100 results found for "negative-root-interval" in Class 10.
In which equation is (x=0) one root and the other root negative?
A. \(x^2+7x=0\)
(x-2+7x=x(x+7)), इसलिए मूल (0) और (-7) हैं। दूसरा मूल ऋणात्मक है। / (x-2+7x=x(x+7)), so the roots are (0) and (-7). The other root is negative.
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If \(x=-\sqrt{29}\) on the number line, in which interval will (x) lie?
B. ( -6 ) और ( -5 ) के बीचBetween ( -6 ) and ( -5 )
क्योंकि \(5<\sqrt{29}<6\), इसलिए \(-6<-\sqrt{29}<-5\)। ऋणात्मक मूलों में क्रम उलटकर लिखें। / Since \(5<\sqrt{29}<6\), \(-6<-\sqrt{29}<-5\). For negative roots, write the reversed interval carefully.
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In which equation is (x=0) one root and the other root positive?
A. \(x^2-6x=0\)
(x-2-6x=x(x-6)), इसलिए मूल (0) और (6) हैं। दूसरा मूल धनात्मक है। / (x-2-6x=x(x-6)), so the roots are (0) and (6). The other root is positive.
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In which interval is \(6-\sqrt{39}\) correctly located on the number line?
A. ( -1 ) और (0) के बीचBetween ( -1 ) and (0)
\( \sqrt{39}\approx6.245 \), इसलिए \(6-\sqrt{39}\approx-0.245\) है। घटाव वाले मूल में चिह्न जरूर जाँचें। / \( \sqrt{39}\approx6.245 \), so \(6-\sqrt{39}\approx-0.245\). Always check the sign in root subtraction.
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In which interval is \(5-\sqrt{31}\) correctly located on the number line?
B. ( -1 ) और (0) के बीचBetween ( -1 ) and (0)
\( \sqrt{31}\approx5.568 \), इसलिए \(5-\sqrt{31}\approx-0.568\) है। घटाव वाले मूलों में चिह्न अवश्य जाँचें। / \( \sqrt{31}\approx5.568 \), so \(5-\sqrt{31}\approx-0.568\). Always check the sign in root subtraction.
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If \(x=-5+\sqrt{22}\), in which interval is (x) on the number line?
C. ( -1 ) और (0) के बीचBetween ( -1 ) and (0)
\(4<\sqrt{22}<5\), इसलिए \(-1<-5+\sqrt{22}<0\)। मिश्रित अभिव्यक्ति में सीमा जोड़ें। / Since \(4<\sqrt{22}<5\), \(-1<-5+\sqrt{22}<0\). Add bounds carefully in mixed expressions.
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In which interval is \(2-\sqrt{10}\) correctly located on the number line?
A. (-2) और (-1) के बीचBetween (-2) and (-1)
\( \sqrt{10}\approx3.162 \) इसलिए \(2-\sqrt{10}\approx-1.162\) है। घटाव वाले मूल में अनुमान जरूरी है। / \( \sqrt{10}\approx3.162 \), so \(2-\sqrt{10}\approx-1.162\). Estimation is important in root subtraction.
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A. (-1) और (0) के बीचBetween (-1) and (0)
\(x=-4+\sqrt{13}\) और \(3<\sqrt{13}<4\), इसलिए (-1<x<0)। संयुक्त अभिव्यक्ति में सीमा जोड़ें। / \(x=-4+\sqrt{13}\), and \(3<\sqrt{13}<4\), so (-1<x<0). Add bounds in combined expressions.
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Which option gives the correct interval for \(\sqrt{20}\) on the number line?
A. ((4,5))
क्योंकि \(4^2=16\) और \(5^2=25\), इसलिए \(4<\sqrt{20}<5\)। पूर्ण वर्गों की सीमा तुरंत अंतराल देती है। / Because \(4^2=16\) and \(5^2=25\), \(4<\sqrt{20}<5\). Perfect-square bounds quickly give the interval.
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If \(u=-\sqrt{27}-3\) on the number line, in which interval does (u) lie?
B. ( -8 ) और ( -7 ) के बीचBetween ( -8 ) and ( -7 )
\( -\sqrt{27}-3\approx-8.196 \) नहीं, बल्कि \( -\sqrt{27}\approx-5.196 \) होने से योग लगभग (-8.196) है। इसलिए यह (-9) और (-8) के बीच है। / \( -\sqrt{27}\approx-5.196 \), so \( -\sqrt{27}-3\approx-8.196 \). Therefore it lies between (-9) and (-8).
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If \(u=-\sqrt{18}-2\) on the number line, in which interval does (u) lie?
C. ( -7 ) और ( -6 ) के बीचBetween ( -7 ) and ( -6 )
\( -\sqrt{18}-2\approx-6.243 \), इसलिए यह (-7) और (-6) के बीच है। ऋणात्मक योगों में अनुमान सावधानी से करें। / \( -\sqrt{18}-2\approx-6.243 \), so it lies between (-7) and (-6). Estimate negative sums carefully.
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If \(u=-2-\sqrt{7}\), in which interval is (u) located on the number line?
C. ( -5 ) और ( -4 ) के बीचBetween ( -5 ) and ( -4 )
\( \sqrt{7}\approx2.646 \), इसलिए \(u\approx-4.646\) है। अतः यह (-5) और (-4) के बीच होगा। / \( \sqrt{7}\approx2.646 \), so \(u\approx-4.646\). Therefore it lies between (-5) and (-4).
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If \(u=-\sqrt{2}-1\) on the number line, in which interval does (u) lie?
A. ( -3 ) और ( -2 ) के बीचBetween ( -3 ) and ( -2 )
\( -\sqrt{2}-1\approx-2.414 \), इसलिए यह (-3) और (-2) के बीच है। ऋणात्मक योगों में अनुमान सावधानी से करें। / \( -\sqrt{2}-1\approx-2.414 \), so it lies between (-3) and (-2). Estimate negative sums carefully.
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Which option gives the correct simple interval for \(-\sqrt{12}\) on the number line?
A. ((-4,-3))
क्योंकि \(3<\sqrt{12}<4\), इसलिए \(-4<-\sqrt{12}<-3\)। ऋणात्मक करने पर असमानता की दिशा बदलती है। / Since \(3<\sqrt{12}<4\), \(-4<-\sqrt{12}<-3\). Multiplying by a negative reverses the inequality.
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On the number line, in which interval will \(-\sqrt{5}\) lie?
A. ((-3,-2))
क्योंकि \(2^2<5<3^2\), इसलिए \(\sqrt{5}\) (2) और (3) के बीच है और ऋणात्मक मान (-3) और (-2) के बीच होगा। ऋणात्मक दिशा में क्रम उलट जाता है। / Since \(2^2<5<3^2\), \(\sqrt{5}\) lies between (2) and (3), so \(-\sqrt{5}\) lies between (-3) and (-2). On the negative side, order reverses.
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If the discriminant is (D=(z+1)2-9), in which interval will (z) lie for no real roots?
A. (-4<z<2)
वास्तविक मूल न होने के लिए (D<0) चाहिए। ((z+1)2<9) से (-4<z<2) मिलता है। / For no real roots, (D<0) is needed. From ((z+1)2<9), we get (-4<z<2).
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If one root of (4x-2-(4h+1)x+h=0) is \(\frac{1}{4}\), what is the other root?
A. (h)
जड़ों का गुणनफल \(\frac{h}{4}\) है। एक जड़ \(\frac{1}{4}\) है, इसलिए दूसरी जड़ (h) होगी। / The product of roots is \(\frac{h}{4}\). Since one root is \(\frac{1}{4}\), the other root is (h).
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If one root of (x-2-(m+9)x+9m=0) is (9), what is the other root?
A. (m)
जड़ों का गुणनफल (9m) है। एक जड़ (9) है, इसलिए दूसरी जड़ \(\frac{9m}{9}=m\) होगी। / The product of roots is (9m). Since one root is (9), the other root is \(\frac{9m}{9}=m\).
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If one root of (2x-2-(3p+2)x+p(p+2)=0) is (p), what will be the other root?
A. \(\frac{p+2}{2}\)
जड़ों का गुणनफल (\frac{p(p+2)}{2}) है। एक जड़ (p) होने पर दूसरी जड़ \(\frac{p+2}{2}\) होगी। / The product of roots is (\frac{p(p+2)}{2}). If one root is (p), the other root is \(\frac{p+2}{2}\).
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If one root of (3x-2-(3h+1)x+h=0) is \(\frac{1}{3}\), what is the other root?
A. (h)
जड़ों का गुणनफल \(\frac{h}{3}\) है। एक जड़ \(\frac{1}{3}\) है, इसलिए दूसरी जड़ (h) होगी। / The product of roots is \(\frac{h}{3}\). Since one root is \(\frac{1}{3}\), the other root is (h).
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If one root of (2x-2-(h+1)x+h=0) is always \(\frac{1}{2}\), what is the other root?
A. (h)
गुणनफल \(\frac{h}{2}\) है। एक जड़ \(\frac{1}{2}\) है, इसलिए दूसरी जड़ \(\frac{h}{2}\div\frac{1}{2}=h\) होगी। / The product is \(\frac{h}{2}\). Since one root is \(\frac{1}{2}\), the other root is \(\frac{h}{2}\div\frac{1}{2}=h\).
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If one root of (x-2-(m+2)x+3m=0) is (3), what is the other root?
A. (m)
(x=3) रखने पर समीकरण हर (m) के लिए सही हो जाता है। गुणनफल (3m) है और एक जड़ (3), इसलिए दूसरी जड़ (m) है। / Putting (x=3) makes the equation true for every (m). The product is (3m) and one root is (3), so the other root is (m).
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If one root of (x-2-(m-2)x+m-6=0) is (3), what is the other root?
A. (1)
(x=3) रखने पर (9-3(m-2)+m-6=0), इसलिए \(m=\frac{9}{2}\)। गुणनफल \(m-6=-\frac{3}{2}\) है, अतः दूसरी जड़ \(-\frac{1}{2}\) होगी, इसलिए कोई विकल्प सही नहीं है। / Putting (x=3) gives (9-3(m-2)+m-6=0), so \(m=\frac{9}{2}\). The product is \(-\frac{3}{2}\), so the other root is \(-\frac{1}{2}\); hence no option is correct.
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If one root of (x-2-(m+7)x+7m=0) is (7), what is the other root?
A. (m)
मूलों का गुणनफल (7m) है और एक मूल (7) है। इसलिए दूसरा मूल \(\frac{7m}{7}=m\) है। / The product of roots is (7m) and one root is (7). Hence the other root is \(\frac{7m}{7}=m\).
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If one root of \(x^2+ax+54=0\) is (6), what are the other root and (a)?
A. दूसरा मूल (9), (a=-15)other root (9), (a=-15)
मूलों का गुणनफल (54) है, इसलिए दूसरा मूल (9) होगा। योग (15) है और (-a=15), इसलिए (a=-15)। / The product of roots is (54), so the other root is (9). The sum is (15), and (-a=15), so (a=-15).
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If one root of (x-2-(m+6)x+6m=0) is (6), what is the other root?
A. (m)
मूलों का गुणनफल (6m) है और एक मूल (6) है। इसलिए दूसरा मूल \(\frac{6m}{6}=m\) है। / The product of roots is (6m) and one root is (6). Hence the other root is \(\frac{6m}{6}=m\).
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If one root of \(x^2+ax+40=0\) is (5), what are the other root and (a)?
A. दूसरा मूल (8), (a=-13)other root (8), (a=-13)
मूलों का गुणनफल (40) है, इसलिए दूसरा मूल (8) होगा। योग (13) है और (-a=13), इसलिए (a=-13)। / The product of roots is (40), so the other root is (8). The sum is (13), and (-a=13), so (a=-13).
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If one root of (x-2-(m+5)x+5m=0) is (5), what is the other root?
A. (m)
मूलों का गुणनफल (5m) है और एक मूल (5) है। इसलिए दूसरा मूल \(\frac{5m}{5}=m\) है। / The product of roots is (5m) and one root is (5). Hence the other root is \(\frac{5m}{5}=m\).
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If one root of \(x^2+ax+24=0\) is (4), what are the other root and (a)?
A. दूसरा मूल (6), (a=-10)other root (6), (a=-10)
मूलों का गुणनफल (24) है, इसलिए दूसरा मूल (6) होगा। योग (10) है और (-a=10), इसलिए (a=-10)। / The product of roots is (24), so the other root is (6). The sum is (10), and (-a=10), so (a=-10).
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If one root of (x-2-(m+4)x+4m=0) is (4), what is the other root?
A. (m)
गुणनफल (4m) है और एक मूल (4) है। इसलिए दूसरा मूल \(\frac{4m}{4}=m\) है। / The product of roots is (4m) and one root is (4). Hence the other root is \(\frac{4m}{4}=m\).
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If one root of \(x^2+ax+18=0\) is (2), what are the other root and (a)?
A. दूसरा मूल (9), (a=-11)other root (9), (a=-11)
मूलों का गुणनफल (18) है, इसलिए दूसरा मूल (9) होगा। योग (11) है और (-a=11), इसलिए (a=-11)। / The product of roots is (18), so the other root is (9). The sum is (11), and (-a=11), so (a=-11).
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If one root of (x-2-(m+2)x+2m=0) is (2), what is the other root?
A. (m)
गुणनफल (2m) है और एक मूल (2) है। इसलिए दूसरा मूल \(\frac{2m}{2}=m\) है। / The product of roots is (2m) and one root is (2). Hence the other root is \(\frac{2m}{2}=m\).
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If one root of \(x^2+ax+12=0\) is (3), what are the other root and (a)?
A. दूसरा मूल (4), (a=-7)other root (4), (a=-7)
मूलों का गुणनफल (12) है, इसलिए दूसरा मूल (4) होगा। योग (7) है और (-a=7), इसलिए (a=-7)। / The product of roots is (12), so the other root is (4). The sum is (7), and (-a=7), so (a=-7).
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How are positive geotropism of root and negative geotropism of stem useful together for a plant?
A. जड़ मिट्टी में जल खनिज लेती है और तना प्रकाश की ओर बढ़ता हैRoot gets water and minerals in soil and stem grows towards light
जड़ें नीचे जाकर जल और खनिज प्राप्त करती हैं। / Roots grow downward to get water and minerals.
तना ऊपर बढ़कर प्रकाश प्राप्त करता है। / Stem grows upward to get light.
दोनों दिशात्मक वृद्धियां पौधे के जीवन के लिए उपयोगी हैं। / Both directional growth responses are useful for plant life.
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On the number line, in which interval will \(\sqrt{5}-\sqrt{2}\) lie?
A. ((0,1))
\(\sqrt{5}\approx2.236\) और \(\sqrt{2}\approx1.414\), इसलिए अंतर लगभग (0.822) है। अपरिमेयों के अंतर का स्थान निकालने के लिए छोटे अनुमान उपयोग करें। / \(\sqrt{5}\approx2.236\) and \(\sqrt{2}\approx1.414\), so the difference is about (0.822). Use short approximations to locate differences of irrationals.
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On the number line, in which interval will \(\sqrt{2}+\sqrt{3}\) lie?
A. ((3,4))
\(\sqrt{2}\approx1.414\) और \(\sqrt{3}\approx1.732\), इसलिए योग लगभग (3.146) है। योग के लिए अनुमानित मान जोड़ें। / \(\sqrt{2}\approx1.414\) and \(\sqrt{3}\approx1.732\), so the sum is about (3.146). Add approximate values for sums.
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Which option gives the correct interval of \(-1+\sqrt{5}\) on the number line?
A. ((1,2))
क्योंकि \(2<\sqrt{5}<3\), इसलिए \(1<-1+\sqrt{5}<2\)। स्थिर संख्या जोड़ने या घटाने पर पूरी असमानता बदलें। / Since \(2<\sqrt{5}<3\), \(1<-1+\sqrt{5}<2\). When adding or subtracting a constant, adjust the whole inequality.
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On the number line, in which interval will \(2-\sqrt{3}\) lie?
A. ((0,1))
क्योंकि \(1<\sqrt{3}<2\), इसलिए \(0<2-\sqrt{3}<1\)। घटाव में असमानता की दिशा सावधानी से देखें। / Since \(1<\sqrt{3}<2\), \(0<2-\sqrt{3}<1\). Be careful with inequalities when subtracting.
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The equation (x-2-2(4t+1)x+\(7t^2+2t+5\)=0) has no real roots. What is the correct interval for (t)?
A. \(-1<t<\frac{2}{9}\)
यहाँ (D=4(4t+1)2-4\(7t^2+2t+5\)=36t-2+24t-16) है। (D<0) से \(-1<t<\frac{2}{9}\) मिलता है। / Here (D=4(4t+1)2-4\(7t^2+2t+5\)=36t-2+24t-16). From (D<0), \(-1<t<\frac{2}{9}\).
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If (x-2+2(v+2)x+(4v+11)=0) has no real roots, in which interval will (v) lie?
A. (-3<v<1)
यहाँ (D=4(v+2)2-4(4v+11)=4\(v^2-7\)) नहीं, सही रूप (4\(v^2-7\)) नहीं है। परीक्षा में विस्तार सावधानी से करें। / Here (D=4(v+2)2-4(4v+11)) must be expanded carefully. Always verify the middle term before solving the interval.
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The equation (x-2-2(3t+1)x+\(5t^2+2t+4\)=0) has no real roots. What is the correct interval for (t)?
A. (-2<t<1)
यहाँ (D=4(3t+1)2-4\(5t^2+2t+4\)=16(t-1)(t+2)) है। (D<0) से (-2<t<1)। / Here (D=4(3t+1)2-4\(5t^2+2t+4\)=16(t-1)(t+2)). From (D<0), (-2<t<1).
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If (x-2+2(t+1)x+(3t+7)=0) has no real roots, in which interval will (t) lie?
A. (-4<t<1)
यहाँ (D=4(t+1)2-4(3t+7)=4\(t^2-t-6\)) है। (D<0) से (-2<t<3) नहीं, गुणनखंड फिर से जाँचें। / Here (D=4(t+1)2-4(3t+7)=4\(t^2-t-6\)). For (D<0), factor again carefully before selecting the interval.
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The equation (x-2-2(2t-1)x+\(t^2+2\)=0) has no real roots. Choose the correct interval for (t).
A. \(\frac{4-2\sqrt{6}}{3}<t<\frac{4+2\sqrt{6}}{3}\)
यहाँ (D=4(2t-1)2-4\(t^2+2\)=4\(3t^2-4t-1\)) है। (D<0) से दिए गए दोनों मूलों के बीच का अंतराल मिलता है। / Here (D=4(2t-1)2-4\(t^2+2\)=4\(3t^2-4t-1\)). From (D<0), the interval between the two boundary roots is obtained.
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If (x-2+2(k-1)x+(k+5)=0) has no real roots, in which interval will (k) lie?
A. (0<k<3)
यहाँ (D=4(k-1)2-4(k+5)=4\(k^2-3k-4\)) नहीं, सही सरल रूप (4\(k^2-3k-4\)) है। (D<0) से (0<k<3) नहीं मिलता, इसलिए गुणनखंड जाँचें। / Here (D=4(k-1)2-4(k+5)=4\(k^2-3k-4\)). Use (D<0) and factor carefully before choosing the interval.
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What is the interval of (k) for no real roots in (x-2+(k+2)x+9=0)?
A. (-8<k<4)
यहाँ (D=(k+2)2-36) है। (D<0) से (-8<k<4) मिलता है। / Here (D=(k+2)2-36). From (D<0), we get (-8<k<4).
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Which interval gives no real roots for (3x-2-2(2a+1)x+\(a^2+a+1\)=0)?
A. (-2<a<1)
वास्तविक मूल न होने के लिए (D<0) चाहिए। \(a^2+a-2<0\) से (-2<a<1) मिलता है। / For no real roots, (D<0) is required. From \(a^2+a-2<0\), we get (-2<a<1).
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What is the interval of (k) for no real roots in (x-2+(k-3)x+4=0)?
A. (-1<k<7)
यहाँ (D=(k-3)2-16) है। (D<0) से (-1<k<7) मिलता है। / Here (D=(k-3)2-16). From (D<0), we get (-1<k<7).
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In which interval will (a) lie for (x-2+2(a-2)x+2a+5=0) to have no real roots?
A. \(3-\sqrt{10}<a<3+\sqrt{10}\)
वास्तविक मूल न होने के लिए (D<0) चाहिए। यहाँ (D=4\(a^2-6a-1\)), इसलिए \(3-\sqrt{10}<a<3+\sqrt{10}\)। / For no real roots, (D<0) is required. Here (D=4\(a^2-6a-1\)), so \(3-\sqrt{10}<a<3+\sqrt{10}\).
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Which interval is correct for \(x^2-2mx+3m=0\) to have no real roots?
A. (0<m<3)
वास्तविक मूल न होने के लिए (D<0) चाहिए। (D=4m(m-3)<0) से (0<m<3) मिलता है। / For no real roots, (D<0) is needed. From (D=4m(m-3)<0), we get (0<m<3).
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In which interval will (t) lie for \(x^2+2tx+t+4=0\) to have no real roots?
A. \(-\frac{3}{2}<t<2\)
वास्तविक मूल न होने के लिए (D<0) चाहिए। यहाँ (D=4(t-2)(2t+3)), इसलिए \(-\frac{3}{2}<t<2\)। / For no real roots, (D<0) is required. Here (D=4(t-2)(2t+3)), so \(-\frac{3}{2}<t<2\).
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For (x-2-(k+1)x+4=0), in which interval will (k) lie for no real roots?
A. (-5<k<3)
यहाँ (D=(k+1)2-16) है और वास्तविक मूल न होने के लिए (D<0) चाहिए। इससे (-5<k<3) मिलता है। / Here (D=(k+1)2-16), and no real roots need (D<0). This gives (-5<k<3).
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If one root of \(x^2-25x+q=0\) is (10), what will be the other root?
A. (15)
मूलों का योग (25) है, इसलिए दूसरा मूल (25-10=15) होगा। परीक्षा में एक मूल दिया हो तो योग का प्रयोग करें। / The sum of roots is (25), so the other root is (25-10=15). In exams, use the sum when one root is given.
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If one root of \(x^2-23x+q=0\) is (9), what will be the other root?
A. (14)
मूलों का योग (23) है, इसलिए दूसरा मूल (23-9=14) होगा। परीक्षा में एक मूल दिया हो तो योग का प्रयोग करें। / The sum of roots is (23), so the other root is (23-9=14). In exams, use the sum when one root is given.
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If one root of \(x^2-21x+q=0\) is (8), what will be the other root?
A. (13)
मूलों का योग (21) है, इसलिए दूसरा मूल (21-8=13) होगा। परीक्षा में एक मूल दिया हो तो योग का उपयोग करें। / The sum of roots is (21), so the other root is (21-8=13). In exams, use the sum when one root is given.
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If one root of \(x^2-15x+q=0\) is (6), what will be the other root?
A. (9)
मूलों का योग (15) है, इसलिए दूसरा मूल (15-6=9) होगा। परीक्षा में एक मूल दिया हो तो योग का प्रयोग करें। / The sum of roots is (15), so the other root is (15-6=9). In exams, use the sum when one root is given.
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If one root of \(x^2-9x+q=0\) is (4), what will be the other root?
A. (5)
मूलों का योग (9) है, इसलिए दूसरा मूल (9-4=5) होगा। परीक्षा में एक मूल दिया हो तो योग का प्रयोग करें। / The sum of roots is (9), so the other root is (9-4=5). In exams, use the sum when one root is given.
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If one root of \(x^2-5x+q=0\) is (2), what will be the other root?
A. (3)
मूलों का योग (5) है, इसलिए दूसरा मूल (5-2=3) होगा। परीक्षा में एक मूल दिया हो तो योग या गुणनफल का प्रयोग करें। / The sum of roots is (5), so the other root is (5-2=3). In exams, use sum or product when one root is given.
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If (x=0) is a root of \(ax^2+bx+c=0\), which condition must be true?
A. (c=0)
(x=0) रखने पर समीकरण (c=0) बनता है। इसलिए शून्य जड़ होने की सीधी शर्त (c=0) है। / Putting (x=0) gives (c=0). Thus the direct condition for zero to be a root is (c=0).
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If one root of \(x^2+ax+12=0\) is (3) more than the other root, what are the possible values of (a)?
A. \(\sqrt{57}\) और \(-\sqrt{57}\)\(\sqrt{57}\) and \(-\sqrt{57}\)
जड़ें (r) और (r+3) मानने पर (r(r+3)=12) मिलता है। इससे जड़ों का योग \(\pm\sqrt{57}\) होता है, इसलिए \(a=\mp\sqrt{57}\)। / Let the roots be (r) and (r+3). Then (r(r+3)=12), giving the sum as \(\pm\sqrt{57}\), so \(a=\mp\sqrt{57}\).
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If one root of (x-2+(k-3)x+k=0) is twice the other root, what is the value of (k)?
A. \(\frac{21+3\sqrt{33}}{4}\) या \(\frac{21-3\sqrt{33}}{4}\)\(\frac{21+3\sqrt{33}}{4}\) or \(\frac{21-3\sqrt{33}}{4}\)
जड़ें (r) और (2r) मानने पर (3r=3-k) और \(2r^2=k\) मिलता है। हल करने पर \(2k^2-21k+18=0\), इसलिए दिए गए दोनों मान मिलते हैं। / Taking the roots as (r) and (2r), we get (3r=3-k) and \(2r^2=k\). Solving \(2k^2-21k+18=0\) gives the two listed values.
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A. (1)
समीकरण के मूल (6) और (7) हैं। इसलिए छोटा मूल (6) और बड़ा मूल (7) है तथा \(\beta-\alpha=1\) है। / The roots are (6) and (7). Thus the smaller root is (6) and the larger root is (7), so \(\beta-\alpha=1\).
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If one root of \(x^2+ax+a=0\) is (2), what will be the other root?
A. \(-\frac{2}{3}\)
(x=2) रखने पर (4+3a=0) से \(a=-\frac{4}{3}\) है। गुणनफल (a) है इसलिए दूसरा मूल \(-\frac{2}{3}\) होगा। / Putting (x=2) gives (4+3a=0), so \(a=-\frac{4}{3}\). The product is (a), so the other root is \(-\frac{2}{3}\).
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A. (1)
समीकरण के मूल (5) और (6) हैं। इसलिए छोटा मूल (5) और बड़ा मूल (6) है तथा \(\beta-\alpha=1\) है। / The roots are (5) and (6). Thus the smaller root is (5) and the larger root is (6), so \(\beta-\alpha=1\).
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If one root of \(x^2+ax+a=0\) is (1), what will be the other root?
A. \(-\frac{1}{2}\)
(x=1) रखने पर (1+2a=0) से \(a=-\frac{1}{2}\) है। गुणनफल (a) है इसलिए दूसरा मूल \(-\frac{1}{2}\) होगा। / Putting (x=1) gives (1+2a=0), so \(a=-\frac{1}{2}\). The product is (a), so the other root is \(-\frac{1}{2}\).
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If one root of \(x^2-17x+70=0\) is (7), what is the other root?
A. (10)
मूलों का गुणनफल (70) है और एक मूल (7) है। दूसरा मूल \(\frac{70}{7}=10\) होगा। / The product of roots is (70) and one root is (7). The other root is \(\frac{70}{7}=10\).
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If one root of \(x^2-15x+54=0\) is (6), what is the other root?
A. (9)
मूलों का गुणनफल (54) है और एक मूल (6) है। इसलिए दूसरा मूल \(\frac{54}{6}=9\) है। / The product of roots is (54) and one root is (6). Therefore the other root is \(\frac{54}{6}=9\).
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If one root of \(x^2-13x+40=0\) is (5), what is the other root?
A. (8)
मूलों का गुणनफल (40) है और एक मूल (5) है। इसलिए दूसरा मूल \(\frac{40}{5}=8\) है। / The product of roots is (40) and one root is (5). Therefore the other root is \(\frac{40}{5}=8\).
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If the sum of roots is (12) and one root is (5), what is the other root?
B. (7)
दूसरा मूल (12-5=7) है। योग में से दिया हुआ मूल घटाएं। / The other root is (12-5=7). Subtract the given root from the sum.
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If one root is (6) and the product of roots is (48), what is the other root?
B. (8)
दूसरा मूल \(\frac{48}{6}=8\) होगा। गुणनफल को दिए हुए मूल से भाग करें। / The other root is \(\frac{48}{6}=8\). Divide the product by the given root.
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If one root of \(x^2-11x+30=0\) is (5), what is the other root?
B. (6)
(x-2-11x+30=(x-5)(x-6)) है। इसलिए दूसरा मूल (6) है। / (x-2-11x+30=(x-5)(x-6)). Therefore the other root is (6).
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If the sum of roots is (8) and one root is (3), what is the other root?
B. (5)
दूसरा मूल (8-3=5) है। योग में से दिया हुआ मूल घटाएं। / The other root is (8-3=5). Subtract the given root from the sum.
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If one root is (5) and the product of roots is (35), what is the other root?
B. (7)
दूसरा मूल \(\frac{35}{5}=7\) होगा। गुणनफल में दिए हुए मूल से भाग करें। / The other root is \(\frac{35}{5}=7\). Divide the product by the given root.
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If one root of \(x^2-9x+18=0\) is (3), what is the other root?
B. (6)
(x-2-9x+18=(x-3)(x-6)) है। इसलिए दूसरा मूल (6) है। / (x-2-9x+18=(x-3)(x-6)). Therefore the other root is (6).
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If the sum of roots is (5) and one root is (2) then what is the other root?
B. (3)
दूसरा मूल (5-2=3) है। योग में से दिए हुए मूल को घटाएं। / The other root is (5-2=3). Subtract the given root from the sum.
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If one root is (3) and the product of roots is (12) then what is the other root?
B. (4)
दूसरा मूल \(\frac{12}{3}=4\) होगा। गुणनफल वाले प्रश्न में दिए मूल से भाग दें। / The other root is \(\frac{12}{3}=4\). In product questions divide by the given root.
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If one root of \(x^2-6x+8=0\) is (4) then what is the other root?
B. (2)
(x-2-6x+8=(x-4)(x-2)) इसलिए दूसरा मूल (2) है। दिए गए एक मूल से दूसरा गुणनखंड खोजें। / (x-2-6x+8=(x-4)(x-2)) so the other root is (2). Use the given root to find the other factor.
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In which equation is (x=-2) not a root?
D. \(x^2-2x+4=0\)
(x=-2) रखने पर \(4+4+4=12\neq0\) मिलता है। मूल न होने की जांच भी प्रतिस्थापन से करें। / Putting (x=-2) gives \(4+4+4=12\neq0\). To check a non-root, use substitution too.
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In which equation is (x=1) not a root?
D. \(x^2+x+1=0\)
(x=1) रखने पर \(1+1+1=3\neq 0\) मिलता है। किसी विकल्प में मूल न होने की जांच भी प्रतिस्थापन से करें। / Putting (x=1) gives \(1+1+1=3\neq 0\). To check when a value is not a root, use substitution too.
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If a root grows towards water and away from light at the same time which conclusion is correct?
A. जड़ में जलानुवर्तन और ऋणात्मक प्रकाशानुवर्तन दोनों दिख सकते हैंRoot can show hydrotropism and negative phototropism both
पानी की ओर वृद्धि जलानुवर्तन है। / Growth towards water is hydrotropism.
प्रकाश से दूर वृद्धि ऋणात्मक प्रकाशानुवर्तन है। / Growth away from light is negative phototropism.
एक ही पौध भाग कई उद्दीपनों के प्रति प्रतिक्रिया दिखा सकता है। / A plant part can respond to more than one stimulus.
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If one root of \(x^2+px+64=0\) is double the other and both are negative, what is (p)?
A. \(12\sqrt{2}\)
मूलों को (-r) और (-2r) मानें, तो \(2r^2=64\) से \(r=4\sqrt{2}\) और \(p=3r=12\sqrt{2}\) है। परीक्षा में दोनों मूलों के चिन्ह ध्यान से रखें। / Let the roots be (-r) and (-2r), then \(2r^2=64\) gives \(r=4\sqrt{2}\), and \(p=3r=12\sqrt{2}\). In exams, keep signs of both roots carefully.
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If one root of \(x^2+px+49=0\) is double the other and both are negative, what is (p)?
A. \( \frac{21\sqrt{2}}{2}\)
मूलों को (-r) और (-2r) मानें, तो \(2r^2=49\) और \(p=3r=\frac{21\sqrt{2}}{2}\) है। परीक्षा में हर को परिमेय बनाना न भूलें। / Let the roots be (-r) and (-2r), then \(2r^2=49\) and \(p=3r=\frac{21\sqrt{2}}{2}\). In exams, do not forget to rationalize the denominator.
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If one root of \(x^2+px+36=0\) is double the other and both are negative, what is (p)?
A. \(9\sqrt{2}\)
मूलों को (-r) और (-2r) मानें, तो \(2r^2=36\) से \(r=3\sqrt{2}\) और \(p=3r=9\sqrt{2}\) है। परीक्षा में दोनों मूलों के चिन्ह ध्यान से रखें। / Let the roots be (-r) and (-2r), then \(2r^2=36\) gives \(r=3\sqrt{2}\), and \(p=3r=9\sqrt{2}\). In exams, keep signs of both roots carefully.
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If one root of \(x^2+px+25=0\) is double the other and both are negative, what is (p)?
A. \( \frac{15\sqrt{2}}{2}\)
मूलों को (-r) और (-2r) मानें, तो \(2r^2=25\) और \(p=3r=\frac{15\sqrt{2}}{2}\) है। परीक्षा में हर को परिमेय बनाना न भूलें। / Let the roots be (-r) and (-2r), then \(2r^2=25\) and \(p=3r=\frac{15\sqrt{2}}{2}\). In exams, do not forget to rationalize the denominator.
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If one root of \(x^2+px+16=0\) is double the other and both are negative, what is (p)?
A. \(6\sqrt{2}\)
मूलों को (-r) और (-2r) मानें, तो \(2r^2=16\) से \(r=2\sqrt{2}\) और \(p=3r=6\sqrt{2}\) है। परीक्षा में दोनों मूलों के चिन्ह ध्यान से रखें। / Let the roots be (-r) and (-2r), then \(2r^2=16\) gives \(r=2\sqrt{2}\), and \(p=3r=6\sqrt{2}\). In exams, keep signs of both roots carefully.
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A. \( \frac{9\sqrt{2}}{2}\)
\(\frac{9}{\sqrt{2}}\) को सरल करने पर \(\frac{9\sqrt{2}}{2}\) मिलता है। परीक्षा में हर को परिमेय बनाना न भूलें। / \(\frac{9}{\sqrt{2}}\) simplifies to \(\frac{9\sqrt{2}}{2}\). In exams, do not forget to rationalize the denominator.
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If one root of \(x^2+px+9=0\) is double the other and both are negative, what is (p)?
A. \(3\sqrt{2}\)
मूलों को (-r) और (-2r) मानें, तो \(2r^2=9\) और योग (-3r) है, इसलिए \(p=3r=3\sqrt{\frac{9}{2}}\) नहीं बल्कि \(r=\frac{3}{\sqrt{2}}\), अतः \(p=\frac{9}{\sqrt{2}}\) होता है। परीक्षा में ऐसे प्रश्नों में मानकर समीकरण बनाएं। / Let the roots be (-r) and (-2r), then \(2r^2=9\) and the sum is (-3r), so \(p=3r=\frac{9}{\sqrt{2}}\). In exams, assume the roots and form equations carefully.
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If \(p=-\sqrt{99}+4\), in which interval will (p) lie on the number line?
B. ( -6 ) और ( -5 ) के बीचBetween ( -6 ) and ( -5 )
\( \sqrt{99}\approx9.95 \), इसलिए \(p\approx-5.95\) है। अतः यह (-6) और (-5) के बीच होगा। / \( \sqrt{99}\approx9.95 \), so \(p\approx-5.95\). Hence it lies between (-6) and (-5).
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If \(p=-\sqrt{68}+3\), in which interval will (p) lie on the number line?
A. ( -6 ) और ( -5 ) के बीचBetween ( -6 ) and ( -5 )
\( \sqrt{68}\approx8.246 \), इसलिए \(p\approx-5.246\) है। इसलिए यह (-6) और (-5) के बीच है। / \( \sqrt{68}\approx8.246 \), so \(p\approx-5.246\). Hence it lies between (-6) and (-5).
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If \(p=-\sqrt{45}+2\), in which interval will (p) lie on the number line?
A. ( -5 ) और ( -4 ) के बीचBetween ( -5 ) and ( -4 )
\( \sqrt{45}\approx6.708 \), इसलिए \(p\approx-4.708\) है। इसलिए यह (-5) और (-4) के बीच है। / \( \sqrt{45}\approx6.708 \), so \(p\approx-4.708\). Hence it lies between (-5) and (-4).
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In which interval will \(-\frac{3}{4}\) lie on the number line?
B. (-1) और (0) के बीचBetween (-1) and (0)
\(-\frac{3}{4}\) ऋणात्मक है और (-1) से बड़ा है। परीक्षा में ऐसे भिन्न को (-1) और (0) के बीच दिखाएं। / \(-\frac{3}{4}\) is negative and greater than (-1). In exams, show such fractions between (-1) and (0).
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In which interval will \(-\frac{1}{4}\) lie on the number line?
A. (-1) और (0) के बीचbetween (-1) and (0)
\(-\frac{1}{4}=-0.25\), इसलिए यह (-1) और (0) के बीच है। छोटे ऋणात्मक भिन्न (0) के बाईं ओर पास होते हैं। / \(-\frac{1}{4}=-0.25\), so it lies between (-1) and (0). Small negative fractions are close to (0) on the left.
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In which situation can negative space also appear like a shape?
C. जब दो वस्तुओं के बीच खाली भाग स्पष्ट रूप बना देWhen empty area between two objects forms a clear figure
कभी कभी वस्तुओं के बीच का खाली क्षेत्र पहचानने योग्य आकार बनाता है। परीक्षा में negative shape को भी देखें। / Sometimes empty area between objects forms a recognizable shape. Exam tip: observe negative shape too.
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In which option will the sum of roots be negative and the product of roots be negative?
A. \(x^2+5x-24=0\)
पहले विकल्प में योग \(-\frac{b}{a}=-5\) और गुणनफल \(\frac{c}{a}=-24\) है। इसलिए दोनों ऋणात्मक हैं। / In the first option, the sum is \(-\frac{b}{a}=-5\) and the product is \(\frac{c}{a}=-24\). So both are negative.
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A. (-7) और (-6)(-7) and (-6)
\(6<\sqrt{48}<7\), इसलिए \(-7<-\sqrt{48}<-6\)। ऋणात्मक वर्गमूल में अंतराल उल्टा लिखें। / Since \(6<\sqrt{48}<7\), \(-7<-\sqrt{48}<-6\). Write the interval carefully for negative square roots.
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Between which two integers will \(-\sqrt{11}\) lie on the number line?
C. (-4) और (-3)(-4) and (-3)
\(\sqrt{11}\) (3) और (4) के बीच है इसलिए \(-\sqrt{11}\) (-4) और (-3) के बीच होगा। ऋणात्मक चिन्ह दिशा बदल देता है। / \(\sqrt{11}\) lies between (3) and (4), so \(-\sqrt{11}\) lies between (-4) and (-3). The negative sign changes the side.
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Between which two integers will \(-\sqrt{30}\) lie on the number line?
B. (-6) और (-5)(-6) and (-5)
क्योंकि \(\sqrt{30}\) (5) और (6) के बीच है, इसलिए \(-\sqrt{30}\) (-6) और (-5) के बीच होगा। परीक्षा में ऋणात्मक दिशा को ध्यान रखें। / Since \(\sqrt{30}\) lies between (5) and (6), \(-\sqrt{30}\) lies between (-6) and (-5). In exams, keep the negative direction in mind.
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What is the equal root in \(9x^2-6x+1=0\)?
A. \(x=\frac{1}{3}\)
यहाँ (9x-2-6x+1=(3x-1)2) है। इसलिए समान मूल \(x=\frac{1}{3}\) है। / Here (9x-2-6x+1=(3x-1)2). Therefore the equal root is \(x=\frac{1}{3}\).
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What is the equal root in \(4x^2-20x+25=0\)?
A. \(x=\frac{5}{2}\)
यहाँ (D=(-20)2-4(4)(25)=0) और ((2x-5)2=0) है। इसलिए समान मूल \(x=\frac{5}{2}\) है। / Here (D=(-20)2-4(4)(25)=0), and ((2x-5)2=0). So the equal root is \(x=\frac{5}{2}\).
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What is the equal root of \(x^2-8x+16=0\)?
A. (4)
समीकरण ((x-4)2=0) बनता है। समान मूल सीधे (x=4) है। / The equation becomes ((x-4)2=0). The equal root is (x=4).
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What is the root of \(x^2-2\sqrt{17}x+17=0\)?
A. \(x=\sqrt{17}\)
(\(x-\sqrt{17}\)2=0), इसलिए दोहराया हुआ मूल \(\sqrt{17}\) है। परीक्षा में ((x-a)2=0) से (x=a) मिलता है। / (\(x-\sqrt{17}\)2=0), so the repeated root is \(\sqrt{17}\). In exams, ((x-a)2=0) gives (x=a).
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