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 numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is not allowed) and are multiple of 3 is :

A

24

B

30

C

36

D

48

Here number should be divisible by 3, that means sum of numbers should be divisible by 3.

Possible 4 digits among 0, 1, 2, 3, 4 which are divisible by 3 are

(1)$$\,\,\,\,$$ (0, 2, 3, 4) Sum of digits = 0 + 2 + 3 +4 = 9 (divisible by 3)

(2) $$\,\,\,\,$$ (0, 1, 2, 3) Sum of digits = 0 + 1 + 2 + 3 = 6 (divisible by 3)

__Case 1__ :

When 4 digits are (0, 2, 3, 4) then

$$\therefore\,\,\,\,$$ Total possible numbers = $$^3{C_1}$$ $$ \times $$ $$^3{C_1}$$ $$ \times $$ $$^2{C_1}$$ $$ \times $$ $$^1{C_1}$$

= 3 $$ \times $$ 3 $$ \times $$ 2 $$ \times $$ 1 = 18

__Case 2__ :

When 4 digits are (0, 1, 2, 3) then,

$$\therefore\,\,\,\,$$ Total possible number in this case = $$^2{C_1}$$ $$ \times $$ $$^3{C_1}$$ $$ \times $$ $$^2{C_1}$$ $$ \times $$ $$^1{C_1}$$

= 2 $$ \times $$ 3 $$ \times $$ 2 $$ \times $$ 1 = 12

$$\therefore\,\,\,\,$$ Total possible numbers will be = 18 + 12 = 30

Possible 4 digits among 0, 1, 2, 3, 4 which are divisible by 3 are

(1)$$\,\,\,\,$$ (0, 2, 3, 4) Sum of digits = 0 + 2 + 3 +4 = 9 (divisible by 3)

(2) $$\,\,\,\,$$ (0, 1, 2, 3) Sum of digits = 0 + 1 + 2 + 3 = 6 (divisible by 3)

When 4 digits are (0, 2, 3, 4) then

$$\therefore\,\,\,\,$$ Total possible numbers = $$^3{C_1}$$ $$ \times $$ $$^3{C_1}$$ $$ \times $$ $$^2{C_1}$$ $$ \times $$ $$^1{C_1}$$

= 3 $$ \times $$ 3 $$ \times $$ 2 $$ \times $$ 1 = 18

When 4 digits are (0, 1, 2, 3) then,

$$\therefore\,\,\,\,$$ Total possible number in this case = $$^2{C_1}$$ $$ \times $$ $$^3{C_1}$$ $$ \times $$ $$^2{C_1}$$ $$ \times $$ $$^1{C_1}$$

= 2 $$ \times $$ 3 $$ \times $$ 2 $$ \times $$ 1 = 12

$$\therefore\,\,\,\,$$ Total possible numbers will be = 18 + 12 = 30

2

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can
be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same
team, is :

A

500

B

350

C

200

D

300

From 5 girls 2 girls can be selected

=^{5}C_{2} ways

From 7 boys 3 boys can be selected

=^{7}C_{3} way

$$ \therefore $$ Total number of ways we can select 2 girls and 3 boys

=^{5}C_{2} $$ \times $$ ^{7}C_{3} ways

When two boys A and B are chosen in a team then one more boy will be chosen from remaining 5 boys.

So, no of ways 3 boys can be chosen when A and B should must be chosen =^{5}C_{1} ways

$$ \therefore $$ Total number of ways a team of 2 girl and 3 boys can be made where boy A and B must be in the team =^{5}^{}C_{1} $$ \times $$ ^{5}C_{2} ways

$$ \therefore $$ Required number of ways

= Total number of ways $$-$$ when A and B are always included.

=^{5}C_{2} $$ \times $$ ^{7}C_{3} $$-$$ ^{5}C_{1} $$ \times $$ ^{5}C_{2}

= 300

=

From 7 boys 3 boys can be selected

=

$$ \therefore $$ Total number of ways we can select 2 girls and 3 boys

=

When two boys A and B are chosen in a team then one more boy will be chosen from remaining 5 boys.

So, no of ways 3 boys can be chosen when A and B should must be chosen =

$$ \therefore $$ Total number of ways a team of 2 girl and 3 boys can be made where boy A and B must be in the team =

$$ \therefore $$ Required number of ways

= Total number of ways $$-$$ when A and B are always included.

=

= 300

3

Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is :

A

9

B

18

C

36

D

32

Area = $${1 \over 2}$$ h. k = 50

h. k = 100

h. k = 2

Total divisors

= (2 + 1) (2 + 1) = 9

if h > 0, k > 0

But $${\matrix{ {h > 0,} & {k < 0} \cr {h < 0,} & {k > 0} \cr {h < 0,} & {k < 0} \cr } }$$

all are possible so that total no. of positive case

9 + 9 + 9 + 9 = 36

4

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

Number in Brackets after Paper Name Indicates No of Questions

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *keyboard_arrow_right*

Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

Permutations and Combinations *keyboard_arrow_right*

Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

Sequences and Series *keyboard_arrow_right*

Matrices and Determinants *keyboard_arrow_right*

Vector Algebra and 3D Geometry *keyboard_arrow_right*

Probability *keyboard_arrow_right*

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Functions *keyboard_arrow_right*

Limits, Continuity and Differentiability *keyboard_arrow_right*

Differentiation *keyboard_arrow_right*

Application of Derivatives *keyboard_arrow_right*

Indefinite Integrals *keyboard_arrow_right*

Definite Integrals and Applications of Integrals *keyboard_arrow_right*

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Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*