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

The number of values of $$\theta $$ $$ \in $$ (0, $$\pi $$) for which the system of linear equations

x + 3y + 7z = 0

$$-$$ x + 4y + 7z = 0

(sin3$$\theta $$)x + (cos2$$\theta $$)y + 2z = 0.

has a non-trival solution, is -

x + 3y + 7z = 0

$$-$$ x + 4y + 7z = 0

(sin3$$\theta $$)x + (cos2$$\theta $$)y + 2z = 0.

has a non-trival solution, is -

A

two

B

one

C

four

D

three

$$\left| {\matrix{
1 & 3 & 7 \cr
{ - 1} & 4 & 7 \cr
{\sin 3\theta } & {\cos 2\theta } & 2 \cr
} } \right| = 0$$

(8 $$-$$ 7 cos 2$$\theta $$) $$-$$ 3($$-$$2 $$-$$ 7 sin 3$$\theta $$)

+7 ($$-$$ cos 2$$\theta $$ $$-$$ 4 sin 3$$\theta $$) = 0

14 $$-$$ 7 cos 2$$\theta $$ + 21 sin 3$$\theta $$ $$-$$ 7 cos 2$$\theta $$

$$-$$ 28 sin 3$$\theta $$ = 0

14 $$-$$ 7 sin 3$$\theta $$ $$-$$ 14 cos 2$$\theta $$ = 0

14 $$-$$ 7 (3 sin $$\theta $$ $$-$$ 4 sin^{3}$$\theta $$ ) $$-$$ 14 (1 $$-$$ 2 sin^{2} $$\theta $$) = 0

$$-$$ 21 sin $$\theta $$ + 28 sin^{3} $$\theta $$ + 28 sin^{2} $$\theta $$ = 0

7 sin $$\theta $$ [$$-$$ 3 + 4 sin^{2} $$\theta $$ + 4 sin $$\theta $$] = 0 sin$$\theta $$,

4 sin^{2} $$\theta $$ + 6 sin $$\theta $$ $$-$$ 2 sin $$\theta $$ $$-$$ 3 = 0

2 sin $$\theta $$(2 sin $$\theta $$ + 3) $$-$$ 1 (2 sin $$\theta $$ + 3) = 0

sin $$\theta $$ = $${{ - 3} \over 2}$$; sin$$\theta $$ = $${1 \over 2}$$

Hence, 2 solutions in (0, $$\pi $$)

(8 $$-$$ 7 cos 2$$\theta $$) $$-$$ 3($$-$$2 $$-$$ 7 sin 3$$\theta $$)

+7 ($$-$$ cos 2$$\theta $$ $$-$$ 4 sin 3$$\theta $$) = 0

14 $$-$$ 7 cos 2$$\theta $$ + 21 sin 3$$\theta $$ $$-$$ 7 cos 2$$\theta $$

$$-$$ 28 sin 3$$\theta $$ = 0

14 $$-$$ 7 sin 3$$\theta $$ $$-$$ 14 cos 2$$\theta $$ = 0

14 $$-$$ 7 (3 sin $$\theta $$ $$-$$ 4 sin

$$-$$ 21 sin $$\theta $$ + 28 sin

7 sin $$\theta $$ [$$-$$ 3 + 4 sin

4 sin

2 sin $$\theta $$(2 sin $$\theta $$ + 3) $$-$$ 1 (2 sin $$\theta $$ + 3) = 0

sin $$\theta $$ = $${{ - 3} \over 2}$$; sin$$\theta $$ = $${1 \over 2}$$

Hence, 2 solutions in (0, $$\pi $$)

2

If the system of linear equations

2x + 2y + 3z = a

3x – y + 5z = b

x – 3y + 2z = c

where a, b, c are non zero real numbers, has more one solution, then :

2x + 2y + 3z = a

3x – y + 5z = b

x – 3y + 2z = c

where a, b, c are non zero real numbers, has more one solution, then :

A

b – c – a = 0

B

a + b + c = 0

C

b – c + a = 0

D

b + c – a = 0

P_{1} : 2x + 2y + 3z = a

P_{2} : 3x $$-$$ y + 5z = b

P_{3} : x $$-$$ 3y + 2z = c

We find

P_{1} + P_{3} = P_{2} $$ \Rightarrow $$ a + c = b

P

P

We find

P

3

Let A = $$\left( {\matrix{
0 & {2q} & r \cr
p & q & { - r} \cr
p & { - q} & r \cr
} } \right).$$ If AA^{T} = I_{3}, then $$\left| p \right|$$ is

A

$${1 \over {\sqrt 2 }}$$

B

$${1 \over {\sqrt 5 }}$$

C

$${1 \over {\sqrt 6 }}$$

D

$${1 \over {\sqrt 3 }}$$

A is orthogonal matrix

$$ \Rightarrow $$ 0^{2} + p^{2} + p^{2} = 1

$$ \Rightarrow $$ $$\left| p \right| = {1 \over {\sqrt 2 }}$$

$$ \Rightarrow $$ 0

$$ \Rightarrow $$ $$\left| p \right| = {1 \over {\sqrt 2 }}$$

4

If $$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a - b} \cr
} } \right|$$

= (a + b + c) (x + a + b + c)^{2}, x $$ \ne $$ 0,

then x is equal to :

= (a + b + c) (x + a + b + c)

then x is equal to :

A

–2(a + b + c)

B

2(a + b + c)

C

abc

D

–(a + b + c)

$$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a - b} \cr
} } \right|$$

R_{1} $$ \to $$ R_{1} + R_{2} + R_{3}

$$ = \left| {\matrix{ {a + b + c} & {a + b + c} & {a + b + c} \cr {2b} & {b - c - a} & {2b} \cr {2c} & {2c} & {c - a - b} \cr } } \right|$$

$$ = \left( {a + b + c} \right)\left| {\matrix{ 1 & 0 & 0 \cr {2b} & { - \left( {a + b + c} \right)} & 0 \cr {2c} & {2c} & {c - a - b} \cr } } \right|$$

$$=$$ (a + b + c) (a + b + c)^{2}

$$ \Rightarrow $$ x $$=$$ $$-$$ 2(a + b + c)

R

$$ = \left| {\matrix{ {a + b + c} & {a + b + c} & {a + b + c} \cr {2b} & {b - c - a} & {2b} \cr {2c} & {2c} & {c - a - b} \cr } } \right|$$

$$ = \left( {a + b + c} \right)\left| {\matrix{ 1 & 0 & 0 \cr {2b} & { - \left( {a + b + c} \right)} & 0 \cr {2c} & {2c} & {c - a - b} \cr } } \right|$$

$$=$$ (a + b + c) (a + b + c)

$$ \Rightarrow $$ x $$=$$ $$-$$ 2(a + b + c)

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (1) *keyboard_arrow_right*

AIEEE 2003 (3) *keyboard_arrow_right*

AIEEE 2004 (3) *keyboard_arrow_right*

AIEEE 2005 (4) *keyboard_arrow_right*

AIEEE 2006 (2) *keyboard_arrow_right*

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JEE Main 2013 (Offline) (1) *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*