Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7, 9 (repitition of digits allowed) is equal to :

A

374

B

372

C

375

D

250

Total no 1 digit numbers possible = 4 (allowed digits 1, 3, 7, 9)

Total no 2 digit numbers possible = 4$$ \times $$5 = 20

Total no 3 digit numbers possible = 4$$ \times $$5$$ \times $$5 = 100

Total no 4 digit numbers possible = 2$$ \times $$5$$ \times $$5$$ \times $$5 = 250

So the number of natural numbers less than 7,000 possible are

= 4 + 20 + 100 + 250 = 374

Total no 2 digit numbers possible = 4$$ \times $$5 = 20

Total no 3 digit numbers possible = 4$$ \times $$5$$ \times $$5 = 100

Total no 4 digit numbers possible = 2$$ \times $$5$$ \times $$5$$ \times $$5 = 250

So the number of natural numbers less than 7,000 possible are

= 4 + 20 + 100 + 250 = 374

2

If $$\sum\limits_{r = 0}^{25} {\left\{ {{}^{50}{C_r}.{}^{50 - r}{C_{25 - r}}} \right\} = K\left( {^{50}{C_{25}}} \right)} ,\,\,$$ then K is equal to -

A

2^{24}

B

2^{25}$$-$$ 1

C

2^{25}

D

(25)^{2}

$$\sum\limits_{r = 0}^{25} {^{50}} {C_r}.{}^{50 - r}{C_{25 - r}}$$

$$ = \sum\limits_{r = 0}^{25} {{{50!} \over {r!\left( {50 - r} \right)!}}} \times {{\left( {50 - r} \right)!} \over {\left( {25} \right)!\left( {25 - r} \right)!}}$$

$$ = \sum\limits_{r = 0}^{25} {{{50!} \over {25!25!}} \times {{25!} \over {\left( {25 - r} \right)!\left( {r!} \right)}}} $$

$$ = {}^{50}{C_{25}}\sum\limits_{r = 0}^{25} {^{25}} {C_r} = \left( {{2^{25}}} \right){}^{50}{C_{25}}$$

$$ \therefore $$ $$K = {2^{25}}$$

$$ = \sum\limits_{r = 0}^{25} {{{50!} \over {r!\left( {50 - r} \right)!}}} \times {{\left( {50 - r} \right)!} \over {\left( {25} \right)!\left( {25 - r} \right)!}}$$

$$ = \sum\limits_{r = 0}^{25} {{{50!} \over {25!25!}} \times {{25!} \over {\left( {25 - r} \right)!\left( {r!} \right)}}} $$

$$ = {}^{50}{C_{25}}\sum\limits_{r = 0}^{25} {^{25}} {C_r} = \left( {{2^{25}}} \right){}^{50}{C_{25}}$$

$$ \therefore $$ $$K = {2^{25}}$$

3

The number of functions f from {1, 2, 3, ...., 20} onto {1, 2, 3, ...., 20} such that f(k) is a multiple of 3,
whenever k is a multiple of 4, is :

A

6^{5} $$ \times $$ (15)!

B

5^{6} $$ \times $$ 15

C

(15)! $$ \times $$ 6!

D

5! $$ \times $$ 6!

f(k) = 3m (3, 6, 9, 12, 15, 18)

for k = 4, 8, 12, 16, 20

6.5.4.3.2 ways

For rest numbers 15! ways

Total ways = 6! (15!)

for k = 4, 8, 12, 16, 20

6.5.4.3.2 ways

For rest numbers 15! ways

Total ways = 6! (15!)

4

Consider three boxes, each containing, 10 balls labelled 1, 2, … , 10. Suppose one ball is randomly drawn from each of the boxes. Denote by n_{i}, the label of the ball drawn from the i^{th} box, (i = 1, 2, 3). Then, the number of ways in which the balls can be chosen such that n_{1} < n_{2} < n_{3} is :

A

164

B

240

C

82

D

120

Number of ways = ^{10}C_{3} = 120

Number in Brackets after Paper Name Indicates No of Questions

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