The co-factor of -7 in the determinant \(\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}\) is________.

If \(\triangle=\begin{vmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{vmatrix}\) then \(\begin{vmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{vmatrix}\) is ________.

The value of the determinant \({\begin{vmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & c \end{vmatrix}}^{2}\)is ________.

If \(\triangle=\begin{vmatrix} {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} \end{vmatrix}\) and A_ij is cofactor of a_ij, then value of \(\triangle\) is given by ________.

If nC₃ = nC₂, then the value of nC₄ is _______.

The value of n, when nP₂ = 20 is _______.

Number of words with or without meaning that can be formed using letters of the word "EQUATION" , with no repetition of letters is _____.

The angle between the pair of straight lines x² - 7xy + 4y² = 0 _______.

If the lines 2x - 3y - 5 = 0 and 3x - 4y - 7 = 0 are the diameters of a circle, then its centre is _______.

The eccentricity of the parabola is _______.

Evaluate \(\begin{vmatrix} 2 &-1 &-2 \\0 & 2 & -1\\3 & -5& 0 \end{vmatrix}.\)

Find the values of x if \(\begin{vmatrix} 2 & 4 \\5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4\\6 & x \end{vmatrix}.\)

Using the property of determinant, evaluate \(\begin{vmatrix} 6 &5 &12 \\ 2 & 4 &4 \\2 & 1 & 4 \end{vmatrix}.\)

Verify that 8C₄ + 8C₃ = 9C₄

How many chords can be drawn through 21 points on a circle?

How many triangles can be formed by joining the vertices of a hexagon?

Show that 10P₃ = 9 P₃ + 3. 9P₂

Find n if 25 C_n+5 = 25 C_2n-1.

Find the centre and radius of the circle x² + y² = 16

Find the center and radius of the circle 5x² + 5y² + 4x - 8y - 16 = 0

Find the center and radius of the circle (x + 2) ( x - 5) + (y - 2 ) ( y - 1) = 0

If A = \(\begin{bmatrix}1 & 1 & 1 \\ 3 & 4 & 7\\1 & -1 & 1 \end{bmatrix}\) verify that A ( adj A ) = ( adj A ) A = |A| I₃.

If A = \(\begin{bmatrix}3 & -1 & 1 \\ -15 & 6 & -5\\5 & -2 & 2 \end{bmatrix}\) then, find the Inverse of A.

Find the adjoint of the matrix \(A=\begin{bmatrix}2&3\\1&4 \end{bmatrix}\)

Using matrix method, solve x + 2y + z = 7, x + 3z = 11 and 2x - 3y =1.

If A =\(\left[ \begin{matrix} -1 & 2 & -2 \\ 4 & -3 & 4 \\ 4 & -4 & 5 \end{matrix} \right] \)then, show that the inverse of A is A itself.

Using co-factors of elements of  second column evaluate \(\left| \begin{matrix} 6 & -1 & 5 \\ 3 & 0 & 4 \\ -2 & 7 & -3 \end{matrix} \right| \)

Decompose into Partial Fractions:\(\frac { 5{ x }^{ 2 }-8x+5 }{ (x-2)({ x }^{ 2 }-x+1) } \)

Evaluate the following : \(\frac { 7! }{ 6! } \)

It the letters of the word are arranged as in dictionary, find the rank of the word "AGAIN".

In how many ways can n prizes be given to n boys, when a boy may receive any number of prizes?

Find the axis, vertex, focus, equation of directrix and the length of latus rectum of the parabola (y - 2)² = 4(x - 1).

For what value of \(\lambda \) are the three lines 2x-5y+3 = 0, 5x-9y+\(\lambda \)=0 and x-2y+1=0 are concurrent?

Find the value of k if the straight line 2x + 3y + 4 + k(6x - y + 12) = 0 is perpendicular to the line 7x + 5y - 4 = 0

Find the locus of a point which moves in such a way that the square of its distance from the point (3, -2) is numerically equal to its distance from the line 5x - 12y = 13

Find the locus of a point such that the sum of its distances from the points (0, 2) and (0, -2) is 6.

Find the slope of the lines which make an angle of 45° with the line 3x - y + 5 = 0.

Determine the values of x for which the matrix A =\(\left[ \begin{matrix} x+1 & -3 & 4 \\ -5 & x+2 & 2 \\ 4 & 1 & x-6 \end{matrix} \right] \)is singular.

Solve by matrix inversion method: 3x - y + 2z = 13 ; 2x + Y - z = 3 ; x + 3y - 5z = - 8.

Expand the following by using binomial theorem.\(\left( x+\frac { 1 }{ y } \right) ^{ 7 }\)

Show that the equation 12x² - 10xy + 2y² + 14x - 5y + 2 = 0 represents a pair of straight lines and also find the separate equations of the straight lines.