If f(x) = vmatrix 1 & x & x^2 \\ x & x^2 & 1 \\ x^2 & 1 & x vmatrix, then the value of f(3√3) is Option 1: -6 Option 2: 6 Option 3: 4 Option 4: -4

Answer: -4. Solution: Expanding gives f(x)=-x^6+2x^3-1=-(x^3-1)^2. With x=3√3 we have x^3=3, so f(x)=-(3-1)^2=-4.

Algebra

Matrices & Determinants

Medium

applied

If $f(x) = \begin{vmatrix} 1 & x & x^2 \\ x & x^2 & 1 \\ x^2 & 1 & x \end{vmatrix}$ , then the value of $f(\sqrt[3]{3})$ is

-6

-4

✅ Correct Option: 4

Expanding gives $f(x)=-x^6+2x^3-1=-(x^3-1)^2$ . With $x=\sqrt[3]{3}$ we have $x^3=3$ , so $f(x)=-(3-1)^2=-4$ .

2022: 7 Aug Shift 2

Match List - I with List - II.

	List - I		List - II
(A)	$A = \begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix}$ , $B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}$ , AB will be	(I)	$\begin{bmatrix} 75 & 117 & 72 \\ 35 & 39 & 24 \end{bmatrix}$
(B)	$P = \begin{bmatrix} 8 & 45 & 30 \\ 17 & 19 & 5 \end{bmatrix}$ , $Q = \begin{bmatrix} 67 & 72 & 42 \\ 8 & 30 & 19 \end{bmatrix}$ , P+Q will be	(II)	$\begin{bmatrix} 75 & 117 & 72 \\ 25 & 39 & 24 \end{bmatrix}$
(C)	$\begin{bmatrix} 85 & 42 & 69 \\ 73 & 42 & 50 \end{bmatrix} - \begin{bmatrix} 10 & -75 & -3 \\ 38 & 3 & 26 \end{bmatrix}$	(III)	$\begin{bmatrix} 75 & 117 & 72 \\ 25 & 49 & 24 \end{bmatrix}$
(D)	$2 \begin{bmatrix} 34 & 36 & 30 \\ 12 & 18 & 20 \end{bmatrix} + \begin{bmatrix} 7 & 55 & 12 \\ 1 & 13 & -16 \end{bmatrix}$	(IV)	$\begin{bmatrix} 75 & 127 & 72 \\ 25 & 49 & 24 \end{bmatrix}$

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2026: 29 May Shift 2

Given that $[a_{ij}]$ is a square matrix of order n.

Match the LIST-I with LIST-II

LIST-I		LIST-II
A.	$A=[a_{ij}]; a_{ij}=\begin{cases}1, & \forall i=j\\0, & \forall i\neq j\end{cases}$	I. Identity Matrix
B.	$B=[a_{ij}]; a_{ij}=\begin{cases}2i-3j, & \forall i=j\\0, & \forall i\neq j\end{cases}$	II. Diagonal Matrix
C.	$C=[a_{ij}]; a_{ij}=\begin{cases}k, & \forall i=j\\0, & \forall i\neq j\end{cases}$	III. Scalar Matrix
D.	$D=[a_{ij}]; a_{ij}=\begin{cases}2i-3j, & \forall i\leq j\\0, & \forall i>j\end{cases}$	IV. Upper Triangular Matrix

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2023: 30 May Shift 3

If A and B are invertible matrices of order 3, $|A| = 2$ and $|(AB)^{-1}| = -\frac{1}{6}$ , then the value of $|B|$ is :

2025: 13 May Shift 1

Match List-I with List-II

List-I	List-II
Type of matrix	Conditions
(A) Square matrix A	(I) $A = [a_{ij}]_{m \times m}$ where $\begin{cases} a_{ij} = 0 & , i \neq j \\ a_{ij} = k & , i = j \end{cases}$ , where $k \neq 0$ is constant.
(B) Scalar Matrix A	(II) $A = [a_{ij}]_{m \times m}$
(C) Diagonal matrix A	(III) $A = [a_{ij}]_{m \times m}$ where $\begin{cases} a_{ij} = 0 & , i \neq j \\ a_{ij} = 1 & , i = j \end{cases}$
(D) Identity matrix A	(IV) $A = [a_{ij}]_{m \times m}$ where $a_{ij} = 0$ , $i \neq j$

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2026: 30 May Shift 1

The value (/s) of $x$ satisfying the matrix equation:

$x\begin{bmatrix}2x & 2\\3 & x\end{bmatrix}+2\begin{bmatrix}8 & 5x\\4 & 4x\end{bmatrix}=2\begin{bmatrix}(x^2+8) & 24\\10 & 6x\end{bmatrix}$ is/are:

2025: 27 May Shift 1

Let $A$ be a non-singular square matrix of order $n$ , then

Match List-I with List-II

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