Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geotechnical Engineering

Transportation Engineering

Irrigation

Engineering Mathematics

Construction Material and Management

Fluid Mechanics and Hydraulic Machines

Hydrology

Environmental Engineering

Engineering Mechanics

Structural Analysis

Reinforced Cement Concrete

Steel Structures

Geomatics Engineering Or Surveying

General Aptitude

1

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 = $${1 \over 3}$$

2

Let A, B and C be three events, which are pair-wise independent and $$\overrightarrow E $$ denotes the completement of an event E. If $$P\left( {A \cap B \cap C} \right) = 0$$ and $$P\left( C \right) > 0,$$ then $$P\left[ {\left( {\overline A \cap \overline B } \right)\left| C \right.} \right]$$ is equal to :

A

$$P\left( {\overline A } \right) - P\left( B \right)$$

B

$$P\left( A \right) + P\left( {\overline B } \right)$$

C

$$P\left( {\overline A } \right) - P\left( {\overline B } \right)$$

D

$$P\left( {\overline A } \right) + P\left( {\overline B } \right)$$

Here, $$P\left( {\overline A \cap \overline B \left| C \right.} \right) = {{P\left( {\overline A \cap \overline B \cap C} \right)} \over {P\left( C \right)}}$$

= $${{P\left[ {\left( {\overline {A \cup B} } \right) \cap C} \right]} \over {P\left( C \right)}}$$

= $${{P\left[ {C - \left( {A \cup B} \right)} \right]} \over {P\left( C \right)}}$$

= $${{P\left( C \right) - P\left( {A \cap C} \right) - P\left( {B \cap C} \right) + P\left( {A \cap B \cap C} \right)} \over {P\left( C \right)}}$$

= $${{P\left( C \right) - P\left( {A \cap C} \right) - P\left( {B \cap C} \right)} \over {P\left( C \right)}}$$ ($$ \because $$$$\left. {P\left( {A \cap B \cap C} \right) = 0} \right)$$

= $${{P\left( C \right) - P\left( A \right).P(C) - P\left( B \right).P(C)} \over {P\left( C \right)}}$$

[$$ \because $$ A, B and C are independent events]

= 1 - P(A) - P(B)

= $$P\left( {\overline A } \right)$$ - P(B) or $$P\left( {\overline B } \right)$$ - P(A)

= $${{P\left[ {\left( {\overline {A \cup B} } \right) \cap C} \right]} \over {P\left( C \right)}}$$

= $${{P\left[ {C - \left( {A \cup B} \right)} \right]} \over {P\left( C \right)}}$$

= $${{P\left( C \right) - P\left( {A \cap C} \right) - P\left( {B \cap C} \right) + P\left( {A \cap B \cap C} \right)} \over {P\left( C \right)}}$$

= $${{P\left( C \right) - P\left( {A \cap C} \right) - P\left( {B \cap C} \right)} \over {P\left( C \right)}}$$ ($$ \because $$$$\left. {P\left( {A \cap B \cap C} \right) = 0} \right)$$

= $${{P\left( C \right) - P\left( A \right).P(C) - P\left( B \right).P(C)} \over {P\left( C \right)}}$$

[$$ \because $$ A, B and C are independent events]

= 1 - P(A) - P(B)

= $$P\left( {\overline A } \right)$$ - P(B) or $$P\left( {\overline B } \right)$$ - P(A)

3

Two different families A and B are blessed with equal numbe of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family B is $${1 \over {12}},$$ then the number of children in each family is :

A

3

B

4

C

5

D

6

Let the number of children in each family be x.

Thus the total number of children in both the families are 2x

Now, it is given that 3 tickets are distributed amongst the children of these two families.

Thus, the probability that all the three tickets go to the children in family B

= $${{{}^x{C_3}} \over {{}^{2x}{C_3}}}$$ = $${1 \over {12}}$$

$$ \Rightarrow $$ $$\,\,\,$$ $${{x\left( {x - 1} \right)\left( {x - 2} \right)} \over {2x\left( {2x - 1} \right)\left( {2x - 2} \right)}}$$ = $${1 \over {12}}$$

$$ \Rightarrow $$ $${{\left( {x - 2} \right)} \over {\left( {2x - 1} \right)}}$$ = $${1 \over 6}$$

Thus, the number of children in each family is 5.

Thus the total number of children in both the families are 2x

Now, it is given that 3 tickets are distributed amongst the children of these two families.

Thus, the probability that all the three tickets go to the children in family B

= $${{{}^x{C_3}} \over {{}^{2x}{C_3}}}$$ = $${1 \over {12}}$$

$$ \Rightarrow $$ $$\,\,\,$$ $${{x\left( {x - 1} \right)\left( {x - 2} \right)} \over {2x\left( {2x - 1} \right)\left( {2x - 2} \right)}}$$ = $${1 \over {12}}$$

$$ \Rightarrow $$ $${{\left( {x - 2} \right)} \over {\left( {2x - 1} \right)}}$$ = $${1 \over 6}$$

Thus, the number of children in each family is 5.

4

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X = 1) + P (X = 2) equals :

A

$$25 \over 169$$

B

$$49\over 169$$

C

$$24 \over 169$$

D

$$52 \over 169$$

P (X = 1) means out of two drawn cards one card is ace.

and P(X = 2) means both the drawn cards are ace.

$$ \therefore $$ P(X = 1) = first card is ace or 2nd card is ace.

= A $$-$$ $$+$$ $$-$$ A

= $${4 \over {52}} \times {{48} \over {52}} + {{48} \over {52}} \times {4 \over {52}}$$

= $$2 \times {4 \over {52}} \times {{48} \over {52}}$$

P(X = 2) =First and second both cards arc ace.

= A A

= $${4 \over {52}} \times {4 \over {52}}$$

$$ \therefore $$ P(X = 1) + P(X = 2)

= $$2 \times {4 \over {52}} \times {{48} \over {52}} + {4 \over {52}} \times {4 \over {52}}$$

= $${{25} \over {169}}$$

and P(X = 2) means both the drawn cards are ace.

$$ \therefore $$ P(X = 1) = first card is ace or 2nd card is ace.

= A $$-$$ $$+$$ $$-$$ A

= $${4 \over {52}} \times {{48} \over {52}} + {{48} \over {52}} \times {4 \over {52}}$$

= $$2 \times {4 \over {52}} \times {{48} \over {52}}$$

P(X = 2) =First and second both cards arc ace.

= A A

= $${4 \over {52}} \times {4 \over {52}}$$

$$ \therefore $$ P(X = 1) + P(X = 2)

= $$2 \times {4 \over {52}} \times {{48} \over {52}} + {4 \over {52}} \times {4 \over {52}}$$

= $${{25} \over {169}}$$

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*

AIEEE 2006 (1) *keyboard_arrow_right*

AIEEE 2007 (2) *keyboard_arrow_right*

AIEEE 2008 (2) *keyboard_arrow_right*

AIEEE 2009 (2) *keyboard_arrow_right*

AIEEE 2010 (2) *keyboard_arrow_right*

AIEEE 2011 (2) *keyboard_arrow_right*

AIEEE 2012 (1) *keyboard_arrow_right*

JEE Main 2013 (Offline) (1) *keyboard_arrow_right*

JEE Main 2014 (Offline) (1) *keyboard_arrow_right*

JEE Main 2015 (Offline) (1) *keyboard_arrow_right*

JEE Main 2016 (Offline) (1) *keyboard_arrow_right*

JEE Main 2016 (Online) 9th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2016 (Online) 10th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2017 (Offline) (3) *keyboard_arrow_right*

JEE Main 2017 (Online) 8th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2017 (Online) 9th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2018 (Offline) (1) *keyboard_arrow_right*

JEE Main 2018 (Online) 15th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2018 (Online) 15th April Evening Slot (1) *keyboard_arrow_right*

JEE Main 2018 (Online) 16th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 11th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 11th January Evening Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th January Evening Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 8th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 8th April Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th April Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th April Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 7th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 7th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 8th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 8th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 9th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 9th January Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 2nd September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 2nd September Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 3rd September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 3rd September Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 4th September Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 6th September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 6th September Evening Slot (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 24th February Morning Slot (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 24th February Evening Slot (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th February Morning Slot (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th February Evening Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th February Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th February Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 16th March Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 16th March Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 17th March Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 17th March Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 18th March Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 20th July Morning Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 20th July Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 22th July Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th July Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 27th July Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th July Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 27th July Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th August Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th August Evening Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 27th August Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 27th August Evening Shift (1) *keyboard_arrow_right*

Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *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*

Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *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*