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

The number of distinct real values of $$\lambda $$ for which the lines

$${{x - 1} \over 1} = {{y - 2} \over 2} = {{z + 3} \over {{\lambda ^2}}}$$ and $${{x - 3} \over 1} = {{y - 2} \over {{\lambda ^2}}} = {{z - 1} \over 2}$$ are coplanar is :

$${{x - 1} \over 1} = {{y - 2} \over 2} = {{z + 3} \over {{\lambda ^2}}}$$ and $${{x - 3} \over 1} = {{y - 2} \over {{\lambda ^2}}} = {{z - 1} \over 2}$$ are coplanar is :

A

4

B

1

C

2

D

3

As planes are coplanar, so

$$\left| {\matrix{ {3 - 1} & {2 - 2} & {1 - \left( { - 3} \right)} \cr 1 & 2 & {{\lambda ^2}} \cr 1 & {{\lambda ^2}} & 2 \cr } } \right| $$ = 0

$$ \Rightarrow $$ $$\left| {\matrix{ 2 & 0 & 4 \cr 1 & 2 & {{\lambda ^2}} \cr 1 & {{\lambda ^2}} & 2 \cr } } \right| $$ = 0

$$ \Rightarrow $$ 2(4 $$-$$ $$\lambda $$^{4}) + 4($$\lambda $$^{2} $$-$$ 2) = 0

$$ \Rightarrow $$ 4 $$-$$ $$\lambda $$^{4} + 2$$\lambda $$^{2} $$-$$ 4 = 0

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

$$ \Rightarrow $$ $$\lambda $$ = 0, $$\sqrt 2 , - \sqrt 2 $$

$$ \therefore $$ 3 distinct real values are possible.

$$\left| {\matrix{ {3 - 1} & {2 - 2} & {1 - \left( { - 3} \right)} \cr 1 & 2 & {{\lambda ^2}} \cr 1 & {{\lambda ^2}} & 2 \cr } } \right| $$ = 0

$$ \Rightarrow $$ $$\left| {\matrix{ 2 & 0 & 4 \cr 1 & 2 & {{\lambda ^2}} \cr 1 & {{\lambda ^2}} & 2 \cr } } \right| $$ = 0

$$ \Rightarrow $$ 2(4 $$-$$ $$\lambda $$

$$ \Rightarrow $$ 4 $$-$$ $$\lambda $$

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

$$ \Rightarrow $$ $$\lambda $$ = 0, $$\sqrt 2 , - \sqrt 2 $$

$$ \therefore $$ 3 distinct real values are possible.

2

Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $${{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}$$ respectively, then the position vector of the orthocentre of this triangle, is :

A

$${\overrightarrow a + \overrightarrow b + \overrightarrow c }$$

B

$$ - \left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}} \right)$$

C

$$\overrightarrow 0 $$

D

$$\left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}} \right)$$

Given,

Position vector of circumcentre, $$\overrightarrow C = {{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}$$

We know, position vector of centroid, $$\overrightarrow G = {{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 3}$$

Now, let $$\overrightarrow R $$ be the orthocentre of the triangle.

We know, $$\overrightarrow G $$ $$ = {{2\overrightarrow C + \overrightarrow R } \over 3}$$

$$ \Rightarrow $$ 3$$\overrightarrow G $$ $$ = 2\overrightarrow C + \overrightarrow R $$

$$ \Rightarrow $$ $$\overrightarrow R = 3\overrightarrow G - 2\overrightarrow C $$

= $$\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) - 2\left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}} \right)$$

= $${{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}$$

3

If the image of the point P(1, –2, 3) in the plane, 2x + 3y – 4z + 22 = 0 measured parallel to the line,

$${x \over 1} = {y \over 4} = {z \over 5}$$ is Q, then PQ is equal to:

$${x \over 1} = {y \over 4} = {z \over 5}$$ is Q, then PQ is equal to:

A

$$2\sqrt {42} $$

B

$$\sqrt {42} $$

C

$$6\sqrt 5 $$

D

$$3\sqrt 5 $$

Equation of line PQ is $${{x - 1} \over 1} = {{y + 2} \over 4} = {{z - 3} \over 5}$$

Let F be ($$\lambda $$ + 1, 4$$\lambda $$ $$-$$ 2, 5$$\lambda $$ + 3)

Since F lies on the plane

$$ \therefore $$ 2($$\lambda $$ + 1) + 3(4$$\lambda $$ $$-$$ 2) $$-$$ 4(5$$\lambda $$ + 3) + 22 $$=$$ 0

$$ \Rightarrow $$ $$-$$ 6$$\lambda $$ + 6 = 0 $$ \Rightarrow $$ $$\lambda $$ = 1

$$ \therefore $$ F is (2, 2, 8)

PQ = 2 PF = 2$$\sqrt {{1^2} + {4^2} + {5^2}} $$ = 2$$\sqrt {42} $$

Let F be ($$\lambda $$ + 1, 4$$\lambda $$ $$-$$ 2, 5$$\lambda $$ + 3)

Since F lies on the plane

$$ \therefore $$ 2($$\lambda $$ + 1) + 3(4$$\lambda $$ $$-$$ 2) $$-$$ 4(5$$\lambda $$ + 3) + 22 $$=$$ 0

$$ \Rightarrow $$ $$-$$ 6$$\lambda $$ + 6 = 0 $$ \Rightarrow $$ $$\lambda $$ = 1

$$ \therefore $$ F is (2, 2, 8)

PQ = 2 PF = 2$$\sqrt {{1^2} + {4^2} + {5^2}} $$ = 2$$\sqrt {42} $$

4

The distance of the point (1, 3, – 7) from the plane passing through the point (1, –1, – 1), having normal
perpendicular to both the lines

$${{x - 1} \over 1} = {{y + 2} \over { - 2}} = {{z - 4} \over 3}$$

and

$${{x - 2} \over 2} = {{y + 1} \over { - 1}} = {{z + 7} \over { - 1}}$$ is

$${{x - 1} \over 1} = {{y + 2} \over { - 2}} = {{z - 4} \over 3}$$

and

$${{x - 2} \over 2} = {{y + 1} \over { - 1}} = {{z + 7} \over { - 1}}$$ is

A

$${{10} \over {\sqrt {83} }}$$

B

$${{5} \over {\sqrt {83} }}$$

C

$${{10} \over {\sqrt {74} }}$$

D

$${{20} \over {\sqrt {74} }}$$

Let the plane be

a(x $$-$$ 1) + b(y + 1) + c (z + 1) = 0

Normal vector

$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & { - 2} & 3 \cr 2 & { - 1} & { - 1} \cr } } \right| = 5\widehat i + 7\widehat j + 3\widehat k$$

So plane is 5(x $$-$$ 1) + 7(y + 1) + 3(z + 1) = 0

$$ \Rightarrow $$ 5x + 7y + 3z + 5 = 0

Distance of point (1, 3, $$-$$ 7) from the plane is

$${{5 + 21 - 21 + 5} \over {\sqrt {25 + 49 + 9} }} = {{10} \over {\sqrt {83} }}$$

a(x $$-$$ 1) + b(y + 1) + c (z + 1) = 0

Normal vector

$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & { - 2} & 3 \cr 2 & { - 1} & { - 1} \cr } } \right| = 5\widehat i + 7\widehat j + 3\widehat k$$

So plane is 5(x $$-$$ 1) + 7(y + 1) + 3(z + 1) = 0

$$ \Rightarrow $$ 5x + 7y + 3z + 5 = 0

Distance of point (1, 3, $$-$$ 7) from the plane is

$${{5 + 21 - 21 + 5} \over {\sqrt {25 + 49 + 9} }} = {{10} \over {\sqrt {83} }}$$

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (9) *keyboard_arrow_right*

AIEEE 2003 (12) *keyboard_arrow_right*

AIEEE 2004 (10) *keyboard_arrow_right*

AIEEE 2005 (11) *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*