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JEE Advanced 2010Paper 1
Exam
JEE Advanced
Year
2010
Paper
Paper 1
Subject
Mathematics
Chapter
Matrices

JEE Advanced 2010 · Paper 1 · Mathematics · Matrices

Paragraph: Let p be an odd prime number and T_p be the following...

Paragraph: Let pp be an odd prime number and TpT_p be the following set of 2×22 \times 2 matrices T_p=\left\{A=\left[\begin{array}{ll}a & b \\ c & a \end{\text{ array }}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\} Question: The number of AA in TpT_p such that det⁡(A)\operatorname{det}(A) is not divisible by p, is

Options

  1. A2p22 p^2
  2. Bp3−5pp^3-5 p
  3. Cp3−3pp^3-3 p
  4. Dp3−p2p^3-p^2

Correct answer

D. p3−p2p^3-p^2

Step-by-step solution

The number of matrices for which pp does not divide Tr⁡(A)=(p−1)p2\operatorname{Tr}(A)=(p-1) p^2 of these (p−1)2(p-1)^2 are such that pp divides ∣A∣|A|. The number of matrices for which pp divides Tr⁡(A)\operatorname{Tr}(A) and p does not divides ∣A∣|A| are (p−1)2(p-1)^2 ∴ Required number =(p−1)p2−(p−1)2+(p−1)2=p3−p2\begin{aligned} & \therefore \text{ Required number } \\ & =(p-1) p^2-(p-1)^2+(p-1)^2 \\ & =p^3-p^2 \end{aligned}
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