Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geotechnical Engineering

Transportation Engineering

Irrigation

Engineering Mathematics

Construction Material and Management

Fluid Mechanics and Hydraulic Machines

Hydrology

Environmental Engineering

Engineering Mechanics

Structural Analysis

Reinforced Cement Concrete

Steel Structures

Geomatics Engineering Or Surveying

General Aptitude

1

The curve amongst the family of curves represented by the differential equation, (x^{2} – y^{2})dx + 2xy dy = 0 which passes through (1, 1) is

A

a circle with centre on the y-axis

B

an ellipse with major axis along the y-axis

C

a circle with centre on the x-axis

D

a hyperbola with transverse axis along the x-axis

(x^{2} $$-$$ y^{2}) dx + 2xy dy = 0

$${{dy} \over {dx}} = {{{y^2} - {x^2}} \over {2xy}}$$

Put $$y = vx \Rightarrow {{dy} \over {dx}} = v + x{{dv} \over {dx}}$$

Solving we get,

$$\int {{{2v} \over {{v^2} + 1}}dv = \int { - {{dx} \over x}} } $$

ln(v^{2} + 1) = $$-$$ ln x + C

(y^{2} + x^{2}) = Cx

1 + 1 = C $$ \Rightarrow $$ C = 2

y^{2} + x^{2} = 2x

$${{dy} \over {dx}} = {{{y^2} - {x^2}} \over {2xy}}$$

Put $$y = vx \Rightarrow {{dy} \over {dx}} = v + x{{dv} \over {dx}}$$

Solving we get,

$$\int {{{2v} \over {{v^2} + 1}}dv = \int { - {{dx} \over x}} } $$

ln(v

(y

1 + 1 = C $$ \Rightarrow $$ C = 2

y

2

Let f be a differentiable function such that f '(x) = 7 - $${3 \over 4}{{f\left( x \right)} \over x},$$ (x > 0) and f(1) $$ \ne $$ 4. Then $$\mathop {\lim }\limits_{x \to 0'} \,$$ xf$$\left( {{1 \over x}} \right)$$

A

does not exist

B

exists and equals $${4 \over 7}$$

C

exists and equals 4

D

exists and equals 0

$$f'(x) = 7 - {3 \over 4}{{f\left( x \right)} \over x}\,\,\,\left( {x > 0} \right)$$

Given f(1) $$ \ne $$ 4 $$\mathop {\lim }\limits_{x \to {0^ + }} \,xf\left( {{1 \over x}} \right)\, = ?$$

$${{dy} \over {dx}} + {3 \over 4}{y \over x} = 7$$ (This is LDE)

IF $$ = {e^{\int {{3 \over {4x}}dx} }} = {e^{{3 \over 4}\ln \left| x \right|}} = {x^{{3 \over 4}}}$$

$$y.{x^{{3 \over 4}}} = \int {7.{x^{{3 \over 4}}}} dx$$

$$y.{x^{{3 \over 4}}} = 7.{{{x^{{7 \over 4}}}} \over {{7 \over 4}}} + C$$

$$f(x) = 4x + C.{x^{ - {3 \over 4}}}$$

$$f\left( {{1 \over 4}} \right) = {4 \over x} + C.{x^{{3 \over 4}}}$$

$$\mathop {\lim }\limits_{x \to {0^ + }} xf\left( {{1 \over x}} \right) = \mathop {\lim }\limits_{x \to {0^ + }} \left( {4 + C.{x^{{7 \over 4}}}} \right) = 4$$

Given f(1) $$ \ne $$ 4 $$\mathop {\lim }\limits_{x \to {0^ + }} \,xf\left( {{1 \over x}} \right)\, = ?$$

$${{dy} \over {dx}} + {3 \over 4}{y \over x} = 7$$ (This is LDE)

IF $$ = {e^{\int {{3 \over {4x}}dx} }} = {e^{{3 \over 4}\ln \left| x \right|}} = {x^{{3 \over 4}}}$$

$$y.{x^{{3 \over 4}}} = \int {7.{x^{{3 \over 4}}}} dx$$

$$y.{x^{{3 \over 4}}} = 7.{{{x^{{7 \over 4}}}} \over {{7 \over 4}}} + C$$

$$f(x) = 4x + C.{x^{ - {3 \over 4}}}$$

$$f\left( {{1 \over 4}} \right) = {4 \over x} + C.{x^{{3 \over 4}}}$$

$$\mathop {\lim }\limits_{x \to {0^ + }} xf\left( {{1 \over x}} \right) = \mathop {\lim }\limits_{x \to {0^ + }} \left( {4 + C.{x^{{7 \over 4}}}} \right) = 4$$

3

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)

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

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

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

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

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

4

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

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (2) *keyboard_arrow_right*

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Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *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*

Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *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*