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### AIEEE 2012

MCQ (Single Correct Answer)

Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000.

Statement-2: $$\sum\limits_{k = 1}^n {\left( {{k^3} - {{(k - 1)}^3}} \right)} = {n^3}$$, for any natural number n.

A
Statement-1 is false, Statement-2 is true.
B
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
C
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
D
Statement-1 is true, Statement-2 is false.

## Explanation

$$n$$th term of the given series

$$= {T_n} = {\left( {n - 1} \right)^2} + \left( {n - 1} \right)n + {n^2}$$

$$= {{\left( {{{\left( {n - 1} \right)}^3} - {n^3}} \right)} \over {\left( {n - 1} \right) - n}}$$

$$= {n^3} - {\left( {n - 1} \right)^3}$$

$$\Rightarrow {S_n} = \sum\limits_{k = 1}^n {\left[ {{k^3} - {{\left( {k - 1} \right)}^3}} \right]}$$

$$\Rightarrow 8000 = {n^3}$$

$$\Rightarrow n = 20\,\,$$ which is a natural number.

Now, put $$n = 1,2,3,.....20$$

$${T_1} = {1^3} - {0^3}$$

$${T_2} = {2^3} - {1^3}$$

.

.

.

$${T_{20}} = {20^3} - {19^3}$$

Now, $${T_1} + {T_2} + ..... + {T_{20}} = {S_{20}}$$

$$\Rightarrow {S_{20}} = {20^3} - {0^3} = 8000$$

Hence, both the given statements are true and statement $$2$$ supports statement $$1.$$
2

### AIEEE 2011

MCQ (Single Correct Answer)
A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 11040 after
A
19 months
B
20 months
C
21 months
D
18 months

## Explanation

Let required number of months $$=n$$

$$\therefore$$ $$200 \times 3 + \left( {240 + 280 + 320 + ...} \right.$$

$$\left. {\,\,\,\,\,\,\,\,\,\,\,\, + {{\left( {n - 3} \right)}^{th}}\,term} \right) = 11040$$

$$\Rightarrow {{n - 3} \over 2}\left[ {2 \times 240 + \left( {n - 4} \right) \times 40} \right]$$

$$\,\,\,\,\,\,\,\,\,\,\,\, = 11040 - 600$$

$$\Rightarrow \left( {n - 3} \right)\left[ {240 + 20n - 80} \right] = 10440$$

$$\Rightarrow \left( {n - 3} \right)\left( {20n + 160} \right) = 10440$$

$$\Rightarrow \left( {n - 3} \right)\left( {n + 8} \right) = 522$$

$$\Rightarrow {n^2} + 5n - 546 = 0$$

$$\Rightarrow \left( {n + 26} \right)\left( {n - 21} \right) = 0$$

$$\therefore$$ $$n = 21$$
3

### AIEEE 2010

MCQ (Single Correct Answer)
A person is to count 4500 currency notes. Let $${a_n}$$ denote the number of notes he counts in the $${n^{th}}$$ minute. If $${a_1}$$ = $${a_2}$$ = ....= $${a_{10}}$$= 150 and $${a_{10}}$$, $${a_{11}}$$,.... are in an AP with common difference - 2, then the time taken by him to count all notes is
A
34 minutes
B
125 minutes
C
135 minutes
D
24 minutes

## Explanation

Till $$10$$th minute number of counted notes $$= 1500$$

$$3000 = {n \over 2}\left[ {2 \times 148 + \left( {n - 1} \right)\left( { - 2} \right)} \right]$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$= n\left[ {148 - n + 1} \right]$$

$$\Rightarrow$$$${n^2} - 149n + 3000 = 0$$

$$\Rightarrow n = 125,24$$

But $$n=125$$ is not possible

$$\therefore$$ total time $$= 24 + 10 = 34$$ minutes.
4

### AIEEE 2009

MCQ (Single Correct Answer)
The sum to infinite term of the series $$1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + .....$$ is
A
3
B
4
C
6
D
2

## Explanation

We have

$$S = 1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + ........\infty \,\,\,\,\,...\left( 1 \right)$$

Multiplying both sides by $${1 \over 3}$$ we get

$${1 \over 3}S = {1 \over 3} + {2 \over {{3^2}}} + {6 \over {{3^3}}} + {{10} \over {{3^4}}} + .......\,\,\,\,\,...\left( 2 \right)$$

Subtracting eqn. $$(2)$$ from eqn. $$(1)$$ we get

$${2 \over 3}S = 1 + {1 \over 3} + {4 \over {{3^2}}} + {4 \over {{3^3}}} + {4 \over {{3^4}}} + .....\infty$$

$$\Rightarrow {2 \over 3}S = {4 \over 3} + {4 \over {{3^2}}} + {4 \over {{3^3}}} + {4 \over {{3^4}}} + .....\infty$$

$$\Rightarrow {2 \over 3}S = {{{4 \over 3}} \over {1 - {1 \over 3}}} = {4 \over 3} \times {3 \over 2}$$

$$\Rightarrow S - 3$$

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