Arts (English Medium) Class 12 - CBSE Important Questions

If ` x in N and |[x+3,-2],[-3x,2x]|=8` , then find the value of x.

Use elementary column operation C₂ → C₂ + 2C₁ in the following matrix equation :

`[[2,1],[2,0]] = [[3,1],[2,0]] [[1,0],[-1,1]]`

Using properties of determinants, show that ΔABC is isosceles if:`|[1,1,1],[1+cosA,1+cosB,1+cosC],[cos^2A+cosA,cos^B+cosB,cos^2C+cosC]|=0​`

For what values of k, the system of linear equations

x + y + z = 2
2x + y – z = 3
3x + 2y + kz = 4

has a unique solution?

Find λ and μ if

`(hati+3hatj+9k)xx(3hati-lambdahatj+muk)=0`

Using the properties of determinants, prove the following:

`|[1,x,x+1],[2x,x(x-1),x(x+1)],[3x(1-x),x(x-1)(x-2),x(x+1)(x-1)]|=6x^2(1-x^2)`

If `A=|[2,0,-1],[5,1,0],[0,1,3]|` , then find A^-1 using elementary row operations

Using the properties of determinants, solve the following for x:

`|[x+2,x+6,x-1],[x+6,x-1,x+2],[x-1,x+2,x+6]|=0`

Using properties of determinants, prove that

`|((x+y)^2,zx,zy),(zx,(z+y)^2,xy),(zy,xy,(z+x)^2)|=2xyz(x+y+z)^3`

Using properties of determinants, prove that

`|[x+y,x,x],[5x+4y,4x,2x],[10x+8y,8x,3x]|=x^3`

Using properties of determinants, prove that :

`|[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc + bc + ca + ab`

Using properties of determinants, prove that 

`|[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]|=2|[a,b,c],[p,q,r],[x,y,z]|`

Prove that the determinant `|(x,sin theta, cos theta),(-sin theta, -x, 1),(cos theta, 1, x)|` is independent of θ.

if A =  `((2,3,10),(4,-6,5),(6,9,-20))`, Find `A^(-1)`. Using `A^(-1)` Solve the system of equation `2/x + 3/y +10/z = 2`; `4/x - 6/y + 5/z = 5`; `6/x + 9/y - 20/z = -4`

If \[a, b\] and c  are all non-zero and 

If \[\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8\] , write the value of x.

Given \[A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}\], compute A⁻¹ and show that \[2 A^{- 1} = 9I - A .\]

If A = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`, find A⁻¹. Using A⁻¹, solve the system of linear equations   x − 2y = 10, 2x − y − z = 8, −2y + z = 7.

Using properties of determinants show that

`[[1,1,1+x],[1,1+y,1],[1+z,1,1]] = xyz+ yz +zx+xy.`

If \[A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix}\] ,find A^–1 and hence solve the system of equations x – 2y = 10, 2x + y + 3z = 8 and –2y + z = 7.