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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 261/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

((1+x)^{11}) में \(x^5\) और \(x^6\) के coefficients क्यों बराबर हैं?

Why are the coefficients of \(x^5\) and \(x^6\) in ((1+x)^{11}) equal?

Explanation opens after your attempt
Correct Answer

A. क्योंकि (5+6=11)Because (5+6=11)

Explanation

Simple Explanation

Coefficients \(^{11}C_5\) और \(^{11}C_6\) हैं जो पूरक indices हैं। परीक्षा में binomial symmetry पहचानें। / The coefficients are \(^{11}C_5\) and \(^{11}C_6\), which have complementary indices. In exams identify binomial symmetry.

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Question 262/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{n}C_1+^{n}C_3+^{n}C_5+\cdots\) का योग किसके बराबर होता है?

The sum \(^{n}C_1+^{n}C_3+^{n}C_5+\cdots\) is equal to what?

Explanation opens after your attempt
Correct Answer

C. \(2^{n-1}\)

Explanation

Simple Explanation

Odd indexed combinations का योग even indexed sum के बराबर \(2^{n-1}\) होता है। परीक्षा में ((1-1)^n) से alternating identity याद रखें। / The sum of odd indexed combinations equals the even indexed sum, \(2^{n-1}\). In exams remember the alternating identity from ((1-1)^n).

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Question 263/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{n}P_r\) के product form में (r=6) हो तो अंतिम factor कौन-सा होगा?

In the product form of \(^{n}P_r\), if (r=6), what is the last factor?

Explanation opens after your attempt
Correct Answer

A. (n-5)

Explanation

Simple Explanation

अंतिम factor (n-r+1) होता है, इसलिए (r=6) पर (n-5) मिलेगा। परीक्षा में (+1) को न भूलें। / The last factor is (n-r+1), so for (r=6) it is (n-5). In exams do not forget the (+1).

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Question 264/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{13}P_6\) के product form में कुल कितने गुणनखंड होंगे?

How many factors are there in the product form of \(^{13}P_6\)?

Explanation opens after your attempt
Correct Answer

C. (6)

Explanation

Simple Explanation

(r=6) स्थान भरने हैं इसलिए (6) गुणनखंड होंगे। परीक्षा में factors की संख्या को positions से जोड़ें। / There are (r=6) positions to fill, so there are (6) factors. In exams connect the number of factors with positions.

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Question 265/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{n}C_2=105\) हो, तो (n) का मान क्या है?

If \(^{n}C_2=105\), what is the value of (n)?

Explanation opens after your attempt
Correct Answer

D. (15)

Explanation

Simple Explanation

(^{n}C_2=\frac{n(n-1)}{2}) और \(\frac{15\times14}{2}=105\) है। परीक्षा में pair formula से (n) पहचानें। / (^{n}C_2=\frac{n(n-1)}{2}), and \(\frac{15\times14}{2}=105\). In exams identify (n) using the pair formula.

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Question 266/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{n}P_2=210\) हो, तो (n) का मान क्या होगा?

If \(^{n}P_2=210\), what will be the value of (n)?

Explanation opens after your attempt
Correct Answer

B. (15)

Explanation

Simple Explanation

(^{n}P_2=n(n-1)) और \(15\times14=210\) है। परीक्षा में ordered pair count में (2) से भाग न दें। / (^{n}P_2=n(n-1)), and \(15\times14=210\). In exams do not divide by (2) in ordered pair count.

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Question 267/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{n}C_5\) के simplified रूप में denominator कौन-सा factorial देता है?

Which factorial gives the denominator in the simplified form of \(^{n}C_5\)?

Explanation opens after your attempt
Correct Answer

C. (5!)

Explanation

Simple Explanation

(^{n}C_5=\frac{n(n-1)(n-2)(n-3)(n-4)}{5!}) होता है। परीक्षा में \(^{n}C_r\) के denominator में (r!) रखें। / (^{n}C_5=\frac{n(n-1)(n-2)(n-3)(n-4)}{5!}). In exams keep (r!) in the denominator of \(^{n}C_r\).

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Question 268/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{n}C_5\) के numerator में कितने लगातार घटते factors आते हैं?

How many consecutive decreasing factors appear in the numerator of \(^{n}C_5\)?

Explanation opens after your attempt
Correct Answer

C. (5)

Explanation

Simple Explanation

((n-5)!) कटने के बाद (n) से (n-4) तक कुल (5) factors बचते हैं। परीक्षा में numerator factors की संख्या (r) होती है। / After cancelling ((n-5)!), there are (5) factors from (n) to (n-4). In exams the number of numerator factors is (r).

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Question 269/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि (10) लोगों में से (5) की समिति और उसी समिति से (1) संयोजक चुनना हो, तो expression क्या होगा?

If a committee of (5) is formed from (10) people and then (1) coordinator is chosen from that committee, what is the expression?

Explanation opens after your attempt
Correct Answer

C. \(^{10}C_5\times5\)

Explanation

Simple Explanation

पहले committee के \(^{10}C_5\) तरीके हैं और संयोजक के (5) विकल्प हैं। परीक्षा में selection के बाद role हो तो multiply करें। / First there are \(^{10}C_5\) ways to form the committee and (5) choices for coordinator. In exams multiply when a role follows selection.

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Question 270/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{10}C_5\times5\) को permutation form में कैसे लिख सकते हैं?

How can \(^{10}C_5\times5\) be written in permutation form?

Explanation opens after your attempt
Correct Answer

A. \(^{10}P_5\div4!\)

Explanation

Simple Explanation

एक व्यक्ति का role अलग है और बाकी (4) सदस्यों का क्रम महत्वहीन है। परीक्षा में special role होने पर remaining members का overcount हटाएँ। / One person has a distinct role and the order of the remaining (4) members is irrelevant. In exams remove overcounting of remaining members when there is a special role.

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