Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

Subjective

Let $$u(x)$$ and $$v(x)$$ satisfy the differential equation $${{du} \over {dx}} + p\left( x \right)u = f\left( x \right)$$ and $${{dv} \over {dx}} + p\left( x \right)v = g\left( x \right),$$ where $$p(x) f(x)$$ and $$g(x)$$ are continuous functions. If $$u\left( {{x_1}} \right) > v\left( {{x_1}} \right)$$ for some $${{x_1}}$$ and $$f(x)>g(x)$$ for all $$x > {x_1},$$ prove that any point $$(x,y)$$ where $$x > {x_1},$$ does not satisfy the equations $$y=u(x)$$ and $$y=v(x)$$

Solve it

2

Subjective

Determine the equation of the curve passing through the origin, in the form $$y=f(x),$$ which satisfies the differential equation $${{dy} \over {dx}} = \sin \left( {10x + 6y} \right).\,$$

$$y = {1 \over 3}\left[ {{{\tan }^{ - 1}}\left( {{{5\tan 4x} \over {4 - 3\tan 4x}}} \right) - 5x} \right]$$

3

Subjective

Let $$y=f(x)$$ be a curve passing through $$(1,1)$$ such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area $$2.$$ From the differential equation and determine all such possible curves.

$$x+y=2$$

and $$xy=1,x,y>0$$

and $$xy=1,x,y>0$$

4

Subjective

A normal is drawn at a point $$P(x,y)$$ of a curve. It meets the $$x$$-axis at $$Q.$$ If $$PQ$$ is of constant length $$k,$$ then show that the differential equation describing such curves is $$y = {{dy} \over {dx}} = \pm \sqrt {{k^2} - {y^2}} $$

Find the equation of such a curve passing through $$(0,k).$$

Solve it.

On those following papers in Subjective

Number in Brackets after Paper Indicates No. of Questions

IIT-JEE 2009 (1)

IIT-JEE 2005 (1)

IIT-JEE 2004 (1)

IIT-JEE 2003 (1)

IIT-JEE 2001 (1)

IIT-JEE 1997 (1)

IIT-JEE 1996 (1)

IIT-JEE 1995 (1)

IIT-JEE 1994 (1)

IIT-JEE 1983 (1)

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

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