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

Suppose A is any 3$$ \times $$ 3 nonsingular matrx and ( A $$-$$ 3I) (A $$-$$ 5I) = O where I = I_{3} and O = O_{3}. If $$\alpha $$A + $$\beta $$A^{-1} = 4I, then $$\alpha $$ + $$\beta $$ is equal to :

A

8

B

7

C

13

D

12

Given,

( A $$-$$ 3I) (A $$-$$ 5I) = O

$$ \Rightarrow $$ A^{2} - 8A + 15I = O

Multiplying both sides by A^{- 1}, we get,

A^{- 1}A.A - 8A^{- 1}A + 15A^{- 1}I = A^{- 1}O

$$ \Rightarrow $$ A - 8I + 15A^{- 1} = O

$$ \Rightarrow $$ A + 15A^{- 1} = 8I

$$ \Rightarrow $$$${A \over 2} + {{15{A^{ - 1}}} \over 2} = 4I$$

Comparing with the equation $$\alpha $$A + $$\beta $$A^{-1} = 4I, we get

$$\alpha $$ = $${1 \over 2}$$ and $$\beta $$ = $${15 \over 2}$$

$$ \therefore $$ $$\alpha $$ + $$\beta $$ = $${1 \over 2}$$ + $${15 \over 2}$$ = $${16 \over 2}$$ = 8

( A $$-$$ 3I) (A $$-$$ 5I) = O

$$ \Rightarrow $$ A

Multiplying both sides by A

A

$$ \Rightarrow $$ A - 8I + 15A

$$ \Rightarrow $$ A + 15A

$$ \Rightarrow $$$${A \over 2} + {{15{A^{ - 1}}} \over 2} = 4I$$

Comparing with the equation $$\alpha $$A + $$\beta $$A

$$\alpha $$ = $${1 \over 2}$$ and $$\beta $$ = $${15 \over 2}$$

$$ \therefore $$ $$\alpha $$ + $$\beta $$ = $${1 \over 2}$$ + $${15 \over 2}$$ = $${16 \over 2}$$ = 8

2

If the system of linear equations

x + ay + z = 3

x + 2y + 2z = 6

x + 5y + 3z = b

has no solution, then :

x + ay + z = 3

x + 2y + 2z = 6

x + 5y + 3z = b

has no solution, then :

A

a = $$-$$ 1, b = 9

B

a = $$-$$ 1, b $$ \ne $$ 9

C

a $$ \ne $$ $$-$$ 1, b = 9

D

a = 1, b $$ \ne $$ 9

As the given system of equations has no solution then

$$\Delta $$ = 0 and at least one of $$\Delta $$_{1}, $$\Delta $$_{2} and $$\Delta $$_{2} should not be zero.

$$ \therefore $$ $$\Delta $$ = $$\left| {\matrix{ 1 & a & 1 \cr 1 & 2 & 2 \cr 1 & 5 & 3 \cr } } \right| = 0$$

$$ \Rightarrow $$ - $$a$$ - 1 = 0

$$ \Rightarrow $$ a = - 1

$$\Delta $$_{2} = $$\left| {\matrix{
1 & 3 & 1 \cr
1 & 6 & 2 \cr
1 & b & 3 \cr
} } \right| \ne 0$$

$$ \Rightarrow $$ b $$ \ne $$ 0

$$\Delta $$ = 0 and at least one of $$\Delta $$

$$ \therefore $$ $$\Delta $$ = $$\left| {\matrix{ 1 & a & 1 \cr 1 & 2 & 2 \cr 1 & 5 & 3 \cr } } \right| = 0$$

$$ \Rightarrow $$ - $$a$$ - 1 = 0

$$ \Rightarrow $$ a = - 1

$$\Delta $$

$$ \Rightarrow $$ b $$ \ne $$ 0

3

The number of values of k for which the system of linear equations,

(k + 2)x + 10y = k

kx + (k +3)y = k -1

has**no solution,** is :

(k + 2)x + 10y = k

kx + (k +3)y = k -1

has

A

1

B

2

C

3

D

infinitely many

System of linear equation have no solution,

$$\therefore\,\,\,$$ determinant of coefficient = 0

$$\left| {\matrix{ {k + 2} & {10} \cr k & {k + 3} \cr } } \right| = 0$$

$$ \Rightarrow $$ $$\,\,\,\,$$ (k + 2) (k + 3) $$-$$ 10 K = 0

$$ \Rightarrow $$ $$\,\,\,\,$$ k^{2} $$-$$ 5k + 6 = 0

$$\therefore\,\,\,\,$$ k = 2, 3

When, k = 2 then equations become,

4x + 10y = 2

and 2x + 5y = 1

It has in finite number of solutions.

When k = 3, equations becomes

5x + 10y = 3

3x + 6y = 2

Those equation has no solutions.

$$\therefore\,\,\,\,$$ When k = 3, then system of equations have no solutions.

$$\therefore\,\,\,$$ determinant of coefficient = 0

$$\left| {\matrix{ {k + 2} & {10} \cr k & {k + 3} \cr } } \right| = 0$$

$$ \Rightarrow $$ $$\,\,\,\,$$ (k + 2) (k + 3) $$-$$ 10 K = 0

$$ \Rightarrow $$ $$\,\,\,\,$$ k

$$\therefore\,\,\,\,$$ k = 2, 3

When, k = 2 then equations become,

4x + 10y = 2

and 2x + 5y = 1

It has in finite number of solutions.

When k = 3, equations becomes

5x + 10y = 3

3x + 6y = 2

Those equation has no solutions.

$$\therefore\,\,\,\,$$ When k = 3, then system of equations have no solutions.

4

Let A = $$\left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right]$$ and B = A^{20}. Then the sum of the elements of the first column of B is :

A

210

B

211

C

231

D

251

=

A = $$\left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right]$$

A^{2} = A.A = $$\left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right] \times \left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right]$$

= $$\left[ {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right]$$

A^{3} = A^{2}.A = $$\left[ {\matrix{
1 & 0 & 0 \cr
2 & 1 & 0 \cr
3 & 2 & 1 \cr
} } \right] \times \left[ {\matrix{
1 & 0 & 0 \cr
1 & 1 & 0 \cr
1 & 1 & 1 \cr
} } \right]$$

= $$\left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 6 & 3 & 1 \cr } } \right]$$

Similarly

A^{4} = $$\left[ {\matrix{
1 & 0 & 0 \cr
4 & 1 & 0 \cr
{10} & 4 & 1 \cr
} } \right]$$

From this we can say,

A^{n} = $$\left[ {\matrix{
1 & 0 & 0 \cr
n & 1 & 0 \cr
{{{n\left( {n + 1} \right)} \over 2}} & n & 1 \cr
} } \right]$$

$$\therefore\,\,\,$$ A^{20} = $$\left[ {\matrix{
1 & 0 & 0 \cr
{20} & 1 & 0 \cr
{210} & {20} & 1 \cr
} } \right]$$

$$\therefore\,\,\,$$ Sum of the first column

= 1 + 20 + 210

= 231

A

= $$\left[ {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right]$$

A

= $$\left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 6 & 3 & 1 \cr } } \right]$$

Similarly

A

From this we can say,

A

$$\therefore\,\,\,$$ A

$$\therefore\,\,\,$$ Sum of the first column

= 1 + 20 + 210

= 231

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