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1

### AIEEE 2003

If $$x$$ is positive, the first negative term in the expansion of $${\left( {1 + x} \right)^{27/5}}$$ is
A
6th term
B
7th term
C
5th term
D
8th term.

## Explanation

General term of $${\left( {1 + x} \right)^{n}}$$ is ($${T_{r + 1}}$$) = $${{n\left( {n - 1} \right).....\left( {n - r + 1} \right)} \over {1.2.3....r}}{x^r}$$

$$\therefore$$ General term of $${\left( {1 + x} \right)^{27/5}}$$ = $${{{{27} \over 5}\left( {{{27} \over 5} - 1} \right).....\left( {{{27} \over 5} - r + 1} \right)} \over {1.2.3....r}}{x^r}$$

For first negative term, $${\left( {{{27} \over 5} - r + 1} \right)}$$ < 0

$$\Rightarrow r > {{27} \over 5} + 1$$

$$\Rightarrow r > {{32} \over 5}$$

$$\Rightarrow r > 6.4$$

$$\therefore$$ r = 7

$${T_{7 + 1}} = {T_8}$$ means 8th term is the first negative term.
2

### AIEEE 2002

If $${a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + .......} } }$$ having $$n$$ radical signs then by methods of mathematical induction which is true
A
$${a_n} > 7\,\,\forall \,\,n \ge 1$$
B
$${a_n} < 7\,\,\forall \,\,n \ge 1$$
C
$${a_n} < 4\,\,\forall \,\,n \ge 1$$
D
$${a_n} > 3\,\,\forall \,\,n \ge 1$$

## Explanation

Given $${a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + .......} } }$$

$$\therefore$$ $${a_n} = \sqrt {7 + {a_n}}$$

$$\Rightarrow$$ $$a_n^2 = 7 + {a_n}$$

$$\Rightarrow$$ $$a_n^2 - {a_n} - 7 = 0$$

$$\Rightarrow {a_n} = {{1 \pm \sqrt {1 - 4 \times 1 \times - 7} } \over 2}$$

$$\Rightarrow {a_n} = {{1 \pm \sqrt {29} } \over 2}$$

As $${a_n}$$ > 0,

$$\therefore$$ $${a_n} = {{1 + \sqrt {29} } \over 2}$$ = 3.19

So $${a_n} > 3\,\,\forall \,\,n \ge 1$$
3

### AIEEE 2002

If the sum of the coefficients in the expansion of $$\,{\left( {a + b} \right)^n}$$ is 4096, then the greatest coefficient in the expansion is
A
1594
B
792
C
924
D
2924

## Explanation

We know, $$\,{\left( {a + b} \right)^n}$$ = $${}^n{C_0}.{a^n} + {}^n{C_1}.{a^{n - 1}}.b + ... + {}^n{C_n}.{b^n}$$

Remember to find sum of coefficient of binomial expansion we ave to put 1 in place of all the variable.

So put $$a$$ = b = 1

$$\therefore$$ 2n = $${}^n{C_0} + {}^n{C_1} + {}^n{C_2}... + {}^n{C_n}$$

According to question, 2n = 4096 = 212

$$\Rightarrow n = 12$$

So $$\,{\left( {a + b} \right)^n}$$ = $$\,{\left( {a + b} \right)^{12}}$$

Here n = 12 is even so formula for greatest term is
$${T_{{n \over 2} + 1}} = {}^n{C_{{n \over 2}}}.{a^{{n \over 2}}}.{b^{{n \over 2}}}$$

For n = 12, greatest term $${T_{6 + 1}} = {}^{12}{C_6}.{a^6}.{b^6}$$

$$\therefore$$ Coefficient of the greatest term = $${}^{12}{C_6}$$ = $${{12!} \over {6!6!}}$$ = 924
4

### AIEEE 2002

The positive integer just greater than $${\left( {1 + 0.0001} \right)^{10000}}$$ is
A
4
B
5
C
2
D
3

## Explanation

$${\left( {1 + 0.0001} \right)^{10000}}$$ = $${\left( {1 + {1 \over {{{10}^4}}}} \right)^{10000}}$$

= 1 + 10000$${ \times {1 \over {{{10}^4}}}}$$ + $${{10000\left( {9999} \right)} \over {2!}} \times {\left( {{1 \over {{{10}^4}}}} \right)^2}$$+......$$\infty$$

< 1 + 1 + $${1 \over {2!}}$$ + $${1 \over {3!}}$$ + ...... $$\infty$$ = e = 2.71828 < 3

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