If x = -4 is a root of vmatrixx & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & xvmatrix = 0, then the sum of the other 2 roots is Option 1: 4 Option 2: -3 Option 3: 2 Option 4: 5

Answer: 4. Solution: Given that x = -4 is a root of vmatrixx & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & xvmatrix = 0 Expanding the determinant using the first row: x vmatrixx & 1 \\ 2 & xvmatrix - 2 vmatrix1 & 1 \\ 3 & xvmatrix + 3 vmatrix1 & x \\ 3 & 2vmatrix = 0 Computing each 2×2 determinant: vmatrixx & 1 \\ 2 & xvmatrix = x^2 - 2 vmatrix1 & 1 \\ 3 & xvmatrix = x - 3 vmatrix1 & x \\ 3 & 2vmatrix = 2 - 3x Substituting back: x(x^2 - 2) - 2(x - 3) + 3(2 - 3x) = 0 x^3 - 2x - 2x + 6 + 6 - 9x = 0 x^3 - 13x + 12 = 0 For a cubic equation x^3 + bx^2 + cx + d = 0, the sum of all roots equals -b. The equation is: x^3 + 0x^2 - 13x + 12 = 0 Sum of all three roots = -0 = 0 Given one root is x = -4: (-4) + (sum of other two roots) = 0 Sum of other two roots = 4

Algebra

Matrices & Determinants

Medium

applied

If $x = -4$ is a root of $\begin{vmatrix}x & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & x\end{vmatrix} = 0$ , then the sum of the other 2 roots is

-3

✅ Correct Option: 1

Given that $x = -4$ is a root of $\begin{vmatrix}x & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & x\end{vmatrix} = 0$

Expanding the determinant using the first row:

$x \begin{vmatrix}x & 1 \\ 2 & x\end{vmatrix} - 2 \begin{vmatrix}1 & 1 \\ 3 & x\end{vmatrix} + 3 \begin{vmatrix}1 & x \\ 3 & 2\end{vmatrix} = 0$

Computing each 2×2 determinant:

$\begin{vmatrix}x & 1 \\ 2 & x\end{vmatrix} = x^2 - 2$

$\begin{vmatrix}1 & 1 \\ 3 & x\end{vmatrix} = x - 3$

$\begin{vmatrix}1 & x \\ 3 & 2\end{vmatrix} = 2 - 3x$

Substituting back:

$x(x^2 - 2) - 2(x - 3) + 3(2 - 3x) = 0$

$x^3 - 2x - 2x + 6 + 6 - 9x = 0$

$x^3 - 13x + 12 = 0$

For a cubic equation $x^3 + bx^2 + cx + d = 0$ , the sum of all roots equals $-b$ .

The equation is: $x^3 + 0x^2 - 13x + 12 = 0$

Sum of all three roots $= -0 = 0$

Given one root is $x = -4$ :

$(-4) + \text{(sum of other two roots)} = 0$

Sum of other two roots $= 4$

2026: 31 May Shift 2

For any non-zero matrix $A = [a_{ij}]_{3\times3}$ and $|A| \neq 0$ , then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\vert adj\,A \vert$	I.	$\dfrac{1}{\vert A \vert}(adj\,A)$
B.	$(adj\,A)A$	II.	$\vert A \vert^2$
C.	$A^{-1}$	III.	$\dfrac{1}{\vert A \vert}$
D.	$\vert A^{-1} \vert$	IV.	$\vert A \vert I$

Choose the correct answer from the options given below:

2025: 13 May Shift 2

Let $A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3 \end{bmatrix}$ and $a_{ij}$ be any element of matrix A, i, j ∈ {1,2,3}, then which of the following are TRUE?

(A) Minor of $a_{23} = 16$

(B) Minor of $a_{23} = -8$

(D) Cofactor of $a_{23} = 8$

(E) Cofactor of $a_{13} = 4$

Choose the correct answer from the options given below:

2025: 22 May Shift 2

If $P\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$ , then matrix P is equal to:

2026: 29 May Shift 2

If A is a skew symmetric matrix, then $A^2$ is a

2025: 2 June Shift 1

If $\begin{bmatrix}3 & 1\\2 & 1\end{bmatrix}A\begin{bmatrix}2 & 1\\1 & 1\end{bmatrix} = \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$ , then matrix 'A' is

2022: 6 Aug Shift 2

If A is a matrix of order $m \times n$ and B is another matrix such that $A'B$ and $BA'$ are both defined, then the order of matrix B is

If x=−4x = -4x=−4 is a root of ∣x231x132x∣=0\begin{vmatrix}x & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & x\end{vmatrix} = 0​x13​2x2​31x​​=0, then the sum of the other 2 roots is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If $x = -4$ is a root of $\begin{vmatrix}x & 2 & 3 \\ 1 & x & 1 \\ 3 & 2 & x\end{vmatrix} = 0$ , then the sum of the other 2 roots is