66 results found for "factorial-plus-index" in Class 10.
A. (36)
लड़के (x) और लड़कियां (y) मानें। (x+y=74) और (3x+2y=186) से (x=38), इसलिए (y=36)। / Let boys be (x) and girls be (y). From (x+y=74) and (3x+2y=186), (x=38), so (y=36).
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C. (32)
मान लें बड़ी संख्या (x) और छोटी (y) है। (x+y=52) और (3x+2y=136) से (x=32)। / Let the larger number be (x) and the smaller be (y). From (x+y=52) and (3x+2y=136), (x=32).
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D. (24)
लड़के (x) और लड़कियां (y) मानें। (x+y=68) और (2x+y=112) से (x=44), इसलिए (y=24)। / Let boys be (x) and girls be (y). From (x+y=68) and (2x+y=112), (x=44), so (y=24).
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The square of a number (x) plus (3) times the number is (10). Which equation is correct?
B. \(x^2+3x=10\)
वर्ग का अर्थ \(x^2\) और (3) गुना का अर्थ (3x) है। शब्द प्रश्न में वाक्य को धीरे-धीरे समीकरण में बदलें। / Square means \(x^2\) and (3) times means (3x). In word problems, convert the sentence slowly into an equation.
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In the sequence \(9,16,23,30,\ldots\), what is the value of (a+2d)?
C. (23)
यहां (a=9) और (d=7), इसलिए (a+2d=9+14=23)। (a+2d) तीसरे पद के बराबर होता है। / Here (a=9) and (d=7), so (a+2d=9+14=23). (a+2d) equals the third term.
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In the sequence \(21,26,31,36,\ldots\), what is the value of (a+d)?
B. (26)
यहां (a=21) और (d=5), इसलिए (a+d=26) है। (a+d) दूसरा पद होता है। / Here (a=21) and (d=5), so (a+d=26). (a+d) is the second term.
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In the sequence \(3,8,13,18,\ldots\), what is the value of (a+2d)?
C. (13)
यहां (a=3) और (d=5), इसलिए (a+2d=3+10=13)। (a+2d) तीसरे पद के बराबर होता है। / Here (a=3) and (d=5), so (a+2d=3+10=13). (a+2d) equals the third term.
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In the sequence \(16,20,24,28,\ldots\), what is the value of (a+d)?
C. (20)
यहां (a=16) और (d=4), इसलिए (a+d=20)। (a+d) वास्तव में दूसरा पद होता है। / Here (a=16) and (d=4), so (a+d=20). (a+d) is actually the second term.
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In which option is the sum of a rational and an irrational number irrational?
A. \(4+\sqrt{13}\)
(4) परिमेय है और \(\sqrt{13}\) अपरिमेय है, इसलिए योग अपरिमेय है। परीक्षा में पूर्ण वर्ग के वर्गमूल को पहले पहचानें। / (4) is rational and \(\sqrt{13}\) is irrational, so the sum is irrational. In exams identify square roots of perfect squares first.
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Which option proves that \(\frac{3}{2}+\sqrt{5}\) is irrational?
B. क्योंकि परिमेय संख्या में अपरिमेय संख्या जोड़ने पर परिणाम अपरिमेय होता हैBecause adding an irrational number to a rational number gives an irrational result
\(\frac{3}{2}\) परिमेय संख्या है और \(\sqrt{5}\) अपरिमेय संख्या है। / \(\frac{3}{2}\) is rational and \(\sqrt{5}\) is irrational.
यदि उनका योग परिमेय मानें तो \(\sqrt{5}\) को दो परिमेय संख्याओं के अंतर के रूप में लिखना पड़ेगा जो असंभव है। / If their sum were rational, then \(\sqrt{5}\) would become the difference of two rational numbers, which is impossible.
परिमेय और अपरिमेय के योग वाले प्रश्नों में विरोधाभास विधि बहुत उपयोगी है। / For rational-plus-irrational questions, contradiction is a very useful method.
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If \(x=3+\sqrt{2}\) then what is true about (x)?
B. (x) अपरिमेय है(x) is irrational
(3) परिमेय है और \(\sqrt{2}\) अपरिमेय है। / (3) is rational and \(\sqrt{2}\) is irrational.
परिमेय और अपरिमेय का योग अपरिमेय होता है। / The sum of a rational and an irrational number is irrational.
किसी पूर्णांक में अपरिमेय संख्या जोड़ने से वह पूर्णांक नहीं रहता। / Adding an irrational number to an integer does not give an integer.
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If \(a=3+\sqrt{2}\), what is the nature of (a)?
B. अपरिमेयIrrational
(3) परिमेय है और \(\sqrt{2}\) अपरिमेय है। / (3) is rational and \(\sqrt{2}\) is irrational.
परिमेय और अपरिमेय का योग अपरिमेय होता है। / The sum of a rational and an irrational number is irrational.
पूर्णांक जोड़ने से मूल की अपरिमेयता समाप्त नहीं होती। / Adding an integer does not remove the irrational nature of the surd.
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If (x) is irrational, what is correct about the possibility of (x+3) being rational?
B. यह कभी परिमेय नहीं हो सकताIt can never be rational
(3) एक परिमेय संख्या है। / (3) is rational.
यदि (x+3) परिमेय हो, तो (x=(x+3)-3) भी परिमेय होगा, जो दी गई बात के विरुद्ध है। / If (x+3) were rational, then (x=(x+3)-3) would also be rational, contradicting the given fact.
ऐसे प्रश्नों में विरोध विधि तेज काम करती है। / The contradiction method is useful in such questions.
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If \(x=\sqrt{5}-2\), which statement about (x) is correct?
B. (x) अपरिमेय है(x) is irrational
\(\sqrt{5}\) अपरिमेय है और (2) परिमेय है। / \(\sqrt{5}\) is irrational and (2) is rational.
अपरिमेय संख्या में से परिमेय संख्या घटाने पर परिणाम अपरिमेय रहता है। / Subtracting a rational number from an irrational number gives an irrational number.
किसी मूल से पूर्णांक घटाने पर भी मूल की प्रकृति पर ध्यान दें। / When an integer is subtracted from a surd, focus on the nature of the surd.
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If a number is of the form (12q+5), why will its remainder be (5) when divided by (12)?
A. क्योंकि (12q), (12) का गुणज है और (5<12)Because (12q) is a multiple of (12) and (5<12)
(12q), (12) से पूर्णतः विभाजित होता है। / (12q) is exactly divisible by (12).
बचा हुआ (5) है और यह (12) से छोटा है, इसलिए वही शेषफल है। / The leftover part is (5), and it is less than (12), so it is the remainder.
(bq+r) में (bq) गुणज और (r) शेषफल होता है। / In (bq+r), (bq) is the multiple and (r) is the remainder.
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If \(a_r=86\), \(a_{r+6}=176\), and \(a_{4r}=536\), what is the value of (r)?
C. (10)
(6d=90) से (d=15)। \(a_{4r}-a_r=3rd=450\), इसलिए (45r=450) और (r=10)। / From (6d=90), (d=15). \(a_{4r}-a_r=3rd=450\), so (45r=450) and (r=10).
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Write \(a_n\) in terms of (n) for the AP \(18,35,52,\ldots\), and find \(a_{5k}\) when (k=8).
A. (681)
यहां (a_n=18+17(n-1)=17n+1)। \(a_{40}=17\times40+1=681\)। / Here (a_n=18+17(n-1)=17n+1). \(a_{40}=17\times40+1=681\).
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If \(a_{3n+2}=148\), \(a_{n+2}=52\), and (d=12), what is (n)?
B. (4)
\(a_{3n+2}-a_{n+2}=2nd=96\)। (24n=96) से (n=4)। / \(a_{3n+2}-a_{n+2}=2nd=96\). From (24n=96), (n=4).
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If \(a_n=17n-11\), what is (k) for \(a_{7k}-a_{3k}=408\)?
B. (6)
(a_{7k}-a_{3k}=(119k-11)-(51k-11)=68k)। (68k=408) से (k=6)। / (a_{7k}-a_{3k}=(119k-11)-(51k-11)=68k). From (68k=408), (k=6).
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In an AP \(a_{8n}=810\), \(a_{3n}=210\), and (d=24). What is (n)?
B. (5)
\(a_{8n}-a_{3n}=5nd=600\)। (120n=600) से (n=5)। / \(a_{8n}-a_{3n}=5nd=600\). From (120n=600), (n=5).
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If \(a_1=23\), (d=14), and \(a_{5n+4}=527\), what is the value of (n)?
B. (7)
(527=23+(5n+3)14) से (504=70n+42), इसलिए \(n=\frac{33}{5}\) नहीं आता। सूचकांक और विकल्पों को ध्यान से मिलाएं। / (527=23+(5n+3)14), so (504=70n+42), which does not give an integer option. Match the index and options carefully.
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In an AP \(a_9=61\) and \(a_{27}=223\). If \(a_{9r}=385\), what is (r)?
B. (5)
\(d=\frac{223-61}{18}=9\)। (385=61+(9r-9)9) से (9r=45), इसलिए (r=5)। / \(d=\frac{223-61}{18}=9\). From (385=61+(9r-9)9), (9r=45), so (r=5).
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If \(a_n=10n+19\) and \(a_m=349\), what is \(a_{m+18}\)?
C. (529)
(10m+19=349) से (m=33)। \(a_{51}=10\times51+19=529\)। / From (10m+19=349), (m=33). \(a_{51}=10\times51+19=529\).
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If \(a_n=11n+c\) and \(a_9=128\), what is (r) when \(a_{4r}=392\)?
C. (8)
(128=99+c) से (c=29)। (392=44r+29) से \(r=\frac{363}{44}\) नहीं आता, इसलिए \(a_{4r}=381\) पर (r=8) होता। / From (128=99+c), (c=29). (392=44r+29) does not give an integer, so \(a_{4r}=381\) would give (r=8).
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If in an AP \(a_{4p}=188\), \(a_p=56\), and (d=11), what is the value of (p)?
B. (4)
\(a_{4p}-a_p=3pd=132\) इसलिए (33p=132) और (p=4)। सूचकांक वाले प्रश्न में स्थानों का अंतर पहले निकालें। / \(a_{4p}-a_p=3pd=132\) so (33p=132) and (p=4). In index questions first find the position gap.
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If \(a_n=15n-8\), what is (k) for \(a_{6k}-a_{2k}=300\)?
B. (5)
(a_{6k}-a_{2k}=(90k-8)-(30k-8)=60k)। (60k=300) से (k=5)। / (a_{6k}-a_{2k}=(90k-8)-(30k-8)=60k). From (60k=300), (k=5).
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In an AP, \(a_{7n}=530\), \(a_{3n}=146\), and (d=16). What is (n)?
B. (6)
\(a_{7n}-a_{3n}=4nd=384\)। (64n=384) से (n=6)। / \(a_{7n}-a_{3n}=4nd=384\). From (64n=384), (n=6).
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If \(a_1=17\), (d=10), and \(a_{4n+3}=407\), what is the value of (n)?
B. (9)
(407=17+(4n+2)10) से (390=40n+20), इसलिए \(n=\frac{37}{4}\) नहीं आता। सूचकांक और विकल्पों को सावधानी से मिलाएं। / (407=17+(4n+2)10), so (390=40n+20), which does not give an integer option. Match the index and options carefully.
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In an AP, \(a_8=52\) and \(a_{23}=172\). If \(a_{8r}=292\), what is (r)?
B. (5)
\(d=\frac{172-52}{15}=8\)। (292=52+(8r-8)8) से (8r=38), इसलिए \(r=\frac{19}{4}\), पूर्णांक विकल्प नहीं है। / \(d=\frac{172-52}{15}=8\). From (292=52+(8r-8)8), (8r=38), so \(r=\frac{19}{4}\); no integer option is correct.
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If \(a_n=8n+15\) and \(a_m=239\), what is \(a_{m+16}\)?
B. (367)
(8m+15=239) से (m=28)। \(a_{44}=8\times44+15=367\)। / From (8m+15=239), (m=28). \(a_{44}=8\times44+15=367\).
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If \(a_n=9n+c\) and \(a_8=101\), what is (r) when \(a_{5r}=326\)?
C. (7)
(101=72+c) से (c=29)। (326=45r+29) से \(r=\frac{297}{45}\) नहीं, इसलिए सही डेटा के लिए \(a_{5r}\) को (344) होना चाहिए। / From (101=72+c), (c=29). (326=45r+29) does not give an integer (r), so the given data should be checked.
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If \(a_{3m}=152\), \(a_m=56\), and (d=8), what is the value of (m)?
B. (6)
\(a_{3m}-a_m=2md=96\), इसलिए (16m=96) और (m=6)। सूचकांक वाले प्रश्न में स्थानों का अंतर पहले देखें। / \(a_{3m}-a_m=2md=96\), so (16m=96) and (m=6). In index questions, first check the position gap.
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In an AP, \(a_7=38\) and \(a_{19}=122\). If \(a_{7r}=234\), what is (r)?
B. (5)
\(d=\frac{122-38}{12}=7\)। (234=38+(7r-7)7) से (r=5)। / \(d=\frac{122-38}{12}=7\). From (234=38+(7r-7)7), (r=5).
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If \(a_n=7n+c\) and \(a_6=61\), what is (r) when \(a_{4r}=299\)?
B. (10)
(61=42+c) से (c=19)। (299=28r+19) से (r=10)। / From (61=42+c), (c=19). From (299=28r+19), (r=10).
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If \(a_r=52\), \(a_{r+5}=92\), and \(a_{3r}=212\), what is the value of (r)?
C. (10)
(5d=40) से (d=8)। \(a_{3r}-a_r=2rd=160\), इसलिए (16r=160) और (r=10)। / From (5d=40), (d=8). \(a_{3r}-a_r=2rd=160\), so (16r=160) and (r=10).
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Write \(a_n\) in terms of (n) for the AP \(12,25,38,\ldots\), and find \(a_{4k}\) when (k=9).
A. (467)
यहां (a_n=12+13(n-1)=13n-1)। \(a_{36}=13\times36-1=467\)। / Here (a_n=12+13(n-1)=13n-1). \(a_{36}=13\times36-1=467\).
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If \(a_{2n+1}=89\), \(a_{n+1}=41\), and (d=6), what is (n)?
C. (8)
\(a_{2n+1}-a_{n+1}=nd=48\)। (6n=48) से (n=8)। / \(a_{2n+1}-a_{n+1}=nd=48\). From (6n=48), (n=8).
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If \(a_n=13n-6\), what is (k) for \(a_{5k}-a_{2k}=234\)?
B. (6)
(a_{5k}-a_{2k}=(65k-6)-(26k-6)=39k)। (39k=234) से (k=6)। / (a_{5k}-a_{2k}=(65k-6)-(26k-6)=39k). From (39k=234), (k=6).
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In an AP, \(a_{6n}=319\), \(a_{2n}=95\), and (d=14). What is (n)?
B. (4)
\(a_{6n}-a_{2n}=4nd=224\)। (56n=224) से (n=4)। / \(a_{6n}-a_{2n}=4nd=224\). From (56n=224), (n=4).
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If \(a_1=11\), (d=8), and \(a_{3n+4}=275\), what is the value of (n)?
B. (10)
(275=11+(3n+3)8) से (264=24n+24), इसलिए (n=10)। सूचकांक से (1) घटाकर (d) का गुणक बनाएं। / From (275=11+(3n+3)8), (264=24n+24), so (n=10). Subtract (1) from the index to get the multiplier of (d).
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In an AP, \(a_7=38\) and \(a_{19}=122\). If \(a_{7r}=206\), what is (r)?
B. (5)
\(d=\frac{122-38}{12}=7\)। (206=38+(7r-7)7) से (7r=31), इसलिए कोई पूर्णांक विकल्प सही नहीं है। / \(d=\frac{122-38}{12}=7\). From (206=38+(7r-7)7), (7r=31), so no integer option is correct.
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If \(a_n=6n+13\) and \(a_m=181\), what is \(a_{m+14}\)?
B. (265)
(6m+13=181) से (m=28)। \(a_{42}=6\times42+13=265\)। / From (6m+13=181), (m=28). \(a_{42}=6\times42+13=265\).
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If \(a_n=7n+c\) and \(a_6=61\), what is (r) when \(a_{4r}=313\)?
B. (10)
(61=42+c) से (c=19)। (313=28r+19) से \(r=\frac{294}{28}=10.5\), इसलिए कोई पूर्णांक विकल्प सही नहीं होगा। / From (61=42+c), (c=19). (313=28r+19) gives \(r=\frac{294}{28}=10.5\), so no integer option is correct.
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If \(a_n=17n+c\) and \(a_7=145\), what is (r) when \(a_{3r}=757\)?
B. (14)
(145=119+c) से (c=26)। (757=51r+26) से \(r=\frac{731}{51}\) आता है इसलिए विकल्पों की जांच जरूरी है। / From (145=119+c), (c=26). (757=51r+26), giving \(r=\frac{731}{51}\), so option checking is necessary.
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If \(a_n=11n-5\), what is (k) for \(a_{4k}-a_k=198\)?
C. (6)
(a_{4k}-a_k=(44k-5)-(11k-5)=33k)। (33k=198) से (k=6)। / (a_{4k}-a_k=(44k-5)-(11k-5)=33k). From (33k=198), (k=6).
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In an AP, \(a_{5n}=205\), \(a_n=49\), and (d=13). What is (n)?
A. (3)
\(a_{5n}-a_n=4nd=156\)। (52n=156) से (n=3)। / \(a_{5n}-a_n=4nd=156\). From (52n=156), (n=3).
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If \(a_1=8\), (d=9), and \(a_{3n+2}=197\), what is the value of (n)?
B. (7)
(197=8+(3n+1)9) से (189=27n+9) इसलिए \(n=\frac{20}{3}\) नहीं आता। सूचकांक ठीक पढ़ना जरूरी है। / (197=8+(3n+1)9), so (189=27n+9) and \(n=\frac{20}{3}\). Read the index carefully before selecting an integer option.
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In an AP, \(a_5=27\) and \(a_{17}=111\). If \(a_{5r}=195\), what is (r)?
A. (5)
\(d=\frac{111-27}{12}=7\)। (195=27+(5r-5)7) से (5r=29) इसलिए \(r=\frac{29}{5}\)। / \(d=\frac{111-27}{12}=7\). From (195=27+(5r-5)7), (5r=29), so \(r=\frac{29}{5}\).
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If \(a_n=4n+9\) and \(a_m=137\), what is \(a_{m+15}\)?
C. (197)
(4m+9=137) से (m=32)। \(a_{47}=4\times47+9=197\)। / From (4m+9=137), (m=32). \(a_{47}=4\times47+9=197\).
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In an AP, \(a_p=42\), \(a_{p+12}=150\), and (p=9). What is \(a_{40}\)?
A. (321)
(12d=108) से (d=9)। \(a_{40}=a_9+31d=42+279=321\)। / From (12d=108), (d=9). \(a_{40}=a_9+31d=42+279=321\).
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In an AP, \(a_{4m}=156\), \(a_m=48\), and (d=6). What is the value of (m)?
B. (6)
\(a_{4m}-a_m=3md=108\) है। (18m=108) से (m=6)। / \(a_{4m}-a_m=3md=108\). From (18m=108), (m=6).
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If \(a_n=13n+c\) and \(a_6=101\), what is (r) when \(a_{4r}=465\)?
B. (9)
(101=78+c) से (c=23)। (465=52r+23) से \(r=\frac{442}{52}=8.5\)। / From (101=78+c), (c=23). (465=52r+23), so \(r=\frac{442}{52}=8.5\).
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If \(a_n=9n+2\), what is (k) for \(a_{3k}-a_k=144\)?
C. (8)
(a_{3k}-a_k=(27k+2)-(9k+2)=18k)। (18k=144) से (k=8)। / (a_{3k}-a_k=(27k+2)-(9k+2)=18k). From (18k=144), (k=8).
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In an AP, \(a_{4n}=140\), \(a_n=32\), and (d=6). What is (n)?
B. (6)
\(a_{4n}-a_n=3nd=108\)। (18n=108) से (n=6)। / \(a_{4n}-a_n=3nd=108\). From (18n=108), (n=6).
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If \(a_1=5\), (d=7), and \(a_{2n+3}=159\), what is the value of (n)?
B. (10)
(159=5+(2n+2)7) से (154=14n+14)। इसलिए (n=10)। / From (159=5+(2n+2)7), (154=14n+14). Therefore (n=10).
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In an AP, \(a_4=18\) and \(a_{16}=78\). If \(a_{4r}=138\), what is (r)?
A. (7)
\(d=\frac{78-18}{12}=5\)। (138=18+(4r-4)5) से (4r=28) इसलिए (r=7)। / \(d=\frac{78-18}{12}=5\). From (138=18+(4r-4)5), (4r=28), so (r=7).
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If \(a_n=3n+4\) and \(a_m=91\), what is \(a_{m+12}\)?
C. (127)
(3m+4=91) से (m=29)। \(a_{41}=3\times41+4=127\)। / From (3m+4=91), (m=29). \(a_{41}=3\times41+4=127\).
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In an AP, \(a_p=29\), \(a_{p+10}=99\), and (p=8). What is \(a_{35}\)?
A. (218)
(10d=70) से (d=7)। \(a_{35}=a_8+27d=29+189=218\)। / From (10d=70), (d=7). \(a_{35}=a_8+27d=29+189=218\).
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In an AP, \(a_{3m}=94\), \(a_m=34\), and (d=5). What is the value of (m)?
B. (6)
\(a_{3m}-a_m=2md=60\) है। (10m=60) से (m=6)। / \(a_{3m}-a_m=2md=60\). From (10m=60), (m=6).
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If \(a_n=11n+c\) and \(a_5=72\), what is (r) when \(a_{5r}=512\)?
B. (9)
(72=55+c) से (c=17)। (512=55r+17) से (r=9)। / From (72=55+c), (c=17). From (512=55r+17), (r=9).
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If \(a_n=7n-4\), what is (k) for \(a_{2k}-a_k=84\)?
C. (12)
(a_{2k}-a_k=(14k-4)-(7k-4)=7k)। (7k=84) से (k=12)। / (a_{2k}-a_k=(14k-4)-(7k-4)=7k). From (7k=84), (k=12).
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In an AP, \(a_{3n}=81\), \(a_n=21\), and (d=5). What is (n)?
B. (6)
\(a_{3n}-a_n=2nd=60\)। (10n=60) से (n=6)। / \(a_{3n}-a_n=2nd=60\). From (10n=60), (n=6).
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If \(a_1=3\), \(a_{2n+1}=123\), and (d=4), what is the value of (n)?
B. (15)
\(123=3+2n\cdot4\) से (120=8n) इसलिए (n=15)। सूचकांक (2n+1) में (1) घटाकर गुणक लें। / From \(123=3+2n\cdot4\), (120=8n), so (n=15). In index (2n+1), subtract (1) to get the multiplier.
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In an AP, \(a_3=11\) and \(a_{13}=51\). If \(a_{3r}=91\), what is (r)?
C. (7)
\(d=\frac{51-11}{10}=4\)। (91=11+(3r-3)4) से (3r=21) इसलिए (r=7)। / \(d=\frac{51-11}{10}=4\). From (91=11+(3r-3)4), (3r=21), so (r=7).
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If \(a_n=2n-1\) and \(a_m=63\), what is \(a_{m+10}\)?
B. (83)
(2m-1=63) से (m=32)। \(a_{42}=2\times42-1=83\)। / From (2m-1=63), (m=32). \(a_{42}=2\times42-1=83\).
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In an AP, \(a_p=17\), \(a_{p+8}=65\), and (p=6). What is \(a_{30}\)?
B. (161)
(8d=48) से (d=6)। \(a_{30}=a_6+24d=17+144=161\)। / From (8d=48), (d=6). \(a_{30}=a_6+24d=17+144=161\).
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