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

If $${\sum\limits_{i = 1}^{20} {\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3} = {k \over {21}}$$ then k is equal to

A

100

B

200

C

50

D

400

$${\sum\limits_{i = 1}^{20} {\left( {{{^{20}{C_{i - 1}}} \over {^{20}{C_i}{ + ^{20}}{C_{i - 1}}}}} \right)} ^3} = {k \over {21}}$$

$$ \Rightarrow \,\,\sum\limits_{i = 1}^{20} {{{\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{21}{C_i}}}} \right)}^3}} = {k \over {21}}$$

$$ \Rightarrow \,\,\sum\limits_{i = 1}^{20} {{{\left( {{i \over {21}}} \right)}^3}} = {k \over {21}}$$

$$ \Rightarrow \,\,{1 \over {{{\left( {21} \right)}^3}}}{\left[ {{{20\left( {21} \right)} \over 2}} \right]^2} = {k \over {21}}$$

$$ \Rightarrow 100 = k$$

$$ \Rightarrow \,\,\sum\limits_{i = 1}^{20} {{{\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{21}{C_i}}}} \right)}^3}} = {k \over {21}}$$

$$ \Rightarrow \,\,\sum\limits_{i = 1}^{20} {{{\left( {{i \over {21}}} \right)}^3}} = {k \over {21}}$$

$$ \Rightarrow \,\,{1 \over {{{\left( {21} \right)}^3}}}{\left[ {{{20\left( {21} \right)} \over 2}} \right]^2} = {k \over {21}}$$

$$ \Rightarrow 100 = k$$

2

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 $${1 \over 4}$$

3

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

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

$$\lambda = \pm 4$$

4

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

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