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

A helicopter is flying along the curve given by y – x^{3/2} = 7, (x $$ \ge $$ 0). A soldier positioned at the point $$\left( {{1 \over 2},7} \right)$$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is -

A

$${1 \over 6}\sqrt {{7 \over 3}} $$

B

$${{\sqrt 5 } \over 6}$$

C

$${1 \over 2}$$

D

$${1 \over 3}$$$$\sqrt {{7 \over 3}} $$

$$y - {x^{3/2}} = 7\left( {x \ge 0} \right)$$

$${{dy} \over {dx}} = {3 \over 2}{x^{1/2}}$$

$$\left( {{3 \over 2}\sqrt x } \right)\left( {{{7 - y} \over {{1 \over 2} - x}}} \right) = - 1$$

$$\left( {{3 \over 2}\sqrt x } \right)\left( {{{ - {x^{3/2}}} \over {{1 \over 2} - x}}} \right) = - 1$$

$${3 \over 2}.{x^2} = {1 \over 2} - x$$

$$3{x^2} = 1 - 2x$$

$$3{x^2} + 2x - 1 = 0$$

$$3{x^2} + 3x - x - 1 = 0$$

$$\left( {x + 1} \right)\left( {3x - 1} \right) = 0$$

$$ \therefore $$ $$x = - 1$$ (rejected)

$$x = {1 \over 3}$$

$$y = 7 + {x^{3/2}} = 7 + {\left( {{1 \over 3}} \right)^{3/2}}$$

$${\ell _{AB}} = \sqrt {{{\left( {{1 \over 2} - {1 \over 3}} \right)}^2} + {{\left( {{1 \over 3}} \right)}^3}} = \sqrt {{1 \over {36}} + {1 \over {27}}} $$

$$ = \sqrt {{{3 + 4} \over {9 \times 12}}} $$

$$ = \sqrt {{7 \over {108}}} = {1 \over 6}\sqrt {{7 \over 3}} $$

2

If xlog_{e}(log_{e}x) $$-$$ x^{2} + y^{2} = 4(y > 0), then $${{dy} \over {dx}}$$ at x = e is equal to :

A

$${{\left( {1 + 2e} \right)} \over {2\sqrt {4 + {e^2}} }}$$

B

$${{\left( {1 + 2e} \right)} \over {\sqrt {4 + {e^2}} }}$$

C

$${{\left( {2e - 1} \right)} \over {2\sqrt {4 + {e^2}} }}$$

D

$${e \over {\sqrt {4 + {e^2}} }}$$

Differentiating with respect to x,

$$x.{1 \over {\ell nx}}.{1 \over x} + \ell n(\ell nx) - 2x + 2y.{{dy} \over {dx}} = 0$$

at $$x = e$$ we get

$$1 - 2e + 2y{{dy} \over {dx}} = 0 \Rightarrow {{dy} \over {dx}} = {{2e - 1} \over {2y}}$$

$$ \Rightarrow {{dy} \over {dx}} = {{2e - 1} \over {2\sqrt {4 + {e^2}} }}\,\,$$

as $$y(e) = \sqrt {4 + {e^2}} $$

$$x.{1 \over {\ell nx}}.{1 \over x} + \ell n(\ell nx) - 2x + 2y.{{dy} \over {dx}} = 0$$

at $$x = e$$ we get

$$1 - 2e + 2y{{dy} \over {dx}} = 0 \Rightarrow {{dy} \over {dx}} = {{2e - 1} \over {2y}}$$

$$ \Rightarrow {{dy} \over {dx}} = {{2e - 1} \over {2\sqrt {4 + {e^2}} }}\,\,$$

as $$y(e) = \sqrt {4 + {e^2}} $$

3

The solution of the differential equation,

$${{dy} \over {dx}}$$ = (x – y)^{2}, when y(1) = 1, is :

$${{dy} \over {dx}}$$ = (x – y)

A

$$-$$ log_{e} $$\left| {{{1 + x - y} \over {1 - x + y}}} \right|$$ = x + y $$-$$ 2

B

log_{e} $$\left| {{{2 - x} \over {2 - y}}} \right|$$ = x $$-$$ y

C

log_{e} $$\left| {{{2 - y} \over {2 - x}}} \right|$$ = 2(y $$-$$ 1)

D

$$-$$ log_{e} $$\left| {{{1 - x + y} \over {1 + x - y}}} \right|$$ = 2(x $$-$$ 1)

x $$-$$ y = t

$$ \Rightarrow $$ $${{dy} \over {dx}} = 1 - {{dt} \over {dx}}$$

$$ \Rightarrow $$ 1 $$-$$ $${{dt} \over {dx}}$$ = t^{2} $$ \Rightarrow $$ $$\int {{{dt} \over {1 - {t^2}}}} $$ = $$\int {1dx} $$

$$ \Rightarrow $$ $${1 \over 2}\ell n\left( {{{1 + t} \over {1 - t}}} \right) = x + \lambda $$

$$ \Rightarrow $$ $${1 \over 2}\ell n\left( {{{1 + x - y} \over {1 - x + y}}} \right) = x + \lambda $$ given y(1) = 1

$$ \Rightarrow $$ $${1 \over 2}\ell n(1) = 1 + \lambda \Rightarrow \lambda = - 1$$

$$ \Rightarrow $$ $$\ell n\left( {{{1 + x - y} \over {1 - x + y}}} \right)$$ = 2(x $$-$$ 1)

$$ \Rightarrow $$ $$ - \ell n\left( {{{1 - x + y} \over {1 + x - y}}} \right)$$ = 2(x $$-$$ 1)

$$ \Rightarrow $$ $${{dy} \over {dx}} = 1 - {{dt} \over {dx}}$$

$$ \Rightarrow $$ 1 $$-$$ $${{dt} \over {dx}}$$ = t

$$ \Rightarrow $$ $${1 \over 2}\ell n\left( {{{1 + t} \over {1 - t}}} \right) = x + \lambda $$

$$ \Rightarrow $$ $${1 \over 2}\ell n\left( {{{1 + x - y} \over {1 - x + y}}} \right) = x + \lambda $$ given y(1) = 1

$$ \Rightarrow $$ $${1 \over 2}\ell n(1) = 1 + \lambda \Rightarrow \lambda = - 1$$

$$ \Rightarrow $$ $$\ell n\left( {{{1 + x - y} \over {1 - x + y}}} \right)$$ = 2(x $$-$$ 1)

$$ \Rightarrow $$ $$ - \ell n\left( {{{1 - x + y} \over {1 + x - y}}} \right)$$ = 2(x $$-$$ 1)

4

Let y = y(x) be the solution of the differential equation, x$${{dy} \over {dx}}$$ + y = x log_{e} x, (x > 1). If 2y(2) = log_{e} 4 $$-$$ 1, then y(e) is equal to :

A

$$ - {e \over 2}$$

B

$$ - {{{e^2}} \over 2}$$

C

$${{{e^2}} \over 4}$$

D

$${e \over 4}$$

$${{dy} \over {dx}} = {y \over x} = \ell nx$$

$${e^{\int {{1 \over x}dx} }} = x$$

$$xy = \int {x\ell nx + C} $$

$$\ell nx{{{x^2}} \over 2} - \int {{1 \over x}.{{{x^2}} \over 2}} $$

$$xy = {x \over 2}\ell nx - {{{x^2}} \over 4} + C,$$

for $$2y\left( 2 \right) = 2\ell n2 - 1$$

$$ \Rightarrow $$ $$C = 0$$

$$y = {x \over 2}\ell nx - {x \over 4}$$

$$y\left( e \right) = {e \over 4}$$

$${e^{\int {{1 \over x}dx} }} = x$$

$$xy = \int {x\ell nx + C} $$

$$\ell nx{{{x^2}} \over 2} - \int {{1 \over x}.{{{x^2}} \over 2}} $$

$$xy = {x \over 2}\ell nx - {{{x^2}} \over 4} + C,$$

for $$2y\left( 2 \right) = 2\ell n2 - 1$$

$$ \Rightarrow $$ $$C = 0$$

$$y = {x \over 2}\ell nx - {x \over 4}$$

$$y\left( e \right) = {e \over 4}$$

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*