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 z$$ \in $$C, the set of complex numbers. Then the equation, 2|z + 3i| $$-$$ |z $$-$$ i| = 0 represents :

A

a circle with radius $${8 \over 3}.$$

B

a circle with diameter $${{10} \over 3}.$$

C

an ellipse with length of major axis $${{16} \over 3}.$$

D

an ellipse with length of minor axis $${{16} \over 9}.$$

Given,

2 $$\,\left| \, \right.$$z + 3i$$\,\left| \, \right.$$ = $$\,\left| \, \right.$$z $$-$$i$$\,\left| \, \right.$$

Let z = x + iy

$$ \Rightarrow $$$$\,\,\,$$ 2 $$\,\left| \, \right.$$ x + iy + 3i $$\,\left| \, \right.$$ = $$\,\left| \, \right.$$ x + iy $$-$$ i $$\,\left| \, \right.$$

$$ \Rightarrow $$$$\,\,\,$$ 2 $$\,\left| \, \right.$$ x + i (y + 3)$$\,\left| \, \right.$$ = $$\,\left| \, \right.$$ x + i (y $$-$$ 1)$$\,\left| \, \right.$$

$$ \Rightarrow $$$$\,\,\,$$ 2 $$\sqrt {{x^2} + {{\left( {y + 3} \right)}^2}} $$ = $$\sqrt {{x^2} + {{\left( {y - 1} \right)}^2}} $$

$$ \Rightarrow $$$$\,\,\,$$ 4 (x^{2} + y^{2} + 6y + 9) = x^{2} + y^{2} $$-$$ 2y + 1

$$ \Rightarrow $$$$\,\,\,$$ 3x^{2} + 3y^{2} + 26y + 35 = 0

$$ \Rightarrow $$$$\,\,\,$$ x^{2} + y^{2} + $${{26} \over 3}$$ y + $${{35} \over 3}$$ = 0

This is a equation of circle with center ($$-$$ $${{13} \over 3}$$, 0)

$$\therefore\,\,\,$$ Radius = $$\sqrt {0 + {{169} \over 9} - {{35} \over 3}} $$

= $$\sqrt {{{64} \over 9}} $$

= $${8 \over 3}$$

2 $$\,\left| \, \right.$$z + 3i$$\,\left| \, \right.$$ = $$\,\left| \, \right.$$z $$-$$i$$\,\left| \, \right.$$

Let z = x + iy

$$ \Rightarrow $$$$\,\,\,$$ 2 $$\,\left| \, \right.$$ x + iy + 3i $$\,\left| \, \right.$$ = $$\,\left| \, \right.$$ x + iy $$-$$ i $$\,\left| \, \right.$$

$$ \Rightarrow $$$$\,\,\,$$ 2 $$\,\left| \, \right.$$ x + i (y + 3)$$\,\left| \, \right.$$ = $$\,\left| \, \right.$$ x + i (y $$-$$ 1)$$\,\left| \, \right.$$

$$ \Rightarrow $$$$\,\,\,$$ 2 $$\sqrt {{x^2} + {{\left( {y + 3} \right)}^2}} $$ = $$\sqrt {{x^2} + {{\left( {y - 1} \right)}^2}} $$

$$ \Rightarrow $$$$\,\,\,$$ 4 (x

$$ \Rightarrow $$$$\,\,\,$$ 3x

$$ \Rightarrow $$$$\,\,\,$$ x

This is a equation of circle with center ($$-$$ $${{13} \over 3}$$, 0)

$$\therefore\,\,\,$$ Radius = $$\sqrt {0 + {{169} \over 9} - {{35} \over 3}} $$

= $$\sqrt {{{64} \over 9}} $$

= $${8 \over 3}$$

2

The locus of the point of intersection of the straight lines,

tx $$-$$ 2y $$-$$ 3t = 0

x $$-$$ 2ty + 3 = 0**(t $$ \in $$ R)**, is :

tx $$-$$ 2y $$-$$ 3t = 0

x $$-$$ 2ty + 3 = 0

A

an ellipse with eccentricity $${2 \over {\sqrt 5 }}$$

B

an ellipse with the length of major axis 6

C

a hyperbola with eccentricity $$\sqrt 5 $$

D

a hyperbola with the length of conjugate axis 3

Here, tx $$-$$ 2y $$-$$ 3t = 0 & x $$-$$ 2ty + 3 = 0

On solving, we get;

y = $${{6t} \over {2{t^2} - 2}}$$ = $${{3t} \over {{t^2} - 1}}$$ & x = $${{3{t^2} + 3} \over {{t^2} - 1}}$$

Put t = tan$$\theta $$

$$ \therefore $$ x = $$-$$ 3 sec 2$$\theta $$ & 2y = 3 ($$-$$ tan 2$$\theta $$)

$$ \because $$ sec^{2}2$$\theta $$ $$-$$ tan^{2}2$$\theta $$ = 1

$$ \Rightarrow $$ $${{{x^2}} \over 9}$$ $$-$$ $${{{y^2}} \over {9/4}}$$ = 1

which represents at hyperbola

$$ \therefore $$ a^{2} = 9 & b^{2} = 9/4

$$\lambda $$(T.A.) = 6; e^{2} = 1 + $${{9/4} \over 9}$$ = 1 + $${1 \over 4}$$ $$ \Rightarrow $$ e = $${{\sqrt 5 } \over 2}$$

On solving, we get;

y = $${{6t} \over {2{t^2} - 2}}$$ = $${{3t} \over {{t^2} - 1}}$$ & x = $${{3{t^2} + 3} \over {{t^2} - 1}}$$

Put t = tan$$\theta $$

$$ \therefore $$ x = $$-$$ 3 sec 2$$\theta $$ & 2y = 3 ($$-$$ tan 2$$\theta $$)

$$ \because $$ sec

$$ \Rightarrow $$ $${{{x^2}} \over 9}$$ $$-$$ $${{{y^2}} \over {9/4}}$$ = 1

which represents at hyperbola

$$ \therefore $$ a

$$\lambda $$(T.A.) = 6; e

3

If the common tangents to the parabola, x^{2} = 4y and the circle, x^{2} + y^{2} = 4 intersect at the point P, then the distance of P from the origin, is :

A

$$\sqrt 2 + 1$$

B

2(3 + 2 $$\sqrt 2 $$)

C

2($$\sqrt 2 $$ + 1)

D

3 + 2$$\sqrt 2 $$

Tangent to x

y = mx $$ \pm $$ 2$$\sqrt {1 + {m^2}} $$

Also, x

x

or x

For D = 0

we have; 16m

$$ \Rightarrow $$ m

$$ \Rightarrow $$ m

$$ \Rightarrow $$ m

$$ \Rightarrow $$ m

$$ \Rightarrow $$ m

$$ \Rightarrow $$ m

$$ \Rightarrow $$ m

4

If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles $${\cos ^{ - 1}}\left( {{1 \over 7}} \right)$$ and sec^{$$-$$1} (7) at the center respectivey, then the distance between these chords, is :

A

$${4 \over {\sqrt 7 }}$$

B

$${8 \over {\sqrt 7 }}$$

C

$${8 \over 7}$$

D

$${16 \over 7}$$

Since cos2$$\theta $$ = 1/7 $$ \Rightarrow $$ 2 cos

$$ \Rightarrow $$ 2 cos

$$ \Rightarrow $$ cos

$$ \Rightarrow $$ cos

$$ \Rightarrow $$ cos

Also, sec

= cos

= 2 cos

= cos$$\phi $$ = $${2 \over {\sqrt 7 }}$$

P

= $${4 \over {\sqrt 7 }} + {4 \over {\sqrt 7 }}$$ = $${8 \over {\sqrt 7 }}$$

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*