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 5, 5r, 5r^{2} are the lengths of the sides of a triangle, then r cannot be equal to -

A

$${7 \over 4}$$

B

$${5 \over 4}$$

C

$${3 \over 4}$$

D

$${3 \over 2}$$

r = 1 is obviously true.

Let 0 < r < 1

$$ \Rightarrow $$ r + r^{2} > 1

$$ \Rightarrow $$ r^{2} + r $$-$$ 1 > 0

$$\left( {r - {{ - 1 - \sqrt 5 } \over 2}} \right)\left( {r - \left( {{{ - 1 + \sqrt 5 } \over 2}} \right)} \right)$$

$$ \Rightarrow r - {{ - 1 - \sqrt 5 } \over 2}$$ or $$r > {{ - 1 + \sqrt 5 } \over 2}$$

$$r \in \left( {{{\sqrt 5 - 1} \over 2},1} \right)$$

$${{\sqrt 5 - 1} \over 2} < r < 1$$

When r > 1

$$ \Rightarrow {{\sqrt 5 + 1} \over 2} > {1 \over r} > 1$$

$$ \Rightarrow r \in \left( {{{\sqrt 5 - 1} \over 2},{{\sqrt 5 + 1} \over 2}} \right)$$

Now check options

Let 0 < r < 1

$$ \Rightarrow $$ r + r

$$ \Rightarrow $$ r

$$\left( {r - {{ - 1 - \sqrt 5 } \over 2}} \right)\left( {r - \left( {{{ - 1 + \sqrt 5 } \over 2}} \right)} \right)$$

$$ \Rightarrow r - {{ - 1 - \sqrt 5 } \over 2}$$ or $$r > {{ - 1 + \sqrt 5 } \over 2}$$

$$r \in \left( {{{\sqrt 5 - 1} \over 2},1} \right)$$

$${{\sqrt 5 - 1} \over 2} < r < 1$$

When r > 1

$$ \Rightarrow {{\sqrt 5 + 1} \over 2} > {1 \over r} > 1$$

$$ \Rightarrow r \in \left( {{{\sqrt 5 - 1} \over 2},{{\sqrt 5 + 1} \over 2}} \right)$$

Now check options

2

A point P moves on the line 2x – 3y + 4 = 0. If Q(1, 4) and R (3, – 2) are fixed points, then the locus of the centroid of $$\Delta $$PQR is a line -

A

parallel to y-axis

B

with slope $${2 \over 3}$$

C

parallel to x-axis

D

with slope $${3 \over 2}$$

Let the centroid of $$\Delta $$PQR is (h, k) & P is ($$\alpha $$, $$\beta $$), then

$${{\alpha + 1 + 3} \over 3} = h\,$$ and $${{\beta + 4 - 2} \over 3} = k$$

$$\alpha = \left( {3h - 4} \right)$$ $$\beta = \left( {3k - 4} \right)$$

Point P($$\alpha $$, $$\beta $$) lies on the line 2x $$-$$ 3y + 4 = 0

$$ \therefore $$ 2(3h $$-$$ 4) $$-$$ 3 (3k $$-$$ 2) + 4 = 0

$$ \Rightarrow $$ locus is 6x $$-$$ 9y + 2 = 0

$${{\alpha + 1 + 3} \over 3} = h\,$$ and $${{\beta + 4 - 2} \over 3} = k$$

$$\alpha = \left( {3h - 4} \right)$$ $$\beta = \left( {3k - 4} \right)$$

Point P($$\alpha $$, $$\beta $$) lies on the line 2x $$-$$ 3y + 4 = 0

$$ \therefore $$ 2(3h $$-$$ 4) $$-$$ 3 (3k $$-$$ 2) + 4 = 0

$$ \Rightarrow $$ locus is 6x $$-$$ 9y + 2 = 0

3

If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is -

A

(3, 4)

B

(2, 2)

C

(4, 4)

D

(4, 3)

$$\left| {{{3r + 4r - 24} \over 5}} \right| = r$$

$$7r - 24 = \pm 5r$$

$$2r = 24$$ or $$12r + 24$$

$$r = 14,\,\,\,r = 2$$

then incentre is $$(2,2)$$

4

Two sides of a parallelogram are along the lines, x + y = 3 & x – y + 3 = 0. If its diagonals intersect at (2, 4), then one of its vertex is -

A

(2, 1)

B

(2, 6)

C

(3, 5)

D

(3, 6)

Solving

$$\matrix{ {x + y = 3} \cr {x - y = - 3} \cr } \,\, > \,\,A\left( {0,3} \right)$$

and $${{{x_1} + 0} \over 2} = 2;\,\,{x_i} = 4$$

similarly y

C $$ \Rightarrow $$ (4, 5)

Now equation of BC is x $$-$$ y = $$-$$ 1

and equation of CD is x + y = 9

Solving x + y = 9 and x $$-$$ y = $$-$$ 3

Point D is (3, 6)

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (4) *keyboard_arrow_right*

AIEEE 2003 (5) *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*