Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

Subjective

A right circular cone with radius $$R$$ and height $$H$$ contains a liquid which eveporates at a rate proportional to its surface area in contact with air (proportionality constant $$ = k > 0$$. Find the time after which the come is empty.

$${\raise0.5ex\hbox{$\scriptstyle H$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle K$}}$$

2

Subjective

A hemispherical tank of radius $$2$$ metres is initially full of water and has an outlet of $$12$$ cm^{2} cross-sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law $$v(t)=0.6$$ $$\sqrt {2gh\left( t \right),} $$ where $$v(t)$$ and $$h(t)$$ are respectively the velocity of the flow through the outlet and the height of water level above the outlet at time $$t,$$ and $$g$$ is the acceleration due to gravity. Find the time it takes to empty the tank. (Hint: From a differential equation by relasing the decreases of water level to the outflow).

$${{14\pi } \over {27\sqrt g }}{\left( {10} \right)^5}$$

3

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

4

Subjective

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

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

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