Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

Subjective

(i) $$PQ$$ is a vertical tower. $$P$$ is the foot and $$Q$$ is the top of the tower. $$A, B, C$$ are three points in the horizontal plane through $$P$$. The angles of elevation of $$Q$$ from $$A$$, $$B$$, $$C$$ are equal, and each is equal to $$\theta $$. The sides of the triangle $$ABC$$ are $$a, b, c$$; and the area of the triangle $$ABC$$ is $$\Delta $$. Show that the height of the tower is $${{abc\tan \theta } \over {4\Delta }}$$.

(ii) $$AB$$ is vertical pole. The end $$A$$ is on the level ground. $$C$$ is the middle point of $$AB$$. $$P$$ is a point on the level ground. The portion $$CB$$ subtends an angle $$\beta $$ at $$P$$. If $$AP = n\,AB,$$ then show that tan$$\beta $$ $$ = {n \over {2{n^2} + 1}}$$

(i) Solve it.

(ii) Solve it.

2

Subjective

$$ABC$$ is a triangle with $$AB=AC$$. $$D$$ is any point on the side $$BC$$. $$E$$ and $$F$$ are points on the side $$AB$$ and $$AC$$, respectively, such that $$DE$$ is parallel to $$AC$$, and $$DF$$ is parallel to $$AB$$. Prove that
$$$DF + FA + AE + ED = AB + AC$$$

Solve it.

3

Subjective

$$ABC$$ is a triangle. $$D$$ is the middle point of $$BC$$. If $$AD$$ is perpendicular to $$AC$$, then prove that
$$$\cos A\,\cos C = {{2\left( {{c^2} - {a^2}} \right)} \over {3ac}}$$$

Solve it.

4

Subjective

(a) If a circle is inscribed in a right angled triangle $$ABC$$ with the right angle at $$B$$, show that the diameter of the circle is equal to $$AB+BC-AC$$.

(b) If a triangle is inscribed in a circle, then the product of any two sides of the triangle is equal to the product of the diameter and the perpendicular distance of the third side from the opposite vertex. Prove the above statement.

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