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

The point represented by 2 + *i* in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $$2\sqrt 2 $$ units in the south-westwardsdirection. Then its new position in the Argand plane is at the point represented by :

A

2 + 2i

B

1 + i

C

$$-$$1 $$-$$ i

D

$$-$$2 $$-$$2i

Here,

z $$-$$ (3 + 3i) = $$2\sqrt 2 $$ (cos($$-$$135

= $$2\sqrt 2 $$ ($$-$$ $${1 \over {\sqrt 2 }}$$ $$-$$$${i \over {\sqrt 2 }}$$)

= $$-$$ 2 $$-$$ 2i

$$ \Rightarrow $$ z = 3 + 3 i $$-$$ 2 $$-$$ 2 i = 1 + i

Polar form of a complex number :

z = r (cos$$\theta $$ + i sin$$\theta $$)

Here r = modulus of z and $$\theta $$ argument of z.

2

Let $$\omega $$ be a complex number such that 2$$\omega $$ + 1 = z where z = $$\sqrt {-3} $$. If

$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$,

then k is equal to

$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$,

then k is equal to

A

z

B

-1

C

1

D

-z

Given 2$$\omega $$ + 1 = z;

z = $$\sqrt 3 i$$

$$ \Rightarrow $$ $$\omega = {{\sqrt 3 i - 1} \over 2}$$

$$ \Rightarrow $$ As $$\omega $$ is complex cube root of unity.

$${\omega ^3} = 1$$

$$1 + \omega + {\omega ^2} = 0$$

$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$

$$ \Rightarrow $$ $$\left| {\matrix{ 1 & 1 & 1 \cr 1 & \omega & {{\omega ^2}} \cr 1 & {{\omega ^2}} & \omega \cr } } \right| = 3k$$

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

$$ \Rightarrow $$ $$\left| {\matrix{ 3 & 0 & 0 \cr 1 & \omega & {{\omega ^2}} \cr 1 & {{\omega ^2}} & \omega \cr } } \right| = 3k$$

$$ \Rightarrow $$ $$3\left( {{\omega ^2} - {\omega ^4}} \right) = 3k$$

$$ \Rightarrow $$ $$\left( {{\omega ^2} - \omega } \right) = k$$

$$ \therefore $$ $$k = \left( {{{ - 1 - \sqrt 3 i} \over 2}} \right) - \left( {{{ - 1 + \sqrt 3 i} \over 2}} \right)$$

$$ \Rightarrow $$ k = $${ - \sqrt 3 i}$$ = -z

z = $$\sqrt 3 i$$

$$ \Rightarrow $$ $$\omega = {{\sqrt 3 i - 1} \over 2}$$

$$ \Rightarrow $$ As $$\omega $$ is complex cube root of unity.

$${\omega ^3} = 1$$

$$1 + \omega + {\omega ^2} = 0$$

$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$

$$ \Rightarrow $$ $$\left| {\matrix{ 1 & 1 & 1 \cr 1 & \omega & {{\omega ^2}} \cr 1 & {{\omega ^2}} & \omega \cr } } \right| = 3k$$

Applying R

$$ \Rightarrow $$ $$\left| {\matrix{ 3 & 0 & 0 \cr 1 & \omega & {{\omega ^2}} \cr 1 & {{\omega ^2}} & \omega \cr } } \right| = 3k$$

$$ \Rightarrow $$ $$3\left( {{\omega ^2} - {\omega ^4}} \right) = 3k$$

$$ \Rightarrow $$ $$\left( {{\omega ^2} - \omega } \right) = k$$

$$ \therefore $$ $$k = \left( {{{ - 1 - \sqrt 3 i} \over 2}} \right) - \left( {{{ - 1 + \sqrt 3 i} \over 2}} \right)$$

$$ \Rightarrow $$ k = $${ - \sqrt 3 i}$$ = -z

3

If $$\alpha ,\beta \in C$$ are the distinct roots of the equation

x^{2} - x + 1 = 0, then $${\alpha ^{101}} + {\beta ^{107}}$$ is equal to

x

A

2

B

-1

C

0

D

1

Given equation,

x^{2} $$-$$ x + 1 = 0

Roots of this equation

x = $${{1 \pm \sqrt 3 i} \over 2}$$

$$\therefore\,\,\,$$ $$ \propto \, = \,{{1 + \sqrt 3 \,i} \over 2}$$

and $$\beta = \,{{1 - \sqrt 3 \,i} \over 2}$$

We know;

$$\omega = {{ - 1 + \sqrt 3 \,i} \over 2} = - \left( {{{1 - \sqrt 3 \,i} \over 2}} \right) = - \beta $$

and $${\omega ^2} = {{ - 1 - \sqrt 3 \,i} \over 2} = - \left( {{{1 + \sqrt 3 \,i} \over 2}} \right) = - \propto $$

$$\therefore\,\,\,$$ $$ \propto \, = - {\omega ^2}$$ and $$\beta \, = \, - \omega $$

$$\therefore\,\,\,$$ $${ \propto ^{101}} + {\beta ^{107}}$$

$$ = {\left( { - {\omega ^2}} \right)^{101}} + {\left( { - \omega } \right)^{107}}$$

$$ = {\left( { - 1} \right)^{101}}.{\left( {{\omega ^2}} \right)^{101}} + {\left( { - 1} \right)^{107}}.{\left( \omega \right)^{107}}$$

$$ = - 1.{\left( {{\omega ^2}} \right)^{101}} - {\omega ^{107}}$$

$$ = - \left( {{\omega ^{202}} + {\omega ^{107}}} \right)$$

$$ = - \left( {{\omega ^{3.67}}.\omega + {\omega ^{3.35}}.{\omega ^2}} \right)$$

$$ = - \left( {\omega + {\omega ^2}} \right)\,\,\,$$ [ as $$\,\,\,$$ $${\omega ^{3n}} = 1$$]

$$ = - \left( { - 1} \right)$$ $$\,\,\,\,\,\,$$ [as $$\,\,\,$$ $$1 + \omega + {\omega ^2} = 0$$ ]

$$ = 1$$

x

Roots of this equation

x = $${{1 \pm \sqrt 3 i} \over 2}$$

$$\therefore\,\,\,$$ $$ \propto \, = \,{{1 + \sqrt 3 \,i} \over 2}$$

and $$\beta = \,{{1 - \sqrt 3 \,i} \over 2}$$

We know;

$$\omega = {{ - 1 + \sqrt 3 \,i} \over 2} = - \left( {{{1 - \sqrt 3 \,i} \over 2}} \right) = - \beta $$

and $${\omega ^2} = {{ - 1 - \sqrt 3 \,i} \over 2} = - \left( {{{1 + \sqrt 3 \,i} \over 2}} \right) = - \propto $$

$$\therefore\,\,\,$$ $$ \propto \, = - {\omega ^2}$$ and $$\beta \, = \, - \omega $$

$$\therefore\,\,\,$$ $${ \propto ^{101}} + {\beta ^{107}}$$

$$ = {\left( { - {\omega ^2}} \right)^{101}} + {\left( { - \omega } \right)^{107}}$$

$$ = {\left( { - 1} \right)^{101}}.{\left( {{\omega ^2}} \right)^{101}} + {\left( { - 1} \right)^{107}}.{\left( \omega \right)^{107}}$$

$$ = - 1.{\left( {{\omega ^2}} \right)^{101}} - {\omega ^{107}}$$

$$ = - \left( {{\omega ^{202}} + {\omega ^{107}}} \right)$$

$$ = - \left( {{\omega ^{3.67}}.\omega + {\omega ^{3.35}}.{\omega ^2}} \right)$$

$$ = - \left( {\omega + {\omega ^2}} \right)\,\,\,$$ [ as $$\,\,\,$$ $${\omega ^{3n}} = 1$$]

$$ = - \left( { - 1} \right)$$ $$\,\,\,\,\,\,$$ [as $$\,\,\,$$ $$1 + \omega + {\omega ^2} = 0$$ ]

$$ = 1$$

4

The set of all $$\alpha $$ $$ \in $$ **R**, for which w = $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$ is purely imaginary number, for all z $$ \in $$ **C** satisfying |z| = 1 and Re z $$ \ne $$ 1, is :

A

an empty set

B

{0}

C

$$\left\{ {0,{1 \over 4}, - {1 \over 4}} \right\}$$

D

equal to **R**

As w = $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$, w is purely imaginary

$$ \therefore w$$ + $$\bar w$$ = 0

$$ \Rightarrow $$ $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$ + $${{1 + \left( {1 - 8\alpha } \right)\bar z} \over {1 - \bar z}}$$ = 0

$$ \Rightarrow $$ [1 + (1 - 8$$\alpha $$)][1 - $$\bar z$$] + [1 + ( 1 - 8$$\alpha $$)][1 - z] = 0

$$ \Rightarrow $$ 2 - (z + $$\bar z$$) + (1 - 8$$\alpha $$)(z + $$\bar z$$) - 2(1 - 8$$\alpha $$) = 0

$$ \Rightarrow $$ 2 - (z + $$\bar z$$) + (z + $$\bar z$$) - 8$$\alpha $$(z + $$\bar z$$) - 2 + 16$$\alpha $$ = 0

$$ \Rightarrow $$ 16$$\alpha $$ = 8$$\alpha $$(z + $$\bar z)$$

z + $$\bar z$$ = 2 or $$\alpha $$ = 0

but z + $$\bar z$$ = 2 is not possible as Re(Z) $$ \ne $$ 1

$$ \therefore $$ $$\alpha $$ = 0

$$ \therefore $$ $$\alpha $$ $$ \in $$ {0}

$$ \therefore w$$ + $$\bar w$$ = 0

$$ \Rightarrow $$ $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$ + $${{1 + \left( {1 - 8\alpha } \right)\bar z} \over {1 - \bar z}}$$ = 0

$$ \Rightarrow $$ [1 + (1 - 8$$\alpha $$)][1 - $$\bar z$$] + [1 + ( 1 - 8$$\alpha $$)][1 - z] = 0

$$ \Rightarrow $$ 2 - (z + $$\bar z$$) + (1 - 8$$\alpha $$)(z + $$\bar z$$) - 2(1 - 8$$\alpha $$) = 0

$$ \Rightarrow $$ 2 - (z + $$\bar z$$) + (z + $$\bar z$$) - 8$$\alpha $$(z + $$\bar z$$) - 2 + 16$$\alpha $$ = 0

$$ \Rightarrow $$ 16$$\alpha $$ = 8$$\alpha $$(z + $$\bar z)$$

z + $$\bar z$$ = 2 or $$\alpha $$ = 0

but z + $$\bar z$$ = 2 is not possible as Re(Z) $$ \ne $$ 1

$$ \therefore $$ $$\alpha $$ = 0

$$ \therefore $$ $$\alpha $$ $$ \in $$ {0}

Number in Brackets after Paper Name Indicates No of Questions

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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*