Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

The conjugate of a complex number is $${1 \over {i - 1}}$$ then that complex number is

A

$${{ - 1} \over {i - 1}}$$

B

$${1 \over {i + 1}}\,$$

C

$${{ - 1} \over {i + 1}}$$

D

$${1 \over {i - 1}}$$

$$\left( {{1 \over {i - 1}}} \right) = {1 \over { - i - 1}} = {{ - 1} \over {i + 1}}$$

2

MCQ (Single Correct Answer)

If $$\,\left| {z + 4} \right|\,\, \le \,\,3\,$$, then the maximum value of $$\left| {z + 1} \right|$$ is

A

6

B

0

C

4

D

10

$$z$$ lies on or inside the circle with center $$(-4,0)$$ and radius $$3$$ units.

From the Argand diagram maximum value of $$\left| {z + 1} \right|$$ is $$6$$

From the Argand diagram maximum value of $$\left| {z + 1} \right|$$ is $$6$$

3

MCQ (Single Correct Answer)

The value of $$\sum\limits_{k = 1}^{10} {\left( {\sin {{2k\pi } \over {11}} + i\,\,\cos {{2k\pi } \over {11}}} \right)} $$ is

A

i

B

1

C

- 1

D

- i

$$\sum\limits_{k = 1}^{10} {\left( {\sin {{2k\pi } \over {11}} + i\cos {{2k\pi } \over {11}}} \right)} $$

$$ = i\sum\limits_{k = 1}^{10} {\left( {\cos {{2k\pi } \over {11}} - i\,\sin {{2k\pi } \over {11}}} \right)} $$

$$ = i\sum\limits_{k = 1}^{10} {{e^{ - {{2k\pi } \over {11}}}}} i = i\left\{ {\sum\limits_{k = 0}^{10} {{e^{ - {{2k\pi } \over {11}}}}} - 1} \right\}$$

$$ = i\left[ {1 + {e^{ - {{2\pi } \over {11}}i}} + e - {{4\pi } \over {11}}i + .....11\,\,terms} \right] - i$$

$$ = i\left[ {{{1 - {{\left( {{e^{ - {{2\pi } \over {11}}}}} \right)}^{11}}} \over {1 - {e^{ - {{2\pi } \over {11}}i}}}}} \right] - i$$

$$ = i\left[ {{{1 - {e^{ - 2\pi i}}} \over {1 - {e^{ - {{2\pi } \over {11}}i}}}}} \right] - i$$

$$ = i \times 0 - i$$

[as $$\,\,\,\,\,\,$$ $${e^{ - 2\pi i}} = 1$$ ]

$$ = - i$$

$$ = i\sum\limits_{k = 1}^{10} {\left( {\cos {{2k\pi } \over {11}} - i\,\sin {{2k\pi } \over {11}}} \right)} $$

$$ = i\sum\limits_{k = 1}^{10} {{e^{ - {{2k\pi } \over {11}}}}} i = i\left\{ {\sum\limits_{k = 0}^{10} {{e^{ - {{2k\pi } \over {11}}}}} - 1} \right\}$$

$$ = i\left[ {1 + {e^{ - {{2\pi } \over {11}}i}} + e - {{4\pi } \over {11}}i + .....11\,\,terms} \right] - i$$

$$ = i\left[ {{{1 - {{\left( {{e^{ - {{2\pi } \over {11}}}}} \right)}^{11}}} \over {1 - {e^{ - {{2\pi } \over {11}}i}}}}} \right] - i$$

$$ = i\left[ {{{1 - {e^{ - 2\pi i}}} \over {1 - {e^{ - {{2\pi } \over {11}}i}}}}} \right] - i$$

$$ = i \times 0 - i$$

[as $$\,\,\,\,\,\,$$ $${e^{ - 2\pi i}} = 1$$ ]

$$ = - i$$

4

MCQ (Single Correct Answer)

If $${z^2} + z + 1 = 0$$, where z is complex number, then value of $${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^2} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^2} + .......... + {\left( {{z^6} + {1 \over {{z^6}}}} \right)^2}$$ is

A

18

B

54

C

6

D

12

$${z^2} + z + 1 = 0 \Rightarrow z = \omega \,\,\,$$ or $$\,\,\,{\omega ^2}$$

So, $$z + {1 \over z} = \omega + {\omega ^2} = - 1$$

$${z^2} + {1 \over {{z^2}}} = {\omega ^2} + \omega = - 1,$$

$${z^3} + {1 \over {{z^3}}} = {\omega ^3} + {\omega ^3} = 2$$

$${z^4} + {1 \over {{z^4}}} = - 1,$$

$${z^5} + {1 \over {{z^5}}} = - 1$$

and $$\,\,\,\,{z^6} + {1 \over {{z^6}}} = 2$$

$$\therefore$$ The given sum $$ = 1 + 1 + 4 + 1 + 1 + 4 = 12$$

So, $$z + {1 \over z} = \omega + {\omega ^2} = - 1$$

$${z^2} + {1 \over {{z^2}}} = {\omega ^2} + \omega = - 1,$$

$${z^3} + {1 \over {{z^3}}} = {\omega ^3} + {\omega ^3} = 2$$

$${z^4} + {1 \over {{z^4}}} = - 1,$$

$${z^5} + {1 \over {{z^5}}} = - 1$$

and $$\,\,\,\,{z^6} + {1 \over {{z^6}}} = 2$$

$$\therefore$$ The given sum $$ = 1 + 1 + 4 + 1 + 1 + 4 = 12$$

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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