Chapter 6

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

The value of the determinant is \(\left|\begin{array}{ccc}\beta \gamma & \beta \gamma^{\prime}+\beta^{\prime} \gamma & \beta^{\prime} \gamma^{\prime} \\\ \gamma \alpha & \gamma \alpha^{\prime}+\gamma^{\prime} \alpha & \gamma^{\prime} \alpha^{\prime} \\ \alpha \beta & \alpha \beta^{\prime}+\alpha^{\prime} \beta & \alpha^{\prime} \beta^{\prime}\end{array}\right|\) (A) \(\left(\alpha \beta^{\prime}-\alpha^{\prime} \beta\right)\left(\beta \gamma^{\prime}-\beta^{\prime} \gamma\right)\left(\gamma \alpha^{\prime}-\gamma^{\prime} \alpha\right)\) (B) \(\alpha \beta \gamma(\alpha+\beta+\gamma)\left(\alpha^{\prime}+\beta^{\prime}+\gamma^{\prime}\right)\) (C) \(\alpha^{\prime} \beta^{\prime} \gamma^{\prime}(\alpha+\beta+\gamma)\left(\alpha^{\prime}+\beta^{\prime}+\gamma\right)\) (D) None of these

If \(a \neq 0, a \neq 1\) and \(\left|\begin{array}{ccc}x+1 & x & x \\ x & x+a & x \\ x & x & x+a^{2}\end{array}\right|=a^{3}+f(x) \cdot a\left(a^{2}+a+1\right)\), then (A) \(f(x)=x\) (B) \(f(x)=x^{2}\) (C) \(f(x)=x^{3}\) (D) None of these

The value of the determinant \(\left|\begin{array}{ccc}-b c & b^{2}+b c & c^{2}+b c \\ a^{2}+a c & -a c & c^{2}+a c \\ a^{2}+a b & b^{2}+a b & -a b\end{array}\right|\) is (A) \(\left(a^{2}+b^{2}+c^{2}\right)^{3}\) (B) \((a b+b c+c a)^{3}\) (C) \(\left(a^{2}+b^{2}+c^{2}\right)(a b+b c+c a)^{2}\) (D) None of these

If \(\left|\begin{array}{ccc}x+a^{2} & a b & a c \\ a b & x+b^{2} & b c \\ a c & b c & x+c^{2}\end{array}\right|=0\) and \(x(\neq 0) \in R\) then \(x\) is equal to (A) \(a^{2}+b^{2}+c^{2}\) (B) \(-\left(a^{2}+b^{2}+c^{2}\right)\) (C) \(2\left(a^{2}+b^{2}+c^{2}\right)\) (D) None of these

The values of \(m\) for which the system of equations \(3 x+m y=m\) and \(2 x-5 y=20\) has a solution satisfying the condition \(x>0, y>0\), are (A) \(m \in\left(-\infty, \frac{-15}{2}\right) \cup(0, \infty)\) (B) \(m \in\left(-\infty, \frac{-15}{2}\right) \cup(30, \infty)\) (C) \(m \in\left(-\infty, \frac{-15}{2}\right) \cup(0,30)\) (D) None of these

If \(a=\cos \theta+i \sin \theta, b=\cos 2 \theta-i \sin 2 \theta, c=\cos 3\) \(\theta+i \sin 3 \theta\) and if \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0\) then \(\theta\) is equal to (A) \(\overline{n \pi}\) (B) \(2 n \pi\) (C) \((2 n+1) \frac{\pi}{2}\) (D) None of these

The value of the determinant \(\left|\begin{array}{ccc}\frac{1}{a} & \frac{1}{a(a+d)} & \frac{1}{(a+d)(a+2 d)} \\ \frac{1}{a+d} & \frac{1}{(a+d)(a+2 d)} & \frac{1}{(a+2 d)(a+3 d)} \\ \frac{1}{a+2 d} & \frac{1}{(a+2 d)(a+3 d)} & \frac{1}{(a+3 d)(a+4 d)}\end{array}\right|\) where \(a, d>0\), is (A) \(-\frac{4 d^{4}}{a(a+d)^{2}(a+2 d)^{3}(a+3 d)^{2}(a+4 d)}\) (B) \(\frac{4 d^{4}}{a(a+d)^{2}(a+2 d)^{3}(a+3 d)^{2}(a+4 d)}\) (C) \(\frac{4 d^{4}}{a(a+d)^{2}(a+2 d)^{3}(a+3 d)^{2}(a+4 d)^{2}}\) (D) None of these

The value of the determinant \(\left|\begin{array}{ccc}(b+c)^{2} & c^{2} & b^{2} \\ c^{2} & (c+a)^{2} & a^{2} \\ b^{2} & a^{2} & (a+b)^{2}\end{array}\right|\) is (A) \(2(a b+b c+c a)^{3}\) (B) \((a b+b c+c a)^{3}\) (C) \(4(a b+b c+c a)^{3}\) (D) None of these

If the equations \((a+1)^{3} x+(a+2)^{3} y=(a+3)^{3},(a+1) x+(a+2) y\) \(=a+3, x+y=1\) are consistent then \(a\) is equal to (A) 1 (B) \(-1\) (C) 2 (D) \(-2\)

If the system of equations \(x \sin \alpha+y \sin \beta+z \sin \gamma=0, x \cos \alpha+y \cos \beta+z \cos \gamma\) \(=0, x+y+z=0\), where \(\alpha, \beta, \gamma\) are angles of a triangle, have a non-trivial solution, then the triangle must be (A) isosceles (B) equilateral (C) right angled (D) None of these

If \(x_{1} \neq 0, x_{2} \neq 0, x_{3} \neq 0\), then the determinant \(\left|\begin{array}{ccc}x_{1}+a_{1} b_{1} & a_{1} b_{2} & a_{1} b_{3} \\\ a_{2} b_{1} & x_{2}+a_{2} b_{2} & a_{2} b_{3} \\ a_{3} b_{1} & a_{3} b_{2} & x_{3}+a_{3} b_{3}\end{array}\right|\) is equal to (A) \(x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)\) (B) \(-x_{1} x_{2} x_{3}\left(1+\frac{a_{1} b_{1}}{x_{1}}+\frac{a_{2} b_{2}}{x_{2}}+\frac{a_{3} b_{3}}{x_{3}}\right)\) (C) \(x_{1} x_{2} x_{3}\left(1-\frac{a_{1} b_{1}}{x_{1}}-\frac{a_{2} b_{2}}{x_{2}}-\frac{a_{3} b_{3}}{x_{3}}\right)\) (D) None of these

If \(\left|\begin{array}{ccc}a & a+d & a+2 d \\ a^{2} & (a+d)^{2} & (a+2 d)^{2} \\ 2 a+3 d & 2(a+d) & 2 a+d\end{array}\right|=0\), then (A) \(a+d=0\) (B) \(d=0\) (C) \(d=0\) or \(a+d=0\) (D) None of these

Let \(\left|\begin{array}{ccc}x+3 & x+2 & (x+2)^{3} \\ x+2 & x+3 & (x+2)^{3} \\\ (x+2)^{3} & x+2 & x+3\end{array}\right|\) \(=a x^{7}+b x^{6}+c x^{5}+d x^{4}+e x^{3}+f x^{2}+g x+h\) be an iden- tity in \(x\), where \(a, b, c, d, e, f, g, h\) are independent of \(x\), then the value of \(g\) is (A) \(-213\) (B) 213 (C) 0 (D) None of these

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

The value of the determinant \(\left|\begin{array}{ccc}\sin \alpha \cos \beta & \cos \alpha \cos \beta & -\sin \alpha \sin \beta \\ \sin \alpha \sin \beta & \cos \alpha \sin \beta & \sin \alpha \cos \beta \\ \cos \alpha & -\sin \alpha & 0\end{array}\right|\) is (A) is independent of \(\alpha\) (B) independent of \(\beta\) (C) independent of \(\alpha\) and \(\beta\) (D) None of these

If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^{3}+p x+q=0\), then the value of the determinant \(\left|\begin{array}{ccc}1+\alpha & 1 & 1 \\ 1 & 1+\beta & 1 \\ 1 & 1 & 1+\gamma\end{array}\right|\) is (A) \(p^{2}-2 q\) (B) \(3 p q\) (C) \(p-q\) (D) None of these

The value of a determinant of third order whose all elements are 1 or \(-1\) is (A) an even number (B) an odd number (C) a prime number (D) cannot be determined

If square matrices \(A\) and \(B\) are such that \(A A^{\theta}=A^{\theta} A\), \(B B^{\theta}=B^{\theta} B\) and \(A B^{\theta}=B^{\theta} A\), then \((A B)(A B)^{\theta}\) is equal to (A) \(B^{\theta} A^{\theta} A B\) (B) \(B A^{\theta} A B\) (C) \(B A^{\theta} A B^{\theta}\) (D) None of these

Let \(\Delta(x)=\left|\begin{array}{ccc}x & 2 & x \\ x^{2} & x & 6 \\ x & x & 6\end{array}\right|=A x^{4}+B x^{3}+C x^{2}+D x+E\)

If \(\Delta_{1}=\) \(\left|\begin{array}{ccc}y^{5} z^{6}\left(z^{3}-y^{3}\right) & x^{4} z^{6}\left(x^{3}-z^{3}\right) & x^{4} z^{5}\left(y^{3}-x^{3}\right) \\ y^{2} z^{3}\left(y^{6}-z^{6}\right) & x z^{3}\left(z^{6}-x^{6}\right) & x y^{2}\left(x^{6}-y^{6}\right) \\ y^{2} z^{3}\left(z^{3}-y^{3}\right) & x z^{3}\left(x^{3}-z^{3}\right) & x y^{2}\left(y^{3}-x^{3}\right)\end{array}\right|\) and, \(\Delta_{2}=\left|\begin{array}{ccc}x & y^{2} & z^{3} \\ x^{4} & y^{5} & z^{6} \\ x^{7} & y^{8} & z^{9}\end{array}\right|\), then \(\Delta_{1} \Delta_{2}=\) (A) \(\Delta_{22}^{2}\) (C) \(\Delta_{2}^{4}\) (B) \(\Delta_{2}^{3}\) (D) None of these

If \(a, b, c\) are the sides of a triangle \(A B C\) such that \(\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \\ (a-1)^{2} & (b-1)^{2} & (c-1)^{2}\end{array}\right|=0\), then \(\Delta A B C\) is (A) a right angled triangle (B) an isosceles triangle (C) an equilateral triangle (D) None of these

The set of equations : \(\lambda x-y+(\cos \theta) z=0 ; 3 x+y+2 z\) \(=0 ;(\cos \theta) x+y+2 z=0,0 \leq \theta<2 \pi\), has non-trivial solutions. (A) for no values of \(\lambda\) and \(\theta\) (B) for all values of \(\lambda\) and \(\theta\) (C) for all values of \(\lambda\) and only two values of \(\theta\) (D) for only one value of \(\lambda\) and all values of \(\theta\)

The value of \(\lambda\) for which the equations \(x+y-3=0\), \((1+\lambda) x+(2+\lambda) y-8=0, x-(1+\lambda) y+(2+\lambda)=0\) are consistent is (A) 1 (B) \(5 / 3\) (C) \(-5 / 3\) (D) None of these

If \(A+B+C=\pi, e^{i \theta}=\cos \theta+i \sin \theta\) 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 \(\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=A x+B\), then (A) \(A=\left|\begin{array}{lll}4 & 0 & 0 \\ 2 & 3 & 3 \\ 4 & 0 & 2\end{array}\right|\) (B) \(B=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & 3 \\ 4 & 0 & -1\end{array}\right|\) (C) \(A=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & -3 \\ 4 & 0 & 2\end{array}\right|\) (D) \(B=\left|\begin{array}{ccc}4 & 0 & 0 \\ 2 & 3 & -3 \\ 4 & 0 & -1\end{array}\right|\)

5\. If \(\left|\begin{array}{ccc}0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0\end{array}\right|=0, a \neq b \neq c\), then (A) \(x=0\) if \(b(a+c) \leq a c\) (B) \(x=\) ? \(\sqrt{b(a+c)-a c}\) if \(b(a+c) \geq a c\) (C) \(x=0, \pm \sqrt{b(a+c)-a c}\) if \(b(a \neq c)>a c\) (D) None of these

If \(\left|\begin{array}{ccc}b c-a^{2} & c a-b^{2} & a b-c^{2} \\ c a-b^{2} & a b-c^{2} & b c-a^{2} \\ a b-c^{2} & b c-a^{2} & c a-b^{2}\end{array}\right|=\left|\begin{array}{ccc}\alpha^{2} & \beta^{2} & \beta^{2} \\ \beta^{2} & \alpha^{2} & \beta^{2} \\ \beta^{2} & \beta^{2} & \alpha^{2}\end{array}\right|\) then (A) \(\alpha^{2}=a^{2}+b^{2}+c^{2}\) (B) \(\beta^{2}=a b+b c+c a\) (C) \(\alpha^{2}=a b+b c+c a\) (D) \(\beta^{2}=a^{2}+b^{2}+c^{2}\)

The determinant \(\left|\begin{array}{ccc}\sin x & \sin y & \sin z \\ \cos x & \cos y & \cos z \\ \cos ^{3} x & \cos ^{3} y & \cos ^{3} z\end{array}\right| ; 0

The value of the determinant \(\left|\begin{array}{ccc}\cos (\theta+\alpha) & -\sin (\theta+\alpha) & \cos 2 \alpha \\ \sin \theta & \cos \theta & \sin \alpha \\ -\cos \theta & \sin \theta & \lambda \cos \alpha\end{array}\right|\) is (A) independent of \(\theta\) for all \(\lambda \in \mathrm{R}\) (B) independent of \(\theta\) and \(\alpha\) when \(\lambda=1\) (C) independent of \(\theta\) and \(\alpha\) when \(\lambda=-1\) (D) None of these

The value of \(\theta\) lying between \(\theta=0\) and \(\theta=\frac{\pi}{2}\) and satisfying the equation \(\left|\begin{array}{ccc}1+\sin ^{2} \theta & \cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & 1+\cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & \cos ^{2} \theta & 1+4 \sin 4 \theta\end{array}\right|=0\) is (A) \(\frac{7 \pi}{24}\) (B) \(\frac{5 \pi}{24}\) (C) \(\frac{11 \pi}{24}\) (D) \(\frac{\pi}{24}\)

If \(a_{n}=\int_{0}^{\pi / 2} \frac{1-\cos 2 n x}{1-\cos 2 x} d x\), then (A) \(a_{n+1}\) is A.M. between \(a_{n}\) and \(a_{n+2}\) (B) \(a_{n+1}\) is G.M between \(a_{n}\) and \(a_{n+2}\) (C) \(a_{n+1}\) is H.M. between \(a_{n}\) and \(a_{n+2}\) (D) \(\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\\ a_{7} & a_{8} & a_{9}\end{array}\right|=0\)

If \(\alpha, \beta, \gamma\) are non-zero real numbers such that \(\left|\begin{array}{ccc}\beta \gamma & \gamma \alpha & \alpha \beta \\\ \gamma \alpha & \alpha \beta & \beta \gamma \\ \alpha \beta & \beta \gamma & \gamma \alpha\end{array}\right|=0\), then (A) \(\frac{1}{\gamma}+\frac{1}{\alpha \omega}+\frac{1}{\beta \omega^{2}}=0\) (B) \(\frac{1}{\beta}+\frac{1}{\alpha \omega}+\frac{1}{\gamma \omega^{2}}=0\) (C) \(\frac{1}{\beta}+\frac{1}{\gamma \omega}+\frac{1}{\alpha \omega^{2}}=0\) (D) \((\alpha \beta)^{3}+(\beta \gamma)^{3}+(\gamma \alpha)^{3}=3 \alpha^{2} \beta^{2} \gamma^{2}\)

If \(f(x)=\left|\begin{array}{ccc}e^{x} & \sin x & 1 \\ \cos x & \log \left(1+x^{2}\right) & 1 \\ x & x^{2} & 1\end{array}\right|=a+b x+c x^{2}\), then (A) \(a=0\) (B) \(a=1\) (C) \(b=-1\) (D) \(b=-2\)

If maximum and minimum values of the determinant \(\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & \sin 2 x \\ \sin ^{2} x & 1+\cos ^{2} x & \sin 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+\sin 2 x\end{array}\right|\) are \(\alpha\) and \(\beta\), then (A) \(\alpha+\beta^{99}=4\) (B) \(\alpha^{3}-\beta^{17}=26\) (C) \(\left(\alpha^{2 n}-\beta^{2 n}\right)\) is always an even integer for \(n \in N\) (D) a triangle can be constructed having its sides as \(\alpha-\beta, \alpha+\beta\) and \(\alpha+3 \beta\)

Let \(A=\left[a_{i j}\right.\) be an \(n \times n\) matrix. The matrix \(A-\lambda I\) is called the characteristics matrix of \(A\), where \(\lambda\) is a scalar and \(I\) is the identity matrix. The determinant \(|A-\lambda I|\) is a non-null polynomial of degree \(n\) in \(\lambda\) and is called the characteristic polynomial of \(A\). The equation \(|A-\lambda I|=0\) is called the characteristic equation of \(A\) and its roots are called the characteristic roots or latent roots or eigen values of \(A\). The set of all eigenvalues of the matrix \(A\) is called the spectrum of A. The product of the eigenvalues of a matrix \(A\) is equal to the determinant \(A\). \(\left.\begin{array}{l}\text { The characteristic roots of the matrix } A=\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 1 & 2 \\ 1 & 2 & 0\end{array}\right]\end{array}\right]\) (A) 1 (B) 2 (C) \(-2\) (D) 3

Let \(A=\left[a_{i j}\right.\) be an \(n \times n\) matrix. The matrix \(A-\lambda I\) is called the characteristics matrix of \(A\), where \(\lambda\) is a scalar and \(I\) is the identity matrix. The determinant \(|A-\lambda I|\) is a non-null polynomial of degree \(n\) in \(\lambda\) and is called the characteristic polynomial of \(A\). The equation \(|A-\lambda I|=0\) is called the characteristic equation of \(A\) and its roots are called the characteristic roots or latent roots or eigen values of \(A\). The set of all eigenvalues of the matrix \(A\) is called the spectrum of A. The product of the eigenvalues of a matrix \(A\) is equal to the determinant \(A\). The given values of the matrix \(A=\left[\begin{array}{lll}1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4\end{array}\right]\) are (A) \(4,-2,-2\), (B) \(-4,2,-2\) (C) \(-4,2,2\) (D) \(4,-4,2\)

Let \(A=\left[a_{i j}\right.\) be an \(n \times n\) matrix. The matrix \(A-\lambda I\) is called the characteristics matrix of \(A\), where \(\lambda\) is a scalar and \(I\) is the identity matrix. The determinant \(|A-\lambda I|\) is a non-null polynomial of degree \(n\) in \(\lambda\) and is called the characteristic polynomial of \(A\). The equation \(|A-\lambda I|=0\) is called the characteristic equation of \(A\) and its roots are called the characteristic roots or latent roots or eigen values of \(A\). The set of all eigenvalues of the matrix \(A\) is called the spectrum of A. The product of the eigenvalues of a matrix \(A\) is equal to the determinant \(A\). Which of the following statements are true? If \(A\) is any \(n \times n\) matrix and \(\lambda\) is a characteristic root of \(A\), then (A) \(A\) and \(A^{\prime}\) have the same characteristic roots (B) \(k \lambda\) is a characteristic root of \(k A\) ( \(k\) being scalar) (C) \(\lambda^{n}\) is a characteristic root of \(A^{n}\) ( \(n\) being positive integer) (D) \(\frac{1}{\lambda}\) is a characteristic root of \(A^{-1}\)

Let \(A=\left[a_{i j}\right.\) be an \(n \times n\) matrix. The matrix \(A-\lambda I\) is called the characteristics matrix of \(A\), where \(\lambda\) is a scalar and \(I\) is the identity matrix. The determinant \(|A-\lambda I|\) is a non-null polynomial of degree \(n\) in \(\lambda\) and is called the characteristic polynomial of \(A\). The equation \(|A-\lambda I|=0\) is called the characteristic equation of \(A\) and its roots are called the characteristic roots or latent roots or eigen values of \(A\). The set of all eigenvalues of the matrix \(A\) is called the spectrum of A. The product of the eigenvalues of a matrix \(A\) is equal to the determinant \(A\). Which of the following statements are correct? (A) If \(A, B\) are \(n\) rowed square matrices and \(A\) is non-singular, then \(A^{-1} B\) and \(B A^{-1}\) has same character-istic roots. (B) If \(A\) and \(P\) are square matrices of same order and \(P\) is non-singular, then \(A\) and \(P^{-1} A P\) have same characteristic roots. (C) If \(A\) and \(B\) be two square matrices of same order, then \(A B\) and \(B A\) have same characteristic roots. (D) All of these

If \(\left|\begin{array}{ccc}1+x & x & x^{2} \\ x & 1+x & x^{2} \\ x^{2} & x & 1+x\end{array}\right|=p x^{5}+q x^{4}+r x^{3}+s x^{2}+t x+w\), then $$ \begin{array}{ll} \hline \text { Column-I } & \text { Column-II } \\ \hline \text { I. } w \text { is equal to } & \text { (A) } 3 \\ \text { II. } t \text { is equal to } & \text { (B) } 1 \\ \text { III. } p+r \text { is equal to } & \text { (C) }-1 \\ \text { IV. } q+s \text { is equal to } & \text { (D) } 0 \\ \hline \end{array} $$