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

For $$x \in R - \left\{ {0,1} \right\}$$, Let f_{1}(x) = $$1\over x$$, f_{2} (x) = 1 – x

and f_{3} (x) = $$1 \over {1 - x}$$
be three given

functions. If a function, J(x) satisfies

(f_{2} o J o f_{1}) (x) = f_{3} (x) then J(x) is equal to :

and f

functions. If a function, J(x) satisfies

(f

A

f_{1} (x)

B

$$1 \over x$$ f_{3} (x)

C

f_{2} (x)

D

f_{3} (x)

Given,

f_{1}(x) = $${1 \over x}$$

f_{2}(x) = 1 $$-$$ x

f_{3}(x) = $${1 \over {1 - x}}$$

(f_{2} $$ \cdot $$ J $$ \cdot $$ f_{1}) (x) = f_{3}(x)

$$ \Rightarrow $$ f_{2} {J(f_{1}(x))} = f_{3}(x)

$$ \Rightarrow $$ f_{2}{J ($${1 \over x}$$)} = $${1 \over {1 - x}}$$

$$ \Rightarrow $$ 1 $$-$$ J($${1 \over x}$$) = $${1 \over {1 - x}}$$

$$ \Rightarrow $$ J($${{1 \over x}}$$) = 1 $$-$$ $${{1 \over {1 - x}}}$$

$$ \Rightarrow $$ J ($${{1 \over { x}}}$$) = $${{ - x} \over {1 - x}}$$ = $${x \over {x - 1}}$$

Put x inplace of $${1 \over x}$$

$$ \therefore $$ J(x) = $${{{1 \over x}} \over {{1 \over x} - 1}}$$

= $${1 \over {1 - x}} = {f_3}\left( x \right)$$

f

f

f

(f

$$ \Rightarrow $$ f

$$ \Rightarrow $$ f

$$ \Rightarrow $$ 1 $$-$$ J($${1 \over x}$$) = $${1 \over {1 - x}}$$

$$ \Rightarrow $$ J($${{1 \over x}}$$) = 1 $$-$$ $${{1 \over {1 - x}}}$$

$$ \Rightarrow $$ J ($${{1 \over { x}}}$$) = $${{ - x} \over {1 - x}}$$ = $${x \over {x - 1}}$$

Put x inplace of $${1 \over x}$$

$$ \therefore $$ J(x) = $${{{1 \over x}} \over {{1 \over x} - 1}}$$

= $${1 \over {1 - x}} = {f_3}\left( x \right)$$

2

Let A = {x$$ \in $$**R** : x is not a positive integer}.

Define a function $$f$$ : A $$ \to $$**R** as $$f(x)$$ $${{2x} \over {x - 1}}$$,

then $$f$$ is :

Define a function $$f$$ : A $$ \to $$

then $$f$$ is :

A

not injective

B

neither injective nor surjective

C

surjective but not injective

D

injective but not surjective

f(x) = $${{2x} \over {x - 1}}$$

f(x) = 2 + $${2 \over {x - 1}}$$

f'(x) = $$-$$ $${2 \over {{{\left( {x - 1} \right)}^2}}}$$ < 0 $$\forall $$ x $$ \in $$ R

Hence f(x) is strictly decreasing

So, f(x) is one-one

**Range : ** Let y = $${{2x} \over {x - 1}}$$

xy $$-$$ y = 2x

$$ \Rightarrow $$ x(y $$-$$ 2) = y

$$ \Rightarrow $$ x = $${y \over {y - 2}}$$

given that x $$ \in $$ R : x is not a +ve integer

$$ \therefore $$ $${y \over {y - 2}} \ne $$ N (N $$ \to $$ Natural number)

$$ \Rightarrow $$ y $$ \ne $$ Ny $$-$$ 2N

$$ \Rightarrow $$ y $$ \ne $$ $${{2N} \over {N - 1}}$$

So range $$ \notin $$ R (in to function)

f(x) = 2 + $${2 \over {x - 1}}$$

f'(x) = $$-$$ $${2 \over {{{\left( {x - 1} \right)}^2}}}$$ < 0 $$\forall $$ x $$ \in $$ R

Hence f(x) is strictly decreasing

So, f(x) is one-one

xy $$-$$ y = 2x

$$ \Rightarrow $$ x(y $$-$$ 2) = y

$$ \Rightarrow $$ x = $${y \over {y - 2}}$$

given that x $$ \in $$ R : x is not a +ve integer

$$ \therefore $$ $${y \over {y - 2}} \ne $$ N (N $$ \to $$ Natural number)

$$ \Rightarrow $$ y $$ \ne $$ Ny $$-$$ 2N

$$ \Rightarrow $$ y $$ \ne $$ $${{2N} \over {N - 1}}$$

So range $$ \notin $$ R (in to function)

3

Let N be the set of natural numbers and two functions f and g be defined as f, g : N $$ \to $$ N such that

f(n) = $$\left\{ {\matrix{ {{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr {{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr } \,\,} \right.$$;

and g(n) = n $$-$$($$-$$ 1)^{n}.

Then fog is -

f(n) = $$\left\{ {\matrix{ {{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr {{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr } \,\,} \right.$$;

and g(n) = n $$-$$($$-$$ 1)

Then fog is -

A

neither one-one nor onto

B

onto but not one-one

C

both one-one and onto

D

one-one but not onto

f(x) = $$\left\{ {\matrix{
{{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr
{{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr
} \,\,} \right.$$;

g(x) = n $$-$$ ($$-$$ 1)^{n} $$\left\{ {\matrix{
{n + 1;\,\,\,\,n\,\,is\,\,odd} \cr
{n - 1;\,\,\,\,n\,\,is\,\,even} \cr
} } \right.$$

f(g(n)) = $$\left\{ {\matrix{ {{n \over 2};\,\,\,\,n\,\,is\,\,even} \cr {{{n + 1} \over 2};\,\,\,\,n\,\,is\,\,odd} \cr } } \right.$$

$$ \therefore $$ many one but onto

g(x) = n $$-$$ ($$-$$ 1)

f(g(n)) = $$\left\{ {\matrix{ {{n \over 2};\,\,\,\,n\,\,is\,\,even} \cr {{{n + 1} \over 2};\,\,\,\,n\,\,is\,\,odd} \cr } } \right.$$

$$ \therefore $$ many one but onto

4

Let f : R $$ \to $$ R be defined by f(x) = $${x \over {1 + {x^2}}},x \in R$$. Then the range of f is :

A

$$\left[ { - {1 \over 2},{1 \over 2}} \right]$$

B

$$R - \left[ { - {1 \over 2},{1 \over 2}} \right]$$

C

($$-$$ 1, 1) $$-$$ {0}

D

R $$-$$ [$$-$$1, 1]

f(0) = 0 & f(x) is odd

Further, if x > 0 then

f(x) = $$f(x) = {1 \over {x + {1 \over x}}} \in \left( {0,{1 \over 2}} \right]$$

Hence, $$f(x) \in \left[ { - {1 \over 2},{1 \over 2}} \right]$$

Further, if x > 0 then

f(x) = $$f(x) = {1 \over {x + {1 \over x}}} \in \left( {0,{1 \over 2}} \right]$$

Hence, $$f(x) \in \left[ { - {1 \over 2},{1 \over 2}} \right]$$

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (1) *keyboard_arrow_right*

AIEEE 2003 (4) *keyboard_arrow_right*

AIEEE 2004 (4) *keyboard_arrow_right*

AIEEE 2005 (3) *keyboard_arrow_right*

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JEE Main 2016 (Online) 9th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2016 (Online) 10th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2017 (Offline) (1) *keyboard_arrow_right*

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JEE Main 2021 (Online) 24th February Morning Slot (1) *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*