Miscellaneous Examples

NCERT solutions for class 11 Maths Miscellaneous Exercise have five questions crafted to push a student to his limits. These are complex sums and require kids to devote their complete concentration. These problems aim to test the foundational understanding of a child’s concepts. Additionally, it teaches kids how to apply the fundamental theories to higher-level sums.

Miscellaneous Examples

1. From the walking sets, select a set that is a subset of one and the other: 

A = {x: x ∈ R and x satisfy x2 8x + 12 = 0} 

B = {2, 4, 6}

C = {2, 4, 6, 8…}

D = { 6}

A = {x: x ∈ R, where x satisfies x2 8x + 12 = 0} 

2 and 6 are the basics of x28 x + 12 = 0, as shown in response to the request. 

Approach:

A = {2, 6} 

B = {2, 4, 6}

C = {2, 4, 6, 8…}

D = {6}  

SoD ⊂ A ⊂ B ⊂ C

SoA ⊂ B, A ⊂ C, B⊂C, D⊂A, D⊂B, D⊂C

2. Determine if the attestation is legitimate or deceiving. If it is legitimate, show it. If it is counterfeit, give a model: 

1. If x ∈ An and A ∈ B, then x ∈ B 

2. If A ⊂ B and B ∈ C, then, A ∈ C 

3. If A ⊂ B and B ⊂ C, then, A ⊂ C 

4. If A ⊄ B and B ⊄ C, then, A ⊄ C 

5. If x ∈ An and A ⊄ B, then x ∈ B 

6. If A ⊂ B and x ∉ B, then, x ∉ A 

Solution:

1. False 

Let: 

A = {1, 2} and B = {1, {1, 2}, {3}} 

We have: 

2 ∈ {1, 2} and {1, 2} ∈ {1, {1, 2}, {3}} 

Consequently, we get: 

A ∈ B 

We know: 

{2} ∉ {1, {1, 2}, {3}} 

2. False 

Let: 

A {2} 

B = {0, 2} 

Besides: 

C = {1, {0, 2}, 3} 

From the request: 

A ⊂ B 

Along these lines: 

B ∈ C 

We know: 

A ∉ C 

3. True 

Let: 

A ⊂ B and B ⊂ C 

If: 

x ∈ A 

Then, we have: 

x ∈ B 

Besides: 

x ∈ C 

Thus: 

A ⊂ C 

4. False 

Let: 

A ⊄ B 

As well: 

B ⊄ C 

If:

A = {1, 2} 

B = {0, 6, 8} 

Besides: 

C = {0, 1, 2, 6, 9} 

∴ A ⊂ C 

5. False 

Let: 

x ∈ A 

Also: 

A ⊄ B 

If:

A = {3, 5, 7} 

As well: 

B = {3, 4, 6} 

We know that: 

A ⊄ B 

∴ 5 ∉ B 

6. True 

Let: 

A ⊂ B 

Also: 

x ∉ B 

If:

x ∈ A, 

We have: 

x ∈ B 

From the request: 

We have x ∉ B 

∴ x ∉ A

3) How many words can be formed by using all the word ‘equation’ letters so that the vowels and consonants are displayed together?

The word ‘equation’ has five vowels, A, E, I, O, and U, and three consonants, Q, T, and N. Each vowel and consonant must occur simultaneously, so they are accepted as separate items (AEIOU) and (QTN). Then, the stages at these two points run at the same time. This number is ²P, ² = 2! Five related to each of these stages and five vowel levels are taken at once. The orders of all three consonants are acquired at the same time. Then, by increasing the standard, the required number of words is = 2! x 51 x 3! = 1440.

4) From the letters of the word ‘daughter’, how many words can each of the two vowels and the three consonants form?

The word ‘daughter’ has exactly three vowels, A, U, and E. Five consonants, D, G, H, T, and R. The number of approaches to select two vowels from 3 vowels is = ³ C₂ = 3 2. The number of approaches to select three consonants from 5 consonants is = $ C₁ = 10. Therefore, the number of mixed two vowels and three consonants is = 3 × 10 = 30. You can set these 30 mixes of 3 vowels and three consonants at 5. Therefore, the number of different words needed is = 30 x 5! = 3600. 

5) If the different permutations of all the letters of the word EXAMINATION are listed in a dictionary, how many words are there in this list before the first word starting with E? 

Given the word, ‘examination’ has 11 letters, of which A, I, and N appear twice, and various other letters appear only once. The words that will be recorded before the words beginning with the letter E in a list will be those beginning with An. To get the number of words beginning with A, the letter An is fixed at the super left position, after which the other ten letters are taken all at once.  Since there are 2 Is and 2 Ns in the remaining ten letters, 10! Several words beginning with A = = 907200 2!2! 

Along these lines, the expected quantities of words are 907200.

Conclusion 

The inquiries in various activities are significant, according to the assessment perspective. They have been asked in numerous past tests, so doing those is likewise significant. We have also included some miscellaneous concepts and miscellaneous properties in FAQs for better understanding and revising of the concepts well before exams.