Tag: mathematics and statistics

If $\Delta =\begin{vmatrix} a _{11} & a _{12} & a _{13}\ a _{21} & a _{22} & a _{23}\ a _{31} & a _{32} & a _{33} \end{vmatrix}$ and $c _{ij}=\left ( -1 \right )^{i+j}$ (determinant obtained by deleting ith row and jth column), then $\begin{vmatrix} c _{11} & c _{12} & c _{13}\ c _{21} & c _{22} & c _{23}\ c _{31} & c _{32} & c _{33} \end{vmatrix}=\Delta ^{2}$

If $\begin{vmatrix} 1 & x & x^{ 2 } \ x & x^{ 2 } & 1 \ x^{ 2 } & 1 & x \end{vmatrix}=7$ and $\Delta =\begin{vmatrix}
x^{3}-1 & 0 & x-x^{4}\
0 & x-x^{4} & x^{3}-1\
x-x^{4} & x^{3}-1 & 0
\end{vmatrix}$, then

Let $\Delta _0=\begin{bmatrix}a _{11} & a _{12}  & a _{13}\a _{21}  & a _{22} &a _{23} \ a _{31} & a _{32} & a _{33}\end{bmatrix}$ (where $\Delta _0 \neq  0$) and let $\Delta _1$ denote the determinant formed by the cofactors of elements of $\Delta _0$ and $\Delta _2$ denote the determinant formed by the cofactor at $\Delta _1$ and so on $\Delta _n$ denotes the determinant formed by the cofactors at $\Delta _{n-1}$ then the determinant value of $\Delta _{n}$ is

Circle on which the coordinates of any point are $(2+4 \cos \theta,-1+4 \sin \theta)$ where $\theta$ is the parameter is given by $(x-2)^2+(y+1)^2=16$.

For any line let $m$ and $b$ represent its slope and $y$ intercept respectively and related by $2m+b=3$. These lines all have a specific common point from where tangents are drawn to $x^{2}+y^{2}=1$.

Slope of $\left{ (x,y)/x=2t+3,y=2t+5,t\epsilon R \right} $ is _______.

The equation of a line which passes through (2,3) and the product of whose intercepts on the coordinate axis is 27, can be

If the straight lines joining the origin and the points of intersection of the curve
$ {5x}^{2} + 12xy-{6y}^{2} +4x -2y+3 =0$ and $x+ky-1=0 $ are equally inclined to the co ordinate axis,then the  value of k-

If the line $AX+BY=1$ passes through point of intersection of $y=x\tan\alpha+p\sec\alpha$,$y\sin(30-\alpha)-x\cos(30^ {o}-\alpha)=p$ and is inclined at $30^ {o}$ with $y=(x\tan\alpha+p\sec\alpha)$ then the value of $a^ {2}+b^ {2}=?$

A line $OP$ through origin $O$ is inclined at $30^{o}$ and $45^{o}$ to $OX$ and $OY$ respectively. The angle at which it is inclined to $OZ$ is-