Joint Entrance Examination

Graduate Aptitude Test in Engineering

Strength of Materials Or Solid Mechanics

Structural Analysis

Construction Material and Management

Reinforced Cement Concrete

Steel Structures

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

Hydrology

Irrigation

Geomatics Engineering Or Surveying

Environmental Engineering

Transportation Engineering

Engineering Mathematics

General Aptitude

1

Let A be a point on the line $$\overrightarrow r = \left( {1 - 3\mu } \right)\widehat i + \left( {\mu - 1} \right)\widehat j + \left( {2 + 5\mu } \right)\widehat k$$ and B(3, 2, 6) be a point in the space. Then the value of $$\mu $$ for which the vector $$\overrightarrow {AB} $$ is parallel to the plane x $$-$$ 4y + 3z = 1 is -

A

$${1 \over 8}$$

B

$${1 \over 2}$$

C

$${1 \over 4}$$

D

$$-$$ $${1 \over 4}$$

Let point A is

$$\left( {1 - 3\mu } \right)\widehat i + \left( {\mu - 1} \right)\widehat j + \left( {2 + 5\mu } \right)\widehat k$$

and point B is (3, 2, 6)

then $$\overrightarrow {AB} = \left( {2 + 3\mu } \right)\widehat i + \left( {3 - \mu } \right)\widehat j + \left( {4 - 5\mu } \right)\widehat k$$

which is parallel to the the plane x $$-$$ 4y + 3z = 1

$$ \therefore $$ 2 + 3$$\mu $$ $$-$$ 12 + 4$$\mu $$ + 12 $$-$$ 15$$\mu $$ = 0

8$$\mu $$ = 2

$$\mu $$ = $${1 \over 4}$$

$$\left( {1 - 3\mu } \right)\widehat i + \left( {\mu - 1} \right)\widehat j + \left( {2 + 5\mu } \right)\widehat k$$

and point B is (3, 2, 6)

then $$\overrightarrow {AB} = \left( {2 + 3\mu } \right)\widehat i + \left( {3 - \mu } \right)\widehat j + \left( {4 - 5\mu } \right)\widehat k$$

which is parallel to the the plane x $$-$$ 4y + 3z = 1

$$ \therefore $$ 2 + 3$$\mu $$ $$-$$ 12 + 4$$\mu $$ + 12 $$-$$ 15$$\mu $$ = 0

8$$\mu $$ = 2

$$\mu $$ = $${1 \over 4}$$

2

On which of the following lines lies the point of intersection of the line, $${{x - 4} \over 2} = {{y - 5} \over 2} = {{z - 3} \over 1}$$ and the plane,
x + y + z = 2 ?

A

$${{x - 4} \over 1} = {{y - 5} \over 1} = {{z - 5} \over { - 1}}$$

B

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

C

$${{x - 1} \over 1} = {{y - 3} \over 2} = {{z + 4} \over { - 5}}$$

D

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

General point on the given line is

x = 2$$\lambda $$ + 4

y = 2$$\lambda $$ + 5

z = $$\lambda $$ + 3

Solving with plane,

2$$\lambda $$ + 4 + 2$$\lambda $$ + 5 + $$\lambda $$ + 3 = 2

5$$\lambda $$ + 12 = 2

5$$\lambda $$ = $$-$$ 10

$$\lambda $$ = $$-$$ 2

x = 2$$\lambda $$ + 4

y = 2$$\lambda $$ + 5

z = $$\lambda $$ + 3

Solving with plane,

2$$\lambda $$ + 4 + 2$$\lambda $$ + 5 + $$\lambda $$ + 3 = 2

5$$\lambda $$ + 12 = 2

5$$\lambda $$ = $$-$$ 10

$$\lambda $$ = $$-$$ 2

3

If $$\overrightarrow \alpha $$ = $$\left( {\lambda - 2} \right)\overrightarrow a + \overrightarrow b $$ and $$\overrightarrow \beta = \left( {4\lambda - 2} \right)\overrightarrow a + 3\overrightarrow b $$ be two given vectors $$\overrightarrow a $$ and $$\overrightarrow b $$ are non-collinear. The value of $$\lambda $$ for which vectors $$\overrightarrow \alpha $$ and $$\overrightarrow \beta $$ are collinear, is -

A

4

B

3

C

$$-$$3

D

$$-$$4

$$\overrightarrow \alpha = \left( {\lambda - 2} \right)\overrightarrow \alpha + \overrightarrow b $$

$$\overrightarrow \beta = \left( {4\lambda - 2} \right)\overrightarrow \alpha + 3\overrightarrow b $$

$${{\lambda - 2} \over {4\lambda - 2}} = {1 \over 3}$$

$$3\lambda - 6 = 4\lambda - 2$$

$$\lambda = - 4$$

$$\overrightarrow \beta = \left( {4\lambda - 2} \right)\overrightarrow \alpha + 3\overrightarrow b $$

$${{\lambda - 2} \over {4\lambda - 2}} = {1 \over 3}$$

$$3\lambda - 6 = 4\lambda - 2$$

$$\lambda = - 4$$

4

The plane which bisects the line segment joining the points (–3, –3, 4) and (3, 7, 6) at right angles, passes through which one of the following points ?

A

(2, 1, 3)

B

(4, $$-$$ 1, 2)

C

(4, 1, $$-$$ 2)

D

($$-$$ 2, 3, 5)

p : 3(x $$-$$ 0) + 5(y $$-$$ 2) + 1 (z $$-$$ 5) = 0

3x + 5y + z = 15

Number in Brackets after Paper Name Indicates No of Questions

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

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Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

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Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

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Vector Algebra and 3D Geometry *keyboard_arrow_right*

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Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

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