Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

Subjective

A straight line $$L$$ through the origin meets the lines $$x + y = 1$$ and $$x + y = 3$$ at $$P $$ and $$Q$$ respectively. Through $$P$$ and $$Q$$ two straight lines $${L_1}$$ and $${L_2}$$ are drawn, parallel to $$2x - y = 5$$ and $$3x + y = 5$$ respectively. Lines $${L_1}$$ and $${L_2}$$ intersect at $$R$$. Show that the locus of $$R$$, as $$L$$ varies is a straight line.

Solve it.

2

Subjective

Let $$a, b, c$$ be real numbers with $${a^2} + {b^2} + {c^2} = 1.$$ Show that

represents a straight line.

the equation $$\left| {\matrix{ {ax - by - c} & {bx + ay} & {cx + a} \cr {bx + ay} & { - ax + by - c} & {cy + b} \cr {cx + a} & {cy + b} & { - ax - by + c} \cr } } \right| = 0$$

represents a straight line.

Solve it.

3

Subjective

Let $$ABC$$ and $$PQR$$ be any two triangles in the same plane. Assume that the prependiculars from the points $$A, B, C$$ to the sides $$QR, RP, PQ$$ respectively are concurrent. Using vector methods or otherwise, prove that the prependiculars from $$P, Q, R $$ to $$BC,$$ $$CA$$, $$AB$$ respectively are also concurrent.

Solve it.

4

Subjective

For points $$P\,\,\, = \left( {{x_1},\,{y_1}} \right)$$ and $$Q\,\,\, = \left( {{x_2},\,{y_2}} \right)$$ of the co-ordinate plane, a new distance $$d\left( {P,\,Q} \right)$$ is defined by $$d\left( {P,\,Q} \right)$$$$ = \left( {{x_2},\,{y_2}} \right)\left| {{x_1} - {x_2}} \right| + \left| {{y_1} - {y_2}} \right|.$$ Let $$O = (0, 0)$$ and $$A = (3, 2)$$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $$O$$ and $$A$$ consists of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.

Solve it.

On those following papers in Subjective

Number in Brackets after Paper Indicates No. of Questions

IIT-JEE 2005 (1)

IIT-JEE 2002 (2)

IIT-JEE 2001 (1)

IIT-JEE 2000 (2)

IIT-JEE 1998 (1)

IIT-JEE 1996 (1)

IIT-JEE 1993 (2)

IIT-JEE 1992 (1)

IIT-JEE 1991 (2)

IIT-JEE 1990 (2)

IIT-JEE 1989 (1)

IIT-JEE 1988 (1)

IIT-JEE 1985 (2)

IIT-JEE 1984 (1)

IIT-JEE 1983 (3)

IIT-JEE 1980 (1)

IIT-JEE 1979 (1)

IIT-JEE 1978 (3)

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