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Determinants Set-1 MCQ Online Practice Test | Class 12 | Quiz Tweek

Determinants Set-1

Practice Important MCQs for Determinants — Mathematics, Class 12.

Determinants are central to Class 12 Mathematics and frequently appear in board and competitive exams because they help test singularity of matrices, solve systems of linear equations, and connect with eigenvalues and transformations. Strong command over determinant properties and computations directly boosts accuracy and speed in both MCQs and higher-level problems.

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Q1. (Evaluate the determinant

\begin{vmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{vmatrix}

(A)

$(x-1)^3$

(B)

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

(C)

$x^3-3x$

(D)

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

Q2. (If

D=\begin{vmatrix} 1+a & 1 & 1 \$$4pt] 1 & 1+b & 1 \$$4pt] 1 & 1 & 1+c \end{vmatrix},

evaluate $D$ in terms of $a,b,c$ .)

(A)

$ab+bc+ca$

(B)

$(1+a)(1+b)(1+c)$

(C)

$abc$

(D)

$ab+bc+ca+abc$

Q3. (Find all real values of $k$ for which $\begin{vmatrix} k & 1 & 1 \$ 4pt] 1 & k & 2 $ $4pt] 1 & 2 & k \end{vmatrix}=0.$ )

(A)

$\{2,\,-1+\sqrt{3},\,-1-\sqrt{3}\}$

(B)

$\{1,\,-2,\,-1\}$

(C)

$\{2,\,1-\sqrt{3},\,1+\sqrt{3}\}$

(D)

$\{-2,\,-1+\sqrt{3},\,-1-\sqrt{3}\}$

Q4. (Let $M$ be the $5\times5$ matrix with diagonal entries $2$ and every off-diagonal entry $1$ , i.e.

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

Then $\det(M)=$ )

(A)

$1$

(B)

$5$

(C)

$6$

(D)

$10$

Q5. (Consider the $3\times3$ integer matrix $A$ with $\det(A)=2$ .
Statement P: Every entry of $\operatorname{adj}(A)$ is even.
Statement Q: From $A\,\operatorname{adj}(A)=\det(A)\,I_3$ it follows that every entry of $\operatorname{adj}(A)$ is divisible by $2$ . Which of the following is correct?)

(A)

Both P and Q are true and Q explains P.

(B)

Both P and Q are true but Q does not explain P.

(C)

P is false but Q is true.

(D)

Both P and Q are false.

Q6. Let $A=\begin{pmatrix}1 & 2 & 3 \$$4pt] 2 & 5 & 8 \$$4pt] 3 & 8 & 14\end{pmatrix}$ . Find $\det(A)$ .

(A)

$0$

(B)

$-1$

(C)

$1$

(D)

$2$

Q7. For real $t$ , let $A=\begin{pmatrix}1+t & 1 & 1 \$$4pt] 1 & 1+t & 1 \$$4pt] 1 & 1 & 1+t\end{pmatrix}$ . The determinant $\det(A)$ as a polynomial in $t$ is

(A)

$(t+1)^3$

(B)

$t^2(t+3)$

(C)

$t(t+1)^2$

(D)

$t^3+3t$

Q8. Let $u,v,w\in\mathbb{R}^3$ . If $\det\bigl[u+v,\; v+w,\; w+u\bigr]=6$ , then $\det\bigl[u,v,w\bigr]$ equals

(A)

$0$

(B)

$-6$

(C)

$3$

(D)

$6$

Q9. Let $A$ be a $3\times3$ real matrix satisfying $A^2=3A-2I$ and $\operatorname{tr}(A)=5$ . Then $\det(A)=$

(A)

$4$

(B)

$8$

(C)

$2$

(D)

$1$

Q10. Let $A$ be a $3\times3$ real matrix such that $\det(I+tA)=1+4t+t^2$ for all real $t$ . Then $\operatorname{rank}(A)=$

(A)

$0$

(B)

$2$

(C)

$1$

(D)

$3$

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