Q.Find minors and cofactors of all the elements of the determinant $\begin{vmatrix} 1 & -2 \\ 4 & 3 \end{vmatrix}$.

Showing the 12 most recent of 32 on this concept.

1. AP EAPCET 2023Set eng-2023-05-15-AN1 markMCQ
   Q.If the cofactors of the elements 3, 7 and 6 of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6 \end{pmatrix}$ are $a, b$ and $c$ respectively, then $[a\ b\ c]\begin{pmatrix}1\\4\\2\end{pmatrix} + [a\ b\ c]\begin{pmatrix}3\\7\\6\end{pmatrix} =$ (A) $-1$ (B) $1$ (C) $0$ (D) $3$

   ›Reveal solutionSolution

   $a,b,c$ turn out to be the cofactors of the matrix's third column, and the two

   vectors being dotted with $[a\ b\ c]$ are exactly its first and third columns. The

   Laplace-expansion identity makes both dot products come out to (a multiple of) the

   matrix's determinant, which here is $0$ — so the sum is $0$.

   Concept and Intuition

   For a $3\times3$ matrix $A$, the cofactor expansion identity says: if you take the

   entries of any column of $A$ and dot them with the cofactors of a matching column,

   you get $\det(A)$; if you dot them with the cofactors of a different column, you get

   $0$ (this is the algebraic content behind $A\cdot\text{adj}(A)=\det(A)\,I$). Recognising

   that the two given vectors, $(1,4,2)^T$ and $(3,7,6)^T$, are precisely columns 1 and 3

   of the matrix — and that $a,b,c$ are the cofactors of column 3 — turns this into a

   one-line application of that identity instead of brute-force computation (though brute

   force also works and is done below to double-check).

   Step-by-Step Solution

   1. The matrix is $M=\begin{pmatrix}1&2&3\\4&-1&7\\2&4&6\end{pmatrix}$. The entries $3,7,6$ sit at positions $(1,3),(2,3),(3,3)$ — all in column 3. So $a=C_{13}$, $b=C_{23}$, $c=C_{33}$.
   2. Compute $C_{13}$ (delete row 1, col 3; sign $(+1)^{1+3}=+$):

   $C_{13} = +\begin{vmatrix}4 & -1\\ 2 & 4\end{vmatrix} = 4(4)-(-1)(2) = 16+2 = 18.$

   3. Compute $C_{23}$ (delete row 2, col 3; sign $(-1)^{2+3}=-$):

   $C_{23} = -\begin{vmatrix}1 & 2\\ 2 & 4\end{vmatrix} = -(1\cdot4-2\cdot2) = -(0) = 0.$

   4. Compute $C_{33}$ (delete row 3, col 3; sign $(+1)^{3+3}=+$):

   $C_{33} = +\begin{vmatrix}1 & 2\\ 4 & -1\end{vmatrix} = 1(-1)-2(4) = -1-8=-9.$

   5. So $[a\ b\ c] = [18\ \ 0\ \ -9]$.
   6. First dot product: $[a\ b\ c]\begin{pmatrix}1\\4\\2\end{pmatrix} = 18(1)+0(4)+(-9)(2) = 18-18 = 0$.
   7. Second dot product: $[a\ b\ c]\begin{pmatrix}3\\7\\6\end{pmatrix} = 18(3)+0(7)+(-9)(6) = 54-54 = 0$. (This equals $\det(M)$ by the same-column expansion rule; direct expansion of $M$ along column 3 confirms $\det(M)=3(18)+7(0)+6(-9)=54-54=0$, i.e. $M$ is singular.)
   8. Sum of both dot products: $0+0 = 0$.

   Common Mistakes

   • Sign errors in the cofactor signs $(-1)^{i+j}$ — easy to drop the minus sign on $C_{23}$ (row+col = odd).
   • Not noticing the two given column vectors are literally columns of $M$, leading to unnecessary/error-prone brute-force arithmetic instead of the quick identity check.
   • Forgetting to add the two dot products together (the question asks for their sum, not just one of them).

   ✓Final answer

   The correct option is (C) — $0$.

   ANSWER: C

2. AP EAPCET 2021Set eng-2021-08-19-AN1 markMCQ
   Q.The value of $\begin{vmatrix} b+c & a & a \\ b & c+a & b \\ c & c & a+b \end{vmatrix}$ is (A) $abc$ (B) $(a+b)(b+c)(c+a)$ (C) $4abc$ (D) $(a-b)(b-c)(c-a)$

   ›Reveal solutionSolution

   This classic 3×3 determinant simplifies via row operations (adding all rows together makes every entry in one row equal to $a+b+c$) to the closed form $4abc$, confirmed by direct numeric substitution.

   Concept and Intuition

   Many "nice" symmetric determinants like this one are best handled by first performing a row or column operation that reveals a common factor (here, adding all three rows makes every entry in the new row equal, exposing an $(a+b+c)$ or similar factor), then simplifying the reduced $2\times2$ structure. When the algebra gets intricate, a quick numeric sanity check with simple values of $a,b,c$ is an efficient way to confirm which answer choice matches.

   Step-by-Step Solution

   1. The determinant is $\begin{vmatrix} b+c & a & a \\ b & c+a & b \\ c & c & a+b \end{vmatrix}$.
   2. Apply $R_1 \to R_1 + R_2 + R_3$: the new first row becomes $(b+c+b+c,\ a+c+a+c,\ a+b+a+b)$... more carefully, summing column-wise: column 1 sum $= (b+c)+b+c = b+2c$... to avoid an error-prone symbolic expansion, verify by direct numeric substitution instead (a clean, reliable check for this type of determinant).
   3. Numeric check: let $a=1, b=2, c=3$. The matrix becomes $\begin{pmatrix}5&1&1\\2&4&2\\3&3&3\end{pmatrix}$.
   4. Expand along Row 1: $\det = 5(4\cdot3-2\cdot3) - 1(2\cdot3-2\cdot3) + 1(2\cdot3-4\cdot3) = 5(12-6) - 1(0) + 1(6-12) = 30 - 0 - 6 = 24$.
   5. Compare with each option at $a=1,b=2,c=3$: (A) $abc=6$ — no. (B) $(a+b)(b+c)(c+a)=3\cdot5\cdot4=60$ — no. (C) $4abc = 4\cdot6=24$ — matches. (D) $(a-b)(b-c)(c-a)=(-1)(-1)(2)=2$ — no.
   6. Only option (C), $4abc$, matches the computed value.

   Common Mistakes

   • Attempting a fully symbolic cofactor expansion without organising the algebra carefully, leading to sign errors — a numeric check with simple distinct values is a fast, robust way to identify the correct closed form among given options.
   • Forgetting that this determinant is NOT antisymmetric in $a,b,c$ (ruling out option D, which vanishes whenever any two variables are equal — but the original determinant does not vanish when, say, $a=b$).

   ✓Final answer

   The correct option is (C) — $4abc$.

   ANSWER: C

3. AP EAPCET 2021Set eng-2021-08-23-FN1 markMCQ
   Q.$\begin{bmatrix} 1 & 2 & 3 \\ -1 & 1 & 2 \\ 3 & 0 & 2 \end{bmatrix} \begin{vmatrix} 2022 & 2024 \\ 2021 & 2023 \end{vmatrix} =$ (A) $\begin{bmatrix} 8 & 4 & 11 \\ 4 & -1 & 3 \\ 9 & 6 & 13 \end{bmatrix}$ (B) $\begin{bmatrix} 8 & 4 & 13 \\ 4 & -1 & 3 \\ 9 & 6 & 12 \end{bmatrix}$ (C) $\begin{bmatrix} 8 & 4 & 13 \\ 4 & -1 & 3 \\ 9 & 6 & 13 \end{bmatrix}$ (D) $\begin{bmatrix} 8 & 4 & 11 \\ 4 & 1 & 13 \\ 9 & 6 & 13 \end{bmatrix}$

   ›Reveal solutionSolution

   Multiplying the given $3\times3$ matrix by itself, entry by entry using the row-times-column rule, produces $\begin{bmatrix}8&4&13\\4&-1&3\\9&6&13\end{bmatrix}$, matching option (C).

   Concept and Intuition

   Matrix multiplication of a square matrix $A$ with itself ($A\times A = A^2$) is computed entry by entry: the $(i,j)$ entry of the product is the dot product of row $i$ of the first matrix with column $j$ of the second matrix. Carrying this out carefully for all 9 entries of a $3\times3$ matrix gives the resulting product matrix.

   Step-by-Step Solution

   Let $A=\begin{bmatrix}1&2&3\\-1&1&2\\3&0&2\end{bmatrix}$. Compute $A^2 = A\cdot A$ entry by entry (row $i$ of $A$ dotted with column $j$ of $A$):

   1. $(1,1)$: $1(1)+2(-1)+3(3) = 1-2+9 = 8$
   2. $(1,2)$: $1(2)+2(1)+3(0) = 2+2+0 = 4$
   3. $(1,3)$: $1(3)+2(2)+3(2) = 3+4+6 = 13$
   4. $(2,1)$: $-1(1)+1(-1)+2(3) = -1-1+6 = 4$
   5. $(2,2)$: $-1(2)+1(1)+2(0) = -2+1+0 = -1$
   6. $(2,3)$: $-1(3)+1(2)+2(2) = -3+2+4 = 3$
   7. $(3,1)$: $3(1)+0(-1)+2(3) = 3+0+6 = 9$
   8. $(3,2)$: $3(2)+0(1)+2(0) = 6+0+0 = 6$
   9. $(3,3)$: $3(3)+0(2)+2(2) = 9+0+4 = 13$

   Assembling these gives $A^2 = \begin{bmatrix}8&4&13\\4&-1&3\\9&6&13\end{bmatrix}$, which is an exact, entry-by-entry match with option (C).

   Common Mistakes

   • Multiplying corresponding entries directly (element-wise/"Hadamard" product) instead of using the proper row-times-column matrix multiplication rule.
   • Sign slip in row 2 (which contains a $-1$), which is where options (A) and (D) diverge from the correct value (they show entry $(2,2)$ or other entries inconsistent with the correct computation) — careful arithmetic through each dot product avoids this.

   ✓Final answer

   The correct option is (C) — $\begin{bmatrix} 8 & 4 & 13 \\ 4 & -1 & 3 \\ 9 & 6 & 13 \end{bmatrix}$.

   ANSWER: C

4. AP EAPCET 2026Set eng-2026-05-12-AN1 markMCQ
   Q.$\begin{vmatrix} -a^2 & ab & ac \\ ab & -b^2 & bc \\ ac & bc & -c^2 \end{vmatrix} =$ (A) $a^2b^2c^2$ (B) $2a^2b^2c^2$ (C) $3a^2b^2c^2$ (D) $4a^2b^2c^2$

   ›Reveal solutionSolution

   Factoring $a$, $b$, $c$ out of the three rows reduces the determinant to a simpler $\pm1$-coefficient determinant that evaluates to $4abc$, giving a total of $4a^2b^2c^2$. Answer: (D).

   Concept and Intuition

   Each row of the given determinant has a common factor: row 1 is $a\cdot(-a,b,c)$, row 2 is $b\cdot(a,-b,c)$, row 3 is $c\cdot(a,b,-c)$. Pulling a common factor out of a row simply multiplies the determinant by that factor (a standard determinant property), so we can simplify before directly expanding the messier original $3\times3$ determinant.

   Step-by-Step Solution

   1. Original determinant: $\begin{vmatrix} -a^2 & ab & ac \\ ab & -b^2 & bc \\ ac & bc & -c^2 \end{vmatrix}$.
   2. Factor $a$ from row 1, $b$ from row 2, $c$ from row 3:

   $= abc\begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix}$

   3. Expand this reduced determinant along the first row:

   $-a\begin{vmatrix}-b&c\\b&-c\end{vmatrix} - b\begin{vmatrix}a&c\\a&-c\end{vmatrix} + c\begin{vmatrix}a&-b\\a&b\end{vmatrix}$

   4. Compute each $2\times2$ minor: $\begin{vmatrix}-b&c\\b&-c\end{vmatrix} = (-b)(-c)-c(b) = bc-bc=0$; $\begin{vmatrix}a&c\\a&-c\end{vmatrix} = a(-c)-c(a) = -2ac$; $\begin{vmatrix}a&-b\\a&b\end{vmatrix} = ab-(-b)(a) = 2ab$.
   5. Substitute: $-a(0) - b(-2ac) + c(2ab) = 0 + 2abc + 2abc = 4abc$.
   6. So the reduced determinant equals $4abc$, and the original determinant is $abc \times 4abc = 4a^2b^2c^2$.

   Common Mistakes

   • Sign errors when expanding the $2\times2$ minors of the reduced matrix (the $-,+,-$ cofactor pattern is easy to mis-apply).
   • Forgetting to multiply the reduced determinant's value back by the $abc$ factored out earlier.

   ✓Final answer

   The correct option is (D) — $4a^2b^2c^2$.

   ANSWER: D

5. AP EAPCET 2026Set eng-2026-05-13-FN1 markMCQ
   Q.For a fixed positive integer $n$, if $D = \begin{vmatrix} n! & (n+1)! & (n+2)! \\ (n+1)! & (n+2)! & (n+3)! \\ (n+2)! & (n+3)! & (n+4)! \end{vmatrix}$, then $\dfrac{D}{n!\,(n+1)!\,(n+2)!} =$ (A) $-4$ (B) $-2$ (C) $2$ (D) $4$

   ›Reveal solutionSolution

   Taking the factorials common from each row collapses the determinant to a constant: $\dfrac{D}{n!\,(n+1)!\,(n+2)!} = 2$ — option (C).

   Working. Factor $n!$, $(n+1)!$, $(n+2)!$ from rows 1, 2, 3 respectively:

   $D = n!\,(n+1)!\,(n+2)!\begin{vmatrix} 1 & n+1 & (n+1)(n+2) \\1 & n+2 & (n+2)(n+3) \\1 & n+3 & (n+3)(n+4) \end{vmatrix}$

   Hence $\dfrac{D}{n!\,(n+1)!\,(n+2)!}$ equals that $3\times3$ determinant. Apply $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$, using $(n+2)(n+3)-(n+1)(n+2)=2(n+2)$ and $(n+3)(n+4)-(n+1)(n+2)=4n+10$:

   $\begin{vmatrix} 1 & n+1 & (n+1)(n+2) \\0 & 1 & 2(n+2) \\0 & 2 & 2(2n+5) \end{vmatrix}$

   Expanding along the first column:

   $1\cdot\big[\,1\cdot 2(2n+5) - 2(n+2)\cdot 2\,\big] = (4n+10) - (4n+8) = 2.$

   ✓Final answer

   $\dfrac{D}{n!\,(n+1)!\,(n+2)!} = 2$ — option (C).

   ANSWER: C

6. AP EAPCET 2024Set eng-2024-05-19-AN1 markMCQ
   Q.$\begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} =$ (A) $(a-b)(b-c)(c-a)(a+b+c)$ (B) $(a-b)(b-c)(c-a)$ (C) $(a-b)(b-c)(a-c)(ab+bc+ca)$ (D) $(a-b)(b-c)(c-a)(ab+bc+ca)$

   ›Reveal solutionSolution

   This determinant is a known generalised-Vandermonde identity that factors as $(a-b)(b-c)(c-a)(ab+bc+ca)$ — verified numerically, option (D).

   Concept and Intuition

   Determinants with rows $1,x,x^2$ (Vandermonde) factor neatly as $(a-b)(b-c)(c-a)$. When the exponent pattern is not consecutive (here $0,2,3$ instead of $0,1,2$), the determinant still factors into the Vandermonde piece times an extra symmetric-polynomial factor — here that extra factor turns out to be $e_2 = ab+bc+ca$ (the second elementary symmetric polynomial), since the exponent set $\{0,2,3\}$ is the base set $\{0,1,2\}$ shifted up by the partition $(0,1,1)$, whose associated Schur polynomial is exactly $e_2$.

   Rather than rely purely on this identity from memory, verifying with actual numbers is the safest exam technique.

   Step-by-Step Solution

   1. Expand the determinant along the first row (all entries 1): $D = (b^2c^3-b^3c^2) - (a^2c^3-a^3c^2) + (a^2b^3-a^3b^2)$ $= b^2c^2(c-b) + a^2c^2(a-c) + a^2b^2(b-a)$.
   2. Rather than fully factor symbolically, test with concrete numbers: let $a=1, b=2, c=3$.
      • Direct determinant: rows $(1,1,1)$, $(1,4,9)$, $(1,8,27)$. $D = 1(4\cdot27-9\cdot8) - 1(1\cdot27-9\cdot1) + 1(1\cdot8-4\cdot1) = 1(108-72)-1(27-9)+1(8-4) = 36-18+4=22$.
   3. Test option (D): $(a-b)(b-c)(c-a)(ab+bc+ca)$ with these values: $(1-2)(2-3)(3-1) = (-1)(-1)(2) = 2$; $ab+bc+ca = 2+6+3=11$; product $=2\times11=22$. Matches $D=22$.
   4. Test option (A): $(a-b)(b-c)(c-a)(a+b+c) = 2\times6=12\ne22$ — rejected.
   5. Test option (C): $(a-b)(b-c)(a-c)(ab+bc+ca)$: note $(a-c)=1-3=-2$ (opposite sign convention to $(c-a)=2$), giving $(-1)(-1)(-2)(11)=-22\ne22$ — rejected (sign mismatch).
   6. Option (D) is confirmed as the correct general factorisation.

   Common Mistakes

   • Sign confusion between $(c-a)$ and $(a-c)$ when comparing to option (C) — a single sign flip changes the whole product's sign.
   • Trying to factor the expression fully symbolically under time pressure instead of the much faster numeric-substitution check.

   ✓Final answer

   The correct option is (D) — $(a-b)(b-c)(c-a)(ab+bc+ca)$.

   ANSWER: D

7. AP EAPCET 2021Set eng-2021-08-23-AN1 markMCQ
   Q.If $\begin{vmatrix} x^3+2x^2+3x-2 & x^2+2x+4 \\ x^3-x^2-2x-1 & 3x^3-2x^2+4x-2 \end{vmatrix} = ax^6+bx^5+cx^4+dx^3+ex^2+fx+g$, then $a+b+c+d+e+f=$ (A) 23 (B) 25 (C) 21 (D) 20

   ›Reveal solutionSolution

   Evaluating the determinant-polynomial at $x=1$ gives the sum of ALL coefficients (including the constant $g$); evaluating at $x=0$ isolates $g$ alone. Subtracting removes $g$, leaving $a+b+c+d+e+f=25$.

   Concept and Intuition

   If $P(x)=ax^6+bx^5+cx^4+dx^3+ex^2+fx+g$, a classic trick to get the sum of coefficients excluding the constant term is: $P(1) = a+b+c+d+e+f+g$ gives the sum of all coefficients (since every power of $1$ is $1$), while $P(0)=g$ isolates just the constant term. So $P(1)-P(0) = a+b+c+d+e+f$. Here $P(x)$ is defined as the given $2\times2$ determinant, so we just need to evaluate that determinant at $x=1$ and $x=0$ directly — no need to expand the full degree-6 polynomial.

   Step-by-Step Solution

   1. The determinant is $\begin{vmatrix} x^3+2x^2+3x-2 & x^2+2x+4 \\ x^3-x^2-2x-1 & 3x^3-2x^2+4x-2 \end{vmatrix}$.
   2. At $x=1$: top-left $=1+2+3-2=4$; top-right $=1+2+4=7$; bottom-left $=1-1-2-1=-3$; bottom-right $=3-2+4-2=3$. Determinant $=4(3)-7(-3)=12+21=33 = P(1) = a+b+c+d+e+f+g$.
   3. At $x=0$: top-left $=-2$; top-right $=4$; bottom-left $=-1$; bottom-right $=-2$. Determinant $=(-2)(-2)-(4)(-1)=4+4=8 = P(0)=g$.
   4. So $a+b+c+d+e+f = P(1)-P(0) = 33-8 = 25$.

   Common Mistakes

   • Trying to fully expand the product of the two cubic polynomials into a degree-6 polynomial to read off coefficients individually — far more error-prone than the substitution trick.
   • Forgetting to subtract $g$ (i.e. reporting $33$, the sum of ALL seven coefficients, instead of the first six).

   ✓Final answer

   The correct option is (B) — 25.

   ANSWER: B

8. AP EAPCET 2021Set eng-2021-08-25-AN1 markMCQ
   Q.If the determinant $\begin{vmatrix} \cos 2x & \sin^2 x & \cos 2x \\ \sin^2 x & \cos 2x & \cos^2 x \\ \cos 2x & \cos^2 x & \cos 2x \end{vmatrix}$ is expanded in powers of $\cos x$, then the constant term in the expansion is (A) 1 (B) $-1$ (C) 0 (D) 2

   ›Reveal solutionSolution

   The constant term of a polynomial in $\cos x$ is exactly the value obtained by setting $\cos x=0$. Answer: (A).

   Concept and Intuition

   If a determinant's entries are polynomials in $c=\cos x$, expanding it produces a polynomial in $c$. Its constant term (the $c^0$ coefficient) equals the value of the whole expression when $c=0$ — a shortcut that avoids expanding the full polynomial in $\cos x$.

   Step-by-Step Solution

   1. Write each entry in terms of $c=\cos x$: $\cos2x = 2c^2-1$, $\sin^2x = 1-c^2$, $\cos^2x = c^2$.
   2. Set $c=0$: $\cos2x \to -1$, $\sin^2x \to 1$, $\cos^2x \to 0$.
   3. The matrix becomes $\begin{bmatrix}-1&1&-1\\1&-1&0\\-1&0&-1\end{bmatrix}$.
   4. Expand: $-1[(-1)(-1)-(0)(0)] - 1[(1)(-1)-(0)(-1)] + (-1)[(1)(0)-(-1)(-1)]$ $= -1(1) - 1(-1) + (-1)(-1) = -1+1+1 = 1$.

   Common Mistakes

   • Trying to fully expand the determinant symbolically in $\cos x$ (needlessly long) instead of using the $c=0$ substitution shortcut for the constant term.

   ✓Final answer

   The correct option is (A) — 1.

   ANSWER: A

9. AP EAPCET 2024Set eng-2024-05-21-AN1 markMCQ
   Q.$\begin{vmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{vmatrix} =$ (A) $(a+b+c)^3$ (B) $2(a+b+c)^3$ (C) $3(a+b+c)^3$ (D) $(a+b+c)$

   ›Reveal solutionSolution

   Writing each diagonal entry as $s+(\text{off-diagonal-like term})$ where $s=a+b+c$ reveals the matrix as $sI$ plus a rank-1 correction, whose determinant works out cleanly to $2(a+b+c)^3$ using the determinant lemma (or row operations).

   Concept and Intuition

   Many "cyclic-symmetric" determinants like this one simplify beautifully once you notice $a+b+2c = (a+b+c)+c$, etc. — subtracting off the common sum $s=a+b+c$ from every diagonal entry turns the matrix into $sI$ (a multiple of the identity) plus a very simple rank-1 matrix whose every row is identical. Determinants of $sI + \text{rank-1}$ matrices have a clean closed form.

   Step-by-Step Solution

   1. Let $s=a+b+c$. Rewrite diagonal entries: $a+b+2c=s+c$, $b+c+2a=s+a$, $c+a+2b=s+b$.
   2. The matrix becomes:

   $M=\begin{bmatrix}s+c & a & b\\ c & s+a & b\\ c & a & s+b\end{bmatrix}$

   3. Subtract $sI$: $M - sI = \begin{bmatrix}c&a&b\\c&a&b\\c&a&b\end{bmatrix}$ — every row is the same vector $(c,a,b)$, so $M-sI$ has rank 1: $M-sI=\mathbf 1 v^T$ where $\mathbf 1=(1,1,1)^T$ and $v=(c,a,b)^T$.
   4. So $M=sI+\mathbf1v^T$. The determinant lemma (matrix determinant lemma) gives:

   $\det(sI+\mathbf1 v^T) = s^3 + s^2\,(v^T\mathbf1)$

   (since $\det(sI)=s^3$ and $\operatorname{adj}(sI)=s^2 I$ for a $3\times3$ matrix).

   5. $v^T\mathbf1 = c+a+b = s$. So $\det(M) = s^3+s^2\cdot s = 2s^3$.

   6. Substituting back $s=a+b+c$: $\det = 2(a+b+c)^3$.

   Common Mistakes

   • Trying to expand the $3\times3$ determinant directly by cofactors without the $s$-substitution — it's algebraically messy and error-prone; the "subtract the common sum" trick is much cleaner and standard for this whole family of problems.
   • Forgetting the factor of $s^2$ multiplying $(v^T\mathbf1)$ and just adding $s^3+v^T\mathbf1$.

   ✓Final answer

   The correct option is (B) — $2(a+b+c)^3$.

   ANSWER: B

10. AP EAPCET 2021Set eng-2021-08-20-FN1 markMCQ
    Q.What is the value of $\begin{vmatrix} a & b & c \\ a-b & b-c & c-a \\ b+c & c+a & a+b \end{vmatrix} = ?$ (A) $a^3 + b^3 + c^3 + 3abc$ (B) $a^3 + b^3 + c^3 - 3abc$ (C) $a^3 + b^3 + c^3 - 6abc$ (D) $a^3 + b^3 + c^3 + 6abc$

    ›Reveal solutionSolution

    Direct cofactor expansion of the determinant shows all cross terms cancel, leaving the

    classical identity $a^3+b^3+c^3-3abc$.

    Concept and Intuition

    This is a disguised version of the well-known factorisation

    $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$, packaged as a determinant. Expanding carefully

    term by term (rather than guessing) confirms which of the four sign variants is correct.

    Step-by-Step Solution

    1. Expand along the first row: $D = a\left[(b-c)(a+b)-(c-a)(c+a)\right] - b\left[(a-b)(a+b)-(c-a)(b+c)\right] + c\left[(a-b)(c+a)-(b-c)(b+c)\right]$.
    2. Compute each bracket:
       • $(b-c)(a+b)-(c-a)(c+a) = ab+b^2-ac-bc-c^2+a^2$
       • $(a-b)(a+b)-(c-a)(b+c) = a^2-b^2-bc-c^2+ab+ac$
       • $(a-b)(c+a)-(b-c)(b+c) = ac+a^2-bc-ab-b^2+c^2$
    3. Multiply through by $a$, $-b$, $c$ respectively and add. All the mixed quadratic-times-linear terms ($a^2b, ab^2, a^2c, ac^2, b^2c, bc^2$) cancel in pairs, leaving only $a^3+b^3+c^3$ from the cubic terms and $-3abc$ from the three $abc$ contributions (one from each bracket).
    4. Verify with a quick numeric check ($a=1,b=0,c=0$): the matrix becomes $\begin{bmatrix}1&0&0\\1&0&-1\\0&1&1\end{bmatrix}$, whose determinant is $1$; and $1^3+0+0-0=1$ — matches.
    5. So $D = a^3+b^3+c^3-3abc$.

    Common Mistakes

    • Sign error on the $3abc$ term (getting $+3abc$ instead of $-3abc$, or $\pm 6abc$) from mis-tracking how many times the $abc$ term appears across the three bracket expansions.

    ✓Final answer

    The correct option is (B) — $a^3+b^3+c^3-3abc$.

    ANSWER: B

11. AP EAPCET 2022Set eng-2022-07-05-AN1 markMCQ
    Q.If b and c are non zero real numbers, $A = \begin{bmatrix} 1 & b & c \\ b & 2 & 3 \\ c & 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & b & c \\ -b & 0 & 2 \\ -c & -2 & 0 \end{bmatrix}$, then $\det(A+B) =$ (A) 3 (B) 1 (C) -1 (D) 0

    ›Reveal solutionSolution

    Adding the symmetric $A$ and skew-symmetric $B$ entrywise cancels the off-diagonal $b,c$ terms below the diagonal, leaving a matrix whose determinant is a fixed number independent of $b,c$: 3.

    Concept and Intuition

    $A$ is symmetric and $B$ is skew-symmetric (its transpose is its negative, with zero diagonal). Adding them entrywise, the upper-triangular parts of $A+B$ pick up $b+b=2b$ and $c+c=2c$, while the lower-triangular parts get $b+(-b)=0$ and $c+(-c)=0$ — so $A+B$ becomes upper triangular in its first column, letting the determinant be computed by a clean cofactor expansion that never involves $b$ or $c$.

    Step-by-Step Solution

    1. $A=\begin{bmatrix}1&b&c\\b&2&3\\c&3&4\end{bmatrix}$, $B=\begin{bmatrix}0&b&c\\-b&0&2\\-c&-2&0\end{bmatrix}$.
    2. Add entrywise: $(A+B)_{11}=1$, $(A+B)_{12}=2b$, $(A+B)_{13}=2c$; $(A+B)_{21}=b-b=0$, $(A+B)_{22}=2$, $(A+B)_{23}=3+2=5$; $(A+B)_{31}=c-c=0$, $(A+B)_{32}=3-2=1$, $(A+B)_{33}=4$.
    3. So $A+B=\begin{bmatrix}1&2b&2c\\0&2&5\\0&1&4\end{bmatrix}$.
    4. Expand the determinant along the first column (only the $(1,1)$ entry is nonzero there): $\det(A+B) = 1\cdot\det\begin{bmatrix}2&5\\1&4\end{bmatrix} = 1\cdot(2\cdot4-5\cdot1) = 8-5=3$.

    Common Mistakes

    • Not recognising $B$ as skew-symmetric and instead trying to compute the full $3\times3$ determinant symbolically in $b,c$ (much more work, and error-prone).
    • Sign slip when subtracting $b-b$ or $c-c$ in the lower-triangular entries.

    ✓Final answer

    The correct option is (A) — 3.

    ANSWER: A

12. AP EAPCET 2025Set eng-2025-05-26-FN1 markMCQ
    Q.If the inverse of $\begin{bmatrix}-x & 14x & 7x\\ 0 & 1 & 0\\ x & -4x & -2x\end{bmatrix}$ is $\begin{bmatrix}2 & 0 & 7\\ 0 & 1 & 0\\ 1 & -2 & 1\end{bmatrix}$, then $\begin{vmatrix}x & x+1 & x+2\\ x+1 & x+2 & x+3\\ x+2 & x+3 & x+4\end{vmatrix} =$ (A) $\dfrac{x}{5}$ (B) $x-5$ (C) $5x-1$ (D) $x+5$

    ›Reveal solutionSolution

    This tests recognizing a determinant with AP-rows (always zero) and pinning down $x$ from a matrix-inverse condition; the answer is whichever option numerically equals $0$ at that $x$.

    Concept and Intuition

    A square matrix whose rows form an arithmetic progression (i.e. $R_1-2R_2+R_3=0$) is always singular: its rows are linearly dependent, so its determinant is identically zero, no matter what parameter sits inside it. Spotting this saves a full cofactor expansion. The matrix-inverse condition is just the defining relation $A\,A^{-1}=I$ applied to one convenient entry.

    Step-by-Step Solution

    1. Let $A=\begin{bmatrix}-x & 14x & 7x\\ 0 & 1 & 0\\ x & -4x & -2x\end{bmatrix}$ and $A^{-1}=\begin{bmatrix}2 & 0 & 7\\ 0 & 1 & 0\\ 1 & -2 & 1\end{bmatrix}$.
    2. Using $A\,A^{-1}=I$, multiply row 1 of $A$ by column 1 of $A^{-1}$: $(-x)(2)+(14x)(0)+(7x)(1)=5x$. This must equal the $(1,1)$ entry of $I$, i.e. $1$. So $5x=1\Rightarrow x=\dfrac15$. (Checking the other products confirms this value is consistent throughout the matrix.)
    3. Now look at $D=\begin{vmatrix}x & x+1 & x+2\\ x+1 & x+2 & x+3\\ x+2 & x+3 & x+4\end{vmatrix}$. Apply $R_1\to R_1-2R_2+R_3$: each entry becomes $x-2(x+1)+(x+2)=0$, turning the first row into $(0,0,0)$.
    4. A zero row forces $D=0$ for every value of $x$ — this is an algebraic identity, independent of the specific $x=1/5$.
    5. Since $D=0$ always, the matching option is the expression that equals $0$ at $x=\dfrac15$: $5x-1 = 5\left(\dfrac15\right)-1 = 1-1=0$. ✓ (Checking the rest: $\dfrac{x}{5}=\dfrac1{25}$, $x-5=-\dfrac{24}{5}$, $x+5=\dfrac{26}{5}$ — all nonzero.)

    Common Mistakes

    • Brute-force expanding the $3\times3$ determinant instead of spotting the AP-row structure, wasting time and inviting slips.
    • Forgetting that "$D=0$ identically" means the answer must be matched numerically against $x=1/5$, not simplified as a formula.
    • Mixing up which matrix is $A$ and which is the given inverse when extracting $x$.

    ✓Final answer

    The correct option is (C) — $5x-1$.

    ANSWER: C