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

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Derivations of formulas and their connections Practice Questions

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Question 491/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

((x+y+z)8) में \(x^3y^2z^3\) का coefficient कौन-सा है?

What is the coefficient of \(x^3y^2z^3\) in ((x+y+z)8)?

Explanation opens after your attempt
Correct Answer

A. \(\frac{8!}{3!2!3!}\)

Explanation

Simple Explanation

Exponents का sum (8) है और coefficient multinomial form से मिलता है। परीक्षा में powers को group sizes मानें। / The exponents sum to (8), and the coefficient comes from the multinomial form. In exams treat powers as group sizes.

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Question 492/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

((1+x)^n) के even-index coefficients का sum कैसे derive होता है?

How is the sum of even-index coefficients of ((1+x)^n) derived?

Explanation opens after your attempt
Correct Answer

A. ((1+1)^n) और ((1-1)^n) को जोड़करBy adding ((1+1)^n) and ((1-1)^n)

Explanation

Simple Explanation

जोड़ने पर odd-index terms cancel हो जाते हैं। परीक्षा में even-odd binomial sums में (x=1) और (x=-1) साथ प्रयोग करें। / On adding, odd-index terms cancel out. In exams use (x=1) and (x=-1) together for even-odd binomial sums.

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Question 493/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\({}^{n}C_0+{}^{n}C_3+{}^{n}C_6+\cdots\) जैसे sums को अलग करने के लिए कौन-सा advanced method उपयोगी है?

Which advanced method is useful for separating sums like \({}^{n}C_0+{}^{n}C_3+{}^{n}C_6+\cdots\)?

Explanation opens after your attempt
Correct Answer

A. Roots of unity filter

Explanation

Simple Explanation

Indices modulo (3) अलग करने के लिए unity roots filter प्रयोग किया जाता है। परीक्षा में ऐसी sums को सामान्य even-odd method से अलग पहचानें। / The roots of unity filter separates indices modulo (3). In exams recognize that such sums differ from the usual even-odd method.

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Question 494/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि \(^{n}C_{r-2}:{}^{n}C_{r-1}:{}^{n}C_r\) दिए हों, तो formula derivation में कौन-सा ratio सबसे पहले उपयोगी होगा?

If \(^{n}C_{r-2}:{}^{n}C_{r-1}:{}^{n}C_r\) is given, which ratio is most useful first in the derivation?

Explanation opens after your attempt
Correct Answer

A. \(\frac{{}^{n}C_{r-1}}{{}^{n}C_{r-2}}=\frac{n-r+2}{r-1}\)

Explanation

Simple Explanation

Consecutive combinations में factorial cancellation से यह ratio मिलता है। परीक्षा में लंबी values निकालने के बजाय adjacent ratio लगाएं। / Factorial cancellation in consecutive combinations gives this ratio. In exams use adjacent ratios instead of calculating long values.

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Question 495/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि \(^{n}C_r\) maximum term है, तो \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}\) के लिए कौन-सी condition उपयोगी है?

If \(^{n}C_r\) is a maximum term, which condition is useful for \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}\)?

Explanation opens after your attempt
Correct Answer

A. Ratio (1) से अधिक होने पर sequence बढ़ती है और (1) से कम होने पर घटती हैThe sequence increases when the ratio is greater than (1) and decreases when it is less than (1)

Explanation

Simple Explanation

Maximum के पास increasing से decreasing transition होता है। परीक्षा में binomial coefficient peak को ratio से locate करें। / Near the maximum, the sequence transitions from increasing to decreasing. In exams locate the binomial coefficient peak by ratios.

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Question 496/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

जब (n) even हो, तो \({}^{n}C_r\) का unique maximum किस (r) पर होता है?

When (n) is even, at which (r) does \({}^{n}C_r\) have a unique maximum?

Explanation opens after your attempt
Correct Answer

A. \(r=\frac{n}{2}\)

Explanation

Simple Explanation

Even (n) में middle index single होता है। परीक्षा में symmetry और ratio से central term पहचानें। / For even (n), the middle index is single. In exams identify the central term by symmetry and ratio.

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Question 497/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

जब (n) odd हो, तो \({}^{n}C_r\) के maximum terms कौन-से होते हैं?

When (n) is odd, which terms are maximum for \({}^{n}C_r\)?

Explanation opens after your attempt
Correct Answer

A. \({}^{n}C_{\frac{n-1}{2}}\) और \({}^{n}C_{\frac{n+1}{2}}\)\({}^{n}C_{\frac{n-1}{2}}\) and \({}^{n}C_{\frac{n+1}{2}}\)

Explanation

Simple Explanation

Odd (n) में दो central complementary indices बराबर maximum देते हैं। परीक्षा में odd case में दो middle terms याद रखें। / For odd (n), the two central complementary indices give equal maxima. In exams remember two middle terms in the odd case.

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Question 498/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि \({}^{20}C_{3r-1}={}^{20}C_{r+5}\) और indices equal नहीं हैं, तो (r) क्या होगा?

If \({}^{20}C_{3r-1}={}^{20}C_{r+5}\) and the indices are not equal, what is (r)?

Explanation opens after your attempt
Correct Answer

B. (4)

Explanation

Simple Explanation

Unequal equal-combination indices complementary होते हैं, इसलिए (3r-1+r+5=20)। परीक्षा में lower indices का sum upper index के बराबर करें। / Unequal equal-combination indices are complementary, so (3r-1+r+5=20). In exams set the sum of lower indices equal to the upper index.

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Question 499/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि \({}^{n}P_3=10{}^{n}P_2\), तो (n) क्या होगा?

If \({}^{n}P_3=10{}^{n}P_2\), what is (n)?

Explanation opens after your attempt
Correct Answer

C. (12)

Explanation

Simple Explanation

({}^{n}P_3=(n-2){}^{n}P_2), इसलिए (n-2=10)। परीक्षा में consecutive permutation relation से जल्दी solve करें। / ({}^{n}P_3=(n-2){}^{n}P_2), so (n-2=10). In exams solve quickly using consecutive permutation relations.

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Question 500/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{2}{3}\), तो (n), (r) के बीच कौन-सा relation बनेगा?

If \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{2}{3}\), what relation is formed between (n) and (r)?

Explanation opens after your attempt
Correct Answer

A. (3n-5r=2)

Explanation

Simple Explanation

Ratio \(\frac{n-r}{r+1}=\frac{2}{3}\) से (3n-3r=2r+2) मिलता है। परीक्षा में consecutive combination ratios को cross multiply करें। / The ratio \(\frac{n-r}{r+1}=\frac{2}{3}\) gives (3n-3r=2r+2). In exams cross-multiply consecutive combination ratios.

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