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Determinants Set-2 MCQ Online Practice Test | Class 12 | Quiz Tweek

Determinants Set-2

Practice Important MCQs for Determinants — Mathematics, Class 12.

Determinants form a core part of Class 12 Mathematics and are frequently used in board as well as competitive exams to solve problems on matrix invertibility, area/volume interpretation, properties of adjoint matrices, and efficient evaluation of determinants using smart transformations and determinant identities.

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Q1. Evaluate the determinant

\begin{vmatrix}2 & 1 & 3\$$4pt]0 & -1 & 4\$$4pt]5 & 2 & 7\end{vmatrix}.

(A)

$-5$

(B)

$0$

(C)

$5$

(D)

$10$

Q2. If $A$ is an invertible $3\times 3$ matrix with $\det(A)=4$ , find $\det\!\bigl(2A^T\operatorname{adj}(A)\bigr)$ .

(A)

$512$

(B)

$64$

(C)

$256$

(D)

$4096$

Q3. Evaluate

D=\begin{vmatrix}1+x & x & x & x\$$4pt]x & 1+x & x & x\$$4pt]x & x & 1+x & x\$$4pt]x & x & x & 1+x\end{vmatrix}.

(A)

$(1+x)^4$

(B)

$1+x^4$

(C)

$1$

(D)

$1+4x$

Q4. Evaluate the determinant

\begin{vmatrix}1 & 1 & 1 & 1\$$4pt]1 & 2 & 2 & 2\$$4pt]1 & 2 & 3 & 3\$$4pt]1 & 2 & 3 & 4\end{vmatrix}.

(A)

$0$

(B)

$1$

(C)

$4$

(D)

$-1$

Q5. Let $A$ be a $3\times 3$ real matrix satisfying $A^T=\operatorname{adj}(A)$ . Then $\det(A)$ equals:

(A)

$0\text{ or }1$

(B)

$1$

(C)

$0$

(D)

$-1$

Q6. If

A=\begin{pmatrix}1 & 2 & 0 & -1\$$2pt]0 & 2 & 3 & 4\$$2pt]0 & 0 & 3 & 5\$$2pt]0 & 0 & 0 & k\end{pmatrix},

find $k$ such that $\det(A)=36$ .

(A)

$4$

(B)

$6$

(C)

$12$

(D)

$-6$

Q7. Let

A=\begin{pmatrix}x & 1 & 1\$$2pt]1 & x & 1\$$2pt]1 & 1 & x\end{pmatrix}.

The value of $\det(A)$ is

(A)

$x^3- x-2$

(B)

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

(C)

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

(D)

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

Q8. Let $A$ be an invertible $3\times 3$ matrix satisfying $\operatorname{adj}(A)=4A$ . Then $\det(A)=$

(A)

$64$

(B)

$16$

(C)

$256$

(D)

$4$

Q9. Assertion (A): Let $A$ be a $3\times 3$ matrix with $\operatorname{rank}A=2$ . Then $\operatorname{adj}(A)\neq 0.$
Reason (R): $\operatorname{adj}(A)=\det(A)\,A^{-1}.$

(A)

Both A and R are true and R is the correct explanation of A

(B)

Both A and R are true but R is not the correct explanation of A

(C)

A is true but R is false

(D)

A is false but R is true

Q10. Let $D=\operatorname{diag}(1,2,3,4)$ , $u=\begin{pmatrix}1\$$2pt]1\$$2pt]1\$$2pt]1\end{pmatrix}$ and $v=\begin{pmatrix}1\$$2pt]-1\$$2pt]1\$$2pt]-1\end{pmatrix}$ . The value of $\det(D+uv^{T})$ is

(A)

$19$

(B)

$38$

(C)

$24$

(D)

$12$

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