IIT-JEE 2010 · Paper 1 · Mathematics Q44

← IIT-JEE 2010 · Paper 1

MathematicsSingle correct+3 / -1Question 44 of 84

Passage

Paragraph for Questions 42 to 44

Let p be an odd prime number and $T_p$ be the following set of $2\times2$ matrices:
$T_p=\left\{A=\begin{bmatrix}a&b\\c&a\end{bmatrix}:a,b,c\in\{0,1,2,\ldots,p-1\}\right\}$

The number of A in $T_p$ such that det (A) is not divisible by p is

2p^2

p^3-5p

p^3-3p

p^3-p^2

Answer and worked solution

Official answer: D

Count matrices with $p\mid\det A$ .

$a=0$ : need $bc\equiv0$ , i.e. $b=0$ or $c=0$ : $2p-1$ matrices.
$a\ne0$ : from the previous question, $(p-1)^2$ matrices.

Total $2p-1+(p-1)^2=p^2$ .

Complement. $p^3-p^2$ matrices have $\det A$ not divisible by $p$ .

(D)

Topics:Determinants and properties Fundamental principle of counting

← Previous question Next question →

IIT-JEE 2010 · Paper 1, Mathematics question 44