Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

If $${}^n{C_r}$$ denotes the number of combination of n things taken r at a time, then the expression $$\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}$$ equals

A

$$\,{}^{n + 1}{C_{r + 1}}$$

B

$${}^{n + 2}{C_r}$$

C

$${}^{n + 2}{C_{r + 1}}$$

D

$$\,{}^{n + 1}{C_r}$$

Arrange it this way,

$$^n{C_{r + 1}} + 2.{}^n{C_r} + {}^n{C_{r - 1}}$$

$$ = {}^n{C_{r + 1}} + {}^n{C_r} + {}^n{C_r} + {}^n{C_{r - 1}}$$

$$\left[ \, \right.$$ Now use the rule,

$$\left. {{}^n{C_r} + {}^n{C_{r - 1}} = {}^{n + 1}{C_r}} \right]$$

$$ = {}^{n + 1}{C_{r + 1}} + {}^{n + 1}Cr$$

$$ = {}^{n + 2}{C_{r + 1}}$$

$$^n{C_{r + 1}} + 2.{}^n{C_r} + {}^n{C_{r - 1}}$$

$$ = {}^n{C_{r + 1}} + {}^n{C_r} + {}^n{C_r} + {}^n{C_{r - 1}}$$

$$\left[ \, \right.$$ Now use the rule,

$$\left. {{}^n{C_r} + {}^n{C_{r - 1}} = {}^{n + 1}{C_r}} \right]$$

$$ = {}^{n + 1}{C_{r + 1}} + {}^{n + 1}Cr$$

$$ = {}^{n + 2}{C_{r + 1}}$$

2

MCQ (Single Correct Answer)

The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by

A

$$7!\, \times 5!\,\,$$

B

$$6!\, \times 5!$$

C

$$30!$$

D

$$5!\, \times 4!$$

6 men can sit at the round table = $$\left( {6 - 1} \right)! = 5!$$ ways

Now at the round table among 6 men there are 6 empty places and 5 women can sit at those 6 empty positions.

So total no of ways 6 men and 5 women can dine at the round table

= $$5!\, \times {}^6{C_5} \times 5!$$

= $$5!\, \times 6 \times 5!$$

= $$5!\, \times 6!$$

3

MCQ (Single Correct Answer)

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

A

346

B

140

C

196

D

280

No of ways student can answer 10 questions = $${}^5{C_4} \times {}^8{C_6}$$ = 140

No of ways student can answer 10 questions = $${}^5{C_5} \times {}^8{C_5}$$ = 56

$$\therefore$$ Total ways = 140 + 56 = 196

4

MCQ (Single Correct Answer)

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

A

3000

B

3050

C

3600

D

3250

According to this question, any number between 1 to 100 should be divisible by 2 or 5 but not by 2$$ \times $$5 = 10.

Possible numbers between 1 to 100 divisible by 2 are 2, 4, 6, .... , 100

This is an A.P where first term = 2, last term = 100 and total terms = 50.

$$ \therefore $$ Sum of the numbers divisible by 2

= $${{50} \over 2}\left[ {2 + 100} \right]$$

= 25$$ \times $$102

= 2550

Possible numbers between 1 to 100 divisible by 5 are 5, 10, 15, .... , 100

$$ \therefore $$ Sum of the numbers divisible by 5

= $${{20} \over 2}\left[ {5 + 100} \right]$$

= 10$$ \times $$105

= 1050

And possible numbers between 1 to 100 divisible by 10 are 10, 20, 30, .... , 100

$$ \therefore $$ Sum of the numbers divisible by 10

= $${{10} \over 2}\left[ {10 + 100} \right]$$

= 5$$ \times $$110

= 550

$$ \therefore $$ Required sum = 2550 + 1050 - 550 = 3050

Possible numbers between 1 to 100 divisible by 2 are 2, 4, 6, .... , 100

This is an A.P where first term = 2, last term = 100 and total terms = 50.

$$ \therefore $$ Sum of the numbers divisible by 2

= $${{50} \over 2}\left[ {2 + 100} \right]$$

= 25$$ \times $$102

= 2550

Possible numbers between 1 to 100 divisible by 5 are 5, 10, 15, .... , 100

$$ \therefore $$ Sum of the numbers divisible by 5

= $${{20} \over 2}\left[ {5 + 100} \right]$$

= 10$$ \times $$105

= 1050

And possible numbers between 1 to 100 divisible by 10 are 10, 20, 30, .... , 100

$$ \therefore $$ Sum of the numbers divisible by 10

= $${{10} \over 2}\left[ {10 + 100} \right]$$

= 5$$ \times $$110

= 550

$$ \therefore $$ Required sum = 2550 + 1050 - 550 = 3050

On those following papers in MCQ (Single Correct Answer)

Number in Brackets after Paper Indicates No. of Questions

JEE Main 2021 (Online) 1st September Evening Shift (1)

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

JEE Main 2021 (Online) 18th March Morning Shift (1)

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

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

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

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

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

JEE Main 2021 (Online) 25th February Morning Shift (1)

JEE Main 2021 (Online) 24th February Morning Shift (1)

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

JEE Main 2020 (Online) 5th September Evening Slot (1)

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

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

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

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

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

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

JEE Main 2019 (Online) 12th April Evening Slot (1)

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

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

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

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

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

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

JEE Main 2019 (Online) 12th January Evening Slot (1)

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

JEE Main 2019 (Online) 11th January Evening Slot (1)

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

JEE Main 2019 (Online) 9th January Evening Slot (2)

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

JEE Main 2018 (Online) 16th April Morning Slot (1)

JEE Main 2018 (Offline) (1)

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

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

JEE Main 2017 (Online) 9th April Morning Slot (1)

JEE Main 2017 (Online) 8th April Morning Slot (1)

JEE Main 2017 (Offline) (1)

JEE Main 2016 (Online) 10th April Morning Slot (2)

JEE Main 2016 (Online) 9th April Morning Slot (2)

JEE Main 2016 (Offline) (1)

JEE Main 2015 (Offline) (1)

JEE Main 2013 (Offline) (2)

AIEEE 2012 (1)

AIEEE 2011 (2)

AIEEE 2010 (1)

AIEEE 2009 (1)

AIEEE 2008 (2)

AIEEE 2007 (1)

AIEEE 2006 (1)

AIEEE 2005 (1)

AIEEE 2004 (2)

AIEEE 2003 (3)

AIEEE 2002 (4)

Complex Numbers

Quadratic Equation and Inequalities

Permutations and Combinations

Mathematical Induction and Binomial Theorem

Sequences and Series

Matrices and Determinants

Vector Algebra and 3D Geometry

Probability

Statistics

Mathematical Reasoning

Trigonometric Functions & Equations

Properties of Triangle

Inverse Trigonometric Functions

Straight Lines and Pair of Straight Lines

Circle

Conic Sections

Functions

Limits, Continuity and Differentiability

Differentiation

Application of Derivatives

Indefinite Integrals

Definite Integrals and Applications of Integrals

Differential Equations