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

If the tangent at a point P, with parameter t, on the curve x = 4t^{2} + 3, y = 8t^{3}−1, *t* $$ \in $$ **R**, meets the curve again at a point Q, then the coordinates of Q are :

A

(t^{2} + 3, − t^{3} −1)

B

(4t^{2} + 3, − 8t^{3} −1)

C

(t^{2} + 3, t^{3} −1)

D

(16t^{2} + 3, − 64t^{3} −1)

Given, x = 4t^{2} + 3 and y = 8t^{3} $$-$$ 1

$$ \therefore $$ P $$ \equiv $$ (4t^{2} + 3, 8t^{3} $$-$$ 1)

$${{dx} \over {dt}} = 8t$$ and $${{dy} \over {dt}}$$ $$=$$ 24t^{2}

Slope of tangent at

P $$=$$ $${{dy} \over {dx}}$$ $$=$$ $${{dy/dt} \over {dx/dt}}$$ $$=$$ 3t

Assume Q $$=$$ (4$$\lambda $$^{2} + 3, 8$$\lambda $$^{3} $$-$$ 1)

$$ \therefore $$ Slope of PQ $$=$$ 3t

$$ \Rightarrow $$ $${{8{\lambda ^3} - 8{t^3}} \over {4{\lambda ^2} - 4{t^2}}} = 3t$$

$$ \Rightarrow $$ $${{8\left( {\lambda - t} \right)\left( {{\lambda ^2} + {t^2} + \lambda t} \right)} \over {4\left( {\lambda - t} \right)\left( {\lambda + t} \right)}} = 3t$$

$$ \Rightarrow $$ 2$$\lambda $$^{2} + 2t^{2} + 2$$\lambda $$t $$=$$ 3t (t + $$\lambda $$)

$$ \Rightarrow $$ t^{2} + $$\lambda $$t $$-$$ 2$$\lambda $$^{2} = 0

$$ \Rightarrow $$ (t $$-$$ $$\lambda $$) (t + 2$$\lambda $$) $$=$$ 0

$$ \therefore $$ $$\lambda $$ $$=$$ t or $$\lambda $$ $$=$$ $$-$$ $${t \over 2}$$

$$ \therefore $$ Q $$=$$ (4t^{2} + 3, 8t^{3} $$-$$ 1) Or

Q $$=$$ $$\left( {4 \times {{\left( { - {t \over 2}} \right)}^2} + 3, - 8 \times {{{t^3}} \over 8} - 1} \right)$$

$$=$$ (t^{2} + 3, $$-$$ t^{3} $$-$$ 1)

$$ \therefore $$ P $$ \equiv $$ (4t

$${{dx} \over {dt}} = 8t$$ and $${{dy} \over {dt}}$$ $$=$$ 24t

Slope of tangent at

P $$=$$ $${{dy} \over {dx}}$$ $$=$$ $${{dy/dt} \over {dx/dt}}$$ $$=$$ 3t

Assume Q $$=$$ (4$$\lambda $$

$$ \therefore $$ Slope of PQ $$=$$ 3t

$$ \Rightarrow $$ $${{8{\lambda ^3} - 8{t^3}} \over {4{\lambda ^2} - 4{t^2}}} = 3t$$

$$ \Rightarrow $$ $${{8\left( {\lambda - t} \right)\left( {{\lambda ^2} + {t^2} + \lambda t} \right)} \over {4\left( {\lambda - t} \right)\left( {\lambda + t} \right)}} = 3t$$

$$ \Rightarrow $$ 2$$\lambda $$

$$ \Rightarrow $$ t

$$ \Rightarrow $$ (t $$-$$ $$\lambda $$) (t + 2$$\lambda $$) $$=$$ 0

$$ \therefore $$ $$\lambda $$ $$=$$ t or $$\lambda $$ $$=$$ $$-$$ $${t \over 2}$$

$$ \therefore $$ Q $$=$$ (4t

Q $$=$$ $$\left( {4 \times {{\left( { - {t \over 2}} \right)}^2} + 3, - 8 \times {{{t^3}} \over 8} - 1} \right)$$

$$=$$ (t

2

The minimum distance of a point on the curve y = x^{2}−4 from the origin is :

A

$${{\sqrt {19} } \over 2}$$

B

$$\sqrt {{{15} \over 2}} $$

C

$${{\sqrt {15} } \over 2}$$

D

$$\sqrt {{{19} \over 2}} $$

Let point on the curve

y = x^{2} $$-$$ 4 is ($$\alpha $$^{2}, $$\alpha $$^{2} $$-$$ 4)

$$ \therefore $$ Distance of the point ($$\alpha $$^{2}, $$\alpha $$^{2} $$-$$ 4) from origin,

D = $$\sqrt {{\alpha ^2} + {{\left( {{\alpha ^2} - 4} \right)}^2}} $$

$$ \Rightarrow $$ D^{2} = $$\alpha $$^{2} + $$\alpha $$^{4} + 16 $$-$$ 8$$\alpha $$^{2}

$$=$$ $$\alpha $$^{4} $$-$$ 7$$\alpha $$^{2} + 16

$$ \therefore $$ $${{d{D^2}} \over {d\alpha }}$$ = 4$$\alpha $$^{3} $$-$$ 14$$\alpha $$

Now, $${{d{D^2}} \over {d\alpha }}$$ = 0

$$ \Rightarrow $$ 4$$\alpha $$^{3} $$-$$ 14$$\alpha $$ = 0

$$ \Rightarrow $$ 2$$\alpha $$ (2$$\alpha $$^{2} $$-$$ 7) = 0

$$\alpha $$ = 0 or $$\alpha $$^{2} = $${7 \over 2}$$

$${{{d^2}{D^2}} \over {d{\alpha ^2}}} = 12{\alpha ^2} - 14$$

$$ \therefore $$ $${\left( {{{{d^2}{D^2}} \over {d{\alpha ^2}}}} \right)_{at\,\,\alpha = 0}} = - 14 < 0$$

$${\left( {{{{d^2}{D^2}} \over {d{\alpha ^2}}}} \right)_{at\,\,{\alpha ^2} = {7 \over 2}}} = 28 > 0$$

$$\therefore\,\,\,$$ Distance is minimum at $$\alpha $$^{2} = $${7 \over 2}$$

$$ \therefore $$ Minimum distance

D = $$\sqrt {{{49} \over 4} - {{49} \over 4} + 16} $$

= $${{\sqrt {15} } \over 2}$$

y = x

$$ \therefore $$ Distance of the point ($$\alpha $$

D = $$\sqrt {{\alpha ^2} + {{\left( {{\alpha ^2} - 4} \right)}^2}} $$

$$ \Rightarrow $$ D

$$=$$ $$\alpha $$

$$ \therefore $$ $${{d{D^2}} \over {d\alpha }}$$ = 4$$\alpha $$

Now, $${{d{D^2}} \over {d\alpha }}$$ = 0

$$ \Rightarrow $$ 4$$\alpha $$

$$ \Rightarrow $$ 2$$\alpha $$ (2$$\alpha $$

$$\alpha $$ = 0 or $$\alpha $$

$${{{d^2}{D^2}} \over {d{\alpha ^2}}} = 12{\alpha ^2} - 14$$

$$ \therefore $$ $${\left( {{{{d^2}{D^2}} \over {d{\alpha ^2}}}} \right)_{at\,\,\alpha = 0}} = - 14 < 0$$

$${\left( {{{{d^2}{D^2}} \over {d{\alpha ^2}}}} \right)_{at\,\,{\alpha ^2} = {7 \over 2}}} = 28 > 0$$

$$\therefore\,\,\,$$ Distance is minimum at $$\alpha $$

$$ \therefore $$ Minimum distance

D = $$\sqrt {{{49} \over 4} - {{49} \over 4} + 16} $$

= $${{\sqrt {15} } \over 2}$$

3

Let C be a curve given by y(x) = 1 + $$\sqrt {4x - 3} ,x > {3 \over 4}.$$ If P is a point
on C, such that the tangent at P has slope $${2 \over 3}$$, then a point through which the normal at P passes, is :

A

(2, 3)

B

(4, $$-$$3)

C

(1, 7)

D

(3, $$-$$ 4),

Given,

y = 1 + $$\sqrt {4x - 3} $$

$$ \therefore $$ $${{dy} \over {dx}}$$ = $${1 \over {2\sqrt {4x - 3} }} \times 4 = {2 \over 3}$$

$$ \Rightarrow $$ 4x $$-$$ 3 = 9

$$ \Rightarrow $$ x = 3

$$ \therefore $$ y = 1 + $$\sqrt {12 - 3} $$ = 4

$$ \therefore $$ Equation of normal at point P(3,4)

y $$-$$ 4 = $$-$$ $${3 \over 2}$$ (x $$-$$ 3)

$$ \Rightarrow $$ 2y $$-$$ 8 = $$-$$ 3x + 9

$$ \Rightarrow $$ 3x + 2y $$-$$ 17 = 0

y = 1 + $$\sqrt {4x - 3} $$

$$ \therefore $$ $${{dy} \over {dx}}$$ = $${1 \over {2\sqrt {4x - 3} }} \times 4 = {2 \over 3}$$

$$ \Rightarrow $$ 4x $$-$$ 3 = 9

$$ \Rightarrow $$ x = 3

$$ \therefore $$ y = 1 + $$\sqrt {12 - 3} $$ = 4

$$ \therefore $$ Equation of normal at point P(3,4)

y $$-$$ 4 = $$-$$ $${3 \over 2}$$ (x $$-$$ 3)

$$ \Rightarrow $$ 2y $$-$$ 8 = $$-$$ 3x + 9

$$ \Rightarrow $$ 3x + 2y $$-$$ 17 = 0

4

The tangent at the point (2, $$-$$2) to the curve, x^{2}y^{2} $$-$$ 2x = 4(1 $$-$$ y) **does not** pass through the point :

A

$$\left( {4,{1 \over 3}} \right)$$

B

(8, 5)

C

($$-$$4, $$-$$9)

D

($$-$$2, $$-$$7)

As, $${{dy} \over {dx}}$$ = $$-$$ $$\left[ {{{{{\delta f} \over {\delta x}}} \over {{{\delta f} \over {\delta y}}}}} \right]$$

$${{{\delta f} \over {\delta x}}}$$ = y^{2} $$ \times $$2x $$-$$ 2

$${{{\delta f} \over {\delta y}}}$$ = x^{2} $$ \times $$ 2y + 4

$$\therefore\,\,\,$$ $${{dy} \over {dx}}$$ = $$-$$ $$\left( {{{2x{y^2} - 2} \over {2{x^2}y + 4}}} \right)$$

$${\left[ {{{dy} \over {dx}}} \right]_{(2, - 2)}}$$ = $$-$$ $$\left( {{{2 \times 2 \times 4 - 2} \over {2 \times 4 \times ( - 2) + 4}}} \right)$$ = $$-$$ $$\left( {{{14} \over { - 12}}} \right)$$ = $${7 \over 6}$$

$$\therefore\,\,\,$$ Slope of tangent to the curve = $${7 \over 6}$$

Equation of tangent passes through (2, $$-$$ 2) is

y + 2 = $${7 \over 6}$$ (x $$-$$ 2)

$$ \Rightarrow $$$$\,\,\,$$ 7x $$-$$ 6y = 26 . . . . .(1)

Now put each option in equation (1) and see which one does not satisfy the equation.

By verifying each points you can see ($$-$$ 2, $$-$$ 7) does not satisfy the equation.

$${{{\delta f} \over {\delta x}}}$$ = y

$${{{\delta f} \over {\delta y}}}$$ = x

$$\therefore\,\,\,$$ $${{dy} \over {dx}}$$ = $$-$$ $$\left( {{{2x{y^2} - 2} \over {2{x^2}y + 4}}} \right)$$

$${\left[ {{{dy} \over {dx}}} \right]_{(2, - 2)}}$$ = $$-$$ $$\left( {{{2 \times 2 \times 4 - 2} \over {2 \times 4 \times ( - 2) + 4}}} \right)$$ = $$-$$ $$\left( {{{14} \over { - 12}}} \right)$$ = $${7 \over 6}$$

$$\therefore\,\,\,$$ Slope of tangent to the curve = $${7 \over 6}$$

Equation of tangent passes through (2, $$-$$ 2) is

y + 2 = $${7 \over 6}$$ (x $$-$$ 2)

$$ \Rightarrow $$$$\,\,\,$$ 7x $$-$$ 6y = 26 . . . . .(1)

Now put each option in equation (1) and see which one does not satisfy the equation.

By verifying each points you can see ($$-$$ 2, $$-$$ 7) does not satisfy the equation.

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*

AIEEE 2006 (2) *keyboard_arrow_right*

AIEEE 2007 (3) *keyboard_arrow_right*

AIEEE 2008 (2) *keyboard_arrow_right*

AIEEE 2009 (2) *keyboard_arrow_right*

AIEEE 2010 (3) *keyboard_arrow_right*

AIEEE 2011 (2) *keyboard_arrow_right*

AIEEE 2012 (3) *keyboard_arrow_right*

JEE Main 2013 (Offline) (1) *keyboard_arrow_right*

JEE Main 2014 (Offline) (1) *keyboard_arrow_right*

JEE Main 2015 (Offline) (1) *keyboard_arrow_right*

JEE Main 2016 (Offline) (2) *keyboard_arrow_right*

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

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

JEE Main 2017 (Online) 8th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2017 (Online) 9th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2018 (Online) 15th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2018 (Online) 16th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th January Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 11th January Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 11th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 8th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 8th April Evening Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th April Morning Slot (3) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th April Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th April Evening Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 9th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 2nd September Morning Slot (3) *keyboard_arrow_right*

JEE Main 2020 (Online) 2nd September Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 3rd September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 3rd September Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 4th September Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 5th September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 5th September Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 6th September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 6th September Evening Slot (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 24th February Morning Slot (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 24th February Evening Slot (3) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th February Morning Slot (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th February Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th February Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th February Evening Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 16th March Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 20th July Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 27th August Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 27th August Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 31st August Morning Shift (1) *keyboard_arrow_right*

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*