Q.Using Cofactors of elements of third column, evaluate $\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}$.

Showing the 12 most recent of 32 on this concept.

1. 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 $D=22$. …

   🔒 Start your 14-day free trial to see the full solution →
2. 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). …

   🔒 Start your 14-day free trial to see the full solution →
3. 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: $\det = 5(4\cdot3-2\cdot3) - 1(2\cdot3-2\cdot3) + 1(2\cdot3-4\cdot3) = 5(12-6) - 1(0) + 1(6-12) = 30 - 0 - 6 = 24$. …

   🔒 Start your 14-day free trial to see the full solution →
4. 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: $\det(sI+\mathbf1 v^T) = s^3 + s^2\,(v^T\mathbf1)$ …

   🔒 Start your 14-day free trial to see the full solution →
5. 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: $-a\begin{vmatrix}-b&c\\b&-c\end{vmatrix} - b\begin{vmatrix}a&c\\a&-c\end{vmatrix} + c\begin{vmatrix}a&-b\\a&b\end{vmatrix}$ …

   🔒 Start your 14-day free trial to see the full solution →
6. 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: $[a\ b\ c]\begin{pmatrix}1\\4\\2\end{pmatrix} = 18(1)+0(4)+(-9)(2) = 18-18 = 0$. …

   🔒 Start your 14-day free trial to see the full solution →
7. AP EAPCET 2026Set eng-2026-05-13-AN1 markMCQ
   Q.If $A = \begin{bmatrix} a & 2b & 3c \\ 2b & c & 3a \\ 3c & 2a & b \end{bmatrix}$ and $\det(A) = pa^3 + qb^3 + rc^3 + s(abc)$, then $p+q+r+s =$ (A) $12$ (B) $20$ (C) $24$ (D) $30$

   ›Reveal solutionSolution

   Expanding the determinant gives $-6a^3-4b^3-9c^3+31abc$, so $p+q+r+s=-6-4-9+31=12$ — option (A).

   $\det(A)=\begin{vmatrix} a & 2b & 3c \\2b & c & 3a \\3c & 2a & b \end{vmatrix} = a(cb-6a^2)-2b(2b^2-9ac)+3c(4ab-3c^2).$

   Expand:

   $= abc-6a^3-4b^3+18abc+12abc-9c^3 = -6a^3-4b^3-9c^3+31abc.$ …

   🔒 Start your 14-day free trial to see the full solution →
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 $\begin{bmatrix}-1&1&-1\\1&-1&0\\-1&0&-1\end{bmatrix}$. …

   🔒 Start your 14-day free trial to see the full solution →
9. 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 $=4(3)-7(-3)=12+21=33 = P(1) = a+b+c+d+e+f+g$. …

   🔒 Start your 14-day free trial to see the full solution →
10. AP EAPCET 2026Set eng-2026-05-15-AN1 markMCQ
    Q.If $f,g,h$ are differentiable functions of $x$, then $\begin{vmatrix} f & g & h \\ (xf)' & (xg)' & (xh)' \\ f' & g' & h' \end{vmatrix} =$ (A) $fg' - gh'$ (B) $gh' + xf'$ (C) $0$ (D) $x(f'+g'+h')$

    ›Reveal solutionSolution

    The determinant simplifies to zero because the second row is a linear combination of the first and third rows, making the rows linearly dependent.

    We are asked to evaluate

    $\Delta = \begin{vmatrix} f & g & h \\ (xf)' & (xg)' & (xh)' \\ f' & g' & h' \end{vmatrix}$

    where $f,g,h$ are differentiable functions of $x$.

    ---

    Concept and Intuition

    The key idea is to use row operations that do not change the value of a determinant, or to notice linear dependence among rows.

    Here, the second row contains derivatives of products $xf, xg, xh$. Using the product rule,

    $(xf)' = f + x f', \quad (xg)' = g + x g', \quad (xh)' = h + x h'.$

    So the second row is actually:

    $\begin{pmatrix} f + x f' & g + x g' & h + x h' \end{pmatrix}.$

    Notice that this is exactly Row 1 plus $x$ times Row 3:

    $\text{Row 2} = \text{Row 1} + x \cdot \text{Row 3}.$

    When one row is a linear combination of the others, the determinant is zero.

    ---

    Step-by-step reasoning

    1. Expand the second row entries using the product rule:

    $(xf)' = f + x f', \quad (xg)' = g + x g', \quad (xh)' = h + x h'.$

    2. Rewrite the determinant with this expanded form:

    $\Delta = \begin{vmatrix} f & g & h \\ f + x f' & g + x g' & h + x h' \\ f' & g' & h' \end{vmatrix}.$

    3. Perform a row operation that does not change the determinant: subtract Row 1 from Row 2.

    $\text{New Row 2} = \text{Row 2} - \text{Row 1} = \begin{pmatrix} x f' & x g' & x h' \end{pmatrix}.$

    So

    $\Delta = \begin{vmatrix} f & g & h \\ x f' & x g' & x h' \\ f' & g' & h' \end{vmatrix}.$ …

    🔒 Start your 14-day free trial to see the full solution →
11. AP EAPCET 2025Set eng-2025-05-23-FN1 markMCQ
    Q.$\begin{vmatrix} 2 & 1 \\ 3 & 1 \end{vmatrix} + \begin{vmatrix} 1 & 1/3 \\ 3 & 1 \end{vmatrix} + \begin{vmatrix} 1/2 & 1/9 \\ 3 & 1 \end{vmatrix} + \begin{vmatrix} 1/4 & 1/27 \\ 3 & 1 \end{vmatrix} + \ldots \infty =$ (A) $0$ (B) $\dfrac{1}{2}$ (C) $-\dfrac{1}{2}$ (D) $-1$

    ›Reveal solutionSolution

    Expand each $2\times2$ determinant, recognise two independent geometric series in the result, and sum them separately. Answer: (C).

    Concept and Intuition

    Every term in the sum is a $2\times2$ determinant with a fixed bottom row $\begin{pmatrix}3&1\end{pmatrix}$, so $\begin{vmatrix}a&b\\3&1\end{vmatrix}=a-3b$ splits linearly into an $a$-part and a $b$-part. Once each part is recognised as its own geometric progression, the infinite sum is just the difference of two standard geometric series sums, $\dfrac{\text{first term}}{1-r}$.

    Step-by-Step Solution

    1. General term: $\begin{vmatrix}a_n & b_n\\3&1\end{vmatrix} = a_n\cdot1 - b_n\cdot3 = a_n - 3b_n$.
    2. First-column values across the given terms: $2, 1, \tfrac12, \tfrac14,\dots$ — a geometric sequence with first term $2$ and ratio $\tfrac12$: $a_n = 2\left(\tfrac12\right)^{n-1}$.
    3. Second-column values: $1, \tfrac13, \tfrac19, \tfrac1{27},\dots$ — geometric with first term $1$ and ratio $\tfrac13$: $b_n=\left(\tfrac13\right)^{n-1}$. …

    🔒 Start your 14-day free trial to see the full solution →
12. AP EAPCET 2021Set eng-2021-08-25-FN1 markMCQ
    Q.If $a, b, c$ are real numbers such that $a^2+b^2+c^2-ab-bc-ac \le 0$, then $\begin{vmatrix} (a-b+1)^5 & b^7-c^7 & c^9-a^9 \\ a^{11}-b^{11} & (b-c+2)^3 & c^{13}-a^{13} \\ a^{15}-b^{15} & b^{17}-c^{17} & (c-a+3)^1 \end{vmatrix} =$ (A) $2abc$ (B) $0$ (C) $24abc$ (D) $24$

    ›Reveal solutionSolution

    The inequality forces $a=b=c$; substituting collapses the matrix to a simple upper-triangular-looking form whose determinant is just the product of its surviving diagonal terms, $24$.

    Concept and Intuition

    The expression $a^2+b^2+c^2-ab-bc-ca$ is a sum of squares in disguise: $\tfrac12[(a-b)^2+(b-c)^2+(c-a)^2]$, which can never be negative. So the given condition "$\le 0$" combined with this built-in "$\ge0$" forces it to be exactly zero, which only happens when $a=b=c$. That single deduction massively simplifies every entry of the determinant, since almost every entry is a difference of equal powers of $a,b,c$ (which all vanish when $a=b=c$), leaving only three surviving diagonal-type terms.

    Step-by-Step Solution

    1. Rewrite the condition: $a^2+b^2+c^2-ab-bc-ca=\tfrac12[(a-b)^2+(b-c)^2+(c-a)^2]$, which is always $\ge0$.
    2. Given this quantity is also $\le0$, it must equal exactly $0$, forcing $(a-b)^2=(b-c)^2=(c-a)^2=0$, i.e. $a=b=c$.
    3. Substitute $a=b=c$ into the matrix: all terms of the form $b^7-c^7$, $c^9-a^9$, $a^{11}-b^{11}$, $c^{13}-a^{13}$, $a^{15}-b^{15}$, $b^{17}-c^{17}$ become $0$ (difference of equal quantities). …

    🔒 Start your 14-day free trial to see the full solution →