Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

A parabola has the origin as its focus and the line $$x=2$$ as the directrix. Then the vertex of the parabola is at

A

$$(0,2)$$

B

$$(1,0)$$

C

$$(0,1)$$

D

$$(2,0)$$

Vertex of a parabola is the mid point of focus and the point

where directrix meets the axis of the parabola.

Here focus is $$O\left( {0,0} \right)$$ and directrix meets the axis at $$B\left( {2,0} \right)$$

$$\therefore$$ Vertex of the parabola is $$(1,0)$$

2

MCQ (Single Correct Answer)

A focus of an ellipse is at the origin. The directrix is the line $$x=4$$ and the eccentricity is $${{1 \over 2}}$$. Then the length of the semi-major axis is

A

$${{8 \over 3}}$$

B

$${{2 \over 3}}$$

C

$${{4 \over 3}}$$

D

$${{5 \over 3}}$$

Perpendicular distance of directrix from focus

$$ = {a \over e} - ae = 4$$

$$ \Rightarrow a\left( {2 - {1 \over 2}} \right) = 4$$

$$ \Rightarrow a = {8 \over 3}$$

$$\therefore$$ Semi major axis $$=8/3$$

3

MCQ (Single Correct Answer)

The normal to a curve at $$P(x,y)$$ meets the $$x$$-axis at $$G$$. If the distance of $$G$$ from the origin is twice the abscissa of $$P$$, then the curve is a

A

circle

B

hyperbola

C

ellipse

D

parabola

Equation of normal at $$P\left( {x,y} \right)$$ is $$Y - y = - {{dx} \over {dy}}\left( {x - x} \right)$$

Coordinate of $$G$$ at $$X$$ axis is $$\left( {X,0} \right)$$ (let)

$$\therefore$$ $$0 - y = - {{dx} \over {dy}}\left( {X - x} \right) \Rightarrow y{{dy} \over {dx}} = X - x$$

$$ \Rightarrow X = x + y{{dy} \over {dx}}$$ $$\therefore$$ Co-ordinate of $$G\left( {x + y{{dy} \over {dx}},0} \right)$$

Given distance of $$G$$ from origin $$=$$ twice of the abscissa of $$P.$$

as distance cannot be $$-ve,$$ therefore abscissa $$x$$ should be $$+ve$$

$$\therefore$$ $$x + y{{dy} \over {dx}} = 2x \Rightarrow y{{dy} \over {dx}} = x \Rightarrow ydx = xdx$$

On Integrating $$ \Rightarrow {{{y^2}} \over 2} = {{{x^2}} \over 2} + {c_1} \Rightarrow {x^2} - {y^2} = - 2{c_1}$$

$$\therefore$$ the curve is a hyperbola

Coordinate of $$G$$ at $$X$$ axis is $$\left( {X,0} \right)$$ (let)

$$\therefore$$ $$0 - y = - {{dx} \over {dy}}\left( {X - x} \right) \Rightarrow y{{dy} \over {dx}} = X - x$$

$$ \Rightarrow X = x + y{{dy} \over {dx}}$$ $$\therefore$$ Co-ordinate of $$G\left( {x + y{{dy} \over {dx}},0} \right)$$

Given distance of $$G$$ from origin $$=$$ twice of the abscissa of $$P.$$

as distance cannot be $$-ve,$$ therefore abscissa $$x$$ should be $$+ve$$

$$\therefore$$ $$x + y{{dy} \over {dx}} = 2x \Rightarrow y{{dy} \over {dx}} = x \Rightarrow ydx = xdx$$

On Integrating $$ \Rightarrow {{{y^2}} \over 2} = {{{x^2}} \over 2} + {c_1} \Rightarrow {x^2} - {y^2} = - 2{c_1}$$

$$\therefore$$ the curve is a hyperbola

4

MCQ (Single Correct Answer)

The equation of a tangent to the parabola $${y^2} = 8x$$ is $$y=x+2$$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

A

$$(2,4)$$

B

$$(-2,0)$$

C

$$(-1,1)$$

D

$$(0,2)$$

Parabola $${y^2} = 8x$$

We know that the locus of point of intersection of two perpendicular tangents to a

parabola is its directrix. Point must be on the directrix of parabola

as equation of directrix $$x + 2 = 0 \Rightarrow x = - 2$$

Hence the point is $$\left( { - 2,0} \right)$$

We know that the locus of point of intersection of two perpendicular tangents to a

parabola is its directrix. Point must be on the directrix of parabola

as equation of directrix $$x + 2 = 0 \Rightarrow x = - 2$$

Hence the point is $$\left( { - 2,0} \right)$$

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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

Statistics

Mathematical Reasoning

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