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Mathematics Derivations of formulas and their connections MCQ Questions for Class 11 General

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Question 541/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\({}^{n}C_r\) के maximum term के लिए ratio test में कौन-सा ratio उपयोग होता है?

Which ratio is used in the ratio test for the maximum term of \({}^{n}C_r\)?

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Correct Answer

A. \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{n-r}{r+1}\)

Explanation

Simple Explanation

Consecutive binomial coefficients में यह ratio बढ़ने और घटने की दिशा बताता है। परीक्षा में peak खोजने के लिए ratio को (1) से compare करें। / This ratio shows the direction of increase and decrease in consecutive binomial coefficients. In exams compare the ratio with (1) to locate the peak.

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Question 542/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \({}^{n}C_{r+1}>{}^{n}C_r\), तो कौन-सी inequality सही है?

If \({}^{n}C_{r+1}>{}^{n}C_r\), which inequality is correct?

Explanation opens after your attempt
Correct Answer

A. (n-r>r+1)

Explanation

Simple Explanation

Ratio \(\frac{n-r}{r+1}>1\) होना चाहिए। परीक्षा में monotonicity के लिए consecutive ratio को (1) से compare करें। / The ratio \(\frac{n-r}{r+1}>1\) must hold. In exams compare consecutive ratios with (1) for monotonicity.

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Question 543/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \({}^{n}C_{r}= {}^{n}C_{r+4}\) और indices बराबर नहीं हैं, तो (r) और (n) का relation क्या है?

If \({}^{n}C_{r}= {}^{n}C_{r+4}\) and the indices are not equal, what is the relation between (r) and (n)?

Explanation opens after your attempt
Correct Answer

A. (2r+4=n)

Explanation

Simple Explanation

Unequal equal-combination indices complementary होते हैं। परीक्षा में lower indices का sum upper index के बराबर करें। / Unequal equal-combination indices are complementary. In exams set the sum of lower indices equal to the upper index.

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Question 544/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \({}^{24}C_{2r-1}={}^{24}C_{r+8}\) और lower indices unequal हैं, तो (r) क्या होगा?

If \({}^{24}C_{2r-1}={}^{24}C_{r+8}\) and the lower indices are unequal, what is (r)?

Explanation opens after your attempt
Correct Answer

B. (6)

Explanation

Simple Explanation

Complementary indices से (2r-1+r+8=24), इसलिए (r=6)। परीक्षा में equal combinations में same-index case और complement case अलग देखें। / Complementary indices give (2r-1+r+8=24), so (r=6). In exams check same-index and complement cases separately in equal combinations.

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Question 545/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \({}^{n}P_4=12{}^{n}P_3\), तो (n) का मान क्या है?

If \({}^{n}P_4=12{}^{n}P_3\), what is the value of (n)?

Explanation opens after your attempt
Correct Answer

B. (15)

Explanation

Simple Explanation

({}^{n}P_4=(n-3){}^{n}P_3), इसलिए (n-3=12)। परीक्षा में consecutive permutation relation सीधे लगाएं। / ({}^{n}P_4=(n-3){}^{n}P_3), so (n-3=12). In exams apply the consecutive permutation relation directly.

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Question 546/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{3}{4}\), तो कौन-सा relation सही है?

If \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{3}{4}\), which relation is correct?

Explanation opens after your attempt
Correct Answer

A. (4n-7r=3)

Explanation

Simple Explanation

\(\frac{n-r}{r+1}=\frac{3}{4}\) से (4n-4r=3r+3) मिलता है। परीक्षा में ratio equations को cross multiply करें। / From \(\frac{n-r}{r+1}=\frac{3}{4}\), we get (4n-4r=3r+3). In exams cross-multiply ratio equations.

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Question 547/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \(\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}}=\frac{5}{2}\), तो relation कौन-सा बनेगा?

If \(\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}}=\frac{5}{2}\), which relation is formed?

Explanation opens after your attempt
Correct Answer

A. (2n-7r+2=0)

Explanation

Simple Explanation

Ratio \(\frac{{}^{n}C_r}{{}^{n}C_{r-1}}=\frac{n-r+1}{r}\) है। परीक्षा में consecutive combination ratio का सही direction रखें। / The ratio is \(\frac{{}^{n}C_r}{{}^{n}C_{r-1}}=\frac{n-r+1}{r}\). In exams keep the direction of consecutive combination ratios correct.

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Question 548/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(n) distinct objects में से (r) चुनने पर कम से कम (2) special objects चाहिए, और special objects (s) हैं। Complement expression कौन-सा है?

When choosing (r) objects from (n) distinct objects, at least (2) special objects are required and there are (s) special objects. Which complement expression is correct?

Explanation opens after your attempt
Correct Answer

A. \(^{n}C_r-{}^{n-s}C_r-s{}^{n-s}C_{r-1}\)

Explanation

Simple Explanation

At least (2) special का complement (0) special या exactly (1) special है। परीक्षा में complement में सभी unwanted cases घटाएं। / The complement of at least (2) special is (0) special or exactly (1) special. In exams subtract all unwanted cases in the complement.

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Question 549/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(10) students में से (5) की committee बनानी है, (A) और (B) दोनों साथ या दोनों बाहर हों। Count क्या है?

A committee of (5) is formed from (10) students, and (A) and (B) are either both included or both excluded. What is the count?

Explanation opens after your attempt
Correct Answer

A. \(^{8}C_3+{}^{8}C_5\)

Explanation

Simple Explanation

Case (1): दोनों शामिल हों, case (2): दोनों बाहर हों। परीक्षा में paired restriction को दो disjoint cases में तोड़ें। / Case (1): both are included, case (2): both are excluded. In exams split paired restrictions into two disjoint cases.

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Question 550/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(9) players में से (4) चुनने हैं, (A) और (B) साथ चयनित न हों। Count कौन-सी है?

(4) players are chosen from (9) players, and (A) and (B) are not selected together. Which count is correct?

Explanation opens after your attempt
Correct Answer

A. \(^{9}C_4-{}^{7}C_2\)

Explanation

Simple Explanation

Total selections से (A,B) दोनों वाले selections घटते हैं। परीक्षा में not together selection में complement सबसे छोटा route है। / Subtract selections containing both (A,B) from total selections. In exams complement is the shortest route for not-together selection.

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