Class CBSE Class 12 Mathematics Matrices and Determinants Q #1678
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If $\begin{vmatrix}-1&-2&5\\-2&a&-1\\0&4&2a\end{vmatrix}=-86$, then the sum of all possible values of a is
(A) 4
(B) 5
(C) -4
(D) 9
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Correct Answer: C

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Detailed Solution

Step 1: Expand the determinant

We expand the determinant along the first row: $$ \Delta = -1 \begin{vmatrix} a & -1 \\ 4 & 2a \end{vmatrix} - (-2) \begin{vmatrix} -2 & -1 \\ 0 & 2a \end{vmatrix} + 5 \begin{vmatrix} -2 & a \\ 0 & 4 \end{vmatrix} = -86 $$

Step 2: Calculate the minors

Calculating the 2x2 determinants: $$ -1(2a^2 - (-4)) + 2(-4a - 0) + 5(-8 - 0) = -86 $$ $$ -1(2a^2 + 4) + 2(-4a) + 5(-8) = -86 $$

Step 3: Simplify the equation

Distribute and combine terms: $$ -2a^2 - 4 - 8a - 40 = -86 $$ $$ -2a^2 - 8a - 44 = -86 $$ $$ -2a^2 - 8a + 42 = 0 $$

Step 4: Solve for a

Divide by -2: $$ a^2 + 4a - 21 = 0 $$ Using the sum of roots formula for a quadratic equation $ax^2 + bx + c = 0$, the sum of roots is $-b/a$. Here, sum $= -4/1 = -4$.

Final Answer: -4

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the expansion rule for determinants and solve a resulting quadratic equation.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a sequence of algebraic steps (expansion, simplification, and root calculation) to reach the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the student's ability to manipulate algebraic expressions within the framework of matrix determinants, a core topic in the Algebra unit.

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