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Determinants Set-6 MCQ Online Practice Test | Class 12 | Quiz Tweek

Determinants Set-6

Practice Important MCQs for Determinants — Mathematics, Class 12.

Determinants are central to Class 12 Mathematics because they connect linear algebra concepts (like invertibility and eigen-structure) with practical computation of areas/volumes and solutions to systems of equations. They are frequently tested in board exams as well as competitive exams through applications (area scaling, consistency of systems) and properties (rank, adjugate, special structured matrices), making a strong command of determinant techniques essential.

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Q1. Evaluate the determinant

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

(A)

$12$

(B)

$-6$

(C)

$18$

(D)

$24$

Q2. A triangle with vertices $A(1,2),\;B(3,4),\;C(5,1)$ is transformed by the linear map represented by the matrix

T=\begin{pmatrix}2 & 1 \$$4pt] 3 & 2\end{pmatrix}.

Find the area of the image triangle.

(A)

$5$

(B)

$10$

(C)

$2.5$

(D)

$15$

Q3. For what value of $k$ does the system

\begin{aligned} x + 2y + z &= 3 \$$4pt] 2x + y - z &= 0 \$$4pt] x + y + kz &= 1 \end{aligned}

have infinitely many solutions?

(A)

$k=1$

(B)

no real value of $k$

(C)

$k=-3$

(D)

$k=0$

Q4. Evaluate the determinant $D=\begin{vmatrix} 5 & 1 & 1 & 1 \$ 4pt] 1 & 5 & 1 & 1 $ $4pt] 1 & 1 & 5 & 1 \$$4pt] 1 & 1 & 1 & 5 \end{vmatrix}.$

(A)

$256$

(B)

$512$

(C)

$1024$

(D)

$625$

Q5. Let $A$ be a $2\times2$ invertible matrix with $\det(A)=-1$ and $\det(A+I)=4$ . Find $\det\!\big(\operatorname{adj}A+I\big)$ .

(A)

$4$

(B)

$2$

(C)

$-4$

(D)

$0$

Q6. Evaluate

\det\begin{bmatrix}1 & 2 & 3 \$$4pt] 2 & 5 & 8 \$$4pt] 3 & 8 & 14\end{bmatrix}

(A)

$0$

(B)

$-1$

(C)

$1$

(D)

$2$

Q7. Find all real $t$ for which

\begin{vmatrix}1+t & 1 & 1 \$$4pt] 1 & 1+t & 1 \$$4pt] 1 & 1 & 1+t\end{vmatrix}=0

(A)

$t=0\ \text{or}\ t=-3$

(B)

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

(C)

$t=-1\ \text{or}\ t=0$

(D)

$t=-3\ \text{only}$

Q8. Assertion (A): If $A$ is a $3\times3$ matrix with $\operatorname{rank}(A)=2$ then $\operatorname{adj}(A)\neq 0$ .
Reason (R): If $\operatorname{rank}(A)=2$ then $\operatorname{rank}(\operatorname{adj}(A))=1$ .

(A)

Both A and R are true but R does not explain A.

(B)

A is true but R is false.

(C)

A is false but R is true.

(D)

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

Q9. Let $A$ be an invertible $3\times3$ real matrix satisfying $\operatorname{adj}(A)=A^{T}$ . Then $\det(A)=$

(A)

$-1$

(B)

$1$

(C)

$0$

(D)

cannot be determined

Q10. Find all real values of $k$ for which the system

\begin{aligned} x+2y-z&=3 \$$4pt] 2x+5y+kz&=7 \$$4pt] 3x+8y+4z&=10 \end{aligned}

has infinitely many solutions.

(A)

$k=\dfrac{3}{2}$

(B)

$k=2$

(C)

No real value of $k$

(D)

$k=0$

Q11. Evaluate the determinant

\begin{vmatrix}2 & 1 & 0 \$$4pt] 1 & 2 & 1 \$$4pt] 0 & 1 & 2\end{vmatrix}.

(A)

$0$

(B)

$4$

(C)

$2$

(D)

$-4$

Q12. Consider the linear system

\begin{aligned} x+2y+z &= 1<br> 2x+3y+2z &= p<br> 3x+5y+3z &= 2 \end{aligned}

For which values of $p$ does the system have a solution?

(A)

The system has a unique solution for every real $p$ .

(B)

The system has a unique solution only when $p=1$ .

(C)

The system has no solution when $p=1$ and infinitely many solutions for $p\ne 1$ .

(D)

The system has infinitely many solutions when $p=1$ and no solution for $p\ne 1$ .

Q13. Let

A=\begin{pmatrix}7&2&2&2\$$4pt]2&7&2&2\$$4pt]2&2&7&2\$$4pt]2&2&2&7\end{pmatrix}.

Compute $\det(A)$ .

(A)

$1625$

(B)

$625$

(C)

$1300$

(D)

$0$

Q14. Assertion (A): For every odd integer $n\ge 1$ , any real skew-symmetric $n\times n$ matrix has determinant $0$ .
Reason (R): If $A$ is skew-symmetric then $A^T=-A$ , and since $\det(A^T)=\det(A)$ we get $\det(A)=\det(-A)=(-1)^n\det(A)$ .

(A)

Both (A) and (R) are true but (R) does not correctly explain (A).

(B)

(A) is true and (R) is false.

(C)

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

(D)

(A) is false and (R) is true.

Q15. Let $T_n$ be the $n\times n$ tridiagonal matrix with $2$ on the main diagonal, $-1$ on the sub- and super-diagonals, and zeros elsewhere. That is,

T_n=\begin{pmatrix} 2 & -1 & 0 & \cdots & 0\$$4pt] -1 & 2 & -1 & \cdots & 0\$$4pt] 0 & -1 & 2 & \ddots & \vdots\$$4pt] \vdots & \vdots & \ddots & \ddots & -1\$$4pt] 0 & 0 & \cdots & -1 & 2 \end{pmatrix}.

The determinant $\det(T_n)$ equals

(A)

$2^n-1$

(B)

$n+1$

(C)

$1$

(D)

$n-1$

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