Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

The locus of the centre of a circle which touches the circle $$\left| {z - {z_1}} \right| = a$$ and$$\left| {z - {z_2}} \right| = b\,$$ externally

($$z,\,{z_1}\,\& \,{z_2}\,$$ are complex numbers) will be

($$z,\,{z_1}\,\& \,{z_2}\,$$ are complex numbers) will be

A

an ellipse

B

a hyperbola

C

a circle

D

none of these

Let the circle be $$\left| {z - {z_3}} \right| = r.$$

Then according to given conditions

$$\left| {{z_3} - {z_1}} \right| = r + a$$ (Shown in the image)

and $$\left| {{z_3} - {z_2}} \right| = r + b.$$ (Shown in the image)

Eliminating $$r,$$ we get

$$\left| {{z_3} - {z_1}} \right| - \left| {{z_3} - {z_2}} \right| = a - b.$$

$$\therefore$$ Locus of center $${z_3}$$ is

$$\left| {z - {z_1}} \right| - \left| {z - {z_2}} \right| = a - b$$ = constant.

Definition of hyperbola says, when difference of distance between two points is constant from a particular point then that particular point will lie on a hyperbola.

Here distance of z_{1} from z_{3} is = $$r + a$$ and distance of z_{2} from z_{3} is = $$r + b$$

Now their difference = ($$r + a$$) - ($$r + b$$) = $$a - b$$ = a constant

$$\therefore$$ Locus of z_{3} is a hyperbola.

Then according to given conditions

$$\left| {{z_3} - {z_1}} \right| = r + a$$ (Shown in the image)

and $$\left| {{z_3} - {z_2}} \right| = r + b.$$ (Shown in the image)

Eliminating $$r,$$ we get

$$\left| {{z_3} - {z_1}} \right| - \left| {{z_3} - {z_2}} \right| = a - b.$$

$$\therefore$$ Locus of center $${z_3}$$ is

$$\left| {z - {z_1}} \right| - \left| {z - {z_2}} \right| = a - b$$ = constant.

Definition of hyperbola says, when difference of distance between two points is constant from a particular point then that particular point will lie on a hyperbola.

Here distance of z

Now their difference = ($$r + a$$) - ($$r + b$$) = $$a - b$$ = a constant

$$\therefore$$ Locus of z

2

MCQ (Single Correct Answer)

If $$\left| {z - 4} \right| < \left| {z - 2} \right|$$, its solution is given by

A

$${\mathop{\rm Re}\nolimits} (z) > 0$$

B

$${\mathop{\rm Re}\nolimits} (z) < 0$$

C

$${\mathop{\rm Re}\nolimits} (z) > 3$$

D

$${\mathop{\rm Re}\nolimits} (z) > 2$$

Given $$\left| {z - 4} \right| < \left| {z - 2} \right|$$

Let $$\,\,\,z = x + iy$$

$$ \Rightarrow \left| {\left. {\left( {x - 4} \right) + iy} \right)} \right| < \left| {\left( {x - 2} \right) + iy} \right|$$

$$ \Rightarrow {\left( {x - 4} \right)^2} + {y^2} < {\left( {x - 2} \right)^2} + {y^2}$$

$$ \Rightarrow {x^2} - 8x + 16 < {x^2} - 4x + 4$$

$$ \Rightarrow 12 < 4x$$

$$ \Rightarrow x > 3$$

$$ \Rightarrow {\mathop{\rm Re}\nolimits} \left( z \right) > 3$$

Let $$\,\,\,z = x + iy$$

$$ \Rightarrow \left| {\left. {\left( {x - 4} \right) + iy} \right)} \right| < \left| {\left( {x - 2} \right) + iy} \right|$$

$$ \Rightarrow {\left( {x - 4} \right)^2} + {y^2} < {\left( {x - 2} \right)^2} + {y^2}$$

$$ \Rightarrow {x^2} - 8x + 16 < {x^2} - 4x + 4$$

$$ \Rightarrow 12 < 4x$$

$$ \Rightarrow x > 3$$

$$ \Rightarrow {\mathop{\rm Re}\nolimits} \left( z \right) > 3$$

3

MCQ (Single Correct Answer)

z and w are two nonzero complex numbers such that $$\,\left| z \right| = \left| w \right|$$ and Arg z + Arg w =$$\pi $$ then z equals

A

$$\overline \omega $$

B

$$ - \overline \omega $$

C

$$\omega $$

D

$$ - \omega $$

Let $$\left| z \right| = \left| \omega \right| = r$$

$$\therefore$$ $$z = r{e^{i\theta }},\omega = r{e^{i\phi }}$$

where $$\,\,\theta + \phi = \pi .$$

$$\therefore$$ $$z = r{e^{i\left( {\pi - \phi } \right)}} = r{e^{i\pi }}.$$ $${e^{ - i\phi }} = - r{e^{ - i\phi }} = - \overline \omega .$$

[as $$\,\,\,\,\overline \omega = r{e^{ - i\phi }}$$ ]

$$\therefore$$ $$z = r{e^{i\theta }},\omega = r{e^{i\phi }}$$

where $$\,\,\theta + \phi = \pi .$$

$$\therefore$$ $$z = r{e^{i\left( {\pi - \phi } \right)}} = r{e^{i\pi }}.$$ $${e^{ - i\phi }} = - r{e^{ - i\phi }} = - \overline \omega .$$

[as $$\,\,\,\,\overline \omega = r{e^{ - i\phi }}$$ ]

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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