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Determinants Set-5 MCQ Online Practice Test | Class 12 | Quiz Tweek

Determinants Set-5

Practice Important MCQs for Determinants — Mathematics, Class 12.

Determinants are a core tool in Class 12 Mathematics because they connect algebraic expressions to geometry (area/volume scaling), help solve systems of linear equations efficiently, and appear repeatedly in board exams as well as competitive tests through rank, eigenvalues, and special matrix patterns. Mastering determinant properties (like row/column operations, adjoint links, and factorization) makes even higher-level problems faster and more reliable.

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Q1. Evaluate the determinant

\begin{vmatrix} a & b & c \\ a+k & b+k & c+k \\ a+2k & b+2k & c+2k \end{vmatrix}

(A)

$k^2(a+b+c)$

(B)

$3k(a+b+c)$

(C)

$0$

(D)

$k(a-b)(b-c)$

Q2. Find all real values of $x$ satisfying

\begin{vmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{vmatrix}=4.

(A)

$x=2$ only

(B)

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

(C)

$x=-1$ only

(D)

$x=1$

Q3. If

D(t)=\begin{vmatrix} 1 & 1 & 1 \\ 1 & t & t^2 \\ 1 & t^2 & t^3 \end{vmatrix},

then $D(t)=$

(A)

$-t(t-1)^2$

(B)

$t(t+1)(t-1)$

(C)

$(t-1)^3$

(D)

$t^2(t-1)$

Q4. For a positive integer $n\ge 1$ , let $I_n$ be the $n\times n$ identity matrix and $J_n$ the $n\times n$ matrix of all ones. Evaluate

\det\begin{pmatrix} I_n & J_n \\ J_n & I_n \end{pmatrix}.

(A)

$(1+n)^2$

(B)

$1+n^2$

(C)

$(-1)^{n}(n^2-1)$

(D)

$1-n^2$

Q5. Compute the determinant

\begin{vmatrix} x & y & z \\ z & x & y \\ y & z & x \end{vmatrix}

in terms of $x,y,z$ .

(A)

$(x-y)(y-z)(z-x)$

(B)

$(x+y+z)\big(x^2+y^2+z^2-xy-yz-zx\big)$

(C)

$(x+y+z)^3$

(D)

$(x+y+z)\big(x^2+y^2+z^2+xy+yz+zx\big)$

Q6. Evaluate the determinant $\begin{vmatrix} 1 & 2 & 3 \$$2pt] 4 & 5 & 6 \$$2pt] 7 & 8 & 10 \end{vmatrix}$

(A)

$-1$

(B)

$-3$

(C)

$0$

(D)

$3$

Q7. For which values of $k$ does the system

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

have a unique solution, no solution, or infinitely many solutions?

(A)

Unique for $k\neq 1,-2$ ; no solution for $k=-2$ ; infinitely many solutions for $k=1$ .

(B)

Unique for all $k\neq -2$ ; no solution for $k=1$ ; infinitely many solutions for $k=-2$ .

(C)

Unique for $k\neq 1,-2$ ; no solution for both $k=1$ and $k=-2$ .

(D)

Unique for $k\neq 1,-2$ ; infinitely many solutions for $k=-2$ ; no solution for $k=1$ .

Q8. Evaluate the determinant $\begin{vmatrix} 2 & 1 & 1 & 1 \$$2pt] 1 & 2 & 1 & 1 \$$2pt] 1 & 1 & 2 & 1 \$$2pt] 1 & 1 & 1 & 2 \end{vmatrix}$

(A)

$5$

(B)

$0$

(C)

$4$

(D)

$8$

Q9. Let $A$ be a $3\times3$ real matrix satisfying $A^{2}=3A-2I$ . The set of all possible values of $\det(A)$ is

(A)

$\{1,8\}$

(B)

$\{2,4,8\}$

(C)

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

(D)

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

Q10. Let $A$ be a $3\times3$ real matrix (not necessarily invertible) such that $\operatorname{adj}(A)=A^{T}$ . Which of the following gives all possible values of $\det(A)$ ?

(A)

$\{1\}$

(B)

$\{0,1\}$

(C)

$\{0\}$

(D)

$\{-1,1\}$

Q11. Evaluate the determinant

\begin{vmatrix}2 & 3 & 5 \$$4pt] 4 & 7 & 11 \$$4pt] 6 & 10 & 16\end{vmatrix}

(A)

$6$

(B)

$-2$

(C)

$0$

(D)

$1$

Q12. Let $A$ be an invertible $3\times 3$ matrix with $\det(A)=4$ . If $B=2A^{T}$ , find $\det\bigl(B^{T}A^{-1}B\bigr)$ .

(A)

$256$

(B)

$64$

(C)

$1024$

(D)

$16$

Q13. For which real values of $k$ is the determinant

\begin{vmatrix}1 & k & k^{2} \$$4pt] k & 1 & k \$$4pt] k^{2} & k & 1\end{vmatrix}

equal to zero?

(A)

$\{0\}$

(B)

$\{1\}$

(C)

$\{-1,0\}$

(D)

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

Q14. Assertion (A): $\displaystyle \operatorname{adj}\bigl(\operatorname{adj}(M)\bigr)=\det(M)\,M$ for every $3\times 3$ matrix $M$ .

Reason (R): For any $n\times n$ matrix $M$ with $n\ge2$ , $\displaystyle \operatorname{adj}\bigl(\operatorname{adj}(M)\bigr)=\det(M)^{\,n-2}\,M$ .

(A)

$A$ is true but $R$ is false.

(B)

Both $A$ and $R$ are true and $R$ is the correct explanation for $A$ .

(C)

Both $A$ and $R$ are true but $R$ is not the correct explanation for $A$ .

(D)

$A$ is false but $R$ is true.

Q15. For real $a$ , evaluate the determinant

\begin{vmatrix}a & 1 & 1 & 1 \$$4pt] 1 & a & 1 & 1 \$$4pt] 1 & 1 & a & 1 \$$4pt] 1 & 1 & 1 & a\end{vmatrix}

as a factorised polynomial in $a$ .

(A)

$(a+1)^{3}(a-3)$

(B)

$(a-1)^{2}(a+2)^{2}$

(C)

$(a-1)^{3}(a+3)$

(D)

$(a+3)^{3}(a-1)$

...and 5 more challenging questions available in the interactive simulator.

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