Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

The value of $$\lambda $$ such that sum of the squares of the roots of the quadratic equation, x^{2} + (3 – $$\lambda $$)x + 2 = $$\lambda $$ has the least value is -

A

1

B

2

C

$${{15} \over 8}$$

D

$${4 \over 9}$$

$$\alpha $$ + $$\beta $$ = $$\lambda $$ $$-$$ 3

$$\alpha $$$$\beta $$ = 2 $$-$$ $$\lambda $$

$$\alpha $$^{2} + $$\beta $$^{2} = ($$\alpha $$ + $$\beta $$)^{2} $$-$$ 2$$\alpha $$$$\beta $$ = ($$\lambda $$ $$-$$ 3)^{2} $$-$$ 2$$\left( {2 - \lambda } \right)$$

= $$\lambda $$^{2} + 9 $$-$$ 6$$\lambda $$ $$-$$ 4 + 2$$\lambda $$

= $$\lambda $$^{2} $$-$$ 4$$\lambda $$ + 5

= ($$\lambda $$ $$-$$ 2)^{2} + 1

$$ \therefore $$ $$\lambda $$ = 2

$$\alpha $$$$\beta $$ = 2 $$-$$ $$\lambda $$

$$\alpha $$

= $$\lambda $$

= $$\lambda $$

= ($$\lambda $$ $$-$$ 2)

$$ \therefore $$ $$\lambda $$ = 2

2

If one real root of the quadratic equation 81x^{2} + kx + 256 = 0 is cube of the other root, then a value of k is

A

$$-$$ 81

B

$$-$$ 300

C

100

D

144

81x^{2} + kx + 256 = 0 ; x = $$\alpha $$, $$\alpha $$^{3}

$$ \Rightarrow $$ $$\alpha $$^{4} = $${{256} \over {81}}$$ $$ \Rightarrow $$ $$\alpha $$ = $$ \pm $$ $${{4} \over {3}}$$

Now $$-$$ $${k \over {81}}$$ = $$\alpha $$ + $$\alpha $$^{3} = $$ \pm $$ $${{100} \over {27}}$$

$$ \Rightarrow $$ k = $$ \pm $$300

$$ \Rightarrow $$ $$\alpha $$

Now $$-$$ $${k \over {81}}$$ = $$\alpha $$ + $$\alpha $$

$$ \Rightarrow $$ k = $$ \pm $$300

3

Let $$\alpha $$ and $$\beta $$ be the roots of the quadratic equation x^{2}
sin $$\theta $$ – x(sin $$\theta $$ cos $$\theta $$ + 1) + cos $$\theta $$ = 0 (0 < $$\theta $$ < 45^{o}), and $$\alpha $$ < $$\beta $$. Then $$\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{{{\left( { - 1} \right)}^n}} \over {{\beta ^n}}}} \right)} $$ is equal to :

A

$${1 \over {1 + \cos \theta }} + {1 \over {1 - \sin \theta }}$$

B

$${1 \over {1 - \cos \theta }} + {1 \over {1 + \sin \theta }}$$

C

$${1 \over {1 - \cos \theta }} - {1 \over {1 + \sin \theta }}$$

D

$${1 \over {1 + \cos \theta }} - {1 \over {1 - \sin \theta }}$$

D = (1 + sin$$\theta $$ cos$$\theta $$)^{2} $$-$$ 4sin$$\theta $$cos$$\theta $$ = (1 $$-$$ sin$$\theta $$ cos$$\theta $$)^{2}

$$ \Rightarrow $$ roots are $$\beta $$ = cosec$$\theta $$ and $$\alpha $$ = cos$$\theta $$

$$\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{\left( { - {1 \over \beta }} \right)}^n}} \right)} = \sum\limits_{n = 0}^\infty {{{\left( {\cos \theta } \right)}^n}} + \sum\limits_{n = 0}^n {{{\left( { - \sin \theta } \right)}^n}} $$

$$ = {1 \over {1 - \cos \theta }} + {1 \over {1 + \sin \theta }}$$

$$ \Rightarrow $$ roots are $$\beta $$ = cosec$$\theta $$ and $$\alpha $$ = cos$$\theta $$

$$\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{\left( { - {1 \over \beta }} \right)}^n}} \right)} = \sum\limits_{n = 0}^\infty {{{\left( {\cos \theta } \right)}^n}} + \sum\limits_{n = 0}^n {{{\left( { - \sin \theta } \right)}^n}} $$

$$ = {1 \over {1 - \cos \theta }} + {1 \over {1 + \sin \theta }}$$

4

If $$\lambda $$ be the ratio of the roots of the quadratic equation in x, 3m^{2}x^{2} + m(m – 4)x + 2 = 0, then the least value of m for which $$\lambda + {1 \over \lambda } = 1,$$ is

A

$$ - 2 + \sqrt 2 $$

B

4$$-$$3$$\sqrt 2 $$

C

2 $$-$$ $$\sqrt 3 $$

D

4 $$-$$ 2$$\sqrt 3 $$

3m^{2}x^{2} + m(m $$-$$ 4) x + 2 = 0

$$\lambda + {1 \over \lambda } = 1,{\alpha \over \beta } + {\beta \over \alpha } = 1,{\alpha ^2} + {\beta ^2} = \alpha \beta $$

($$\alpha $$ + $$\beta $$)^{2} = 3$$\alpha $$$$\beta $$

$${\left( { - {{m\left( {m - 4} \right)} \over {3{m^2}}}} \right)^2} = {{3\left( 2 \right)} \over {3{m^2}}},{{{{\left( {m - 4} \right)}^2}} \over {9{m^2}}} = {6 \over {3m}}$$

$${\left( {m - 4} \right)^2} = 18,m = 4 \pm \sqrt {18,} \,\,4 \pm 3\sqrt 2 $$

$$\lambda + {1 \over \lambda } = 1,{\alpha \over \beta } + {\beta \over \alpha } = 1,{\alpha ^2} + {\beta ^2} = \alpha \beta $$

($$\alpha $$ + $$\beta $$)

$${\left( { - {{m\left( {m - 4} \right)} \over {3{m^2}}}} \right)^2} = {{3\left( 2 \right)} \over {3{m^2}}},{{{{\left( {m - 4} \right)}^2}} \over {9{m^2}}} = {6 \over {3m}}$$

$${\left( {m - 4} \right)^2} = 18,m = 4 \pm \sqrt {18,} \,\,4 \pm 3\sqrt 2 $$

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (5) *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*