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 k for which the function

$$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\,\,,} & {0 < x < {\pi \over 2}} \cr {k + {2 \over 5}\,\,\,,} & {x = {\pi \over 2}} \cr } } \right.$$

is continuous at x = $${\pi \over 2},$$ is :

$$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\,\,,} & {0 < x < {\pi \over 2}} \cr {k + {2 \over 5}\,\,\,,} & {x = {\pi \over 2}} \cr } } \right.$$

is continuous at x = $${\pi \over 2},$$ is :

A

$${{17} \over {20}}$$

B

$${{2} \over {5}}$$

C

$${{3} \over {5}}$$

D

$$-$$ $${{2} \over {5}}$$

2

Let f be a polynomial function such that

f (3x) = f ' (x) . f '' (x), for all x $$ \in $$**R**. Then :

f (3x) = f ' (x) . f '' (x), for all x $$ \in $$

A

f (2) + f ' (2) = 28

B

f '' (2) $$-$$ f ' (2) = 0

C

f '' (2) $$-$$ f (2) = 4

D

f (2) $$-$$ f ' (2) + f '' (2) = 10

3

For each t $$ \in R$$, let [t] be the greatest integer less than or equal to t.

Then $$\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right] + ..... + \left[ {{{15} \over x}} \right]} \right)$$

Then $$\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right] + ..... + \left[ {{{15} \over x}} \right]} \right)$$

A

does not exist in R

B

is equal to 0

C

is equal to 15

D

is equal to 120

Given,

$$\mathop {lim}\limits_{x \to {0^ + }} \,\,x\,\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right]} \right. + $$ $$\left. {\,.\,.\,.\,.\,.\, + \left[ {{{15} \over x}} \right]} \right)$$

as we know that

$${1 \over x} = \left[ {{1 \over x}} \right] + \left\{ {{1 \over x}} \right\}$$

$$ \Rightarrow \,\,\,\,\left[ {{1 \over x}} \right] = {1 \over x} - \left\{ {{1 \over x}} \right\}$$

$$ = \,\,\,\,\mathop {\lim }\limits_{x \to {0^ + }} \,\,x\left[ {{1 \over x} - \left\{ {{1 \over x}} \right\} + {2 \over 2} - \left\{ {{2 \over x}} \right\} + ........{{15} \over x} - \left\{ {{{15} \over x}} \right\}} \right]$$

$$ = \,\,\,\,\mathop {\lim }\limits_{x \to {0^ + }} \,\left[ {x.{1 \over x} + x.{2 \over x} + .....x.{{15} \over x}} \right]$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \,\left[ {x.\left\{ {{1 \over 2}} \right\} + .. + x.\left\{ {{{15} \over x}} \right\}} \right]$$

We know $$\left\{ {{1 \over x}} \right\}$$ is fractional part of $${1 \over x}.$$

So, the range of $$\,\left\{ {{1 \over x}} \right\}$$ is $$0 \le \left\{ {{1 \over x}} \right\} < 1$$

So, $$\mathop {lim}\limits_{x \to {0^ + }} \,\,x\,\left\{ {{1 \over x}} \right\} = 0.$$ (finite no) $$=0$$

Similarly $$\mathop {\lim }\limits_{x \to {0^ + }} x.\left\{ {{2 \over x}} \right\} = 0$$

$$ = \,\,\,\,\left( {1 + 2 + ... + 15} \right) - \left( {0 + 0...} \right)$$

$$ = \,\,\,\,{{15 \times 16} \over 2}$$

$$ = \,\,\,\,120$$

$$\mathop {lim}\limits_{x \to {0^ + }} \,\,x\,\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right]} \right. + $$ $$\left. {\,.\,.\,.\,.\,.\, + \left[ {{{15} \over x}} \right]} \right)$$

as we know that

$${1 \over x} = \left[ {{1 \over x}} \right] + \left\{ {{1 \over x}} \right\}$$

$$ \Rightarrow \,\,\,\,\left[ {{1 \over x}} \right] = {1 \over x} - \left\{ {{1 \over x}} \right\}$$

$$ = \,\,\,\,\mathop {\lim }\limits_{x \to {0^ + }} \,\,x\left[ {{1 \over x} - \left\{ {{1 \over x}} \right\} + {2 \over 2} - \left\{ {{2 \over x}} \right\} + ........{{15} \over x} - \left\{ {{{15} \over x}} \right\}} \right]$$

$$ = \,\,\,\,\mathop {\lim }\limits_{x \to {0^ + }} \,\left[ {x.{1 \over x} + x.{2 \over x} + .....x.{{15} \over x}} \right]$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \,\left[ {x.\left\{ {{1 \over 2}} \right\} + .. + x.\left\{ {{{15} \over x}} \right\}} \right]$$

We know $$\left\{ {{1 \over x}} \right\}$$ is fractional part of $${1 \over x}.$$

So, the range of $$\,\left\{ {{1 \over x}} \right\}$$ is $$0 \le \left\{ {{1 \over x}} \right\} < 1$$

So, $$\mathop {lim}\limits_{x \to {0^ + }} \,\,x\,\left\{ {{1 \over x}} \right\} = 0.$$ (finite no) $$=0$$

Similarly $$\mathop {\lim }\limits_{x \to {0^ + }} x.\left\{ {{2 \over x}} \right\} = 0$$

$$ = \,\,\,\,\left( {1 + 2 + ... + 15} \right) - \left( {0 + 0...} \right)$$

$$ = \,\,\,\,{{15 \times 16} \over 2}$$

$$ = \,\,\,\,120$$

4

If $$f\left( x \right) = \left| {\matrix{
{\cos x} & x & 1 \cr
{2\sin x} & {{x^2}} & {2x} \cr
{\tan x} & x & 1 \cr
} } \right|,$$ then $$\mathop {\lim }\limits_{x \to 0} {{f'\left( x \right)} \over x}$$

A

does not exist.

B

exists and is equal to 2.

C

existsand is equal to 0.

D

exists and is equal to $$-$$ 2.

Given,

$$f\left( x \right) = \left| {\matrix{ {\cos x} & x & 1 \cr {2\sin x} & {{x^2}} & {2x} \cr {\tan x} & x & 1 \cr } } \right|$$

= cosx(x^{2} - 2x^{2}) - x(2 sinx - 2x tanx) + (2x sinx - x^{2} tanx)

= x^{2} (tanx - cosx)

$$ \therefore $$ $${f^{'}}(x)$$ = 2x (tanx - cosx) + x^{2}(sec^{2}x + sinx)

$$ \therefore $$ $$\mathop {\lim }\limits_{x \to 0} {{f'\left( x \right)} \over x}$$

= $$\mathop {\lim }\limits_{x \to o} {{2x(\tan x - \cos x) + {x^2}({{\sec }^2}x + \sin x)} \over x}$$

= $$\mathop {\lim }\limits_{x \to o} \,\,2(\tan x - \cos x) + x({\sec ^2}x + \sin x)$$

= 2 (0-1) + 0

= -2

$$f\left( x \right) = \left| {\matrix{ {\cos x} & x & 1 \cr {2\sin x} & {{x^2}} & {2x} \cr {\tan x} & x & 1 \cr } } \right|$$

= cosx(x

= x

$$ \therefore $$ $${f^{'}}(x)$$ = 2x (tanx - cosx) + x

$$ \therefore $$ $$\mathop {\lim }\limits_{x \to 0} {{f'\left( x \right)} \over x}$$

= $$\mathop {\lim }\limits_{x \to o} {{2x(\tan x - \cos x) + {x^2}({{\sec }^2}x + \sin x)} \over x}$$

= $$\mathop {\lim }\limits_{x \to o} \,\,2(\tan x - \cos x) + x({\sec ^2}x + \sin x)$$

= 2 (0-1) + 0

= -2

Number in Brackets after Paper Name Indicates No of Questions

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Trigonometric Functions & Equations *keyboard_arrow_right*

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Definite Integrals and Applications of Integrals *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*