JEE Advanced 2025 · Paper 2 · Mathematics Q5

← JEE Advanced 2025 · Paper 2

MathematicsMultiple correct+4 / -2Question 5 of 48

Let $I = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$ and $P = \begin{pmatrix} 2 & 0\\ 0 & 3 \end{pmatrix}$ . Let $Q = \begin{pmatrix} x & y\\ z & 4 \end{pmatrix}$ for some non-zero real numbers $x$ , $y$ , and $z$ , for which there is a $2\times2$ matrix $R$ with all entries being non-zero real numbers, such that $QR = RP$ .

Then which of the following statements is (are) TRUE?

The determinant of

Q - 2I

is zero

The determinant of

Q - 6I

is 12

The determinant of

Q - 3I

is 15

yz = 2

Answer and worked solution

Official answer: A, B

Eigenvectors. $QR=RP$ means the first column of $R$ is an eigenvector of $Q$ for eigenvalue 2, and the second column for eigenvalue 3. Both columns are nonzero, so $Q$ has eigenvalues 2 and 3.

Entries of Q.

Trace: $x+4=5$ , so $x=1$ .
Determinant: $4x-yz=6$ , so $yz=-2$ .

Options. $\det(Q-\lambda I)=(2-\lambda)(3-\lambda)$ .

(A) $\lambda=2$ : $0$ . ✓
(B) $\lambda=6$ : $(-4)(-3)=12$ . ✓
(C) $\lambda=3$ : $0$ , not 15. ✗
(D) $yz=-2$ , not 2. ✗

Topics:Matrix algebra Determinants and properties

← Previous question Next question →

JEE Advanced 2025 · Paper 2, Mathematics question 5