Joint Entrance Examination

Graduate Aptitude Test in Engineering

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General Aptitude

1

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

2

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

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

3

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)

4

Let A and B be two invertible matrices of order 3 $$ \times $$ 3. If det(ABA^{T}) = 8 and det(AB^{–1}) = 8,

then det (BA^{–1} B^{T}) is equal to :

then det (BA

A

$${1 \over 4}$$

B

16

C

$${1 \over {16}}$$

D

1

$${\left| A \right|^2}.\left| B \right| = 8$$

and $${{\left| A \right|} \over {\left| B \right|}} = 8 \Rightarrow \left| A \right| = 4$$

and $$\left| B \right| = {1 \over 2}$$

$$ \therefore $$ det(BA^{$$-$$1}. B^{T}) $$ = {1 \over 4} \times {1 \over 4} = {1 \over {16}}$$

and $${{\left| A \right|} \over {\left| B \right|}} = 8 \Rightarrow \left| A \right| = 4$$

and $$\left| B \right| = {1 \over 2}$$

$$ \therefore $$ det(BA

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