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1

JEE Main 2016 (Offline)

MCQ (Single Correct Answer)
English
Hindi
If $$A = \left[ {\matrix{ {5a} & { - b} \cr 3 & 2 \cr } } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$$ then $$5a+b$$ is equal to :
A
$$4$$
B
$$13$$
C
$$-1$$
D
$$5$$

Explanation

$$A\left( {Adj\,\,A} \right) = A\,{A^T}$$

$$ \Rightarrow {A^{ - 1}}A\left( {adj\,\,A} \right) = {A^{ - 1}}A\,{A^T}$$

$$Adj\,\,A = {A^T}$$

$$ \Rightarrow \left[ {\matrix{ 2 & b \cr { - 3} & {5a} \cr } } \right] = \left[ {\matrix{ {5a} & 3 \cr { - b} & 2 \cr } } \right]$$

$$ \Rightarrow a = {2 \over 5}\,\,$$ and $$\,\,b = 3$$

$$ \Rightarrow 5a + b = 5$$

यदि $$\mathrm{A}=\left[\begin{array}{cc}5 a & -b \\ 3 & 2\end{array}\right]$$ तथा $$\mathrm{A}$$ $$\mathrm{adj}$$ $$\mathrm{A}=\mathrm{A} ~\mathrm{A}^{\mathrm{T}}$$ हैं, तो $$5 a+b$$ बराबर है :

A
$$4$$
B
$$13$$
C
$$-1$$
D
$$5$$
2

JEE Main 2015 (Offline)

MCQ (Single Correct Answer)
English
Hindi
If $$A = \left[ {\matrix{ 1 & 2 & 2 \cr 2 & 1 & { - 2} \cr a & 2 & b \cr } } \right]$$ is a matrix satisfying the equation

$$A{A^T} = 9\text{I},$$ where $$I$$ is $$3 \times 3$$ identity matrix, then the ordered

pair $$(a, b)$$ is equal to :
A
$$(2, 1)$$
B
$$(-2, -1)$$
C
$$(2, -1)$$
D
$$(-2, 1)$$

Explanation

$$\left[ {\matrix{ 1 & 2 & 2 \cr 2 & 1 & { - 2} \cr a & 2 & b \cr } } \right]\left[ {\matrix{ 1 & 2 & a \cr 2 & 1 & 2 \cr 2 & { - 2} & b \cr } } \right] = \left[ {\matrix{ 9 & 0 & 0 \cr 0 & 9 & 0 \cr 0 & 0 & 9 \cr } } \right]$$

$$ \Rightarrow \left[ {\matrix{ {1 + 4 + 4} & {2 + 2 - 4} & {a + 4 + 2b} \cr {2 + 2 - 4} & {4 + 1 + 4} & {2a + 2 - 2b} \cr {a + 4 + 2b} & {2a + 2 - 2b} & {{a^2} + 4 + {b^2}} \cr } } \right]$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left[ {\matrix{ 9 & 0 & 0 \cr 0 & 9 & 0 \cr 0 & 0 & 9 \cr } } \right]$$

$$ \Rightarrow a + 4 + 2b = 0$$ $$ \Rightarrow a + 2b = - 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

$$2a + 2 - 2b = 0 \Rightarrow 2a - 2b = - 2$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\,\,\,\,\,\,\,\,\\,$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow a - b = - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$$

On solving $$(i)$$ and $$(ii)$$ we get

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\,\,\,\,\,\,\,\,$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 1 + b + 2b = - 4\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$$$b=-1$$ and $$a=-2$$

$$\left( {a,b} \right) = \left( { - 2, - 1} \right)$$

यदि $$A = \left[ {\matrix{ 1 & 2 & 2 \cr 2 & 1 & { - 2} \cr a & 2 & b \cr } } \right]$$ एक ऐसा आव्यूह है जो आव्यूह समीकरण $$\mathrm{AA}^{\mathrm{T}}=9 \mathrm{I}$$, को संतुष्ट करता है, जहाँ $$\mathrm{I}, 3 \times 3$$ का तत्समक आव्यूह है, तो क्रमित युग्म $$(a, b)$$ का मान है :

A
$$(2, 1)$$
B
$$(-2, -1)$$
C
$$(2, -1)$$
D
$$(-2, 1)$$
3

JEE Main 2015 (Offline)

MCQ (Single Correct Answer)
English
Hindi
The set of all values of $$\lambda $$ for which the system of linear equations:

$$\matrix{ {2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}} \cr {2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}} \cr { - {x_1} + 2{x_2} = \lambda {x_3}} \cr } $$

has a non-trivial solution
A
contains two elements
B
contains more than two elements
C
in an empty set
D
is a singleton

Explanation

$$\left. {\matrix{ {2{x_1} - 2{x_2} + {x^3} = \lambda {x_1}} \cr {2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}} \cr {\,\,\,\,\,\,\,\,\,\, - {x_1} + 2{x_2} = \lambda {x_3}} \cr } } \right\}$$

$$\eqalign{ & \Rightarrow \,\,\,\,\,\,\,\left( {2 - \lambda } \right){x_1} - 2{x_2} + {x_3} = 0 \cr & \,\,\,\,\,\,\,\,\,\,\,\,2{x_1} - \left( {3 + \lambda } \right){x_2} + 2{x_3} = 0 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - {x_1} + 2{x_2} - \lambda {x_3} = 0 \cr} $$

For non-trivial solution, $$\Delta = 0$$

i.e. $$\,\,\,\left| {\matrix{ {2 - \lambda } & { - 2} & 1 \cr 2 & { - \left( {3 + \lambda } \right)} & 2 \cr { - 1} & 2 & { - \lambda } \cr } } \right| = 0$$

$$ \Rightarrow \left( {2 - \lambda } \right)\left[ {\lambda \left( {3 + \lambda } \right) - 4} \right] + $$

$$\,\,\,\,\,\,\,\,\,2\left[ { - 2\lambda + 2} \right] + 1\left[ {4 - \left( {3 + \lambda } \right)} \right] = 0$$

$$ \Rightarrow {\lambda ^3} + {\lambda ^2} - 5\lambda + 3 = 0$$

$$ \Rightarrow \lambda = 1,1,3$$

Hence, $$\lambda $$ has $$2$$ values.

$$\lambda$$ के सभी मानों का समुच्चय, जिनके लिए रैखिक समीकरण निकाय

$$2 x_{1}-2 x_{2}+x_{3}=\lambda x_{1}$$

$$2 x_{1}-3 x_{2}+2 x_{3}=\lambda x_{2}$$

$$-x_{1}+2 x_{2} =\lambda x_{3}$$

का एक अतुच्छ हल है,

A
में दो अवयव हैं।
B
में दो से अधिक अवयव हैं।
C
एक रिक्त समुच्चय है।
D
एक एकल समुच्चय है।
4

JEE Main 2014 (Offline)

MCQ (Single Correct Answer)
If $$A$$ is a $$3 \times 3$$ non-singular matrix such that $$AA'=A'A$$ and
$$B = {A^{ - 1}}A',$$ then $$BB'$$ equals:
A
$${B^{ - 1}}$$
B
$$\left( {{B^{ - 1}}} \right)'$$
C
$$I+B$$
D
$$I$$

Explanation

$$BB' = B\left( {{A^{ - 1}}A'} \right)' = B\left( {A'} \right)'\left( {{A^{ - 1}}} \right)' = BA\left( {{A^{ - 1}}} \right)'$$

$$ = \left( {{A^{ - 1}}A'} \right)\left( {A\left( {{A^{ - 1}}} \right)'} \right)$$

$$ = {A^{ - 1}}A.A'.\left( {{A^{ - 1}}} \right)'\,\,\,\,\,\,$$ $$\left\{ {} \right.$$ as $$\,\,\,\,\,\,$$ $$AA' = A'A$$ $$\left. \, \right\}$$

$$ = I\left( {{A^{ - 1}}A} \right)'$$

$$ = I.I = {I^2} = I$$

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