Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geotechnical Engineering

Transportation Engineering

Irrigation

Engineering Mathematics

Construction Material and Management

Fluid Mechanics and Hydraulic Machines

Hydrology

Environmental Engineering

Engineering Mechanics

Structural Analysis

Reinforced Cement Concrete

Steel Structures

Geomatics Engineering Or Surveying

General Aptitude

1

Equation of a common tangent to the parabola y^{2} = 4x and the hyperbola xy = 2 is

A

x + y + 1 = 0

B

4x + 2y + 1 = 0

C

x – 2y + 4 = 0

D

x + 2y + 4 = 0

Let the equation of tangent to parabola

y^{2} = 4x be y = mx + $${1 \over m}$$

It is also a tangent to hyperbola xy = 2

$$ \Rightarrow $$ x$$\left( {mx + {1 \over m}} \right)$$ = 2

$$ \Rightarrow $$ x^{2}m + $${x \over m}$$ $$-$$ 2 = 0

D = 0 $$ \Rightarrow $$ m = $$-$$ $${1 \over 2}$$

So tangent is 2y + x + 4 = 0

y

It is also a tangent to hyperbola xy = 2

$$ \Rightarrow $$ x$$\left( {mx + {1 \over m}} \right)$$ = 2

$$ \Rightarrow $$ x

D = 0 $$ \Rightarrow $$ m = $$-$$ $${1 \over 2}$$

So tangent is 2y + x + 4 = 0

2

If tangents are drawn to the ellipse x2^{} + 2y^{2} = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :

A

$${{{x^2}} \over 2} + {{{y^2}} \over 4} = 1$$

B

$${1 \over {2{x^2}}} + {1 \over {4{y^2}}} = 1$$

C

$${1 \over {4{x^2}}} + {1 \over {2{y^2}}} = 1$$

D

$${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$$

Equation of general tangent on ellipse

$${x \over {a\,\sec \theta }} + {y \over {b\cos ec\theta }} = 1$$

$$a = \sqrt 2 ,\,\,b = 1$$

$$ \Rightarrow {x \over {\sqrt 2 \sec \theta }} + {y \over {\cos ec\theta }} = 1$$

Let the midpoint be (h, k)

$$h = {{\sqrt 2 \sec \theta } \over 2} \Rightarrow \cos \theta = {1 \over {\sqrt 2 h}}$$

and $$k = {{\cos ec\theta } \over 2} \Rightarrow \sin \theta = {1 \over {2k}}$$

$$ \because $$ $${\sin ^2}\theta + {\cos ^2}\theta = 1$$

$$ \Rightarrow $$ $${1 \over {2{h^2}}} + {1 \over {4{k^2}}} = 1$$

$$ \Rightarrow $$ $${1 \over {2{x^2}}} + {1 \over {4{y^2}}} = 1$$

$${x \over {a\,\sec \theta }} + {y \over {b\cos ec\theta }} = 1$$

$$a = \sqrt 2 ,\,\,b = 1$$

$$ \Rightarrow {x \over {\sqrt 2 \sec \theta }} + {y \over {\cos ec\theta }} = 1$$

Let the midpoint be (h, k)

$$h = {{\sqrt 2 \sec \theta } \over 2} \Rightarrow \cos \theta = {1 \over {\sqrt 2 h}}$$

and $$k = {{\cos ec\theta } \over 2} \Rightarrow \sin \theta = {1 \over {2k}}$$

$$ \because $$ $${\sin ^2}\theta + {\cos ^2}\theta = 1$$

$$ \Rightarrow $$ $${1 \over {2{h^2}}} + {1 \over {4{k^2}}} = 1$$

$$ \Rightarrow $$ $${1 \over {2{x^2}}} + {1 \over {4{y^2}}} = 1$$

3

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the
eccentricity of the hyperbola is :

A

$${{13} \over 6}$$

B

2

C

$${{13} \over 12}$$

D

$${{13} \over 8}$$

2b = 5 and 2ae = 13

b^{2} = a^{2}(e^{2} $$-$$ 1) $$ \Rightarrow $$ $${{25} \over 4}$$ = $${{169} \over 4}$$ $$-$$ a^{2}

$$ \Rightarrow $$ a $$=$$ 6 $$ \Rightarrow $$ e $$=$$ $${{13} \over {12}}$$

b

$$ \Rightarrow $$ a $$=$$ 6 $$ \Rightarrow $$ e $$=$$ $${{13} \over {12}}$$

4

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it ?

A

$$\left( {4\sqrt 2 ,2\sqrt 3 } \right)$$

B

$$\left( {4\sqrt 3 ,2\sqrt 3 } \right)$$

C

$$\left( {4\sqrt 3 ,2\sqrt 2 } \right)$$

D

$$\left( {4\sqrt 2 ,2\sqrt 2 } \right)$$

$${{2{b^2}} \over a} = 8$$ and 2ae $$=$$ 2b

$$ \Rightarrow $$ $${b \over a}$$ = e and 1 $$-$$ e^{2} = e^{2} $$ \Rightarrow $$ e $$=$$ $${1 \over {\sqrt 2 }}$$

$$ \Rightarrow $$ b = 4$$\sqrt 2 $$ and a $$=$$ 8

So equation of ellipse is $${{{x^2}} \over {64}} + {{{y^2}} \over {32}} = 1$$

$$ \Rightarrow $$ $${b \over a}$$ = e and 1 $$-$$ e

$$ \Rightarrow $$ b = 4$$\sqrt 2 $$ and a $$=$$ 8

So equation of ellipse is $${{{x^2}} \over {64}} + {{{y^2}} \over {32}} = 1$$

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (1) *keyboard_arrow_right*

AIEEE 2003 (2) *keyboard_arrow_right*

AIEEE 2004 (2) *keyboard_arrow_right*

AIEEE 2005 (3) *keyboard_arrow_right*

AIEEE 2006 (3) *keyboard_arrow_right*

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Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*

Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

Permutations and Combinations *keyboard_arrow_right*

Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

Sequences and Series *keyboard_arrow_right*

Matrices and Determinants *keyboard_arrow_right*

Vector Algebra and 3D Geometry *keyboard_arrow_right*

Probability *keyboard_arrow_right*

Statistics *keyboard_arrow_right*

Mathematical Reasoning *keyboard_arrow_right*

Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *keyboard_arrow_right*

Functions *keyboard_arrow_right*

Limits, Continuity and Differentiability *keyboard_arrow_right*

Differentiation *keyboard_arrow_right*

Application of Derivatives *keyboard_arrow_right*

Indefinite Integrals *keyboard_arrow_right*

Definite Integrals and Applications of Integrals *keyboard_arrow_right*

Differential Equations *keyboard_arrow_right*