains irrational.
Step 2
Why this answer is correct
The correct answer is A. यह \(3\sqrt{2}\) है / It is \(3\sqrt{2}\). \(\sqrt{8}=2\sqrt{2}\), so the sum is \(3\sqrt{2}\). A non zero rational multiple of \(\sqrt{2}\) remains irrational.
Step 3
Exam Tip
\(\sqrt{8}=2\sqrt{2}\), इसलिए योग \(3\sqrt{2}\) है। गैर शून्य परिमेय गुणक के साथ \(\sqrt{2}\) अपरिमेय रहता है।
Multiplying by the conjugate makes the denominator (1). The numerator is (\(\sqrt{3}+\sqrt{2}\)2=5+2\sqrt{6}).
Step 2
Why this answer is correct
The correct answer is A. \(5+2\sqrt{6}\). Multiplying by the conjugate makes the denominator (1). The numerator is (\(\sqrt{3}+\sqrt{2}\)2=5+2\sqrt{6}).
Step 3
Exam Tip
हर के संयुग्मी से गुणा करने पर हर (1) बनता है। अंश (\(\sqrt{3}+\sqrt{2}\)2=5+2\sqrt{6}) है।
Multiplying by the conjugate gives denominator (5-3=2) and numerator \(8+2\sqrt{15}\). The simplified form is \(4+\sqrt{15}\).
Step 2
Why this answer is correct
The correct answer is A. \(4+\sqrt{15}\). Multiplying by the conjugate gives denominator (5-3=2) and numerator \(8+2\sqrt{15}\). The simplified form is \(4+\sqrt{15}\).
Step 3
Exam Tip
संयुग्मी से गुणा करने पर हर (5-3=2) और अंश \(8+2\sqrt{15}\) बनता है। सरल रूप \(4+\sqrt{15}\) है।
After some digits, (45) repeats, so it is a recurring decimal. A recurring decimal is rational.
Step 2
Why this answer is correct
The correct answer is A. परिमेय संख्या / Rational number. After some digits, (45) repeats, so it is a recurring decimal. A recurring decimal is rational.
Step 3
Exam Tip
कुछ अंकों के बाद (45) दोहरता है, इसलिए यह आवर्ती दशमलव है। आवर्ती दशमलव परिमेय होता है।
A. (m) पूर्ण वर्ग होना चाहिए/(m) must be a perfect square
Step 1
Concept
The square root of a positive integer is rational only when it is a perfect square. This is the key rule for roots.
Step 2
Why this answer is correct
The correct answer is A. (m) पूर्ण वर्ग होना चाहिए / (m) must be a perfect square. The square root of a positive integer is rational only when it is a perfect square. This is the key rule for roots.
Step 3
Exam Tip
धनात्मक पूर्णांक की वर्गमूल परिमेय तभी होती है जब वह पूर्ण वर्ग हो। यह जड़ों की प्रकृति का मुख्य नियम है।
\(\sqrt{289}=17\) is rational and \(\sqrt{290}\) is irrational. A rational plus an irrational is irrational.
Step 2
Why this answer is correct
The correct answer is A. अपरिमेय संख्या / Irrational number. \(\sqrt{289}=17\) is rational and \(\sqrt{290}\) is irrational. A rational plus an irrational is irrational.
Step 3
Exam Tip
\(\sqrt{289}=17\) परिमेय है और \(\sqrt{290}\) अपरिमेय है। परिमेय और अपरिमेय का योग अपरिमेय होता है।
(180) is not a perfect square so \(\sqrt{180}\) is irrational. Subtracting an irrational from a rational gives an irrational result.
Step 2
Why this answer is correct
The correct answer is A. (m=180). (180) is not a perfect square so \(\sqrt{180}\) is irrational. Subtracting an irrational from a rational gives an irrational result.
Step 3
Exam Tip
(180) पूर्ण वर्ग नहीं है इसलिए \(\sqrt{180}\) अपरिमेय है। परिमेय से अपरिमेय घटाने पर परिणाम अपरिमेय होता है।
\(\sqrt{8}=2\sqrt{2}\), so \(\sqrt{2}+\sqrt{8}=3\sqrt{2}\). The unlike root \(\sqrt{5}\) remains separate.
Step 2
Why this answer is correct
The correct answer is A. \(3\sqrt{2}+\sqrt{5}\). \(\sqrt{8}=2\sqrt{2}\), so \(\sqrt{2}+\sqrt{8}=3\sqrt{2}\). The unlike root \(\sqrt{5}\) remains separate.
Step 3
Exam Tip
\(\sqrt{8}=2\sqrt{2}\) इसलिए \(\sqrt{2}+\sqrt{8}=3\sqrt{2}\) होता है। असमान जड़ \(\sqrt{5}\) अलग रहती है।
\(\frac{\sqrt{3}}{\sqrt{12}}=\sqrt{\frac{3}{12}}=\sqrt{\frac{1}{4}}=\frac{1}{2}\). Simplify the ratio inside the roots.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{1}{2}\). \(\frac{\sqrt{3}}{\sqrt{12}}=\sqrt{\frac{3}{12}}=\sqrt{\frac{1}{4}}=\frac{1}{2}\). Simplify the ratio inside the roots.
Step 3
Exam Tip
\(\frac{\sqrt{3}}{\sqrt{12}}=\sqrt{\frac{3}{12}}=\sqrt{\frac{1}{4}}=\frac{1}{2}\) है। जड़ों के भाग में अनुपात सरल करें।
Both sides are positive and (\(\sqrt{2}+\sqrt{3}\)2=5+2\sqrt{6}>5). So the first side is larger.
Step 2
Why this answer is correct
The correct answer is A. \(\sqrt{2}+\sqrt{3}>\sqrt{5}\). Both sides are positive and (\(\sqrt{2}+\sqrt{3}\)2=5+2\sqrt{6}>5). So the first side is larger.
Step 3
Exam Tip
दोनों पक्ष धनात्मक हैं और (\(\sqrt{2}+\sqrt{3}\)2=5+2\sqrt{6}>5) है। इसलिए पहला पक्ष बड़ा है।
The conjugate of the denominator is \(5+\sqrt{6}\), and the denominator becomes (25-6=19). Hence the first option is correct.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{5+\sqrt{6}}{19}\). The conjugate of the denominator is \(5+\sqrt{6}\), and the denominator becomes (25-6=19). Hence the first option is correct.
Step 3
Exam Tip
हर का संयुग्मी \(5+\sqrt{6}\) है और हर (25-6=19) बनता है। इसलिए पहला विकल्प सही है।
The terms become \(\sqrt{5}+2\sqrt{5}+3\sqrt{5}+4\sqrt{5}\). The total is \(10\sqrt{5}\), so check the options carefully.
Step 2
Why this answer is correct
The correct answer is A. \(12\sqrt{5}\). The terms become \(\sqrt{5}+2\sqrt{5}+3\sqrt{5}+4\sqrt{5}\). The total is \(10\sqrt{5}\), so check the options carefully.
Step 3
Exam Tip
ये पद \(\sqrt{5}+2\sqrt{5}+3\sqrt{5}+4\sqrt{5}\) बनते हैं। कुल \(10\sqrt{5}\) नहीं बल्कि \(10\sqrt{5}\) है, विकल्पों को ध्यान से जाँचें।
