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Determinants Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Determinants Set-4

Practice Important MCQs for Determinants — Mathematics, Class 12.

Determinants are central to Class 12 Mathematics and appear in many competitive exams because they connect linear algebra (systems of equations, eigenvalues) with algebraic tools (properties, rank, adjugate). Mastery of determinant identities and parameter-based reasoning helps you solve both computation and concept-heavy MCQs quickly and accurately.

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Q1. Let $A$ be the $3\times3$ matrix with entries $a_{ij}=u_i+v_j$ , where $u=(1,2,3)$ and $v=(4,5,6)$ . Evaluate $\det(A)$ .

A=\begin{pmatrix}5&6&7\\6&7&8\\7&8&9\end{pmatrix}

(A)

$54$

(B)

$0$

(C)

$-54$

(D)

$18$

Q2. Let $A$ be a real $3\times3$ matrix satisfying

A^2=3A.

Then $\det(A)$ is equal to:

(A)

$0\ \text{or}\ 27$

(B)

$0$

(C)

$27$

(D)

$\text{no restriction (any real number)}$

Q3. Let $M$ be the $4\times4$ matrix with $m_{ii}=x+1$ for all $i$ and $m_{ij}=1$ for $i\ne j$ . Compute $\det(M)$ as a polynomial in $x$ .

(A)

$(x+1)^3(x-1)$

(B)

$x^4+3x^3$

(C)

$(x+1)^2(x^2+2)$

(D)

$x^3(x+4)$

Q4. For an arbitrary $3\times3$ real matrix $A$ consider the statements:
Statement (A): $\operatorname{adj}(\operatorname{adj}(A))=\det(A)\,A.$
Statement (R): $\operatorname{adj}(A)=\det(A)\,A^{-1}.$
Which one is correct?

(A)

Both (A) and (R) are true and (R) is a correct explanation of (A)

(B)

Both (A) and (R) are true but (R) is not a correct explanation of (A)

(C)

(A) is true and (R) is false

(D)

(A) is false and (R) is true

Q5. For real parameter $k$ , consider the linear system

\begin{aligned} x+y+z&=1<br> x+ky+2z&=k<br> x+2y+kz&=k \end{aligned}

Which option correctly describes the nature of solutions?

(A)

Unique solution for every real $k$

(B)

Unique solution for $k\ne 0,2$ ; infinitely many solutions for $k=2$ ; no solution for $k=0$

(C)

Unique solution for $k\ne 0$ ; infinitely many solutions for $k=0$ ; no solution for $k=2$

(D)

Unique solution for $k\ne 2$ ; infinitely many solutions for $k=0$ ; no solution for $k=2$

Q6. Let $x$ be a real number such that $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & x \end{vmatrix} = 0$ . Then $x=$

(A)

$6$

(B)

$9$

(C)

$-3$

(D)

$3$

Q7. Evaluate the determinant $D(x)=\begin{vmatrix} 1 & x & x^2 \\ 1 & x+1 & (x+1)^2 \\ 1 & x+2 & (x+2)^2 \end{vmatrix}$ for real $x$ .

(A)

$0$

(B)

$2x$

(C)

$x(x+1)(x+2)$

(D)

$2$

Q8. For which real values of $\alpha$ is the matrix $A=\begin{pmatrix} \alpha & 1 & 1 \$$4pt] 1 & \alpha & 1 \$$4pt] 1 & 1 & \alpha \end{pmatrix}$ invertible?

(A)

$\alpha\in\mathbb{R}\setminus\{1,-2\}$

(B)

$\alpha=1\ \text{or}\ \alpha=-2$

(C)

$\alpha>1$

(D)

$\alpha\ge 0$

Q9. Let $A$ be a real $3\times 3$ matrix satisfying $A^2=3A-2I$ . The possible values of $\det(A)$ are

(A)

$\{1,2\}$

(B)

$\{0,1,2,4,8\}$

(C)

$\{1,2,4,8\}$

(D)

$\{1\}$

Q10. Let $v_1,v_2,v_3\in\mathbb{R}^3$ and let $D=\det(v_1,v_2,v_3)$ . Find $\det(v_1+v_2,\;v_2+v_3,\;v_3+v_1)$ in terms of $D$ .

(A)

$2D$

(B)

$-2D$

(C)

$D$

(D)

$0$

Q11. Let $A$ be an invertible $3\times3$ matrix with $\det(A)=4$ . Evaluate $\det\big(3A^{-1}A^{T}\big)$ .

(A)

$4$

(B)

$27$

(C)

$108$

(D)

$1$

Q12. Let $A$ be an invertible $3\times3$ matrix with column vectors $\mathbf{c}_1,\mathbf{c}_2,\mathbf{c}_3$ . Let $B$ be the $3\times3$ matrix whose columns are $\mathbf{c}_1+\mathbf{c}_2,\ \mathbf{c}_2+\mathbf{c}_3,\ \mathbf{c}_3+\mathbf{c}_1$ . Then $\det(B)$ equals:

(A)

$0$

(B)

$\det(A)$

(C)

$3\det(A)$

(D)

$2\det(A)$

Q13. Let $A$ be a $3\times3$ matrix with $\det(A)=5$ . Matrix $B$ is obtained from $A$ by simultaneously replacing the first row by the sum of the three original rows and the second row by (original second row minus original first row), leaving the third row unchanged. Compute $\det(B)$ .

(A)

$10$

(B)

$5$

(C)

$-5$

(D)

$0$

Q14. Let $A$ be a $3\times 3$ real matrix satisfying $\operatorname{adj}(A)=A^{T}$ . Then $\det(A)$ must be equal to:

(A)

$\{-1,0\}$

(B)

$\{-1,1\}$

(C)

$\{0,1\}$

(D)

$\{0\}$

Q15. Let $J$ denote the $5\times5$ matrix with every entry $1$ , and let $A=I_{5}+J$ . Compute $\det(A)$ .

(A)

$5$

(B)

$6$

(C)

$1$

(D)

$0$

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