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 A = $$\left( {\matrix{
0 & {2q} & r \cr
p & q & { - r} \cr
p & { - q} & r \cr
} } \right).$$ If AA^{T} = I_{3}, then $$\left| p \right|$$ is

A

$${1 \over {\sqrt 2 }}$$

B

$${1 \over {\sqrt 5 }}$$

C

$${1 \over {\sqrt 6 }}$$

D

$${1 \over {\sqrt 3 }}$$

A is orthogonal matrix

$$ \Rightarrow $$ 0^{2} + p^{2} + p^{2} = 1

$$ \Rightarrow $$ $$\left| p \right| = {1 \over {\sqrt 2 }}$$

$$ \Rightarrow $$ 0

$$ \Rightarrow $$ $$\left| p \right| = {1 \over {\sqrt 2 }}$$

2

If $$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a - b} \cr
} } \right|$$

= (a + b + c) (x + a + b + c)^{2}, x $$ \ne $$ 0,

then x is equal to :

= (a + b + c) (x + a + b + c)

then x is equal to :

A

–2(a + b + c)

B

2(a + b + c)

C

abc

D

–(a + b + c)

$$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a - b} \cr
} } \right|$$

R_{1} $$ \to $$ R_{1} + R_{2} + R_{3}

$$ = \left| {\matrix{ {a + b + c} & {a + b + c} & {a + b + c} \cr {2b} & {b - c - a} & {2b} \cr {2c} & {2c} & {c - a - b} \cr } } \right|$$

$$ = \left( {a + b + c} \right)\left| {\matrix{ 1 & 0 & 0 \cr {2b} & { - \left( {a + b + c} \right)} & 0 \cr {2c} & {2c} & {c - a - b} \cr } } \right|$$

$$=$$ (a + b + c) (a + b + c)^{2}

$$ \Rightarrow $$ x $$=$$ $$-$$ 2(a + b + c)

R

$$ = \left| {\matrix{ {a + b + c} & {a + b + c} & {a + b + c} \cr {2b} & {b - c - a} & {2b} \cr {2c} & {2c} & {c - a - b} \cr } } \right|$$

$$ = \left( {a + b + c} \right)\left| {\matrix{ 1 & 0 & 0 \cr {2b} & { - \left( {a + b + c} \right)} & 0 \cr {2c} & {2c} & {c - a - b} \cr } } \right|$$

$$=$$ (a + b + c) (a + b + c)

$$ \Rightarrow $$ x $$=$$ $$-$$ 2(a + b + c)

3

Let A and B be two invertible matrices of order 3 $$ \times $$ 3. If det(ABA^{T}) = 8 and det(AB^{–1}) = 8,

then det (BA^{–1} B^{T}) is equal to :

then det (BA

A

$${1 \over 4}$$

B

16

C

$${1 \over {16}}$$

D

1

$${\left| A \right|^2}.\left| B \right| = 8$$

and $${{\left| A \right|} \over {\left| B \right|}} = 8 \Rightarrow \left| A \right| = 4$$

and $$\left| B \right| = {1 \over 2}$$

$$ \therefore $$ det(BA^{$$-$$1}. B^{T}) $$ = {1 \over 4} \times {1 \over 4} = {1 \over {16}}$$

and $${{\left| A \right|} \over {\left| B \right|}} = 8 \Rightarrow \left| A \right| = 4$$

and $$\left| B \right| = {1 \over 2}$$

$$ \therefore $$ det(BA

4

An ordered pair ($$\alpha $$, $$\beta $$) for which the system of linear equations

(1 + $$\alpha $$) x + $$\beta $$y + z = 2

$$\alpha $$x + (1 + $$\beta $$)y + z = 3

$$\alpha $$x + $$\beta $$y + 2z = 2

has a unique solution, is :

(1 + $$\alpha $$) x + $$\beta $$y + z = 2

$$\alpha $$x + (1 + $$\beta $$)y + z = 3

$$\alpha $$x + $$\beta $$y + 2z = 2

has a unique solution, is :

A

(–3, 1)

B

(1, –3)

C

(–4, 2)

D

(2, 4)

For unique solution

$$\Delta $$ $$ \ne $$ 0 $$ \Rightarrow $$ $$\left| {\matrix{ {1 + \alpha } & \beta & 1 \cr \alpha & {1 + \beta } & 1 \cr \alpha & \beta & 2 \cr } } \right| \ne 0$$

$$\left| {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr \alpha & \beta & 2 \cr } } \right| \ne 0 \Rightarrow \alpha + \beta \ne - 2$$

$$\Delta $$ $$ \ne $$ 0 $$ \Rightarrow $$ $$\left| {\matrix{ {1 + \alpha } & \beta & 1 \cr \alpha & {1 + \beta } & 1 \cr \alpha & \beta & 2 \cr } } \right| \ne 0$$

$$\left| {\matrix{ 1 & { - 1} & 0 \cr 0 & 1 & { - 1} \cr \alpha & \beta & 2 \cr } } \right| \ne 0 \Rightarrow \alpha + \beta \ne - 2$$

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*

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AIEEE 2011 (2) *keyboard_arrow_right*

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

JEE Main 2014 (Offline) (2) *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*