Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

The remainder left out when $${8^{2n}} - {\left( {62} \right)^{2n + 1}}$$ is divided by 9 is :

A

2

B

7

C

8

D

0

$${8^{2n}} - {\left( {62} \right)^{2n + 1}}$$

= $${\left( {{8^2}} \right)^n} - {\left( {62} \right)^{2n + 1}}$$

= $${\left( {1 + 63} \right)^n} - {\left( {1 - 63} \right)^{2n + 1}}$$

= $$\left( {1 + n.63 + {}^n{C_2}{{.63}^2} + ......} \right)$$

+ $$\left( {1 + {}^{2n + 1}{C_1}.\left( { - 63} \right) + {}^{2n + 1}{C_2}.{{\left( { - 63} \right)}^2} + ......} \right)$$

= 2 + 63$$\left[ {\left( {n + {}^n{C_2} + ....} \right) + \left( { - {}^{2n + 1}{C_1} + {}^{2n + 1}{C_2}.63 + ......} \right)} \right]$$

= 63$$ \times $$[Some integral value] + 2

63$$ \times $$[Some integral value] + 2 by dividing with 9 we will get 2 as remainder as 63 is multiple of 9.

= $${\left( {{8^2}} \right)^n} - {\left( {62} \right)^{2n + 1}}$$

= $${\left( {1 + 63} \right)^n} - {\left( {1 - 63} \right)^{2n + 1}}$$

= $$\left( {1 + n.63 + {}^n{C_2}{{.63}^2} + ......} \right)$$

+ $$\left( {1 + {}^{2n + 1}{C_1}.\left( { - 63} \right) + {}^{2n + 1}{C_2}.{{\left( { - 63} \right)}^2} + ......} \right)$$

= 2 + 63$$\left[ {\left( {n + {}^n{C_2} + ....} \right) + \left( { - {}^{2n + 1}{C_1} + {}^{2n + 1}{C_2}.63 + ......} \right)} \right]$$

= 63$$ \times $$[Some integral value] + 2

63$$ \times $$[Some integral value] + 2 by dividing with 9 we will get 2 as remainder as 63 is multiple of 9.

2

MCQ (Single Correct Answer)

A

Statement - 1 is false, Statement - 2 is true

B

Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1

C

Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1

D

Statement - 1 is true, Statement - 2 is false

$$\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}}$$

= $$\sum\limits_{r = 0}^n {r.{}^n{C_r}} $$ + $$\sum\limits_{r = 0}^n {{}^n{C_r}} $$

= $$\sum\limits_{r = 1}^n {r.{n \over r}{}^{n - 1}{C_{r - 1}}} $$ $$ + {2^n}$$

= $$n\sum\limits_{r = 1}^n {{}^{n - 1}{C_{r - 1}}} $$ $$ + {2^n}$$

= $$n \times {2^{n - 1}}$$$$ + {2^n}$$

= $${2^{n - 1}}\left[ {n + 2} \right]$$

$$\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}{x^r}}$$

= $$\sum\limits_{r = 0}^n r .{}^n{C_r}.{x^r}$$ + $$\sum\limits_{r = 0}^n {{}^n{C_r}.{x^r}} $$

= $$n\sum\limits_{r = 1}^n {{}^{n - 1}{C_{r - 1}}.{x^r}} $$ $$ + {\left( {1 + x} \right)^n}$$

= $$nx\sum\limits_{r = 1}^n {{}^{n - 1}{C_{r - 1}}.{x^{r - 1}}} + {\left( {1 + x} \right)^n}$$

= $$nx{\left( {1 + x} \right)^{n - 1}} + {\left( {1 + x} \right)^n}$$

Substitude x = 1 in the statement 2 and we get,

$$\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r} = \left( {n + 2} \right){2^{n - 1}}.} $$

So Option B is correct.

3

MCQ (Single Correct Answer)

In the binomial expansion of $${\left( {a - b} \right)^n},\,\,\,n \ge 5,$$ the sum of $${5^{th}}$$ and $${6^{th}}$$ terms is zero, then $$a/b$$ equals

A

$${{n - 5} \over 6}$$

B

$${{n - 4} \over 5}$$

C

$${5 \over {n - 4}}$$

D

$${6 \over {n - 5}}$$

According to the question,

t_{5} + t_{6} = 0

$$\therefore$$ $${}^n{C_4}.{a^{n - 4}}.{b^4}$$ + $$\left( { - {}^n{C_5}.{a^{n - 5}}.{b^5}} \right)$$ = 0

By solving we get,

$${a \over b} = {{n - 4} \over 5}$$

t

$$\therefore$$ $${}^n{C_4}.{a^{n - 4}}.{b^4}$$ + $$\left( { - {}^n{C_5}.{a^{n - 5}}.{b^5}} \right)$$ = 0

By solving we get,

$${a \over b} = {{n - 4} \over 5}$$

4

MCQ (Single Correct Answer)

The sum of the series $${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .....\, - \,.....\, + {}^{20}{C_{10}}$$ is

A

$$0$$

B

$${}^{20}{C_{10}}$$

C

$$ - {}^{20}{C_{10}}$$

D

$${1 \over 2}{}^{20}{C_{10}}$$

We know

$${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... + {}^{20}{C_{10}} - {}^{20}{C_{11}}+ ...... + {}^{20}{C_{20}} = 0$$

$$ \Rightarrow $$ $$({}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}})$$ $$+{}^{20}{C_{10}}$$ $$(-{}^{20}{C_{9}}+ {}^{20}{C_{8}}+...... + {}^{20}{C_{0}})$$

(As $${}^{20}{C_{11}} = {}^{20}{C_9}$$)

$$ \Rightarrow $$ $$2({}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}})$$ $$+{}^{20}{C_{10}}$$ = 0

$$ \Rightarrow $$ $${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}}$$ = $$ - {1 \over 2}{}^{20}{C_{10}}$$

Adding $${}^{20}{C_{10}}$$ both sides,

$$ \Rightarrow $$ $${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}}$$ $$ + {}^{20}{C_{10}}$$ = $$ - {1 \over 2}{}^{20}{C_{10}}$$ $$ + {}^{20}{C_{10}}$$

$$ \Rightarrow $$ $${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}}$$ $$ + {}^{20}{C_{10}}$$ = $$ {1 \over 2}{}^{20}{C_{10}}$$

$${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... + {}^{20}{C_{10}} - {}^{20}{C_{11}}+ ...... + {}^{20}{C_{20}} = 0$$

$$ \Rightarrow $$ $$({}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}})$$ $$+{}^{20}{C_{10}}$$ $$(-{}^{20}{C_{9}}+ {}^{20}{C_{8}}+...... + {}^{20}{C_{0}})$$

(As $${}^{20}{C_{11}} = {}^{20}{C_9}$$)

$$ \Rightarrow $$ $$2({}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}})$$ $$+{}^{20}{C_{10}}$$ = 0

$$ \Rightarrow $$ $${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}}$$ = $$ - {1 \over 2}{}^{20}{C_{10}}$$

Adding $${}^{20}{C_{10}}$$ both sides,

$$ \Rightarrow $$ $${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}}$$ $$ + {}^{20}{C_{10}}$$ = $$ - {1 \over 2}{}^{20}{C_{10}}$$ $$ + {}^{20}{C_{10}}$$

$$ \Rightarrow $$ $${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}}$$ $$ + {}^{20}{C_{10}}$$ = $$ {1 \over 2}{}^{20}{C_{10}}$$

On those following papers in MCQ (Single Correct Answer)

Number in Brackets after Paper Indicates No. of Questions

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

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

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

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

JEE Main 2021 (Online) 25th July Evening Shift (3)

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

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

JEE Main 2021 (Online) 20th July Morning 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 Morning Shift (1)

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

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

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

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

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

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

JEE Main 2020 (Online) 4th September Morning Slot (1)

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

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

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

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

JEE Main 2020 (Online) 8th January Evening 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 (2)

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 Evening 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 (2)

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 (2)

JEE Main 2019 (Online) 11th January Morning Slot (2)

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

JEE Main 2019 (Online) 10th January Morning Slot (2)

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

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 (1)

JEE Main 2016 (Offline) (1)

JEE Main 2015 (Offline) (1)

JEE Main 2014 (Offline) (1)

JEE Main 2013 (Offline) (1)

AIEEE 2012 (1)

AIEEE 2011 (1)

AIEEE 2010 (1)

AIEEE 2009 (1)

AIEEE 2008 (1)

AIEEE 2007 (2)

AIEEE 2006 (2)

AIEEE 2005 (5)

AIEEE 2004 (4)

AIEEE 2003 (2)

AIEEE 2002 (5)

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