(A) Let $\Delta = \left| \begin{array}{ccc} 1 & 1 & x \\ p+1 & p+1 & p+x \\ 3 & x+1 & x+2 \end{array} \right| = 0$. Applying column operations $C_2 \t

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(A) Let $\Delta = \left| \begin{array}{ccc} 1 & 1 & x \\ p+1 & p+1 & p+x \\ 3 & x+1 & x+2 \end{array} \right| = 0$. Applying column operations $C_2 \to C_2 - C_1$: $\Delta = \left| \begin{array}{ccc} 1 & 0 & x \\ p+1 & 0 & p+x \\ 3 & x-2 & x+2 \end{array} \right| = 0$. Expanding along the second column: $-(x-2) \left| \begin{array}{cc} 1 & x \\ p+1 & p+x \end{array} \right| = 0$. $-(x-2) [ (p+x) - (p+1)x ] = 0$. $-(x-2) [ p + x - px - x ] = 0$. $-(x-2) [ p(1-x) ] = 0$. $p(x-2)(x-1) = 0$. Thus,the solutions are $x = 1$ and $x = 2$.

Class 12 Mathematics 3-4. Matrices and Determinants Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

Easy

The solutions of the equation $\left| \begin{array}{ccc} 1 & 1 & x \\ p+1 & p+1 & p+x \\ 3 & x+1 & x+2 \end{array} \right| = 0$ are:

A
$x = 1, 2$
B
$x = 2, 3$
C
$x = 1, p, 2$
D
$x = 1, 2, -p$

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3-4. Matrices and Determinants

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Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

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Medium

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MediumTS EAMCET 2024

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Let $S = \left\{ A = \begin{bmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{bmatrix} : a, b, c, d, e \in \{0, 1\} \text{ and } |A| \in \{-1, 1\} \right\}$,where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is:

MediumIIT 2024

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The solutions of the equation $\left| \begin{array}{ccc} 1 & 1 & x \\ p+1 & p+1 & p+x \\ 3 & x+1 & x+2 \end{array} \right| = 0$ are:

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If matrix $A = \begin{bmatrix} \sin \theta & \csc \theta & 1 \\ \csc \theta & 1 & \sin \theta \\ 1 & \sin \theta & \csc \theta \end{bmatrix}$ is a non-invertible matrix,then the possible value of $\theta$ is $(n \in \mathbb{Z})$

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Let $S = \left\{ A = \begin{bmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{bmatrix} : a, b, c, d, e \in \{0, 1\} \text{ and } |A| \in \{-1, 1\} \right\}$,where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is:

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