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Mathematics · Sets and Relations

JEE Main 2025 — 28 January, Morning Shift — Question 21

Let M denote the set of all real matrices of order 3×33 \times 3 and let S={−3,−2,−1,1,2}S=\{-3,-2,-1,1,2\}. Let

S1={A=[aij]∈M:A=ATS_{1}=\left\{A=\left[a_{i j}\right] \in M: A=A^{T}\right. and aij∈S,∀i,j}\left.a_{i j} \in S, \forall i, j\right\}

S2={A=[aij]∈M:A=−AT\mathrm{S}_{2}=\left\{\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}}\right. and aij∈S,∀i,j}\left.\mathrm{a}_{\mathrm{ij}} \in \mathrm{S}, \forall \mathrm{i}, \mathrm{j}\right\}

S3={A=[aij]∈M:a11+a22+a33=0\mathrm{S}_{3}=\left\{\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{a}_{11}+\mathrm{a}_{22}+\mathrm{a}_{33}=0\right. and aij∈S,∀i,j}\left.\mathrm{a}_{\mathrm{ij}} \in \mathrm{S}, \forall \mathrm{i}, \mathrm{j}\right\}

If n(S1∪S2∪S3)=125αn\left(S_{1} \cup S_{2} \cup S_{3}\right)=125 \alpha, then α\alpha equals.

Answer: 1613

Numerical answer — enter this value.

Step-by-step solution

[a11a12a13a21a22a23a31a32a33]\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} No. of elements in S1:A=AT⇒53×53\text{No. of elements in } S_1 : A = A^T \Rightarrow 5^3 \times 5^3 No. of elements in A=−AT⇒0\text{No. of elements in } A = -A^T \Rightarrow 0 since no. zero in 5\text{since no. zero in 5} No. of elements in S3⇒\text{No. of elements in } S_3 \Rightarrow a11+a22+a33=0⇒(1,2,−3)⇒3!a_{11} + a_{22} + a_{33} = 0 \Rightarrow (1,2,-3) \Rightarrow 3! or (1,1,−2)⇒3\text{or } (1,1,-2) \Rightarrow 3 or (−1,−1,2)⇒3\text{or } (-1,-1,2) \Rightarrow 3 ⇒12×56\Rightarrow 12 \times 5^6 n(S1∩S3)=12×53n(S_1 \cap S_3) = 12 \times 5^3 n(S1∪S2∪S3)=n(S1∪S3)=n(S1)+n(S3)−n(S1∩S3)=56(1+12)−12×53n(S_1 \cup S_2 \cup S_3) = n(S_1 \cup S_3) = n(S_1) + n(S_3) - n(S_1 \cap S_3) = 5^6 (1+12) - 12 \times 5^3 ⇒53×[13×53−12]=125α\Rightarrow 5^3 \times [13 \times 5^3 - 12] = 125 \alpha α=1613\alpha = 1613

Answer key and solution verified before publishing.

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Exam
JEE Main 2025
Paper
28 January, Morning Shift
Subject
Mathematics
Chapter
Sets and Relations
Topic
Sets

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