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 circles

x^{2} + y^{2} $$-$$ 16x $$-$$ 20y + 164 = r^{2}

and (x $$-$$ 4)^{2} + (y $$-$$ 7)^{2} = 36

intersect at two distinct points, then :

x

and (x $$-$$ 4)

intersect at two distinct points, then :

A

r > 11

B

0 < r < 1

C

r = 11

D

1 < r < 11

Circles are x^{2} + y^{2} $$-$$ 16x $$-$$ 20y + 164 = r^{2} $$ \Rightarrow $$ c_{1} (8, 10)

and (x $$-$$ 4)^{2} + (y $$-$$ 7)^{2} = 36

they intersect at two distinct points

$$\left| {{r_1} - {r_2}} \right| < {c_1}{c_2} < {r_1} + {r_2}\left\{ {{c_1}{c_2} = \sqrt {16 + 9} = 5} \right\}$$

Now $$\left| {r - 6} \right| < 5 < r + 6$$

$$\left| {r - 6} \right| < 5$$

$$ \Rightarrow $$ $$ - 5 < r - 6 < 5$$

$$ \Rightarrow $$ $$1 < r < 11\,\,\,\,\,\,\,\,\,...(i)$$

$$5 < r + 6$$

$$ - 1 < r\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)$$

from (i) and (ii)

r $$ \in $$ (1, 11)

and (x $$-$$ 4)

they intersect at two distinct points

$$\left| {{r_1} - {r_2}} \right| < {c_1}{c_2} < {r_1} + {r_2}\left\{ {{c_1}{c_2} = \sqrt {16 + 9} = 5} \right\}$$

Now $$\left| {r - 6} \right| < 5 < r + 6$$

$$\left| {r - 6} \right| < 5$$

$$ \Rightarrow $$ $$ - 5 < r - 6 < 5$$

$$ \Rightarrow $$ $$1 < r < 11\,\,\,\,\,\,\,\,\,...(i)$$

$$5 < r + 6$$

$$ - 1 < r\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)$$

from (i) and (ii)

r $$ \in $$ (1, 11)

2

If a circle C passing through the point (4, 0) touches the circle x^{2} + y^{2} + 4x – 6y = 12 externally at the point (1, – 1), then the radius of C is -

A

5

B

2$$\sqrt {5} $$

C

4

D

$$\sqrt {37} $$

x^{2} + y^{2} + 4x $$-$$ 6y $$-$$ 12 = 0

Equation of tangent at (1, $$-$$ 1)

x $$-$$ y + 2(x + 1) $$-$$ 3(y $$-$$ 1) $$-$$ 12 = 0

3x $$-$$ 4y $$-$$ 7 = 0

$$ \therefore $$ Equation of circle is

(x^{2} + y^{2} + 4x $$-$$ 6y $$-$$ 12) + $$\lambda $$ (3x $$-$$ 4y $$-$$ 7) = 0

It passes through (4, 0) :

(16 + 16 $$-$$ 12) + $$\lambda $$ (12 $$-$$ 7) = 0

$$ \Rightarrow $$ 20 + $$\lambda $$(5) = 0

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

$$ \therefore $$ (x^{2} + y^{2} + 4x $$-$$ 6y $$-$$ 12) $$-$$ 4(3x $$-$$ 4y $$-$$ 7) = 0

or x^{2} + y^{2} $$-$$ 8x + 10y + 16 = 0

Radius = $$\sqrt {16 + 25 - 16} = 5$$

Equation of tangent at (1, $$-$$ 1)

x $$-$$ y + 2(x + 1) $$-$$ 3(y $$-$$ 1) $$-$$ 12 = 0

3x $$-$$ 4y $$-$$ 7 = 0

$$ \therefore $$ Equation of circle is

(x

It passes through (4, 0) :

(16 + 16 $$-$$ 12) + $$\lambda $$ (12 $$-$$ 7) = 0

$$ \Rightarrow $$ 20 + $$\lambda $$(5) = 0

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

$$ \therefore $$ (x

or x

Radius = $$\sqrt {16 + 25 - 16} = 5$$

3

If the area of an equilateral triangle inscribed in the circle x^{2} + y^{2}
+ 10x + 12y + c = 0 is $$27\sqrt 3 $$ sq units then c is equal to

A

20

B

25

C

$$-$$ 25

D

13

$$3\left( {{1 \over 2}{r^2}.\sin {{120}^o}} \right) = 27\sqrt 3 $$

$${{{r^2}} \over 2}{{\sqrt 3 } \over 2} = {{27\sqrt 3 } \over 3}$$

$${r^2} = {{108} \over 3} = 36$$

Radius $$ = \sqrt {25 + 36 - C} = \sqrt {36} $$

$$C = 25$$

4

The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is

A

$$4\sqrt 5 $$

B

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

C

$$2\sqrt 5 $$

D

$${{\sqrt 5 } \over 4}$$

Equation of circle

(x $$-$$ 1) (x $$-$$ 0) + (y $$-$$ 0) (y $$-$$ $${1 \over 2}$$) = 0

$$ \Rightarrow $$ x

Equation of tangent of region is 2x + y = 0

$$\ell $$

= $${{4 + 1} \over {2\sqrt 5 }} = {{\sqrt 5 } \over 2}$$

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (4) *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*