Chapter 6

Let \(\alpha_{1}, \alpha_{2}\) and \(\beta_{1}, \beta_{2}\), be the roots of \(a x^{2}+b x+c=0\) and \(p x^{2}+q x+r=0\) respectively. If the system of equations \(\alpha_{1} y+\alpha_{2} z=0\) and \(\beta_{1} y+\beta_{2} z=0\) has a non- trivial solution, then (A) \(\frac{b^{2}}{q^{2}}=\frac{a c}{p r}\) (B) \(\frac{c^{2}}{r^{2}}=\frac{a b}{p q}\) (C) \(\frac{a^{2}}{p^{2}}=\frac{b c}{q r}\) (D) None of these

If the value of a third order determinant is 11 , then the value of the determinant formed by its cofactors will be (A) 11 (B) 121 (C) 1331 (D) 14641

The value of the determinant \(\left|\begin{array}{ccc}\sqrt{x}+\sqrt{y} & 2 \sqrt{2} & \sqrt{z} \\ \sqrt{y z}+\sqrt{2 x} & z & \sqrt{2 z} \\ y+\sqrt{x z} & \sqrt{y z} & z\end{array}\right|\) where \(x, y, z\) are positive real numbers, is (A) \(z(\sqrt{2} y-z \sqrt{y)}\) (B) \(y(\sqrt{2} z-y \sqrt{z)}\) (C) \(x(\sqrt{2} y-z \sqrt{y})\) (D) None of these

Let \(D_{k}=\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ 2.3^{k} & 16.9^{k} & 26.27^{k} \\ \left(3^{10}-1\right) & 2\left(9^{10}-1\right) & \left(27^{10}-1\right)\end{array}\right|\) then the value of \(\sum_{k=1}^{10} D_{k}\) is (A) \(2(\alpha+\beta+\eta)\) (B) \(\alpha \beta+\alpha \gamma+\beta \gamma\) (C) \(\alpha \beta \gamma\) (D) 0

If \(M\) is a \(3 \times 3\) matrix, where \(M^{\prime} M=I\) and \(\operatorname{det}(M)=1\), then \(\operatorname{det}(M-I)=\) (A) 1 (B) 0 (C) \(-1\) (D) None of these

If \([x]\) denotes the greatest integer less than or equal to \(x\), then the value of the determinant \(\left|\begin{array}{ccc}{[e]} & {[\pi]} & {\left[\pi^{2}-6\right]} \\ {[\pi]} & {\left[\pi^{2}-6\right]} & {[e]} \\\ {\left[\pi^{2}-6\right]} & {[e]} & {[\pi]}\end{array}\right|\), then (A) \(-8\) (B) 8 (C) 0 (D) None of these

If \(a_{i}, b_{i}, c_{i} \in R(i=1,2,3)\) and \(x \in R\) and \(\left|\begin{array}{ccc}a_{1}+b_{1} x & a_{1} x+b_{1} & c_{1} \\ a_{2}+b_{2} x & a_{2} x+b_{2} & c_{2} \\ a_{3}+b_{3} x & a_{3} x+b_{3} & c_{3}\end{array}\right|=0\), then (A) \(\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\\ a_{3} & b_{3} & c_{3}\end{array}\right|=4\) (B) \(x=\pm 1\) (C) \(x=2\) (D) None of these

The value of the determinant \(\left|\begin{array}{lcc}\sin \theta & \cos \theta & \sin 2 \theta \\ \sin \left(\theta+\frac{2 \pi}{3}\right) & \cos \left(\theta+\frac{2 \pi}{3}\right) & \sin \left(2 \theta+\frac{4 \pi}{3}\right) \\ \sin \left(\theta-\frac{2 \pi}{3}\right) & \cos \left(\theta-\frac{2 \pi}{3}\right) & \sin \left(2 \theta-\frac{4 \pi}{3}\right)\end{array}\right|\) (A) 0 (B) \(\sin \theta\) (C) \(\cos \theta\) (D) independent of \(\theta\)

If \(D_{k}=\left|\begin{array}{ccc}1 & n & n \\ 2 k & n^{2}+n+2 & n^{2}+n \\\ 2 k-1 & n^{2} & n^{2}+n+2\end{array}\right|\) and \(\sum_{k=1}^{n} D_{k}=48\), then \(n\) equals (A) 4 (B) 6 (C) 8 (D) None of these

If \(A, B, C\) are the angles of a triangle and \(\left|\begin{array}{ccc}1 & 1 & 1 \\ 1+\sin A & 1+\sin B & 1+\sin C \\ \sin A+\sin ^{2} A & \sin B+\sin ^{2} B & \sin C+\sin ^{2} C\end{array}\right|=0\) then the triangle is \(\mathrm{a} /\) an (A) equilaterral (B) isosceles (C) right-angled triangle (D) any triangle

If \(a_{0}, a_{1} a_{2}, a_{3}, a_{4}\) are in A.P with the common difference \(d\), the value of \(\left|\begin{array}{lll}a_{1} a_{2} & a_{1} & a_{0} \\\ a_{2} a_{3} & a_{2} & a_{1} \\ a_{3} a_{4} & a_{3} & a_{2}\end{array}\right|\) is (A) \(2 d^{4}\) (B) \(2 d^{3}\) (C) \(2 d^{2}\) (D) \(2 d\)

If \(\alpha, \beta, \gamma\) are different from and are the roots of \(a x^{3}+\) \(b x^{2}+c x+d=0\) and \((\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)=\frac{25}{2}\), then the determinant \(\Delta=\left|\begin{array}{ccc}\frac{\alpha}{1-\alpha} & \frac{\beta}{1-\beta} & \frac{\gamma}{1-\gamma} \\ \alpha & \beta & \gamma \\\ \alpha^{2} & \beta^{2} & \gamma^{2}\end{array}\right|\) equals (A) \(\frac{25 d}{2 a}\) (B) \(\frac{25 d}{a}\) (C) \(\frac{-25 d}{a+b+c+d}\) (D) None of these

Let \(\left\\{\Delta_{1}, \Delta_{2}, \Delta_{3}, \ldots, \Delta_{k}\right\\}\) be the set of third order determinants that can be made with the distinct nonzero real numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{9}\). Then (A) \(k=9 !\) (B) \(\sum_{i=1}^{k} \Delta_{i}=0\) (C) at least one \(\Delta_{i}=0\) (D) None of these

If \(f(x)=\left|\begin{array}{ccc}(1+x)^{a} & (1+2 x)^{b} & 1 \\ 1 & (1+x)^{a} & (1+2 x)^{b} \\ (1+2 x)^{b} & 1 & (1+x)^{a}\end{array}\right|, a, b\) being positive integers, then (A) constant term of \(f(x)\) is 4 (B) coefficieent of \(x\) in \(f(x)\) is 0 (C) constant term in \(f(x)\) is \(a-b\) (D) constant term in \(f(x)\) is \(a+b\)

If \(P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\) and \(Q=P A P^{\prime}\), then \(p^{\prime} Q^{2005} P\) is (A) \(\left[\begin{array}{cc}1 & 1 \\ 2005 & 1\end{array}\right]\) (B) \(\left[\begin{array}{cc}1 & 2005 \\ 0 & 1\end{array}\right]\) (C) \(\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) (D) \(\left[\begin{array}{cc}1 & 2005 \\ 2005 & 1\end{array}\right]\)

If \(\left|\begin{array}{lll}x^{n} & x^{n+2} & x^{n+3} \\ y^{n} & y^{n+2} & y^{n+3} \\ z^{n} & z^{n+2} & z^{n+3}\end{array}\right|\) \(=(x-y)(y-z)(z-x),\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\) then \(n=\) (A) \(-2\) (B) \(-1\) (C) 0 (D) 1

If \(\alpha, \beta, \gamma\) are the roots of the equation \(a x^{3}+b x^{2}+c\) \(=0\), then the value of the determinant \(\left|\begin{array}{ccc}\alpha \beta & \beta \gamma & \gamma \alpha \\ \beta \gamma & \gamma \alpha & \alpha \beta \\\ \gamma \alpha & \alpha \beta & \beta \gamma\end{array}\right|\) (A) \(a\) (B) \(\underline{b}\) (C) 0 (D) \(c\)

If \(p+q+r=0=a+b+c\), then the value of the determinant \(\left|\begin{array}{ccc}p a & q b & n c \\ q c & n a & p b \\ r b & p c & q a\end{array}\right|\) is (A) 0 (B) \(p q+q b+r c\) (C) 1 (D) None of these

A determinant of second order is made with the elements 0 and \(1 .\) The number of determinants with non-negative values is (A) 3 (B) 10 (C) 11 (D) 13

If \(f_{j}=\sum_{i=0}^{2} a_{i j} x^{i}, j=1,2,3\) and if \(f_{j}^{\prime}, f_{j}^{\prime \prime}\) denote \(\frac{d f_{j}}{d x}, \frac{d^{2} f_{j}}{d x^{2}}\) respectively, then \(g(x)=\left|\begin{array}{lll}f_{1} & f_{2} & f_{3} \\\ f_{1}^{\prime} & f_{2}^{\prime} & f_{3}^{\prime} \\ f_{1}^{\prime \prime} & f_{2}^{\prime \prime} & f_{3}^{\prime \prime}\end{array}\right|\) is (A) a cubic in \(x\) (B) a quadratic in \(x\) (C) linear in \(x\) (D) a constant

The value of the determinant \(\Delta=\left|\begin{array}{ccc}2 a_{1} b_{1} & a_{1} b_{2}+a_{2} b_{1} & a_{1} b_{3}+a_{3} b_{1} \\ a_{1} b_{2}+a_{2} b_{1} & 2 a_{2} b_{2} & a_{2} b_{3}+a_{3} b_{2} \\ a_{1} b_{3}+a_{3} b_{1} & a_{3} b_{2}+a_{2} b_{3} & 2 a_{3} b_{3}\end{array}\right|\) is (A) 1 (B) \(-1\) (C) 0 (D) \(a_{1} a_{2} a_{3} b_{1} b_{2} b_{3}\)

If \(A+B+C=\pi, e^{\mathrm{i} \theta}=\cos \theta+i \sin \theta\) and \(z=\left|\begin{array}{lll}e^{2 i A} & e^{-i C} & e^{-i B} \\ e^{-i C} & e^{2 i B} & e^{-i A} \\ e^{-i B} & e^{-i A} & e^{2 i C}\end{array}\right|\) then (A) \(\operatorname{Re}(z)=4\) (B) \(\operatorname{Im}(z)=0\) (C) \(\operatorname{Re}(z)=-4\) (D) \(\operatorname{Im}(z)=-1\)

If the system of equations \(a x+b y+c=0, b x+c y+a\) \(=0, c x+a y+b=0\) has a solution then the system of equations \((b+c) x+(c+a) y+(a+b) z=0\) \((c+a) x+(a+b) y+(b+c) z=0\) \((a+b) x+(b+c) y+(c+a) z=0\) has (A) only one solution (B) no solution (C) infinite number of solutions (D) None of these

\((b+c)(y+z)-a x=b-c\), \((c+a)(z+x)-b y=c-a\), \((a+b)(x+y)-c z=a-b\), where \(a+b+c \neq 0\), then \(x=\) (A) \(\frac{c-b}{a+b+c}\) (B) \(\frac{a-c}{a+b+c}\) (C) \(\frac{b-a}{a+b+c}\) (D) \(\frac{1}{a+b+c}\)

The equations \(x+y+z=6, x+2 y+3 z=10, x+2 y+\) \(m z=n\) give infinite number of values of the triplet \((x,\), \(y, z)\) if (A) \(m=3, n \in R\) (B) \(m=3, n \neq 10\) (C) \(m=3, n=10\) (D) None of these

If \(x \neq 0, y \neq 0, z \neq 0\) and \(\left|\begin{array}{ccc}1+x & 1 & 1 \\\ 1+y & 1+2 y & 1 \\ 1+z & 1+z & 1+3 z\end{array}\right|=0\), then \(x^{-1}+y^{-1}+z^{-1}\) is equal to (A) \(-1\) (B) \(-2\) (C) \(-3\) (D) None of these

If \(\Delta(x)=\left|\begin{array}{ccc}x & 1+x^{2} & x^{3} \\ \log \left(1+x^{2}\right) & e^{x} & \sin x \\ \cos x & \tan x & \sin ^{2} x\end{array}\right|\) then (A) \(\Delta(x)\) is divisible by \(x\) (B) \(\Delta(x)=0\) (C) \(\Delta^{\prime}(x)=0\) (D) None of these

The number of values of \(k\) for which the linear equations $$ \begin{array}{r} 4 x+k y+2 z=0 \\ k x+4 y+z=0 \\ 2 x+2 y+z=0 \end{array} $$ possess a non-zero solution is (A) 0 (B) 3 (C) 2 (D) 1

Let \(P\) and \(Q\) be \(3 \times 3\) matrices \(P \neq Q\). If \(P^{3}=Q^{3}\) and \(P^{2} Q=Q^{2} P\), then determinant of \(\left(P^{2}+Q^{2}\right)\) is equal to: (A) \(-2\) (B) 1 (C) 0 (D) \(-1\)

The value of the determinant \(\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & 10 \\ 3+\sqrt{65} & \sqrt{15} & 5\end{array}\right|\) is equal to: (A) \(5 \sqrt{3}(\sqrt{6}-5)\) (B) \(5 \sqrt{3}(\sqrt{6}-\sqrt{5})\) (C) \(5(\sqrt{6}-5)\) (D) \(\sqrt{3}(\sqrt{6}-\sqrt{5})\)

Let \(a, b, c\) be any real numbers. Suppose that there are real numbers \(x, y, z\) not all zero such that \(x=c y+b z, y\) \(=a z+c x\) and \(z=b x+a y\). Then \(a^{2}+b^{2}+c^{2}+2 a b c\) is equal to (A) 2 (B) \(-1\) (C) 0 (D) 1

Let \(a, b, c\) be such that \(b(a+c) \neq 0\). If \(\left|\begin{array}{ccc}a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1\end{array}\right|\) \(+\left|\begin{array}{rrr}a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^{n} c\end{array}\right|=0\), then the value of ' \(n\) ' is (A) zero (B) any even integer (C) any odd integer (D) any integer

Let \(A\) be a \(2 \times 2\) matrix Statement-1: \(\operatorname{adj}(\operatorname{adj} A)=A\) Statement-2: \(|\operatorname{adj} A|=|A|\) (A) Statement-1 is true, Statement-2 is true; Statement- 2 is a correct explanation for Statement- 1 (B) Statement-1 is true, Statement-2 is true; Statement- 2 is not a correct explanation for Statement-1 (C) Statement- 1 is true, Statement- 2 is false (D) Statement- 1 is false, Statement- 2 is true

If \(a, b, c, d>0 ; x \in R\) and \(\left(a^{2}+b^{2}+c^{2}\right) x^{2}-2(a b+b c+c d) x+b^{2}+c^{2}+d^{2} \leq 0\), then \(\left|\begin{array}{ccc}33 & 14 & \log a \\ 65 & 27 & \log b \\ 97 & 40 & \log c\end{array}\right|=\) (A) 1 (B) \(-1\) (C) 0 (D) None of these

The value of the determinant \(\left|\begin{array}{ccc}\sqrt{13}+\sqrt{3} & \sqrt[2]{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & \sqrt{10} \\ 3+\sqrt{65} & \sqrt{15} & 5\end{array}\right|\) is (A) \(-5 \sqrt{3}(5-\sqrt{6})\) (B) \(-5 \sqrt{3}(5+\sqrt{6})\) (C) \(-5 \sqrt{3}(\sqrt{6}-5)\) (D) None of these

If \(A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}\) and \(A_{3} B_{3} C_{3}\) are three three-digit numbers, each of which is divisible by \(k\), then \(\Delta=\left|\begin{array}{ccc}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|\) is (A) divisible by \(k\) (B) divisible by \(k^{2}\) (C) divisible by \(2 k\) (D) None of these

The value of the determinant of \(n\)th order, being given by \(\left|\begin{array}{cccc}x & 1 & 1 & \ldots \\ 1 & x & 1 & \ldots \\ 1 & 1 & x & \ldots \\ \ldots & \ldots & \ldots & \ldots\end{array}\right|\), is (A) \((x-1)^{n-1}(x+n-1)\) (B) \((x-1)^{n}(x+n-1)\) (C) \((1-x)^{n-1}(x+n-1)\) (D) None of thes

The value of the determinant \(\left|\begin{array}{ccc}\sqrt{x}+\sqrt{y} & 2 \sqrt{z} & \sqrt{z} \\ \sqrt{y z}+\sqrt{2 x} & z & \sqrt{2 z} \\ y+\sqrt{x z} & \sqrt{y z} & z\end{array}\right|\) where \(x, y, z\) are positive real numbers, is (A) \(z(\sqrt{2} y-z \sqrt{y)}\) (B) \(y(\sqrt{2} z-y \sqrt{z)}\) (C) \(x(\sqrt{2} y-z \sqrt{y})\) (D) None of these

If \(f_{j}=\sum_{i=0}^{2} a_{i j} x^{i}, j=1,2,3\) and if \(f_{j}^{\prime} f_{j}^{\prime \prime}\) denote \(\frac{d f_{j}}{d x}, \frac{d^{2} f_{j}}{d x^{2}}\) respectively, then \(g(x)=\left|\begin{array}{lll}f_{1} & f_{2} & f_{3} \\\ f_{1}^{\prime} & f_{2}^{\prime} & f_{3}^{\prime} \\ f_{1}^{\prime \prime} & f_{2}^{\prime \prime} & f_{3}^{\prime \prime}\end{array}\right|\) is (A) a cubic in \(x\) (B) a quadratic in \(x\) (C) linear in \(x\) (D) a constant

\((b+c)(y+z)-a x=b-c\), \((c+a)(z+x)-b y=c-a\), \((a+b)(x+y)-c z=a-b\) where \(a+b+c \neq 0\), then \(x=\) (A) \(\frac{c-b}{a+b+c}\) (B) \(\frac{a-c}{a+b+c}\) (C) \(\frac{b-a}{a+b+c}\) (D) \(\frac{1}{a+b+c}\)

If \(x \neq 0, y \neq 0, z \neq 0\) and \(\left|\begin{array}{ccc}1+x & 1 & 1 \\\ 1+y & 1+2 y & 1 \\ 1+z & 1+z & 1+3 z\end{array}\right|=0\), then \(x^{-1}+y^{-1}+z^{-1}\) is equal to (A) \(-1\) (B) \(-2\) (C) \(-3\) (D) None of these

If \(2 s=a+b+c\) and \(\left|\begin{array}{ccc}a^{2} & (s-a)^{2} & (s-a)^{2} \\\ (s-b)^{2} & b^{2} & (s-b)^{2} \\ (s-c)^{2} & (s-c)^{2} & c^{2}\end{array}\right|=\) \(k(s-a)(s-b)(s-c)\), then \(k\) is equal to (A) 2 (B) \(2 s\) (C) \(2 s^{2}\) (D) \(2 s^{3}\)

Let \(\alpha, \beta\) be the roots of the equation \(a x^{2}+b x+c=0\). Let \(s_{n}=\alpha^{n}+\beta^{n}\) for \(n \geq 1\). Then, the value of the determinant \(\left|\begin{array}{ccc}3 & 1+s_{1} & 1+s_{2} \\ 1+s_{1} & 1+s_{2} & 1+s_{3} \\ 1+s_{2} & 1+s_{3} & 1+s_{4}\end{array}\right|\) is (A) \(\frac{(a+b+c)\left(b^{2}-4 a c\right)}{a^{4}}\) (B) \(\frac{(a+b+c)^{2}\left(b^{2}-4 a c\right)}{a^{4}}\) (C) \(\frac{(a+b+c)^{2}\left(b^{2}-4 a c\right)}{a^{2}}\) (D) None of these

The value of the determinant \(\left|\begin{array}{ccc}a & b-c & c+b \\ \text { (A) } a^{2}+b^{2}+c^{2}\end{array}\right| \begin{array}{ccc}a+c & b & c-a \\\ a-b & a+b & c\end{array} \mid\) is (B) \(a b c(a+b+c)\) (C) \(\left(a^{2}+b^{2}+c^{2}\right)(a+b+c)\) (D) None of these

If \(\left|\begin{array}{ccc}\operatorname{cosec} \alpha & 1 & 0 \\ 1 & 2 \operatorname{cosec} \alpha & 1 \\ 0 & 1 & 2 \operatorname{cosec} \alpha\end{array}\right|=\frac{1}{2}\left(z^{3}+\frac{1}{z^{3}}\right)\) then \(z\) is equal to (A) \(\sin \alpha / 2\) (B) \(\cos \alpha / 2\) (C) \(\tan \alpha / 2\) (D) None of these

If \(\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \\ (a-1)^{2} & (b-1)^{2} & (c-1)^{2}\end{array}\right|=k(a-b)(b-c)\) \((c-a)\), then \(k\) is equal to (A) 4 (B) \(-4\) (C) 2 (D) \(-2\)

The value of the determinant \(\left|\begin{array}{ccc}\left(a-a_{1}\right)^{-2} & \left(a-a_{1}\right)^{-1} & a_{1}^{-1} \\ \left(a-a_{2}\right)^{-2} & \left(a-a_{2}\right)^{-1} & a_{2}^{-1} \\ \left(a-a_{3}\right)^{-2} & \left(a-a_{3}\right)^{-1} & a_{3}^{-1}\end{array}\right|\) is (A) \(\frac{a^{2} \Pi\left(a_{i}-a_{j}\right)}{\pi a_{i} \Pi\left(a-a_{i}\right)^{2}}\) (B) \(\frac{-a^{2} \Pi\left(a_{i}-a_{j}\right)}{\Pi a_{i} \Pi\left(a-a_{i}\right)^{2}}\) (C) \(\frac{\Pi a_{i} \Pi\left(a-a_{i}\right)^{2}}{a^{2} \Pi\left(a_{i}-a_{j}\right)}\) (D) \(-\frac{\Pi a_{i} \Pi\left(a-a_{i}\right)^{2}}{a^{2} \Pi\left(a_{i}-a_{j}\right)}\)

If \(\left|\begin{array}{ccc}\frac{1}{a+x} & \frac{1}{b+x} & \frac{1}{c+x} \\\ \frac{1}{a+y} & \frac{1}{b+y} & \frac{1}{c+y} \\ \frac{1}{a+z} & \frac{1}{b+z} & \frac{1}{c+z}\end{array}\right|=\frac{P}{Q}\), where \(Q\) is the product of denominators, then \(P\) is equal to (A) \((a-b)(b-c)(c-a)\) (B) \((x-y)(y-z)(z-x)\) (C) \((a-b)(b-c)(c-a)(x-y)(y-z)(z-x)\) (D) None of these

If \(a, b, c, d\) are the roots of the equation \(\alpha x^{4}+\beta x^{3}+\gamma x^{2}\) \(+\delta x+\xi=0\), then the value of the determinant \(\left|\begin{array}{cccc}1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1+d\end{array}\right|\) is (A) \(\frac{\delta-\gamma}{\alpha}\) (B) \(\frac{\xi-\delta}{\alpha}\) (C) \(\frac{\alpha-\beta}{\alpha}\) (D) \(\frac{\beta-\alpha}{\alpha}\)

The value of the determinant \(\left|\begin{array}{cccc}0 & x & y & z \\ -x & 0 & c & b \\ -y & -c & 0 & a \\ -z & -b & -a & 0\end{array}\right|\) is (A) \((a x+b y+c z)^{2}\) (B) \((a x-b y+c z)^{2}\) (C) \((a x+b y-c z)^{2}\) (D) None of these