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Questions >> JEEMAIN and NEET >> Mathematics >> Class12 >> Determinants

If a square matrix A is such that $AA^T=I=A^TA$ then $|A|$ is equal to

answered Apr 25, 2014 by sreemathi.v

1 answer

If $A$ is square matrix of order $n\times n$ then adj(adj(A)) is equal to

answered Apr 25, 2014 by sreemathi.v

1 answer

The number of non trivial solutions of the system $x-y+z=0,x+2y-z=0,2x+y+3z=0$ is

answered Apr 25, 2014 by sreemathi.v

1 answer

If $ax^3+bx^2+cx+d=\begin{vmatrix}x^2&(x-1)^2&(x-2)\\(x-)^2&(x-2)^2&(x-3)^2\\(x-2)^2&(x-3)^2&(x-4)^2\end{vmatrix}$ then

answered Apr 25, 2014 by sreemathi.v

1 answer

The values of $\lambda$ and $\mu$ for which the equations $x+y+z=3,x+3y+2z=6,x+\lambda y+3z=\mu$ have

answered Apr 25, 2014 by sreemathi.v

1 answer

If $1+\sin x+\cos x\neq 0$ the value of $x$ for which $\begin{vmatrix}1&\sin x&\cos x\\\sin x&1&\cos x\\\cos x&\sin x&1\end{vmatrix}=0$ is

answered Apr 25, 2014 by sreemathi.v

1 answer

If $f(x)=\begin{vmatrix}\cos x&1&0\\1&2\cos x&1\\0&1&2\cos x\end{vmatrix}$ then $\int\limits_0^{\large\frac{\pi}{2}}2f(x)dx$ is equal to

answered Apr 25, 2014 by sreemathi.v

1 answer

If $A=\begin{vmatrix}a&b&c\\x&y&z\\p&q&r\end{vmatrix}$ and $B=\begin{vmatrix}q&-b&y\\-p&a&-x\\r&-c&z\end{vmatrix}$ then

answered Apr 24, 2014 by sreemathi.v

1 answer

If $A$ is a square matrix of order $n\times n$ then adj.(adj A) is equal to

answered Apr 24, 2014 by sreemathi.v

1 answer

If $D_r=\begin{vmatrix}2^{r-1}&2.3^{r-1}&4.5^{r-1}\\\alpha&\beta&\gamma\\2^n-1&3^n-1&5^n-1\end{vmatrix}$ then the value of $\sum\limits_{r=1}^n D_r$ is

answered Apr 24, 2014 by sreemathi.v

1 answer

If the system of linear equations : $x+2ay+az=0,x+3by+bz=0,x+4cy+cz=0$ has a zero solutions then $a,b,c$

answered Apr 24, 2014 by sreemathi.v

1 answer

Value of the determinant $\begin{vmatrix} 10!&11!&12!\\11!&12!&13!\\12!&13!&14!\end{vmatrix}$ is

answered Apr 24, 2014 by sreemathi.v

1 answer

The value of $\theta$ in the first quadrant satisfying the equation $\begin{vmatrix}1+\cos^2\theta&\sin^2\theta&4\sin 4\theta\\\cos^2\theta&1+\sin^2\theta&4\sin4\theta\\\cos^2\theta&\sin^2\theta&1+4\sin 4\theta\end{vmatrix}=0$ is

answered Apr 23, 2014 by sreemathi.v

1 answer

If $C=2\cos \theta$ then the value of the determinant $4\Delta=\begin{vmatrix}c&1&0\\1&c&1\\0&1&c\end{vmatrix}$ is

answered Apr 23, 2014 by sreemathi.v

1 answer

If $A=\begin{bmatrix}1&2\\3&-5\end{bmatrix}$ then $A^{-1}$ =

answered Apr 23, 2014 by sreemathi.v

1 answer

If $x^ay^b=e^m,x^cy^d=e^n$ $\Delta_1=\begin{vmatrix}m&b\\n&d\end{vmatrix}$ , $\Delta_2=\begin{vmatrix}a&m\\c&n\end{vmatrix}$ , $\Delta_3=\begin{vmatrix}a&b\\c&d\end{vmatrix}$ then the values of $x$ and $y$ are respectively

answered Apr 23, 2014 by sreemathi.v

1 answer

If 5 is one root of the equation $\begin{vmatrix}x&3&7\\2&x&-2\\7&8&x\end{vmatrix}=0$ then the other two roots of the equation are

answered Apr 23, 2014 by sreemathi.v

1 answer

If $f(x)=\begin{vmatrix}\sin x&\cos x&\tan x\\x^3&x^2&x\\2x&1&1\end{vmatrix}$ then $\lim\limits_{x\to 0}\large\frac{f(x)}{x^2}$ is

answered Apr 23, 2014 by sreemathi.v

1 answer

The number of solutions of the system of equations $2x+y-z=7,x-3y+2z=1,x+4y-3z=5$ is

answered Apr 23, 2014 by sreemathi.v

1 answer

If $\omega$ is cube root of unity then $\Delta=\begin{vmatrix}x+1&\omega&\omega^2\\\omega&x+\omega^2&1\\\omega^2&1&x+\omega^2\end{vmatrix}=$

answered Apr 23, 2014 by sreemathi.v

1 answer

If $A=\begin{bmatrix}1&\tan\large\frac{\theta}{2}\\-\tan\large\frac{\theta}{2}&1\end{bmatrix}$ and $AB=I$ then $B$ =

answered Apr 23, 2014 by sreemathi.v

1 answer

The value of a for which the system of equations $a^3x+(a+1)^3y+(a+2)^3z=0,ax+(a+1)y+(a+2)z=0,x+y+z=0$ has a non-zero solution is

answered Apr 23, 2014 by sreemathi.v

1 answer

If A is singular matrix then adj.A is

answered Apr 23, 2014 by sreemathi.v

1 answer

Suppose $D=\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}$ and $D'=\begin{vmatrix}a_1+pb_1&b_1+qc_1&c_1+ra_1\\a_2+pb_2&b_2+qc_2&c_2+ra_2\\a_3+pb_3&b_3+qc_3&c_3+ra_3\end{vmatrix}$ then

answered Apr 23, 2014 by sreemathi.v

1 answer

From the matrix equation $AB=AC$ we can conclude $B=C$ provided.

answered Apr 23, 2014 by sreemathi.v

1 answer

If $\alpha,\beta,\gamma$ are real numbers then, $\begin{vmatrix} 1& \cos(\beta-\alpha)&\cos(\gamma-\alpha)\\\cos(\alpha-\beta)&1&\cos(\gamma-\beta)\\\cos(\alpha-\gamma)&\cos(\beta-\gamma)&1\end{vmatrix}$ is equal to

answered Apr 23, 2014 by sreemathi.v

1 answer

If $\Delta_r=\begin{vmatrix}2^{r-1}&\large\frac{(r+1)!}{1+\large\frac{1}{r}}&2r\\a&b&c\\2^n-1&(n+1)!-1&n(n+1)\end{vmatrix}$ then value of $\sum\limits_{r=1}^n\Delta_r$ is

answered Apr 23, 2014 by sreemathi.v

1 answer

If $\Delta=\begin{vmatrix} 1+x_1y_1&1+x_1y_2&1+x_1y_3\\1+x_2y_1&1+x_2y_2&1+x_2y_3\\1+x_3y_1&1+x_3y_2&1+x_3y_3\end{vmatrix}$ then $\Delta$ is

answered Apr 22, 2014 by sreemathi.v

1 answer

If $f(x)=\begin{vmatrix}1&a&a^2\\\sin(n-1) x&\sin nx&\sin(n+1)x\\\cos(n-1)x&\cos nx&\cos(n+1)x\end{vmatrix}$ then $\int_0^{\large\frac{\pi}{2}}f(x)dx$ is equal to

answered Apr 22, 2014 by sreemathi.v

1 answer

If the system of equations $x+ay+az=0,bx+y+bz=0$ and $cx+cy+z=0$ where $a,b,c$ are non-zero ,non unity has a non-trivial solution then the value of $\large\frac{a}{1-a}+\frac{b}{1-b}+\frac{c}{1-c}$ is

answered Apr 22, 2014 by sreemathi.v

1 answer

In a triangle ABC $\begin{vmatrix} a^2& b\sin A & C\sin A\\b\sin A&1&\cos(B-C)\\C\sin A&\cos (B-C) & 1\end{vmatrix}$ equals

answered Apr 22, 2014 by sreemathi.v

1 answer

If $A,B,C$ are angles of a triangle and $\begin{vmatrix} 1&1&1\\1+\sin A&1+\sin B&1+\sin C\\\sin A+\sin ^2A&\sin B+\sin^2B& \sin C+\sin^2C\end{vmatrix}=0$

answered Apr 22, 2014 by sreemathi.v

1 answer

If $\alpha,\beta,\gamma$ are such that $\alpha+\beta+\gamma=0$ then $\begin{vmatrix}1&\cos \gamma&\cos \beta\\\cos \alpha& 1&\cos\alpha\\\cos \beta&\cos \alpha&1\end{vmatrix}$ is equal to

answered Apr 22, 2014 by sreemathi.v

1 answer

If $F(\alpha)=\begin{bmatrix}\cos \alpha&-\sin \alpha&0\\\sin \alpha&\cos \alpha&0\\0&0&1\end{bmatrix}$ $\alpha \in R$ then $[F(\alpha)]^{-1}$ is equal to

answered Apr 22, 2014 by sreemathi.v

1 answer

The value of $\begin{vmatrix}1&a&a^2-bc\\1&b& b^2-ca\\1&c&c^2-ab\end{vmatrix}$ is

answered Apr 22, 2014 by sreemathi.v

1 answer

If $A=\begin{bmatrix}\large\frac{1}{a}&a^2&bc\\\large\frac{1}{b}&b^2&ca\\\large\frac{1}{c}&c^2&ab\end{bmatrix}$ then $|A|$ is

answered Apr 22, 2014 by sreemathi.v

1 answer

If $\alpha, \beta \neq 0$ , and $f(n) = \alpha^n+\beta^n$ and $\begin{vmatrix} 3 & 1+f(1) & 1+f(2)\\ 1+f(1)& 1+f(2) &1+f(3) \\ 1+f(2)&1+f(3) & 1+f(4) \end{vmatrix}$ $=K(1-\alpha)^2\;(1-\beta)^2\;(\alpha-\beta)^2$ , then $K$ is equal to

asked Apr 6, 2014 by balaji.thirumalai

0 answers

If $A$ is an $3\times 3$ non-singular matrix such that $A A' = A' A$ and $B = A^{-1} A'$ , then $B B'$ equals

asked Apr 6, 2014 by balaji.thirumalai

0 answers

If $\Delta=\begin{vmatrix}x^n&x^{n+2}&x^{2n}\\1&x^p&p\\x^{n+5}&x^{p+6}&x^{2n+5}\end{vmatrix}=0$ then $p$ is given by

answered Nov 26, 2013 by sreemathi.v

1 answer

Find the value of determinant $\begin{vmatrix}1&x&y+z\\1&y&z+x\\1&z&x+y\end{vmatrix}$

answered Nov 26, 2013 by sreemathi.v

1 answer

If the system of equations $x+2y-3z=1$ , $(p+2)z=3$ , $(2p+1)y+z=2$ is inconsistent, then the value of $p$ is

answered Nov 26, 2013 by sreemathi.v

1 answer

If the system of equations $ax+y+z=0$ , $x+by+z=0$ , $x+y+cz=0$ , where $(a,b,c\neq 1)$ has a non-trivial solutions, then the value of $\large\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

answered Nov 26, 2013 by sreemathi.v

1 answer

If $A,B,C$ are the angles of a $\Delta$ le then the value of the determinant $\small\begin{vmatrix}-1+\cos B&\cos C+\cos B&\cos B\\\cos C+\cos A&-1+\cos A&\cos A\\-1+\cos B&-1+\cos B&-1\end{vmatrix}$ is

answered Nov 25, 2013 by sreemathi.v

1 answer

For all values of $A,B,C$ and $P,Q,R$ the value of the determinant

$(x+a)^3\small\begin{vmatrix}\cos(A-P)&\cos (A-Q)&\cos(A-R)\\\cos(B-P)&\cos(B-Q)&\cos(B-R)\\\cos(C-P)&\cos(C-Q)&\cos(C-R)\end{vmatrix}$ is

answered Nov 25, 2013 by sreemathi.v

1 answer

If $A^k=0$ for some value of k. $(I-A)^P=I+A+A^2+....+A^{k-1}$ thus P is

answered Nov 25, 2013 by sreemathi.v

1 answer

If $A,B$ and $C$ are the angles of a $\Delta$ le then $\begin{vmatrix}\sin 2A&\sin C&\sin B\\\sin C&\sin 2B&\sin A\\\sin B&\sin A&\sin 2C\end{vmatrix}$ = $\lambda\sin A\sin B\sin C$ , then $\lambda$ is

answered Nov 25, 2013 by sreemathi.v

1 answer

If $A=\begin{bmatrix}3&4\\2&4\end{bmatrix},B=\begin{bmatrix}-2&-2\\0&-2\end{bmatrix}$ then $(A+B)^{-1}$ is equal to

answered Nov 25, 2013 by sreemathi.v

1 answer

If $\alpha,\beta,\gamma$ are the cube roots of unity,then the value of the determinant $\begin{vmatrix}e^{\alpha}&e^{2\alpha}&e^{3\alpha}-1\\e^{\beta}&e^{2\beta}&e^{3\beta}-1\\e^{\gamma}&e^{2\gamma}&e^{3\gamma}-1\end{vmatrix}$ is equal to

answered Nov 25, 2013 by sreemathi.v

1 answer

If $A,B,C$ are angles of a triangle then $\begin{vmatrix}e^{2iA}&e^{-iC}&e^{-iB}\\e^{-iC}&e^{2iB}&e^{-iA}\\e^{-iB}&e^{-iA}&e^{2iC}\end{vmatrix}$ is equal to

answered Nov 25, 2013 by sreemathi.v

1 answer

If $E(\theta)=\begin{bmatrix}\cos^2\theta&\cos \theta\sin \theta\\\cos \theta\sin\theta&\sin^2\theta\end{bmatrix}$ and $\theta$ and $\phi$ differ by an odd multiple of $\large\frac{\pi}{2}$ then $E(\theta)E(\phi)$ is a

answered Nov 25, 2013 by sreemathi.v

1 answer

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