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 :

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.]

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 :

A block of mass m is on an inclined plane of angle θ. The coefficient of friction between the block and the plane is μ and tan θ > μ. The block is held stationary by applying a force P parallel to the plane. The direction of force pointing up the plane is taken to be positive. As P is varied from P₁ = mg(sinθ − μ cosθ) to P₂ = mg(sinθ + μ cosθ), the frictional force f versus P graph will look like