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

The equation of a tangent to the hyperbola 4x^{2} – 5y^{2} = 20 parallel to the line x – y = 2 is -

A

x $$-$$ y + 9 = 0

B

x $$-$$ y $$-$$ 3 = 0

C

x $$-$$ y + 1 = 0

D

x $$-$$ y + 7 = 0

Hyperbola $${{{x^2}} \over 5} - {{{y^2}} \over 4} = 1$$

slope of tangent = 1

equation of tangent y = x $$ \pm $$ $$\sqrt {5 - 4} $$

$$ \Rightarrow $$ y = x $$ \pm $$ 1

$$ \Rightarrow $$ y = x + 1

or

$$ \Rightarrow $$ y = x $$-$$ 1

slope of tangent = 1

equation of tangent y = x $$ \pm $$ $$\sqrt {5 - 4} $$

$$ \Rightarrow $$ y = x $$ \pm $$ 1

$$ \Rightarrow $$ y = x + 1

or

$$ \Rightarrow $$ y = x $$-$$ 1

2

The length of the chord of the parabola x^{2} $$=$$ 4y having equation x – $$\sqrt 2 y + 4\sqrt 2 = 0$$ is -

A

$$8\sqrt 2 $$

B

$$6\sqrt 3 $$

C

$$3\sqrt 2 $$

D

$$2\sqrt {11} $$

x^{2} = 4y

x $$-$$ $$\sqrt 2 $$y + 4$$\sqrt 2 $$ = 0

Solving together we get

x^{2} = 4$$\left( {{{x + 4\sqrt 2 } \over {\sqrt 2 }}} \right)$$

$$\sqrt 2 $$x^{2} + 4x + 16$$\sqrt 2 $$

$$\sqrt 2 $$x^{2} $$-$$ 4x $$-$$ 16$$\sqrt 2 $$ = 0

x_{1} + x_{2} = 2$$\sqrt 2 $$; x_{1}x_{2} = $${{ - 16\sqrt 2 } \over {\sqrt 2 }}$$ = $$-$$ 16

Similarly,

($$\sqrt 2 $$y $$-$$ 4$$\sqrt 2 $$)^{2} = 4y

2y^{2} + 32 $$-$$ 16y = 4y

$$\ell $$_{AB} = $$\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $$

$$ = \sqrt {{{\left( {2\sqrt 2 } \right)}^2} + 64 + {{\left( {10} \right)}^2} - 4\left( {16} \right)} $$

$$ = \sqrt {8 + 64 + 100 - 64} $$

$$ = \sqrt {108} = 6\sqrt 3 $$

x $$-$$ $$\sqrt 2 $$y + 4$$\sqrt 2 $$ = 0

Solving together we get

x

$$\sqrt 2 $$x

$$\sqrt 2 $$x

x

Similarly,

($$\sqrt 2 $$y $$-$$ 4$$\sqrt 2 $$)

2y

$$\ell $$

$$ = \sqrt {{{\left( {2\sqrt 2 } \right)}^2} + 64 + {{\left( {10} \right)}^2} - 4\left( {16} \right)} $$

$$ = \sqrt {8 + 64 + 100 - 64} $$

$$ = \sqrt {108} = 6\sqrt 3 $$

3

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

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

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

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

4

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

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (1) *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*