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

If A = $$\left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$,

then the determinant of the matrix (A^{2016} − 2A^{2015} − A^{2014}) is :

then the determinant of the matrix (A

A

2014

B

$$-$$ 175

C

2016

D

$$-$$ 25

Given,

$$A = \left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$

$${A^2} = \left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]\left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$

$$ = \left[ {\matrix{ {13} & 3 \cr { - 9} & { - 2} \cr } } \right]$$

A^{2} $$-$$ 2A $$-$$ I

$$ = \left[ {\matrix{ {13} & 3 \cr { - 9} & { - 2} \cr } } \right] - \left[ {\matrix{ { - 8} & { - 2} \cr 6 & 2 \cr } } \right] - \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right]$$

$$ = \left[ {\matrix{ {20} & 5 \cr { - 15} & { - 5} \cr } } \right]$$

$$\left| A \right| = \left| {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right|$$ $$=$$ $$-$$ 4 + 3 $$=$$ $$-$$ 1

Now,

$$\left| {{A^{2016}} - 2{A^{2015}} - {A^{2014}}} \right|$$

$$=$$ $${\left| A \right|^{2014}}\left| {{A^2} - 2A - {\rm I}} \right|$$

$$ = {\left( { - 1} \right)^{2014}}\left| {\matrix{ {20} & 5 \cr { - 15} & { - 5} \cr } } \right|$$

$$=$$ 1 $$ \times $$ ($$-$$ 100 + 75)

$$=$$ $$-$$ 25

$$A = \left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$

$${A^2} = \left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]\left[ {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right]$$

$$ = \left[ {\matrix{ {13} & 3 \cr { - 9} & { - 2} \cr } } \right]$$

A

$$ = \left[ {\matrix{ {13} & 3 \cr { - 9} & { - 2} \cr } } \right] - \left[ {\matrix{ { - 8} & { - 2} \cr 6 & 2 \cr } } \right] - \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right]$$

$$ = \left[ {\matrix{ {20} & 5 \cr { - 15} & { - 5} \cr } } \right]$$

$$\left| A \right| = \left| {\matrix{ { - 4} & { - 1} \cr 3 & 1 \cr } } \right|$$ $$=$$ $$-$$ 4 + 3 $$=$$ $$-$$ 1

Now,

$$\left| {{A^{2016}} - 2{A^{2015}} - {A^{2014}}} \right|$$

$$=$$ $${\left| A \right|^{2014}}\left| {{A^2} - 2A - {\rm I}} \right|$$

$$ = {\left( { - 1} \right)^{2014}}\left| {\matrix{ {20} & 5 \cr { - 15} & { - 5} \cr } } \right|$$

$$=$$ 1 $$ \times $$ ($$-$$ 100 + 75)

$$=$$ $$-$$ 25

2

If $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$,

then adj(3A^{2} + 12A) is equal to

then adj(3A

A

$$\left[ {\matrix{
{51} & {63} \cr
{84} & {72} \cr
} } \right]$$

B

$$\left[ {\matrix{
{51} & {84} \cr
{63} & {72} \cr
} } \right]$$

C

$$\left[ {\matrix{
{72} & {-63} \cr
{-84} & {51} \cr
} } \right]$$

D

$$\left[ {\matrix{
{72} & {-84} \cr
{-63} & {51} \cr
} } \right]$$

We have, $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$

$$ \therefore $$ A^{2} = A.A = $$\left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]\left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$

= $$\left[ {\matrix{ {4 + 12} & { - 6 - 3} \cr { - 8 - 4} & {12 + 1} \cr } } \right]$$

= $$\left[ {\matrix{ {16} & { - 9} \cr { - 12} & {13} \cr } } \right]$$

Now, 3A^{2} + 12A

= $$3\left[ {\matrix{ {16} & { - 9} \cr { - 12} & {13} \cr } } \right] + 12\left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$

= $$\left[ {\matrix{ {48} & { - 27} \cr { - 36} & {39} \cr } } \right] + \left[ {\matrix{ {24} & { - 36} \cr { - 48} & {12} \cr } } \right]$$

= $$\left[ {\matrix{ {72} & { - 63} \cr { - 84} & {51} \cr } } \right]$$

$$ \therefore $$ adj(3A^{2} + 12A) = $$\left[ {\matrix{
{51} & {63} \cr
{84} & {72} \cr
} } \right]$$

$$ \therefore $$ A

= $$\left[ {\matrix{ {4 + 12} & { - 6 - 3} \cr { - 8 - 4} & {12 + 1} \cr } } \right]$$

= $$\left[ {\matrix{ {16} & { - 9} \cr { - 12} & {13} \cr } } \right]$$

Now, 3A

= $$3\left[ {\matrix{ {16} & { - 9} \cr { - 12} & {13} \cr } } \right] + 12\left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$

= $$\left[ {\matrix{ {48} & { - 27} \cr { - 36} & {39} \cr } } \right] + \left[ {\matrix{ {24} & { - 36} \cr { - 48} & {12} \cr } } \right]$$

= $$\left[ {\matrix{ {72} & { - 63} \cr { - 84} & {51} \cr } } \right]$$

$$ \therefore $$ adj(3A

3

If S is the set of distinct values of 'b' for which the following system of linear equations

x + y + z = 1

x + ay + z = 1

ax + by + z = 0

has no solution, then S is :

x + y + z = 1

x + ay + z = 1

ax + by + z = 0

has no solution, then S is :

A

an empty set

B

an infinite set

C

a finite set containing two or more elements

D

a singleton

$$\left| {\matrix{
1 & 1 & 1 \cr
1 & a & 1 \cr
a & b & 1 \cr
} } \right| = 0$$

$$ \Rightarrow $$ 1 [a – b] – 1 [1 – a] + 1 [b – a^{2}] = 0

$$ \Rightarrow $$ (a - 1)^{2} = 0

$$ \Rightarrow $$ a = 1

For a = 1, the equations become

x + y + z = 1

x + y + z = 1

x + by + z = 0

These equations give no solution for b = 1

$$ \Rightarrow $$ S is singleton set.

$$ \Rightarrow $$ 1 [a – b] – 1 [1 – a] + 1 [b – a

$$ \Rightarrow $$ (a - 1)

$$ \Rightarrow $$ a = 1

For a = 1, the equations become

x + y + z = 1

x + y + z = 1

x + by + z = 0

These equations give no solution for b = 1

$$ \Rightarrow $$ S is singleton set.

4

Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is **not** always true ?

A

adj (A) = $$\left| \right.$$A$$\left| \right.$$.A^{$$-$$1}

B

adj (adj(A)) = $$\left| \right.$$A$$\left| \right.$$.A

C

adj (adj(A)) = $$\left| \right.$$A$$\left| \right.$$^{2}.(adj(A))^{$$-$$1}

D

adj (adj(A)) = $$\left| \, \right.$$A $$\left| \, \right.$$.(adj(A))^{$$-$$1}

We know, the formula

A^{-1} = $${{adj\left( A \right)} \over {\left| A \right|}}$$

$$ \therefore $$ adj (A) = $$\left| \right.$$A$$\left| \right.$$.A^{$$-$$1}

**So, Option (A) is true.**

We know, the formula

adj (adj (A)) = $${\left| A \right|^{n - 2}}.A$$

Now if we put n = 3 as given that A is a 3 $$ \times $$ 3 matrix, we get

adj (adj (A)) = $${\left| A \right|^{3 - 2}}.A$$ = $$\left| A \right|.A$$

**So, Option (B) is also true.**

We know, the formula

adj (adj (A)) = $${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$

Now if we put n = 3 as given that A is a 3 $$ \times $$ 3 matrix, we get

adj (adj (A)) = $${\left| A \right|^{3 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$ = $${\left| A \right|^{2}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$

**So, Option (C) is also true.**

Now in this formula

adj (adj (A)) = $${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$

if we put n = 2, we get

adj (adj (A)) = $${\left| A \right|^{2 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$ = $${\left| A \right|}{\left( {adj\left( A \right)} \right)^{ - 1}}$$

But as A is a 3 $$ \times $$ 3 matrix so we can not take n = 2, so we can say for a 3 $$ \times $$ 3 matrix option (D) is not true.

**So, Option (D) is false.**

A

$$ \therefore $$ adj (A) = $$\left| \right.$$A$$\left| \right.$$.A

We know, the formula

adj (adj (A)) = $${\left| A \right|^{n - 2}}.A$$

Now if we put n = 3 as given that A is a 3 $$ \times $$ 3 matrix, we get

adj (adj (A)) = $${\left| A \right|^{3 - 2}}.A$$ = $$\left| A \right|.A$$

We know, the formula

adj (adj (A)) = $${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$

Now if we put n = 3 as given that A is a 3 $$ \times $$ 3 matrix, we get

adj (adj (A)) = $${\left| A \right|^{3 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$ = $${\left| A \right|^{2}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$

Now in this formula

adj (adj (A)) = $${\left| A \right|^{n - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$

if we put n = 2, we get

adj (adj (A)) = $${\left| A \right|^{2 - 1}}{\left( {adj\left( A \right)} \right)^{ - 1}}$$ = $${\left| A \right|}{\left( {adj\left( A \right)} \right)^{ - 1}}$$

But as A is a 3 $$ \times $$ 3 matrix so we can not take n = 2, so we can say for a 3 $$ \times $$ 3 matrix option (D) is not true.

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (1) *keyboard_arrow_right*

AIEEE 2003 (3) *keyboard_arrow_right*

AIEEE 2004 (3) *keyboard_arrow_right*

AIEEE 2005 (4) *keyboard_arrow_right*

AIEEE 2006 (2) *keyboard_arrow_right*

AIEEE 2007 (2) *keyboard_arrow_right*

AIEEE 2008 (3) *keyboard_arrow_right*

AIEEE 2009 (2) *keyboard_arrow_right*

AIEEE 2010 (3) *keyboard_arrow_right*

AIEEE 2011 (2) *keyboard_arrow_right*

AIEEE 2012 (2) *keyboard_arrow_right*

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

JEE Main 2014 (Offline) (2) *keyboard_arrow_right*

JEE Main 2015 (Offline) (2) *keyboard_arrow_right*

JEE Main 2016 (Offline) (3) *keyboard_arrow_right*

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

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

JEE Main 2017 (Offline) (2) *keyboard_arrow_right*

JEE Main 2017 (Online) 8th April Morning Slot (3) *keyboard_arrow_right*

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

JEE Main 2018 (Offline) (2) *keyboard_arrow_right*

JEE Main 2018 (Online) 15th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2018 (Online) 15th April Evening Slot (2) *keyboard_arrow_right*

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

JEE Main 2019 (Online) 9th January Morning Slot (2) *keyboard_arrow_right*

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

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

JEE Main 2019 (Online) 10th January Evening Slot (2) *keyboard_arrow_right*

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

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

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

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

JEE Main 2019 (Online) 8th April Morning Slot (2) *keyboard_arrow_right*

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

JEE Main 2019 (Online) 9th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th April Evening Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th April Evening Slot (2) *keyboard_arrow_right*

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

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

JEE Main 2020 (Online) 7th January Morning Slot (2) *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 (2) *keyboard_arrow_right*

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

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

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

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

JEE Main 2020 (Online) 4th September Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 5th September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 5th September Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 6th September Morning Slot (2) *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 (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th February Evening Shift (3) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th February Morning Shift (3) *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) 17th March Morning Shift (2) *keyboard_arrow_right*

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

JEE Main 2021 (Online) 18th March Morning Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 18th March Evening Shift (2) *keyboard_arrow_right*

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

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

JEE Main 2021 (Online) 22th July Evening Shift (2) *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 (2) *keyboard_arrow_right*

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

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

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

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

JEE Main 2021 (Online) 27th August Evening Shift (3) *keyboard_arrow_right*

JEE Main 2021 (Online) 31st August Morning Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 31st August Evening Shift (1) *keyboard_arrow_right*

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*