Joint Entrance Examination

Graduate Aptitude Test in Engineering

Strength of Materials Or Solid Mechanics

Structural Analysis

Construction Material and Management

Reinforced Cement Concrete

Steel Structures

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

Hydrology

Irrigation

Geomatics Engineering Or Surveying

Environmental Engineering

Transportation Engineering

Engineering Mathematics

General Aptitude

1

Let S = $$\left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm 1.$$ Then S represents

A

an ellipse whose eccentricity is $${1 \over {\sqrt {r + 1} }},$$ where r > 1

B

an ellipse whose eccentricity is $${2 \over {\sqrt {r + 1} }},$$ where 0 < r < 1

C

an ellipse whose eccentricity is $${2 \over {\sqrt {r - 1} }},$$ where 0 < r < 1

D

an ellipse whose eccentricity is $${2 \over {\sqrt {r + 1} }},$$ where r > 1

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

for r > 1, $${{{y^2}} \over {1 + r}} + {{{x^2}} \over {1 - r}} = 1$$

$$e = \sqrt {1 - \left( {{{r - 1} \over {r + 2}}} \right)} $$

$$ = \sqrt {{{\left( {r + 1} \right) - \left( {r - 1} \right)} \over {\left( {r + 1} \right)}}} $$

$$ = \sqrt {{2 \over {r + 1}}} = \sqrt {{2 \over {r + 1}}} $$

for r > 1, $${{{y^2}} \over {1 + r}} + {{{x^2}} \over {1 - r}} = 1$$

$$e = \sqrt {1 - \left( {{{r - 1} \over {r + 2}}} \right)} $$

$$ = \sqrt {{{\left( {r + 1} \right) - \left( {r - 1} \right)} \over {\left( {r + 1} \right)}}} $$

$$ = \sqrt {{2 \over {r + 1}}} = \sqrt {{2 \over {r + 1}}} $$

2

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

3

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

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

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

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

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*

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *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*

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*

Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*