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Determinants Set-3 MCQ Online Practice Test | Class 12 | Quiz Tweek

Determinants Set-3

Practice Important MCQs for Determinants — Mathematics, Class 12.

Determinants are central to Class 12 Mathematics and appear frequently in CBSE and competitive exams because they connect algebraic properties (like invertibility and eigenvalues) with geometric meaning (volume/area scaling). Strong command of determinant identities, row/column operations, and determinant formulas for special matrices helps you solve both numerical and theory-based problems quickly and accurately.

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Q1. Let $A$ be a $3\times3$ matrix with $\det(A)=4$ . Find $\det(2A^{T})$ .

(A)

$16$

(B)

$8$

(C)

$32$

(D)

$64$

Q2. Evaluate the determinant

\begin{vmatrix} 5 & 2 & 2 & 2 \\ 2 & 5 & 2 & 2 \\ 2 & 2 & 5 & 2 \\ 2 & 2 & 2 & 5 \end{vmatrix}

(A)

$297$

(B)

$125$

(C)

$256$

(D)

$99$

Q3. For which real values of $k$ does the determinant

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

vanish?

(A)

$\{1\}$

(B)

$\{-2\}$

(C)

$\{1,-1\}$

(D)

$\{1,-2\}$

Q4. If $A$ and $B$ are $3\times3$ matrices with $\det(A)=4$ and $\det(B)=2$ , evaluate $\det\big(\operatorname{adj}(A)\,B^{2}\big)$ .

(A)

$16$

(B)

$64$

(C)

$256$

(D)

$8$

Q5. Let $A$ be a $3\times3$ matrix whose eigenvalues are $1,2,3$ . Find $\det\big(\operatorname{adj}(A)+2I\big)$ .

(A)

$160$

(B)

$144$

(C)

$120$

(D)

$180$

Q6. Evaluate the determinant

\begin{vmatrix}1 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 6 & 10\end{vmatrix}

(A)

$0$

(B)

$-1$

(C)

$1$

(D)

$2$

Q7. For which value of $k$ does the system of equations have infinitely many solutions?

\begin{aligned} x+y+z=1 <br> 2x+3y+4z=k <br> 3x+5y+7z=2 \end{aligned}

(A)

$\dfrac{3}{2}$

(B)

$1$

(C)

$0$

(D)

$\text{No such real }k$

Q8. Find all real values of $k$ for which

\begin{vmatrix}1 & k & k^2 \\ k & 1 & k \\ k^2 & k & 1\end{vmatrix}=0

(A)

$k=1\text{ only}$

(B)

$k=-1\text{ only}$

(C)

$k=0$

(D)

$k=\pm 1$

Q9. Evaluate the determinant

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

as a polynomial in $a$ .

(A)

$(a-1)^4$

(B)

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

(C)

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

(D)

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

Q10. Let $A$ be a real $3\times3$ matrix satisfying $\operatorname{adj}(A)=A$ . Then $\det(A)$ equals

(A)

$-1$

(B)

$1$

(C)

$0\text{ or }1$

(D)

$0\text{ only}$

Q11. Let $A$ and $B$ be $3\times3$ matrices with $\det(A)=4$ and $\det(B)=-2$ . The value of $\det\big(3A^{T}B^{-1}\big)$ is

(A)

$54$

(B)

$-108$

(C)

$-54$

(D)

$-27$

Q12. Find all real values of $k$ for which

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

(A)

$k=2,\ -1\pm\sqrt{3}$

(B)

$k=2$

(C)

$k=1,\ -2\pm\sqrt{3}$

(D)

$k=0,\ 2$

Q13. Let $a,b,c$ be real numbers. Evaluate

D=\begin{vmatrix}1 & 1 & 1 \$$4pt] a & b & c \$$4pt] a^3 & b^3 & c^3\end{vmatrix}

in terms of $a,b,c$ .

(A)

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

(B)

$(a+b+c)(a-b)(b-c)$

(C)

$3abc\,(a-b)(b-c)(c-a)$

(D)

$(a+b+c)(a-b)(b-c)(c-a)$

Q14. Let $A$ be a $4\times4$ matrix with $\det(A)=5$ . Matrix $B$ is obtained from $A$ by performing the following operations in order: (i) replace column $C_3$ by $C_3+C_1$ , (ii) multiply row $R_4$ by $2$ , (iii) swap rows $R_2$ and $R_3$ , (iv) replace column $C_2$ by $C_2-3C_1$ . The value of $\det(B)$ is

(A)

$10$

(B)

$-10$

(C)

$-5$

(D)

$20$

Q15. If $A$ is an invertible $3\times3$ real matrix with $\det(A)=2$ , then $\det\big(\operatorname{adj}(A^{T}A)\big)$ equals

(A)

$8$

(B)

$256$

(C)

$16$

(D)

$4$

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