Joint Entrance Examination

Graduate Aptitude Test in Engineering

Strength of Materials Or Solid Mechanics

Structural Analysis

Construction Material and Management

Reinforced Cement Concrete

Steel Structures

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

Hydrology

Irrigation

Geomatics Engineering Or Surveying

Environmental Engineering

Transportation Engineering

Engineering Mathematics

General Aptitude

1

Let $${\left( { - 2 - {1 \over 3}i} \right)^3} = {{x + iy} \over {27}}\left( {i = \sqrt { - 1} } \right),\,\,$$ where x and y are real numbers, then y $$-$$ x equals :

A

$$-$$ 85

B

85

C

$$-$$ 91

D

91

$${\left( { - 2 - {i \over 3}} \right)^3} = - {{{{\left( {6 + i} \right)}^3}} \over {27}}$$

$$ = {{ - 198 - 107i} \over {27}} = {{x + iy} \over {27}}$$

Hence, $$y - x = 198 - 107 = 91$$

$$ = {{ - 198 - 107i} \over {27}} = {{x + iy} \over {27}}$$

Hence, $$y - x = 198 - 107 = 91$$

2

Let z be a complex number such that |z| + z = 3 + i (where i = $$\sqrt { - 1} $$). Then |z| is equal to

A

$${{\sqrt {34} } \over 3}$$

B

$${5 \over 3}$$

C

$${5 \over 4}$$

D

$${{\sqrt {41} } \over 4}$$

$$\left| z \right| + z = 3 + i$$

$$z = 3 - \left| z \right| + i$$

Let $$3 - \left| z \right| = a \Rightarrow \left| z \right| = \left( {3 - a} \right)$$

$$ \Rightarrow z = a + i \Rightarrow \left| z \right| = \sqrt {{a^2} + 1} $$

$$ \Rightarrow 9 + {a^2} - 6a = {a^2} + 1 \Rightarrow a = {8 \over 6} = {4 \over 3}$$

$$ \Rightarrow \left| z \right| = 3 - {4 \over 3} = {5 \over 3}$$

$$z = 3 - \left| z \right| + i$$

Let $$3 - \left| z \right| = a \Rightarrow \left| z \right| = \left( {3 - a} \right)$$

$$ \Rightarrow z = a + i \Rightarrow \left| z \right| = \sqrt {{a^2} + 1} $$

$$ \Rightarrow 9 + {a^2} - 6a = {a^2} + 1 \Rightarrow a = {8 \over 6} = {4 \over 3}$$

$$ \Rightarrow \left| z \right| = 3 - {4 \over 3} = {5 \over 3}$$

3

If $${{z - \alpha } \over {z + \alpha }}\left( {\alpha \in R} \right)$$ is a purely imaginary number and | z | = 2, then a value of $$\alpha $$ is :

A

$${1 \over 2}$$

B

$$\sqrt 2 $$

C

2

D

1

$${{z - \alpha } \over {z + \alpha }} + {{\overline z - \alpha } \over {\overline z + \alpha }} = 0$$

$$z\overline z + z\alpha - \alpha \overline z - {\alpha ^2} + z\overline z - z\alpha + \overline z \alpha - {\alpha ^2} = 0$$

$${\left| z \right|^2} = {\alpha ^2},$$ $$a = \pm 2$$

$$z\overline z + z\alpha - \alpha \overline z - {\alpha ^2} + z\overline z - z\alpha + \overline z \alpha - {\alpha ^2} = 0$$

$${\left| z \right|^2} = {\alpha ^2},$$ $$a = \pm 2$$

4

Let z_{1} and z_{2} be two complex numbers satisfying | z_{1} | = 9 and | z_{2} – 3 – 4i | = 4. Then the minimum value of
| z_{1} – z_{2} | is :

A

0

B

1

C

2

D

$$\sqrt 2 $$

$$\left| {{z_1}} \right| = 9,\,\,\left| {{z_2} - \left( {3 + 4i} \right)} \right| = 4$$

$${C_1},(0,0)$$ radius r_{1} = 9

C_{2} (3, 4), radius r_{2} = 4

C_{1}C_{2} = $$\left| {{r_1} - {r_2}} \right| = 5$$

$$ \therefore $$ Circle touches internally

$$ \therefore $$ $${\left| {{z_1} - {z_2}} \right|_{\min }} = 0$$

$${C_1},(0,0)$$ radius r

C

C

$$ \therefore $$ Circle touches internally

$$ \therefore $$ $${\left| {{z_1} - {z_2}} \right|_{\min }} = 0$$

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*