Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

Subjective

Let $$ABC$$ be a triangle having $$O$$ and $$I$$ as its circumcenter and in centre respectively. If $$R$$ and $$r$$ are the circumradius and the inradius, respectively, then prove that $${\left( {IO} \right)^2} = {R^2} - 2{\mathop{\rm Rr}\nolimits} $$. Further show that the triangle BIO is a right-angled triangle if and only if $$b$$ is arithmetic mean of $$a$$ and $$c$$.

Solve it.

2

Subjective

Prove that a triangle $$ABC$$ is equilateral if and only if $$\tan A + \tan B + \tan C = 3\sqrt 3 $$.

Solve it.

3

Subjective

A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose $${60^ \circ }$$ and $${30^ \circ }$$ are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is at the points $$P$$ and $$Q$$ respectively on its path. Let $$\theta $$ be the angle of elevation of the bird when it is a point on the are of the circle exactly midway between $$P$$ and $$Q$$. Find the numerical value of $${\tan ^2}\theta $$. (Assume that the observer is not inside the vertical projection of the path of the bird.)

$${3 \over 5}$$

4

Subjective

A tower $$AB$$ leans towards west making an angle $$\alpha $$ with the vertical. The angular elevation of $$B$$, the topmost point of the tower is $$\beta $$ as observed from a point $$C$$ due west of $$A$$ at a distance $$d$$ from $$A$$. If the angular elevation of $$B$$ from a point $$D$$ due east of $$C$$ at a distance $$2d$$ from $$C$$ is $$\gamma $$, then prove that $$2$$ tan $$\alpha = - \cot \beta + \cot \gamma $$.

Solve it.

On those following papers in Subjective

Number in Brackets after Paper Indicates No. of Questions

IIT-JEE 2003 (1)

IIT-JEE 2001 (1)

IIT-JEE 2000 (1)

IIT-JEE 1999 (1)

IIT-JEE 1998 (2)

IIT-JEE 1994 (3)

IIT-JEE 1993 (1)

IIT-JEE 1992 (1)

IIT-JEE 1991 (3)

IIT-JEE 1990 (1)

IIT-JEE 1989 (1)

IIT-JEE 1988 (1)

IIT-JEE 1986 (1)

IIT-JEE 1985 (2)

IIT-JEE 1984 (2)

IIT-JEE 1983 (1)

IIT-JEE 1982 (1)

IIT-JEE 1981 (1)

IIT-JEE 1980 (3)

IIT-JEE 1979 (2)

IIT-JEE 1978 (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