1
JEE Main 2016 (Offline)
+4
-1

The system of linear equations

$$\matrix{ {x + \lambda y - z = 0} \cr {\lambda x - y - z = 0} \cr {x + y - \lambda z = 0} \cr }$$

has a non-trivial solution for :
A
infinitely many values of $$\lambda .$$
B
exactly one value of $$\lambda .$$
C
exactly two values of $$\lambda .$$
D
exactly three values of $$\lambda .$$
2
JEE Main 2015 (Offline)
+4
-1
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
3
JEE Main 2015 (Offline)
+4
-1
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
$$(-2, 1)$$
4
JEE Main 2014 (Offline)
+4
-1
If $$\alpha ,\beta \ne 0,$$ and $$f\left( n \right) = {\alpha ^n} + {\beta ^n}$$ and $$\left| {\matrix{ 3 & {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} \cr {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} \cr {1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} & {1 + f\left( 4 \right)} \cr } } \right|$$\$
$$= K{\left( {1 - \alpha } \right)^2}{\left( {1 - \beta } \right)^2}{\left( {\alpha - \beta } \right)^2},$$ then $$K$$ is equal to :
A
$$1$$
B
$$-1$$
C
$$\alpha \beta$$
D
$${1 \over {\alpha \beta }}$$
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