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

Let E and F be two independent events. The probability that both E and F happen is $${1 \over {12}}$$ and the probability that neither E nor F happens is $${1 \over {2}}$$, then a value of $${{P\left( E \right)} \over {P\left( F \right)}}$$ is :

A

$${4 \over 3}$$

B

$${3 \over 2}$$

C

$${1 \over 3}$$

D

$${5 \over 12}$$

2

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and
this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at
random from the bag, then the probability that this drawn ball is red, is :

A

$${3 \over 4}$$

B

$${3 \over 10}$$

C

$${2 \over 5}$$

D

$${1 \over 5}$$

If we follow path 1, then probability of getting 1st ball black $$ = {6 \over {10}}$$ and probability of getting 2nd ball red when there is 4 R and 8 B balls = $${4 \over {12}}$$.

So, the probability of getting 1st ball black and 2nd ball red = $${6 \over {10}} \times {4 \over {12}}$$.

If we follow path 2, then the probability of getting 1st ball red $$ = {4 \over {10}}$$ and probability of getting 2nd ball red when in the bag there is 6 red and 6 black balls = $${6 \over {12}}$$

$$\therefore\,\,\,$$ Probability of getting 2nd ball as red

$$ = {6 \over {10}} \times {4 \over {12}} + {4 \over {10}} \times {6 \over {12}}$$

$$ = {1 \over 5} + {1 \over 5}$$

$$ = {2 \over 5}$$

3

A box 'A' contains $$2$$ white, $$3$$ red and $$2$$ black balls. Another box 'B' contains $$4$$ white, $$2$$ red and $$3$$ black balls. If two balls are drawn at random, without eplacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box 'B' is :

A

$${9 \over {16}}$$

B

$${7 \over {16}}$$

C

$${9 \over {32}}$$

D

$${7 \over {8}}$$

Probability of drawing a white ball and then a red ball

from bag B is given by $${{{}^4{C_1} \times {}^2{C_1}} \over {{}^9{C_2}}}$$ = $${2 \over 9}$$

Probability of drawing a white ball and then a red ball

from bag A is given by $${{{}^2{C_1} \times {}^3{C_1}} \over {{}^7{C_2}}}$$ = $${2 \over 7}$$

Hence, the probability of drawing a white ball and then

a red ball from bag B = $${{{2 \over 9}} \over {{2 \over 7} + {2 \over 9}}}$$ = $${{2 \times 7} \over {18 + 14}}$$ = $${7 \over {16}}$$

from bag B is given by $${{{}^4{C_1} \times {}^2{C_1}} \over {{}^9{C_2}}}$$ = $${2 \over 9}$$

Probability of drawing a white ball and then a red ball

from bag A is given by $${{{}^2{C_1} \times {}^3{C_1}} \over {{}^7{C_2}}}$$ = $${2 \over 7}$$

Hence, the probability of drawing a white ball and then

a red ball from bag B = $${{{2 \over 9}} \over {{2 \over 7} + {2 \over 9}}}$$ = $${{2 \times 7} \over {18 + 14}}$$ = $${7 \over {16}}$$

4

A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If X starts the game, and the probability of winning the game by both the players is equal, then the value of 'p' is :

A

$${1 \over 5}$$

B

$${1 \over 3}$$

C

$${2 \over 5}$$

D

$${1 \over 4}$$

P(X getting head) = p

$$ \therefore $$ P(X getting tail) = 1 - p

P(Y getting head) = P(Y getting tail) = $${1 \over 2}$$

P(X wins) = p + (1 - p)$${1 \over 2}$$p + (1 - p)$${1 \over 2}$$(1 - p)$${1 \over 2}$$p + ...

= $${p \over {1 - \left( {{{1 - p} \over 2}} \right)}}$$

= $${{2p} \over {1 + p}}$$

P(Y win) = (1 - p)$${1 \over 2}$$ + (1 - p)$${1 \over 2}$$(1 - p)$${1 \over 2}$$ + ...

= $$\left( {{{1 - p} \over 2}} \right).{p \over {1 - \left( {{{1 - p} \over 2}} \right)}} = {{1 - p} \over {1 + p}}$$

According to question,

P(X wins) = P(Y wins)

$$ \therefore $$ $${{2p} \over {1 + p}}$$ = $${{1 - p} \over {1 + p}}$$

$$ \Rightarrow $$ 3p = 1

$$ \Rightarrow $$ p = $${1 \over 3}$$

$$ \therefore $$ P(X getting tail) = 1 - p

P(Y getting head) = P(Y getting tail) = $${1 \over 2}$$

P(X wins) = p + (1 - p)$${1 \over 2}$$p + (1 - p)$${1 \over 2}$$(1 - p)$${1 \over 2}$$p + ...

= $${p \over {1 - \left( {{{1 - p} \over 2}} \right)}}$$

= $${{2p} \over {1 + p}}$$

P(Y win) = (1 - p)$${1 \over 2}$$ + (1 - p)$${1 \over 2}$$(1 - p)$${1 \over 2}$$ + ...

= $$\left( {{{1 - p} \over 2}} \right).{p \over {1 - \left( {{{1 - p} \over 2}} \right)}} = {{1 - p} \over {1 + p}}$$

According to question,

P(X wins) = P(Y wins)

$$ \therefore $$ $${{2p} \over {1 + p}}$$ = $${{1 - p} \over {1 + p}}$$

$$ \Rightarrow $$ 3p = 1

$$ \Rightarrow $$ p = $${1 \over 3}$$

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (3) *keyboard_arrow_right*

AIEEE 2003 (3) *keyboard_arrow_right*

AIEEE 2004 (2) *keyboard_arrow_right*

AIEEE 2005 (3) *keyboard_arrow_right*

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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*

Statistics *keyboard_arrow_right*

Mathematical Reasoning *keyboard_arrow_right*

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*

Differential Equations *keyboard_arrow_right*

Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*