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

If the third term in the binomial expansion

of $${\left( {1 + {x^{{{\log }_2}x}}} \right)^5}$$ equals 2560, then a possible value of x is -

of $${\left( {1 + {x^{{{\log }_2}x}}} \right)^5}$$ equals 2560, then a possible value of x is -

A

$$2\sqrt 2 $$

B

$$4\sqrt 2 $$

C

$${1 \over 8}$$

D

$${1 \over 4}$$

$${\left( {1 + {x^{{{\log }_2}x}}} \right)^5}$$

$${T_3} = {}^5{C_2}.{\left( {{x^{{{\log }_2}x}}} \right)^2} = 2560$$

$$ \Rightarrow \,\,10.{x^{2{{\log }_2}x}} = 2560$$

$$ \Rightarrow \,\,{x^{2\log 2x}} = 256$$

$$ \Rightarrow \,\,2{({\log _2}x)^2} = {\log _2}256$$

$$ \Rightarrow 2{({\log _2}x)^2} = 8$$

$$ \Rightarrow \,\,{({\log _2}x)^2} = 4$$

$$ \Rightarrow \,\,{\log _2}x = 2$$ or $$-$$ 2

$$x = 4$$ or $${1 \over 4}$$

$${T_3} = {}^5{C_2}.{\left( {{x^{{{\log }_2}x}}} \right)^2} = 2560$$

$$ \Rightarrow \,\,10.{x^{2{{\log }_2}x}} = 2560$$

$$ \Rightarrow \,\,{x^{2\log 2x}} = 256$$

$$ \Rightarrow \,\,2{({\log _2}x)^2} = {\log _2}256$$

$$ \Rightarrow 2{({\log _2}x)^2} = 8$$

$$ \Rightarrow \,\,{({\log _2}x)^2} = 4$$

$$ \Rightarrow \,\,{\log _2}x = 2$$ or $$-$$ 2

$$x = 4$$ or $${1 \over 4}$$

2

The positive value of $$\lambda $$ for which the co-efficient of x^{2}
in the expression x^{2} $${\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}}$$ is 720, is -

A

4

B

$$2\sqrt 2 $$

C

3

D

$$\sqrt 5 $$

$${x^2}\left( {{}^{10}{C_r}{{\left( {\sqrt x } \right)}^{10 - r}}{{\left( {{\lambda \over {{x^2}}}} \right)}^r}} \right)$$

$${x^2}\left[ {{}^{10}{C_r}{{\left( x \right)}^{{{10 - r} \over 2}}}{{\left( \lambda \right)}^r}{{\left( x \right)}^{ - 2r}}} \right]$$

$${x^2}\left[ {{}^{10}{C_r}{\lambda ^r}{x^{{{10 - r} \over 2}}}} \right]$$

$$ \therefore $$ r = 2

Hence, $${}^{10}{C_2}{\lambda ^2} = 720$$

$${\lambda ^2} = 16$$

$$\lambda = \pm 4$$

$${x^2}\left[ {{}^{10}{C_r}{{\left( x \right)}^{{{10 - r} \over 2}}}{{\left( \lambda \right)}^r}{{\left( x \right)}^{ - 2r}}} \right]$$

$${x^2}\left[ {{}^{10}{C_r}{\lambda ^r}{x^{{{10 - r} \over 2}}}} \right]$$

$$ \therefore $$ r = 2

Hence, $${}^{10}{C_2}{\lambda ^2} = 720$$

$${\lambda ^2} = 16$$

$$\lambda = \pm 4$$

3

The value of r for which ^{20}C_{r} ^{20}C_{0} + ^{20}C_{r$$-$$1} ^{20}C_{1} + ^{20}C_{r$$-$$2} ^{20}C_{2} + . . . . .+ ^{20}C_{0} ^{20}C_{r} is maximum, is

A

20

B

15

C

10

D

11

Given sum = coefficient of x^{r} in the expansion of

(1 + x)^{20}(1 + x)^{20},

Which is equal to^{40}C_{r}

It is maximum when r = 20

(1 + x)

Which is equal to

It is maximum when r = 20

4

The sum of the real values of x for which the middle term in the binomial expansion of $${\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8}$$ equals 5670 is :

A

0

B

8

C

6

D

4

$${T_5} = {}^8{C_4}{{{x^{12}}} \over {81}} \times {{81} \over {{x^4}}} = 5670$$

$$ \Rightarrow 70{x^8} = 5670$$

$$ \Rightarrow x = \pm \sqrt 3 $$

$$ \Rightarrow 70{x^8} = 5670$$

$$ \Rightarrow x = \pm \sqrt 3 $$

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*