Class 12 Math Important Questions

Combination & Permutation

1. If nC10 = nC12 determine nC2.
2. How many three different digit numbers less than 500 can be formed from the integers 1,2,3,4,5,6,7?
3. In how many ways can 4 letters be posted in six letter boxes?
4. How many different 4-digit numbers can be formed from the digits 2,3,4,5,6? How many of them are divisible by 5? How many of these are not divisible by 5?
5. In how many ways can the letters of the word 'COMPUTER' be arranged so that:
   (a) all vowels are always together?
   (b) the relative positions of the vowels and consonants are not changed?
   (c) the vowels may occupy only the odd positions?
6. From 6 gentlemen and 4 ladies a committee of 5 is to be formed. In how many ways can this be done so as to include at least 3 ladies?
7. In an examination, a candidate has to pass in each of the 4 subjects. In how many ways can he fail?
8. A person has got 12 acquaintances of whom 8 are relatives. In how many ways can he invite 7 guests so that 5 of them may be relatives?
9. From 10 persons, in how many ways can a selection of 4 be made:
   (a) when one particular person is always included?
   (b) when two particular persons are excluded?

Language Based Questions

1. Write the relation between Combination & Permutation.
2. Write a difference between Combination & Permutation.
3. Write the formula for circular permutation when clockwise & anticlockwise arrangements are different.
4. Write the formula for circular permutation when clockwise & anticlockwise arrangements are same.

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Binomial Theorem & Log & Exponential Series

1. Find the term independent of x in the expansion of \( \left( \frac{x}{3} + \frac{\sqrt{3}}{2x^2} \right)^{10} \).
2. Find the number of terms in the expansion of \( (1 + 2x + x^2)^{11} \).
3. In the expansion of \( (1 + x)^{43} \), coefficient of (2r + 1)th term and (r + 2)th term are equal. Find r.
4. Write the expansion of \( \log_e(1 + x) \).
5. \( C_0C_n + C_1C_{n-1} + \cdots + C_nC_0 = \frac{2n!}{n!n!} \)
6. Show that the middle term in the expansion of \( (1 + x)^{2n} \) is \( \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{n!} 2^n x^n \)
7. If the three consecutive coefficients in the expansion of \( (1 + x)^n \) be 165, 330, 462; find n.
8. \( \frac{1}{1!} + \frac{4}{3!} + \frac{6}{4!} + \cdots = e \)
9. \( 1 + \frac{1+2}{2!} + \frac{1+2+3}{3!} + \frac{1+2+3+4}{4!} + \cdots = \frac{3e}{2} \)
10. Show that \( \sum_{n=1}^{\infty} \frac{n^2}{(n+1)!} = e - 1 \)
11. Prove that: \( \frac{1}{2 \cdot 3} + \frac{1}{4 \cdot 5} + \frac{1}{6 \cdot 7} + \cdots = \log_e 2 \)
12. \( \frac{1}{1 \cdot 2} + \frac{1}{3 \cdot 4} + \frac{1}{5 \cdot 6} + \cdots = 1 - \log_e 2 \)
13. If \( y = \frac{x^2}{2} + \frac{x^3}{3} + \cdots \), show that \( x = y + \frac{y^2}{2} + \frac{y^3}{3} + \cdots \)

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