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QUESTION: 1

The sum of coefficients of (1 + x - 3x^{2})^{2134} is

Solution:

For the sum of coefficient, put x = 1, to obtain the sum is (1 + 1 - 3)^{2134} = 1

QUESTION: 2

The sum ^{r}C_{r} + ^{r+1}C_{r} + ^{r+2}C_{r} + .....+ ^{n}C_{r} (n __>__ r) equals

Solution:

C(n, r) + c(n -1, r) + C(n - 2, r) + ... + C(r, r)

= ^{r+1}C_{r+1} + ^{r+1}C_{r} + ^{r+2}C_{r} + .... + ^{n-1}C_{r} + ^{n}C_{r}

= ^{n+1}C_{r+1} (applying same rule again and again ) (∴ ^{n}C_{r} + ^{n}C_{r-1} = ^{n+1}C_{r})

QUESTION: 3

The expansion [x^{2} + (x^{6} - 1)^{1/2}]^{5} + [x^{2} -(x^{6} - 1)^{1/2}]^{5} is a polynomial of degree

Solution:

Here last term is of 14 degree.

QUESTION: 4

The term independent of x in

Solution:

The general term

The term independent of x, (or the constant term) corresponds to x^{18-3r} being x^{0} or 18 - 3r = 0 ⇒ r = 6 .

QUESTION: 5

The value of the greatest term in the expansion of

Solution:

Hence, t_{8} is the greatest term and its value is

QUESTION: 6

9^{n+1} - 8n- 9 is divisible by

Solution:

QUESTION: 7

The first integral term in the expansion of

Solution:

For first integral term for r = 3;

QUESTION: 8

The number of irrational terms in the expansion of (2^{1/5 }+3^{1/10})^{55} is

Solution:

(2^{1/5} + ^{31/10})^{55}

Total terms = 55 + 1 = 56

Here r = 0, 10, 20, 30, 40, 50

Number of rational terms = 6;

Number of irrational terms = 56 - 6 = 50

QUESTION: 9

The number of terms in the expansion of (2x + 3y- 4z)^{n} is

Solution:

We have, (2x + 3y - 4z)^{n} = {2 + (3 - 4)}^{n}

Clearly, the first term in the above expansion gives one term, second term gives two terms, third term gives three terms and so on.

So, Total number of term = 1 +2+3+...+n+(n+1) =

QUESTION: 10

In the expansion of the coefficient of x^{-10} will be

Solution:

Given expansion is

∴ General term

Since, we have to find coefficient of x^{-10 }∴ -12 + 2r = -10 ⇒ r = 1

Now, then coefficient of x^{-10} is ^{12}C_{1}(a)^{11}(b)^{1} = 12a^{11}b

QUESTION: 11

If (1 + ax)^{n} = 1 + 8x + 24x^{2} + ….., then the values of a and n are equal to

Solution:

∴ n = 4,a = 2

QUESTION: 12

The product of middle terms in the expansion of is equal to

Solution:

it has 12 terms in it’s expansion , so there are two middle terms (6th and 7th);

QUESTION: 13

The middle term in the expansion of (1 – 2x + x^{2})^{n} is

Solution:

Here 2n is even integer, therefore, term will be the middle term.

Now, (n + 1)^{th} term in (1 - x)^{2n}

QUESTION: 14

The sum of the binomial coefficients in the expansion of (x^{-3/4} + ax^{5/4})^{n} lies between 200 and 400 and the term independent of x equals 448. The value of a is

Solution:

QUESTION: 15

^{23}C_{0} + ^{23}C_{2} + ^{23}C_{4} + ... ^{23}C_{22} equals

Solution:

Given sum = sum of odd terms

QUESTION: 16

Solution:

QUESTION: 17

The greatest coefficient in the expansion of (1 + x)^{2n + 2} is

Solution:

Here 2n + 2 is even

Greatest coefficient

QUESTION: 18

(^{n}C_{0})^{2} + (^{n}C_{1})^{2} + (^{n}C_{2})^{2} + .....+ (^{n}C_{n})^{2} equals

Solution:

Multiply (i) and (ii) and consider the coefficient x n of both sides, we have

QUESTION: 19

The value of C_{1} + 3C_{3} + 5C_{5} + 7C_{7} + ...., where C_{0}, C_{3}, C_{5}, C_{7},..... are binomial coefficients is

Solution:

QUESTION: 20

Fractional part of

Solution:

2^{78} = 8 + an integer multiple of 31;

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