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

A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point :

A

$$\left( {{1 \over 3},24} \right)$$

B

$$\left( {{1 \over 2},4} \right)$$

C

$$\left( {2,{1 \over 8}} \right)$$

D

$$\left( {3,{1 \over 28}} \right)$$

2

The function f defined by

f(x) = x^{3} $$-$$ 3x^{2} + 5x + 7 , is :

f(x) = x

A

increasing in **R**.

B

decreasing in **R**.

C

decreasing in (0, $$\infty $$) and increasing in ($$-$$ $$\infty $$, 0)

D

increasing in (0, $$\infty $$) and decreasing in ($$-$$ $$\infty $$, 0)

3

If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm^{2}) of this cone is :

A

$$6\sqrt 2 \pi $$

B

$$6\sqrt 3 \pi $$

C

$$8\sqrt 2 \pi $$

D

$$8\sqrt 3 \pi $$

Sphere of radius r = 3 cm

Let b, h be base radius and height of cone respectively.

So, volume of cone = $${1 \over 2}$$ $$\pi $$b^{2}h

In right $$\Delta $$ABC by Pythagoras theorem

(h $$-$$ r)^{2} + b^{2} = r^{2}

$$ \Rightarrow $$ b^{2} = r^{2} $$-$$ (h $$-$$ r)^{2} = r^{2} $$-$$ (h^{2} $$-$$ 2hr + r^{2}) = 2hr $$-$$ h^{2}

$$\therefore\,\,\,$$ Volume (v) = $${1 \over 3}$$ $$\pi $$h[2hr $$-$$ h^{2}] = $${1 \over 3}$$ [ 2h^{2}r $$-$$ h^{3}]

$${{dv} \over {dh}}$$ = $${1 \over 3}$$ [4hr $$-$$ 3h^{2}] = 0

$$ \Rightarrow $$ h (4r $$-$$ 3h) = 0

$${{{d^2}v} \over {d{h^2}}}$$ = $${1 \over 3}$$ [4r $$-$$ 6h]

At h = $${{4r} \over 3}$$, $${{{d^2}y} \over {d{h^2}}}$$ = $${1 \over 3}$$ $$\left[ {4r - {{4r} \over 3} \times 6} \right] = {1 \over 3}\left[ {4r - 8r} \right] < 0$$

$$ \Rightarrow $$ maximum volume cours at h = $${{4r} \over 3}$$ = $${4 \over 3}$$ $$ \times $$ 3 = 4 cm

As from (1),

(h $$-$$ r)^{2} + b^{2} = r^{2}

$$ \Rightarrow $$ b^{2} = 2hr $$-$$ h^{2} = 2.$${{4r} \over 3}$$ r $$-$$ $${{16{r^2}} \over 9}$$ = $${{8{r^2}} \over 3}$$ $$-$$ $${{16{r^2}} \over 9}$$

= $${{\left( {24 - 16} \right){r^2}} \over 9}$$ = $${{8{r^2}} \over 9}$$

$$ \Rightarrow $$ b = $${{2\sqrt 2 } \over 3}$$ r = 2 $$\sqrt 2 \,\,m$$

Therefore curved surface area = $$\pi bl$$

= $$\pi $$b$$\sqrt {{h^2} + {r^2}} $$ = $$\pi $$2$$\sqrt 2 $$ $$\sqrt {{4^2} + 8} $$ = 8$$\sqrt 3 $$$$\pi $$ cm^{2}

Let b, h be base radius and height of cone respectively.

So, volume of cone = $${1 \over 2}$$ $$\pi $$b

In right $$\Delta $$ABC by Pythagoras theorem

(h $$-$$ r)

$$ \Rightarrow $$ b

$$\therefore\,\,\,$$ Volume (v) = $${1 \over 3}$$ $$\pi $$h[2hr $$-$$ h

$${{dv} \over {dh}}$$ = $${1 \over 3}$$ [4hr $$-$$ 3h

$$ \Rightarrow $$ h (4r $$-$$ 3h) = 0

$${{{d^2}v} \over {d{h^2}}}$$ = $${1 \over 3}$$ [4r $$-$$ 6h]

At h = $${{4r} \over 3}$$, $${{{d^2}y} \over {d{h^2}}}$$ = $${1 \over 3}$$ $$\left[ {4r - {{4r} \over 3} \times 6} \right] = {1 \over 3}\left[ {4r - 8r} \right] < 0$$

$$ \Rightarrow $$ maximum volume cours at h = $${{4r} \over 3}$$ = $${4 \over 3}$$ $$ \times $$ 3 = 4 cm

As from (1),

(h $$-$$ r)

$$ \Rightarrow $$ b

= $${{\left( {24 - 16} \right){r^2}} \over 9}$$ = $${{8{r^2}} \over 9}$$

$$ \Rightarrow $$ b = $${{2\sqrt 2 } \over 3}$$ r = 2 $$\sqrt 2 \,\,m$$

Therefore curved surface area = $$\pi bl$$

= $$\pi $$b$$\sqrt {{h^2} + {r^2}} $$ = $$\pi $$2$$\sqrt 2 $$ $$\sqrt {{4^2} + 8} $$ = 8$$\sqrt 3 $$$$\pi $$ cm

4

If $$\beta $$ is one of the angles between the normals to the ellipse, x^{2} + 3y^{2} = 9 at the points (3 cos $$\theta $$, $$\sqrt 3 \sin \theta $$) and ($$-$$ 3 sin $$\theta $$, $$\sqrt 3 \,\cos \theta $$); $$\theta \in \left( {0,{\pi \over 2}} \right);$$ then $${{2\,\cot \beta } \over {\sin 2\theta }}$$ is equal to :

A

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

B

$${1 \over {\sqrt 3 }}$$

C

$$\sqrt 2 $$

D

$${{\sqrt 3 } \over 4}$$

Since, x^{2} + 3y^{2} = 9

$$ \Rightarrow $$ 2x + 6y $${{dy} \over {dx}}$$ = 0

$$ \Rightarrow $$ $${{dy} \over {dx}}$$ = $${{ - x} \over {3y}}$$

Slope of normal is $$-$$ $${{dx} \over {dy}}$$ = $${{3y} \over x}$$

$$ \Rightarrow $$ $${\left( { - {{dx} \over {dy}}} \right)_{\left( {3\cos \theta ,\sqrt 3 \sin \theta } \right)}}$$

= $${{3\sqrt 3 \sin \theta } \over {3\cos \theta }}$$ = $$\sqrt 3 \tan \theta $$ = m_{1}

& $${\left( { - {{dx} \over {dy}}} \right)_{\left( { - 3\sin \theta ,\sqrt 3 \cos \theta } \right)}}$$

= $${{3\sqrt 3 \cos \theta } \over { - 3\sin \theta }}$$ = $$ - \sqrt 3 \cot \theta $$ = m_{2}

As, $$\beta $$ is the angle between the normals to the given ellipse then

tan$$\beta $$ = $$\left| {{{{m_1} - {m_2}} \over {1 + {m_1}{m_2}}}} \right|$$

= $$\left| {{{\sqrt 3 \tan \theta + \sqrt 3 \cot \theta } \over {1 - 3\tan \theta \cot \theta }}} \right|$$ = $$\left| {{{\sqrt 3 \tan \theta + \sqrt 3 \cot \theta } \over {1 - 3}}} \right|$$

So, tan $$\beta $$ = $${{\sqrt 3 } \over 2}$$ $$\left| {\tan \theta + \cot \theta } \right|$$

$$ \Rightarrow $$ $${1 \over {\cot \beta }} = {{\sqrt 3 } \over 2}\left| {{{\sin \theta } \over {\cos \theta }} + {{\cos \theta } \over {\sin \theta }}} \right|$$

$$ \Rightarrow $$ $${1 \over {\cot \beta }} = {{\sqrt 3 } \over 2}$$ $$\left| {{1 \over {\sin \theta \cos \theta }}} \right|$$

$$ \Rightarrow $$ $${1 \over {\cot \beta }} = {{\sqrt 3 } \over {\sin 2\theta }}$$

$$ \Rightarrow $$ $${{2\cot \beta } \over {\sin 2\theta }}$$ = $${2 \over {\sqrt 3 }}$$

$$ \Rightarrow $$ 2x + 6y $${{dy} \over {dx}}$$ = 0

$$ \Rightarrow $$ $${{dy} \over {dx}}$$ = $${{ - x} \over {3y}}$$

Slope of normal is $$-$$ $${{dx} \over {dy}}$$ = $${{3y} \over x}$$

$$ \Rightarrow $$ $${\left( { - {{dx} \over {dy}}} \right)_{\left( {3\cos \theta ,\sqrt 3 \sin \theta } \right)}}$$

= $${{3\sqrt 3 \sin \theta } \over {3\cos \theta }}$$ = $$\sqrt 3 \tan \theta $$ = m

& $${\left( { - {{dx} \over {dy}}} \right)_{\left( { - 3\sin \theta ,\sqrt 3 \cos \theta } \right)}}$$

= $${{3\sqrt 3 \cos \theta } \over { - 3\sin \theta }}$$ = $$ - \sqrt 3 \cot \theta $$ = m

As, $$\beta $$ is the angle between the normals to the given ellipse then

tan$$\beta $$ = $$\left| {{{{m_1} - {m_2}} \over {1 + {m_1}{m_2}}}} \right|$$

= $$\left| {{{\sqrt 3 \tan \theta + \sqrt 3 \cot \theta } \over {1 - 3\tan \theta \cot \theta }}} \right|$$ = $$\left| {{{\sqrt 3 \tan \theta + \sqrt 3 \cot \theta } \over {1 - 3}}} \right|$$

So, tan $$\beta $$ = $${{\sqrt 3 } \over 2}$$ $$\left| {\tan \theta + \cot \theta } \right|$$

$$ \Rightarrow $$ $${1 \over {\cot \beta }} = {{\sqrt 3 } \over 2}\left| {{{\sin \theta } \over {\cos \theta }} + {{\cos \theta } \over {\sin \theta }}} \right|$$

$$ \Rightarrow $$ $${1 \over {\cot \beta }} = {{\sqrt 3 } \over 2}$$ $$\left| {{1 \over {\sin \theta \cos \theta }}} \right|$$

$$ \Rightarrow $$ $${1 \over {\cot \beta }} = {{\sqrt 3 } \over {\sin 2\theta }}$$

$$ \Rightarrow $$ $${{2\cot \beta } \over {\sin 2\theta }}$$ = $${2 \over {\sqrt 3 }}$$

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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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*

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

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

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

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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*

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

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

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

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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*

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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*

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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*