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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 591/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(\(1+x+x^2\)^n) में \(x^4\) coefficient के cases में कौन-सा option सही है?

Which option correctly lists cases for the coefficient of \(x^4\) in (\(1+x+x^2\)^n)?

Explanation opens after your attempt
Correct Answer

A. चार (x), या दो (x) और एक \(x^2\), या दो \(x^2\)Four (x)'s, or two (x)'s and one \(x^2\), or two \(x^2\)'s

Explanation

Simple Explanation

Exponent (4) बनाने वाले सभी disjoint choices जोड़ने पड़ते हैं। परीक्षा में polynomial coefficient में exponent partitions बनाएं। / All disjoint choices that form exponent (4) must be added. In exams make exponent partitions for polynomial coefficients.

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Question 592/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

((1+x)^n) में indices \(0,3,6,\ldots\) वाले coefficients अलग करने के लिए कौन-सा method use होता है?

Which method is used to separate coefficients with indices \(0,3,6,\ldots\) in ((1+x)^n)?

Explanation opens after your attempt
Correct Answer

A. Roots of unity filter

Explanation

Simple Explanation

Modulo (3) classes अलग करने के लिए cube roots of unity filter उपयोगी है। परीक्षा में three-step coefficient sums को advanced filter से पहचानें। / The cube roots of unity filter is useful for separating modulo (3) classes. In exams identify three-step coefficient sums with an advanced filter.

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Question 593/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

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

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

Explanation opens after your attempt
Correct Answer

A. (n-r>r+1)

Explanation

Simple Explanation

Ratio \(\frac{n-r}{r+1}\) को (1) से बड़ा होना चाहिए। परीक्षा में increasing region ratio से identify करें। / The ratio \(\frac{n-r}{r+1}\) must be greater than (1). In exams identify the increasing region by ratio.

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Question 594/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

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

Consecutive ratio (1) से कम होने पर coefficients घटने लगते हैं। परीक्षा में peak के बाद inequality उलटी हो जाती है। / When the consecutive ratio is less than (1), the coefficients start decreasing. In exams the inequality reverses after the peak.

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Question 595/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \({}^{n}C_{r}= {}^{n}C_{r+6}\) और indices unequal हैं, तो relation कौन-सा है?

If \({}^{n}C_{r}= {}^{n}C_{r+6}\) and the indices are unequal, which relation is correct?

Explanation opens after your attempt
Correct Answer

A. (2r+6=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 596/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \({}^{30}C_{3r-2}={}^{30}C_{r+8}\) और lower indices unequal हैं, तो (r) क्या है?

If \({}^{30}C_{3r-2}={}^{30}C_{r+8}\) and the lower indices are unequal, what is (r)?

Explanation opens after your attempt
Correct Answer

A. (6)

Explanation

Simple Explanation

Complementary condition से (3r-2+r+8=30), इसलिए (r=6)। परीक्षा में same-index और complement cases अलग solve करें। / The complementary condition gives (3r-2+r+8=30), so (r=6). In exams solve same-index and complement cases separately.

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Question 597/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \({}^{n}P_5=15{}^{n}P_4\), तो (n) का मान क्या होगा?

If \({}^{n}P_5=15{}^{n}P_4\), what is the value of (n)?

Explanation opens after your attempt
Correct Answer

A. (19)

Explanation

Simple Explanation

({}^{n}P_5=(n-4){}^{n}P_4), इसलिए (n-4=15)। परीक्षा में consecutive permutation relation सीधे लगाएं। / ({}^{n}P_5=(n-4){}^{n}P_4), so (n-4=15). In exams apply the consecutive permutation relation directly.

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Question 598/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{4}{5}\), तो कौन-सा relation सही है?

If \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{4}{5}\), which relation is correct?

Explanation opens after your attempt
Correct Answer

A. (5n-9r=4)

Explanation

Simple Explanation

Ratio \(\frac{n-r}{r+1}=\frac{4}{5}\) से (5n-5r=4r+4) मिलता है। परीक्षा में combination ratio को cross multiply करें। / The ratio \(\frac{n-r}{r+1}=\frac{4}{5}\) gives (5n-5r=4r+4). In exams cross-multiply combination ratios.

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Question 599/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}}=\frac{7}{3}\), तो कौन-सा relation बनेगा?

If \(\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}}=\frac{7}{3}\), which relation is formed?

Explanation opens after your attempt
Correct Answer

A. (3n-10r+3=0)

Explanation

Simple Explanation

\(\frac{{}^{n}C_r}{{}^{n}C_{r-1}}=\frac{n-r+1}{r}\) होता है और cross multiplication से relation मिलता है। परीक्षा में ratio direction सही रखें। / \(\frac{{}^{n}C_r}{{}^{n}C_{r-1}}=\frac{n-r+1}{r}\), and cross multiplication gives the relation. In exams keep the ratio direction correct.

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Question 600/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(12) candidates में से (5) की team बनानी है और (A,B,C) में से कम से कम (2) selected हों। Count कौन-सी है?

A team of (5) is formed from (12) candidates and at least (2) of (A,B,C) are selected. Which count is correct?

Explanation opens after your attempt
Correct Answer

A. \(^{3}C_2{}^{9}C_3+{}^{3}C_3{}^{9}C_2\)

Explanation

Simple Explanation

At least (2) special के cases exactly (2) और exactly (3) हैं। परीक्षा में small special group में direct cases साफ रहते हैं। / The cases for at least (2) special are exactly (2) and exactly (3). In exams direct cases are clear for a small special group.

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