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

Let f : R $$ \to $$ R be a function such that f(x) = x^{3} + x^{2}f'(1) + xf''(2) + f'''(3), x $$ \in $$ R. Then f(2) equals -

A

30

B

$$-$$ 2

C

$$-$$ 4

D

8

f(x) = x^{3} + x^{2}f '(1) + xf ''(2) + f '''(3)

$$ \Rightarrow $$ f '(x) = 3x^{2} + 2xf '(1) + f ''(x) . . . . . (1)

$$ \Rightarrow $$ f ''(x) = 6x + 2f '(1) . . . . . . (2)

$$ \Rightarrow $$ f '''(x) = 6 . . . . . .(3)

put x = 1 in equation (1) :

f '(1) = 3 + 2f '(1) + f ''(2) . . . . .(4)

put x = 2 in equation (2) :

f ''(2) = 12 + 2f '(1) . . . . .(5)

from equation (4) & (5) :

$$-$$3 $$-$$ f '(1) = 12 + 2f'(1)

$$ \Rightarrow $$ 3f '(1) = $$-$$ 15

$$ \Rightarrow $$ f '(1) = $$-$$ 5 $$ \Rightarrow $$ f ''(2) = 2 . . . . .(2)

put x = 3 in equation (3) :

f ''' (3) = 6

$$ \therefore $$ f(x) = x^{3} $$-$$ 5x^{2} + 2x + 6

f(2) = 8 $$-$$ 20 + 4 + 6 = $$-$$ 2

$$ \Rightarrow $$ f '(x) = 3x

$$ \Rightarrow $$ f ''(x) = 6x + 2f '(1) . . . . . . (2)

$$ \Rightarrow $$ f '''(x) = 6 . . . . . .(3)

put x = 1 in equation (1) :

f '(1) = 3 + 2f '(1) + f ''(2) . . . . .(4)

put x = 2 in equation (2) :

f ''(2) = 12 + 2f '(1) . . . . .(5)

from equation (4) & (5) :

$$-$$3 $$-$$ f '(1) = 12 + 2f'(1)

$$ \Rightarrow $$ 3f '(1) = $$-$$ 15

$$ \Rightarrow $$ f '(1) = $$-$$ 5 $$ \Rightarrow $$ f ''(2) = 2 . . . . .(2)

put x = 3 in equation (3) :

f ''' (3) = 6

$$ \therefore $$ f(x) = x

f(2) = 8 $$-$$ 20 + 4 + 6 = $$-$$ 2

2

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

Then $$\mathop {\lim }\limits_{x \to 1 + } {{\left( {1 - \left| x \right| + \sin \left| {1 - x} \right|} \right)\sin \left( {{\pi \over 2}\left[ {1 - x} \right]} \right)} \over {\left| {1 - x} \right|.\left[ {1 - x} \right]}}$$

Then $$\mathop {\lim }\limits_{x \to 1 + } {{\left( {1 - \left| x \right| + \sin \left| {1 - x} \right|} \right)\sin \left( {{\pi \over 2}\left[ {1 - x} \right]} \right)} \over {\left| {1 - x} \right|.\left[ {1 - x} \right]}}$$

A

equals $$-$$ 1

B

equals 1

C

equals 0

D

does not exist

$$\mathop {\lim }\limits_{x \to {1^ + }} {{\left( {1 - \left| x \right| + \sin \left| {1 - x} \right|} \right)\sin \left( {{\pi \over 2}\left[ {1 - x} \right]} \right)} \over {\left| {1 - x} \right|\left[ {1 - x} \right]}}$$

$$=$$ $$\mathop {\lim }\limits_{x \to {1^ + }} {{\left( {1 - x} \right) + \sin \left( {x - 1} \right)} \over {\left( {x - 1} \right)\left( { - 1} \right)}}$$ $$\sin \left( {{\pi \over 2}\left( { - 1} \right)} \right)$$

$$=$$ $$\mathop {\lim }\limits_{x \to {1^ + }} \left( {1 - {{\sin \left( {x - 1} \right)} \over {\left( {x - 1} \right)}}} \right)\left( { - 1} \right) = \left( {1 - 1} \right)\left( { - 1} \right) = 0$$

$$=$$ $$\mathop {\lim }\limits_{x \to {1^ + }} {{\left( {1 - x} \right) + \sin \left( {x - 1} \right)} \over {\left( {x - 1} \right)\left( { - 1} \right)}}$$ $$\sin \left( {{\pi \over 2}\left( { - 1} \right)} \right)$$

$$=$$ $$\mathop {\lim }\limits_{x \to {1^ + }} \left( {1 - {{\sin \left( {x - 1} \right)} \over {\left( {x - 1} \right)}}} \right)\left( { - 1} \right) = \left( {1 - 1} \right)\left( { - 1} \right) = 0$$

3

Let $$f\left( x \right) = \left\{ {\matrix{
{\max \left\{ {\left| x \right|,{x^2}} \right\}} & {\left| x \right| \le 2} \cr
{8 - 2\left| x \right|} & {2 < \left| x \right| \le 4} \cr
} } \right.$$

Let S be the set of points in the interval (– 4, 4) at which f is not differentiable. Then S

Let S be the set of points in the interval (– 4, 4) at which f is not differentiable. Then S

A

equals $$\left\{ { - 2, - 1,1,2} \right\}$$

B

equals $$\left\{ { - 2, - 1,0,1,2} \right\}$$

C

equals $$\left\{ { - 2,2} \right\}$$

D

is an empty set

$$f\left( x \right) = \left\{ {\matrix{
{8 + 2x,} & { - 4 \le x \le - 2} \cr
{{x^2},} & { - 2 \le x \le - 1} \cr
{\left| x \right|,} & { - 1 < x < 1} \cr
{{x^2},} & {1 \le x \le 2} \cr
{8 - 2x,} & {2 < x \le 4} \cr
} } \right.$$

f(x) is not differentiable at

x = $$\left\{ { - 2, - 1,0,1,2} \right\}$$

$$ \Rightarrow $$ S = {$$-$$2, $$-$$ 1, 0, 1, 2}

f(x) is not differentiable at

x = $$\left\{ { - 2, - 1,0,1,2} \right\}$$

$$ \Rightarrow $$ S = {$$-$$2, $$-$$ 1, 0, 1, 2}

4

Let f : ($$-$$1, 1) $$ \to $$ R be a function defined by f(x) = max $$\left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} } \right\}.$$ If K be the set of all points at which f is not differentiable, then K has exactly -

A

one element

B

three elements

C

five elements

D

two elements

f : ($$-$$ 1, 1) $$ \to $$ R

f(x) = max {$$-$$ $$\left| x \right|, - \sqrt {1 - {x^2}} $$}

Non-derivable at 3 points in ($$-$$1, 1)

f(x) = max {$$-$$ $$\left| x \right|, - \sqrt {1 - {x^2}} $$}

Non-derivable at 3 points in ($$-$$1, 1)

Number in Brackets after Paper Name Indicates No of Questions

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Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

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