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

The tangent to the curve, y = xe^{x2} passing through the point (1, e) also passes through the point

A

$$\left( {{4 \over 3},2e} \right)$$

B

(3, 6e)

C

(2, 3e)

D

$$\left( {{5 \over 3},2e} \right)$$

y = xe^{x2}

$${\left. {{{dy} \over {dx}}} \right|_{(1,e)}}{\left. { = \left( {e.e{x^2}.2x + {e^{{x^2}}}} \right)} \right|_{(1,e)}}$$

$$ = 2 \cdot e + e = 3e$$

T : y $$-$$ e = 3e (x $$-$$ 1)

y = 3ex $$-$$ 3e + e

y = $$\left( {3e} \right)x - 2e$$

$$\left( {{4 \over 3},2e} \right)$$ lies on it

$${\left. {{{dy} \over {dx}}} \right|_{(1,e)}}{\left. { = \left( {e.e{x^2}.2x + {e^{{x^2}}}} \right)} \right|_{(1,e)}}$$

$$ = 2 \cdot e + e = 3e$$

T : y $$-$$ e = 3e (x $$-$$ 1)

y = 3ex $$-$$ 3e + e

y = $$\left( {3e} \right)x - 2e$$

$$\left( {{4 \over 3},2e} \right)$$ lies on it

2

The maximum value of the function f(x) = 3x^{3} – 18x^{2} + 27x – 40 on the set S = $$\left\{ {x\, \in R:{x^2} + 30 \le 11x} \right\}$$ is :

A

$$-$$ 222

B

$$-$$ 122

C

$$122$$

D

222

S = {x $$ \in $$ R, x^{2} + 30 $$-$$ 11x $$ \le $$ 0}

= {x $$ \in $$ R, 5 $$ \le $$ x $$ \le $$ 6}

Now f(x) = 3x^{3} $$-$$ 18x^{2} + 27x $$-$$ 40

$$ \Rightarrow $$ f '(x) = 9(x $$-$$ 1)(x $$-$$ 3),

which is positive in [5, 6]

$$ \Rightarrow $$ f(x) increasing in [5, 6]

Hence maximum value = f(6) = 122

= {x $$ \in $$ R, 5 $$ \le $$ x $$ \le $$ 6}

Now f(x) = 3x

$$ \Rightarrow $$ f '(x) = 9(x $$-$$ 1)(x $$-$$ 3),

which is positive in [5, 6]

$$ \Rightarrow $$ f(x) increasing in [5, 6]

Hence maximum value = f(6) = 122

3

If y(x) is the solution of the differential equation $${{dy} \over {dx}} + \left( {{{2x + 1} \over x}} \right)y = {e^{ - 2x}},\,\,x > 0,\,$$ where $$y\left( 1 \right) = {1 \over 2}{e^{ - 2}},$$ then

A

y(log_{e}2) = log_{e}4

B

y(x) is decreasing in (0, 1)

C

y(log_{e}2) = $${{{{\log }_e}2} \over 4}$$

D

y(x) is decreasing in $$\left( {{1 \over 2},1} \right)$$

$${{dy} \over {dx}} + \left( {{{2x + 1} \over x}} \right)y = {e^{ - 2x}}$$

I.F. $$ = {e^{\int {\left( {{{2x + 1} \over x}} \right)dx} }} = {e^{\int {\left( {2 + {1 \over x}} \right)dx} }} = {e^{2x + \ell nx}} = {e^{2x}}.x$$

So, $$y\left( {x{e^{2x}}} \right) = \int {{e^{ - 2x}}.x{e^{2x}} + C} $$

$$ \Rightarrow xy{e^{2x}} = \int {xdx + C} $$

$$ \Rightarrow 2xy{e^{2x}} = {x^2} + 2C$$

It passes through $$\left( {1,{1 \over 2}{e^{ - 2}}} \right)$$ we get C $$=$$ 0

$$y = {{x{e^{ - 2x}}} \over 2}$$

$$ \Rightarrow {{dy} \over {dx}} = {1 \over 2}{e^{ - 2x}}\left( { - 2x + 1} \right)$$

$$ \Rightarrow f(x)$$ is decreasing in $$\left( {{1 \over 2},1} \right)$$

$$y\left( {{{\log }_e}2} \right) = {{\left( {{{\log }_e}2} \right){e^{ - 2({{\log }_e}2)}}} \over 2}$$

$$ = {1 \over 8}{\log _e}2$$

I.F. $$ = {e^{\int {\left( {{{2x + 1} \over x}} \right)dx} }} = {e^{\int {\left( {2 + {1 \over x}} \right)dx} }} = {e^{2x + \ell nx}} = {e^{2x}}.x$$

So, $$y\left( {x{e^{2x}}} \right) = \int {{e^{ - 2x}}.x{e^{2x}} + C} $$

$$ \Rightarrow xy{e^{2x}} = \int {xdx + C} $$

$$ \Rightarrow 2xy{e^{2x}} = {x^2} + 2C$$

It passes through $$\left( {1,{1 \over 2}{e^{ - 2}}} \right)$$ we get C $$=$$ 0

$$y = {{x{e^{ - 2x}}} \over 2}$$

$$ \Rightarrow {{dy} \over {dx}} = {1 \over 2}{e^{ - 2x}}\left( { - 2x + 1} \right)$$

$$ \Rightarrow f(x)$$ is decreasing in $$\left( {{1 \over 2},1} \right)$$

$$y\left( {{{\log }_e}2} \right) = {{\left( {{{\log }_e}2} \right){e^{ - 2({{\log }_e}2)}}} \over 2}$$

$$ = {1 \over 8}{\log _e}2$$

4

Let f(x) = $${x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,$$ x $$\, \in $$ R, where a, b and d are non-zero real constants. Then :

A

f is an increasing function of x

B

f is neither increasing nor decreasing function of x

C

f ' is not a continuous function of x

D

f is a decreasing function of x

$$f\left( x \right) = {x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }}$$

$$f'\left( x \right) = {{{a^2}} \over {{{\left( {{a^2} + {x^2}} \right)}^{3/2}}}} + {{{b^2}} \over {{{\left( {{b^2} + {{\left( {d - x} \right)}^2}} \right)}^{3/2}}}} > 0\forall x \in R$$

f(x) is an increasing function.

$$f'\left( x \right) = {{{a^2}} \over {{{\left( {{a^2} + {x^2}} \right)}^{3/2}}}} + {{{b^2}} \over {{{\left( {{b^2} + {{\left( {d - x} \right)}^2}} \right)}^{3/2}}}} > 0\forall x \in R$$

f(x) is an increasing function.

Number in Brackets after Paper Name Indicates No of Questions

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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*

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Vector Algebra and 3D Geometry *keyboard_arrow_right*

Probability *keyboard_arrow_right*

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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*

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Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*