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

Let $$\overrightarrow a = \widehat i + 2\widehat j + 4\widehat k,$$ $$\overrightarrow b = \widehat i + \lambda \widehat j + 4\widehat k$$ and $$\overrightarrow c = 2\widehat i + 4\widehat j + \left( {{\lambda ^2} - 1} \right)\widehat k$$ be coplanar vectors. Then the non-zero vector $$\overrightarrow a \times \overrightarrow c $$ is :

A

$$ - 10\widehat i - 5\widehat j$$

B

$$ - 10\widehat i + 5\widehat j$$

C

$$ - 14\widehat i + 5\widehat j$$

D

$$ - 14\widehat i - 5\widehat j$$

$$\left[ {\overrightarrow a \,\,\overrightarrow b \,\,\overrightarrow c } \right] = 0$$

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

$$ \Rightarrow {\lambda ^3} - 2{\lambda ^2} - 9\lambda + 18 = 0$$

$$ \Rightarrow {\lambda ^2}\left( {\lambda - 2} \right) - 9\left( {\lambda - 2} \right) = 0$$

$$ \Rightarrow \left( {\lambda - 3} \right)\left( {\lambda + 3} \right)\left( {\lambda - 2} \right) = 0$$

$$ \Rightarrow \lambda = 2,3, - 3$$

So, $$\lambda $$ = 2 (as $$\overrightarrow a $$ is parallel to $$\overrightarrow c $$ for $$\lambda $$ = $$ \pm $$3)

Hence $$\overrightarrow a \times \overrightarrow c = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & 2 & 4 \cr 2 & 4 & 3 \cr } } \right|$$

$$ = - 10\widehat i + 5\widehat j$$

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

$$ \Rightarrow {\lambda ^3} - 2{\lambda ^2} - 9\lambda + 18 = 0$$

$$ \Rightarrow {\lambda ^2}\left( {\lambda - 2} \right) - 9\left( {\lambda - 2} \right) = 0$$

$$ \Rightarrow \left( {\lambda - 3} \right)\left( {\lambda + 3} \right)\left( {\lambda - 2} \right) = 0$$

$$ \Rightarrow \lambda = 2,3, - 3$$

So, $$\lambda $$ = 2 (as $$\overrightarrow a $$ is parallel to $$\overrightarrow c $$ for $$\lambda $$ = $$ \pm $$3)

Hence $$\overrightarrow a \times \overrightarrow c = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & 2 & 4 \cr 2 & 4 & 3 \cr } } \right|$$

$$ = - 10\widehat i + 5\widehat j$$

2

The plane containing the line $${{x - 3} \over 2} = {{y + 2} \over { - 1}} = {{z - 1} \over 3}$$ and also containing its projection on the plane 2x + 3y $$-$$ z = 5, contains which one of the following points ?

A

($$-$$ 2, 2, 2)

B

(2, 2, 0)

C

(2, 0, $$-$$ 2)

D

(0, $$-$$ 2, 2)

The normal vector of required plane

$$ = \left( {2\widehat i - \widehat j + 3\widehat k} \right) \times \left( {2\widehat i + 3\widehat j - \widehat k} \right)$$

$$ = - 8\widehat i + 8\widehat j + 8\widehat k$$

So, direction ratio of normal is $$\left( { - 1,1,1} \right)$$

So required plane is

$$ - \left( {x - 3} \right) + \left( {y + 2} \right) + \left( {z - 1} \right) = 0$$

$$ \Rightarrow - x + y + z + 4 = 0$$

Which is satisfied by $$\left( {2,0, - 2} \right)$$

$$ = \left( {2\widehat i - \widehat j + 3\widehat k} \right) \times \left( {2\widehat i + 3\widehat j - \widehat k} \right)$$

$$ = - 8\widehat i + 8\widehat j + 8\widehat k$$

So, direction ratio of normal is $$\left( { - 1,1,1} \right)$$

So required plane is

$$ - \left( {x - 3} \right) + \left( {y + 2} \right) + \left( {z - 1} \right) = 0$$

$$ \Rightarrow - x + y + z + 4 = 0$$

Which is satisfied by $$\left( {2,0, - 2} \right)$$

3

The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and making an angle $${\pi \over 4}$$ with the plane y $$-$$ z + 5 = 0 are :

A

2, $$-$$1, 1

B

$$2\sqrt 3 ,1, - 1$$

C

$$\sqrt 2 ,1, - 1$$

D

$$\sqrt 2 , - \sqrt 2 $$

Let the equation of plane be

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

It passes through (0, 0, 1) then

b + c = 0 . . . . (1)

Now cos $${\pi \over 4}$$ = $${{a\left( 0 \right) + b\left( 1 \right) + c\left( { - 1} \right)} \over {\sqrt 2 \sqrt {{a^2} + {b^2} + {c^2}} }}$$

$$ \Rightarrow $$ a^{2} $$=$$ $$-$$ 2bc and b $$=$$ $$-$$ c

we get a^{2} $$=$$ 2c^{2}

$$ \Rightarrow $$ a $$=$$ $$ \pm $$ $$\sqrt 2 $$ c

$$ \Rightarrow $$ direction ratio (a, b, c) = ($$\sqrt 2 $$, $$-$$1, 1) or ($$\sqrt 2 $$, 1, $$-$$ 1)

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

It passes through (0, 0, 1) then

b + c = 0 . . . . (1)

Now cos $${\pi \over 4}$$ = $${{a\left( 0 \right) + b\left( 1 \right) + c\left( { - 1} \right)} \over {\sqrt 2 \sqrt {{a^2} + {b^2} + {c^2}} }}$$

$$ \Rightarrow $$ a

we get a

$$ \Rightarrow $$ a $$=$$ $$ \pm $$ $$\sqrt 2 $$ c

$$ \Rightarrow $$ direction ratio (a, b, c) = ($$\sqrt 2 $$, $$-$$1, 1) or ($$\sqrt 2 $$, 1, $$-$$ 1)

4

Two lines $${{x - 3} \over 1} = {{y + 1} \over 3} = {{z - 6} \over { - 1}}$$ and $${{x + 5} \over 7} = {{y - 2} \over { - 6}} = {{z - 3} \over 4}$$ intersect at the point R. The reflection of R in the xy-plane has coordinates :

A

(2, 4, 7)

B

(2, $$-$$ 4, $$-$$7)

C

(2, $$-$$ 4, 7)

D

($$-$$ 2, 4, 7)

Point on L_{1} ($$\lambda $$ + 3, 3$$\lambda $$ $$-$$ 1, $$-$$$$\lambda $$ + 6)

Point on L_{2} (7$$\mu $$ $$-$$ 5, $$-$$6$$\mu $$ + 2, 4$$\mu $$ + 3

$$ \Rightarrow $$ $$\lambda $$ + 3 = 7$$\mu $$ $$-$$ 5 . . . . (i)

3$$\lambda $$ $$-$$ 1 = $$-$$6$$\mu $$ + 2 . . . .(ii)

$$ \Rightarrow $$ $$\lambda $$ = $$-$$1, $$\mu $$ = 1

point R(2, $$-$$ 4, 7)

Reflection is (2, $$-$$4, $$-$$ 7)

Point on L

$$ \Rightarrow $$ $$\lambda $$ + 3 = 7$$\mu $$ $$-$$ 5 . . . . (i)

3$$\lambda $$ $$-$$ 1 = $$-$$6$$\mu $$ + 2 . . . .(ii)

$$ \Rightarrow $$ $$\lambda $$ = $$-$$1, $$\mu $$ = 1

point R(2, $$-$$ 4, 7)

Reflection is (2, $$-$$4, $$-$$ 7)

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (9) *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*