IIT-JEE 2010 Paper 1 Offline | Matrices and Determinants Question 46 | Mathematics

IIT-JEE 2010 Paper 1 Offline

MCQ (Single Correct Answer)

-1

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system

$\mathrm{A}\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solutions, is

$2^9-1$

168

IIT-JEE 2010 Paper 1 Offline

MCQ (Single Correct Answer)

-1

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices :

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname{det}(\mathrm{A}) \operatorname{divisible}$ by $p$ is :

$(p-1)^2$

$2(p-1)$

$(p-1)^2+1$

$2 p-1$

IIT-JEE 2010 Paper 1 Offline

MCQ (Single Correct Answer)

-1

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices :

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

The number of A in $\mathrm{T}_p$ such that the trace of A is not divisible by $p$ but $\operatorname{det}(\mathrm{A})$ is divisible by $p$ is

[Note : The trace of a matrix is the sum of its diagonal entries.]

$(p-1)\left(p^2-p+1\right)$

$p^3-(p-1)^2$

$(p-1)^2$

$(p-1)\left(p^2-2\right)$

IIT-JEE 2010 Paper 1 Offline

MCQ (Single Correct Answer)

-1

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices :

$$ \mathrm{T}_{\mathrm{p}}=\left\{\mathrm{A}=\left[\begin{array}{ll} a & b \\ c & a \end{array}\right]: a, b, c \in\{0,1,2, \ldots, p-1\}\right\} $$

The number of A in $\mathrm{T}_p$ such that $\operatorname{det}(\mathrm{A})$ is not divisible by $p$ is :

$2 p^2$

$p^3-5 p$

$p^3-3 p$

$p^3-p^2$

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