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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 551/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{k=r}^{n}{}^{k}C_r={}^{n+1}C_{r+1}\) किस counting idea से सिद्ध होती है?

The identity \(\sum_{k=r}^{n}{}^{k}C_r={}^{n+1}C_{r+1}\) is proved by which counting idea?

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A. सबसे बड़े चुने हुए element को fix करनाFixing the largest chosen element

Explanation

Simple Explanation

यह hockey-stick identity है और सबसे बड़ा selected element cases बनाता है। परीक्षा में ऐसी staircase sums में last element method लगाएं। / This is the hockey-stick identity and the largest selected element creates cases. In exams use the last-element method for such staircase sums.

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Question 552/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{k=0}^{r}{}^{m+k}C_k={}^{m+r+1}C_r\) को पहचानने का सबसे अच्छा तरीका क्या है?

What is the best way to recognize \(\sum_{k=0}^{r}{}^{m+k}C_k={}^{m+r+1}C_r\)?

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Correct Answer

B. इसे hockey-stick identity का shifted रूप माननाTreat it as a shifted form of the hockey-stick identity

Explanation

Simple Explanation

Upper और lower indices साथ बढ़ रहे हैं इसलिए यह diagonal sum है। परीक्षा में diagonal combination sum दिखे तो hockey-stick सोचें। / The upper and lower indices increase together so it is a diagonal sum. In exams think of hockey-stick when a diagonal combination sum appears.

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Question 553/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{r=0}^{n}{}^{n}C_r{}^{r}C_2\) का combinatorial अर्थ क्या है?

What is the combinatorial meaning of \(\sum_{r=0}^{n}{}^{n}C_r{}^{r}C_2\)?

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B. एक subset चुनकर उसमें unordered marked pair चुननाChoosing a subset and selecting an unordered marked pair inside it

Explanation

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पहले marked pair चुनें और बाकी elements freely choose करें। परीक्षा में \({}^{r}C_2\) दिखे तो pair marking सोचें। / Choose the marked pair first and choose the remaining elements freely. In exams think of pair marking when \({}^{r}C_2\) appears.

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Question 554/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{r=0}^{n}{}^{n}C_r{}^{r}C_3\) का simplified form कौन-सा है?

What is the simplified form of \(\sum_{r=0}^{n}{}^{n}C_r{}^{r}C_3\)?

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Correct Answer

C. \(^{n}C_3 2^{n-3}\)

Explanation

Simple Explanation

पहले (3) marked members चुनते हैं और बाकी (n-3) members freely चुने जाते हैं। परीक्षा में inner combination को पहले count करें। / Choose the (3) marked members first and freely choose the remaining (n-3) members. In exams count the inner combination first.

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Question 555/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{r=0}^{n}r{}^{n}C_r{}^{r}C_2\) में \(r{}^{r}C_2\) को किस रूप में बदलना सबसे उपयोगी है?

In \(\sum_{r=0}^{n}r{}^{n}C_r{}^{r}C_2\), what is the most useful form of \(r{}^{r}C_2\)?

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Correct Answer

D. \(2{}^{r}C_2+3{}^{r}C_3\)

Explanation

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पहले pair mark हो और फिर एक member mark हो तो cases pair के अंदर या बाहर बनते हैं। परीक्षा में product of marks को cases में तोड़ें। / If a pair is marked and then one member is marked, cases occur inside or outside the pair. In exams split products of marks into cases.

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Question 556/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\({}^{n}C_a{}^{a}C_b{}^{b}C_c\) को सही order बदलकर कैसे लिखा जा सकता है?

How can \({}^{n}C_a{}^{a}C_b{}^{b}C_c\) be written by changing the order correctly?

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Correct Answer

A. \(^{n}C_c{}^{n-c}C_{b-c}{}^{n-b}C_{a-b}\)

Explanation

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सबसे अंदर के (c) elements पहले चुनें और फिर layers जोड़ें। परीक्षा में nested choices को अंदर से बाहर count करें। / Choose the innermost (c) elements first and then add layers. In exams count nested choices from inside to outside.

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Question 557/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\({}^{n}C_r{}^{r}C_s\) को \({}^{n}C_s{}^{n-s}C_{r-s}\) लिखने में कौन-सा शर्त जरूरी है?

What condition is necessary to write \({}^{n}C_r{}^{r}C_s\) as \({}^{n}C_s{}^{n-s}C_{r-s}\)?

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Correct Answer

A. \(s\leq r\leq n\)

Explanation

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छोटा selected set बड़े selected set के अंदर होना चाहिए। परीक्षा में nested combination में valid index order पहले check करें। / The smaller selected set must lie inside the larger selected set. In exams first check valid index order in nested combinations.

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Question 558/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(\sum_{r=0}^{n}(-1)^r r{}^{n}C_r) का मान (n>1) के लिए क्या है?

What is the value of (\sum_{r=0}^{n}(-1)^r r{}^{n}C_r) for (n>1)?

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Correct Answer

B. (0)

Explanation

Simple Explanation

((1+x)^n) को differentiate करके (x=-1) रखने पर यह zero मिलता है। परीक्षा में alternating weighted sums में derivative plus (x=-1) लगाएं। / Differentiate ((1+x)^n) and put (x=-1) to get zero. In exams use derivative plus (x=-1) for alternating weighted sums.

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Question 559/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(\sum_{r=0}^{n}(-1)^r r(r-1){}^{n}C_r) का मान (n>2) के लिए क्या होगा?

What is (\sum_{r=0}^{n}(-1)^r r(r-1){}^{n}C_r) for (n>2)?

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Correct Answer

C. (0)

Explanation

Simple Explanation

Second derivative के बाद (x=-1) रखने पर ((1-1)^{n-2}) आता है। परीक्षा में degree से कम falling factor हो तो zero check करें। / After the second derivative and putting (x=-1), ((1-1)^{n-2}) appears. In exams check zero when the falling factor is below the degree.

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Question 560/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{r=0}^{n}r^2{}^{n}C_r\) को derive करने में \(r^2\) का कौन-सा split सही है?

Which split of \(r^2\) is correct for deriving \(\sum_{r=0}^{n}r^2{}^{n}C_r\)?

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Correct Answer

A. (r-2=r(r-1)+r)

Explanation

Simple Explanation

यह split standard first और second weighted sums जोड़ देता है। परीक्षा में powers को falling factorials में बदलें। / This split connects first and second weighted sums. In exams convert powers into falling factorials.

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