ISC (Arts) Class 12 - CISCE Important Questions for Mathematics

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`

Three schools X, Y, and Z organized a fete (mela) for collecting funds for flood victims in which they sold hand-helds fans, mats and toys made from recycled material, the sale price of each being Rs. 25, Rs. 100 andRs. 50 respectively. The following table shows the number of articles of each type sold:

Using matrices, find the funds collected by each school by selling the above articles and the total funds collected. Also write any one value generated by the above situation.

if A =`((5,a),(b,0))` is symmetric matrix show that a = b

Without expanding at any stage, find the value of:

`|(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|`

Use properties of determinants to solve for x:

`|(x+a, b, c),(c, x+b, a),(a,b,x+c)| = 0` and `x != 0` 

Using matrices, solve the following system of equations :

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y - 2z = -3

Using elementary transformation, find the inverse of the matrix :

`[(0,1,2),(1,2,3),(3,1,1)]`

Find the value of k if M  = `[(1,2),(2,3)]` and `M^2 - km - I_2 = 0`

Using properties of determinants, prove that:

`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^2+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1 + a^2 + b^2)^3`

Given two matrices A and B 

`A = [(1,-2,3),(1,4,1),(1,-3, 2)]  and B  = [(11,-5,-14),(-1, -1,2),(-7,1,6)]`

find AB and use this result to solve the following system of equations:

x - 2y + 3z = 6, x + 4x + z = 12, x - 3y + 2z = 1

 Using determinants, find the values of k, if the area of triangle with vertices (–2, 0), (0, 4) and (0, k) is 4 square units. 

Show that (A + A') is symmetric matrix, if `A = ((2,4),(3,5))`

Using propertiesof determinants prove that:
`|(x , x(x^2), x+1), (y, y(y^2 + 1), y+1),( z, z(z^2 + 1) , z+1) | = (x-y) (y - z)(z - x)(x + y+ z)`

Solve the following system of linear equation using matrix method: 
`1/x + 1/y +1/z = 9`

`2/x + 5/y+7/z = 52`

`2/x+1/y-1/z=0`

Write the negation of the following statements :
(a) Radha likes tea or coffee.
(b) `∃x cc` R such that x + 3 ≥ 10.

If A = `[(1,2), (1,3)]`, find A² - 3A

If the matrix `((6,-"x"^2),(2"x"-15 , 10))` is symmetric, find the value of x.

Find the matrix X for which:
`[(5, 4),(1,1)]` X=`[(1,-2),(1,3)]`

Using properties of determinants, prove that:

`|(a,b,b+c),(c,a,c+a),(b,c,a+b)|` = (a+b+c)(a-c)² 

Given that:  A = `((1,-1,0),(2,3,4),(0,1,2))`  and B = `((2,2,-4),(-4,2,-4),(2,-1,5))`  , find AB. Using this result, solve the following system of equation: x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7.