JEE Advanced · Mathematics · 18. Matrices

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

A
$2 p^2$
B
$p^3-5 p$
C
$p^3-3 p$
D
$p^3-p^2$

Verified Solution

Answer & Solution

Correct Answer

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

Step-by-step Solution

Detailed explanation

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

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Let

α, β

and

γ

be real numbers. Consider the following system of linear equations

x + 2 y + z = 7

x + α z = 11

2 x - 3 y + β z = γ

Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$(P)$	If $β = \frac{1}{2} (7 α - 3)$ and $γ = 28$ then the system has	$(1)$	a unique solution
$(Q)$	If $β = \frac{1}{2} (7 α - 3)$ and $γ \neq 28$ , then the system has	$(2)$	no solution
$(R)$	If $β \neq \frac{1}{2} (7 α - 3)$ where $α = 1$ and $γ \neq 28$ , then the system has	$(3)$	infinitely many solutions
$(S)$	If $β \neq \frac{1}{2} (7 α - 3)$ where $α = 1$ and $γ = 28$ , then the system has	$(4)$	$x = 11, y = - 2$ and $z = 0$ as a solution
		$(5)$	$x = - 15, y = 4$ and $z = 0$ as a solution

The correct option is:

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From JEE Advanced

Columns 1, 2 and 3 contain starting materials, reaction conditions and type of reactions respectively.

Column 1	Column 2	Column 3
(I) Toulene	(i) $NaOH$ $ /Br _2$	(P) Condensation
(II) Acetophenone	(ii) $B r_2 / h v$	(Q) Carboxylation
(III) Benzaldehyde	(iii) $\left( CH _3 CO \right)_2$ $O / CH _3$ $COOK$	(R) Substitution
(IV) Phenol	(iv) $NaOH$ $/CO _2$	(S) Haloform

The only CORRECT combination in which the reaction proceeds through radical mechanism is

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