Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geotechnical Engineering

Transportation Engineering

Irrigation

Engineering Mathematics

Construction Material and Management

Fluid Mechanics and Hydraulic Machines

Hydrology

Environmental Engineering

Engineering Mechanics

Structural Analysis

Reinforced Cement Concrete

Steel Structures

Geomatics Engineering Or Surveying

General Aptitude

1

If $${{dy} \over {dx}} + {3 \over {{{\cos }^2}x}}y = {1 \over {{{\cos }^2}x}},\,\,x \in \left( {{{ - \pi } \over 3},{\pi \over 3}} \right)$$ and $$y\left( {{\pi \over 4}} \right) = {4 \over 3},$$ then $$y\left( { - {\pi \over 4}} \right)$$ equals -

A

$${1 \over 3} + {e^6}$$

B

$${1 \over 3}$$

C

$${1 \over 3}$$ + e^{3}

D

$$-$$ $${4 \over 3}$$

$${{dy} \over {dx}} + 3{\sec ^2}x.y = {\sec ^2}x$$

I.F. = $${e^{3\int {{{\sec }^2}xdx} }} = {e^{3\tan x}}$$

or $$y.e{}^{3\tan x} = \int {{{\sec }^2}x.{e^{3\tan x}}} $$

or $$y.{e^{3\tan x}} = {1 \over 3}{e^{3\tan x}} + C$$

Given

$$y\left( {{\pi \over 4}} \right) = {4 \over 3}$$

$$ \therefore $$ $${4 \over 3}.{e^3} = {1 \over 3}{e^3} + C$$

$$ \therefore $$ C = e^{3}

I.F. = $${e^{3\int {{{\sec }^2}xdx} }} = {e^{3\tan x}}$$

or $$y.e{}^{3\tan x} = \int {{{\sec }^2}x.{e^{3\tan x}}} $$

or $$y.{e^{3\tan x}} = {1 \over 3}{e^{3\tan x}} + C$$

Given

$$y\left( {{\pi \over 4}} \right) = {4 \over 3}$$

$$ \therefore $$ $${4 \over 3}.{e^3} = {1 \over 3}{e^3} + C$$

$$ \therefore $$ C = e

2

The shortest distance between the point $$\left( {{3 \over 2},0} \right)$$ and the curve y = $$\sqrt x $$, (x > 0), is -

A

$${{\sqrt 3 } \over 2}$$

B

$${5 \over 4}$$

C

$${3 \over 2}$$

D

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

Let points $$\left( {{3 \over 2},0} \right),\left( {{t^2},t} \right),t > 0$$

Distance = $$\sqrt {{t^2} + {{\left( {{t^2} - {3 \over 2}} \right)}^2}} $$

= $$\sqrt {{t^4} - 2{t^2} + {9 \over 4}} = \sqrt {{{\left( {{t^2} - 1} \right)}^2} + {5 \over 4}} $$

So minimum distance is $$\sqrt {{5 \over 4}} = {{\sqrt 5 } \over 2}$$

Distance = $$\sqrt {{t^2} + {{\left( {{t^2} - {3 \over 2}} \right)}^2}} $$

= $$\sqrt {{t^4} - 2{t^2} + {9 \over 4}} = \sqrt {{{\left( {{t^2} - 1} \right)}^2} + {5 \over 4}} $$

So minimum distance is $$\sqrt {{5 \over 4}} = {{\sqrt 5 } \over 2}$$

3

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

4

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

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (2) *keyboard_arrow_right*

AIEEE 2003 (1) *keyboard_arrow_right*

AIEEE 2004 (4) *keyboard_arrow_right*

AIEEE 2005 (4) *keyboard_arrow_right*

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JEE Main 2016 (Online) 9th April Morning Slot (2) *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*

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*

Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *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*