ISC (Arts) कक्षा १२ - CISCE Important Questions for Mathematics

Solve the following system of linear equations using matrix method: 
3x + y + z = 1
2x + 2z = 0
5x + y + 2z = 2

If A is a square matrix of order 3, then |2A| is equal to ______.

For what value of k the matrix `[(0, k),(-6, 0)]` is a skew symmetric matrix?

Evaluate the following determinant without expanding:

`|(5, 5, 5),(a, b, c),(b + c, c + a, a + b)|`

Assertion: Let the matrices A = `((-3, 2),(-5, 4))` and B = `((4, -2),(5, -3))` be such that A¹⁰⁰B = BA¹⁰⁰

Reason: AB = BA implies AB = BA for all positive integers n.

If A and B are symmetric matrices of the same order, then AB – BA is ______.

Find the value of the determinant given below, without expanding it at any stage.

`|(βγ, 1, α(β + γ)),(γα, 1, β(γ + α)),(αβ, 1, γ(α + β))|`

A furniture factory uses three types of wood namely, teakwood, rosewood and satinwood for manufacturing three types of furniture, that are, table, chair and cot.

The wood requirements (in tonnes) for each type of furniture are given below:

It is found that 29 tonnes of teakwood, 13 tonnes of rosewood and 16 tonnes of satinwood are available to make all three types of furniture.

Using the above information, answer the following questions:

1. Express the data given in the table above in the form of a set of simultaneous equations.
2. Solve the set of simultaneous equations formed in subpart (i) by matrix method.
3. Hence, find the number of table(s), chair(s) and cot(s) produced.

A matrix which is both symmetric and skew symmetric matrix is a ______.

if `|(a, b, c),(m, n, p),(x, y, z)| = k`, then what is the value of `|(6a, 2b, 2c),(3m, n, p),(3x, y, z)|`?

In a third order matrix a_ij denotes the element of the i^th row and the j^th column.

A = `a_(ij) = {(0",", for, i = j),(1",", f or, i > j),(-1",", f or, i < j):}`

Assertion: Matrix ‘A’ is not invertible.

Reason: Determinant A = 0

Which of the following is correct?

The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of C²?

Answer the following question:

1. Translate the problem into a system of equations.
2. Solve the system of equation by using matrix method.
3. Hence, find the cost of one paper bag, one scrap book and one pastel sheet.

If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`

If x=α sin 2t (1 + cos 2t) and y=β cos 2t (1−cos 2t), show that `dy/dx=β/αtan t`

If x cos(a+y)= cosy then prove that `dy/dx=(cos^2(a+y)/sina)`

Hence show that `sina(d^2y)/(dx^2)+sin2(a+y)(dy)/dx=0`

Find the value of `dy/dx " at " theta =pi/4 if x=ae^theta (sintheta-costheta) and y=ae^theta(sintheta+cos theta)`

If x = cos t (3 – 2 cos² t) and y = sin t (3 – 2 sin² t), find the value of dx/dy at t =4/π.

Find the value of constant ‘k’ so that the function f (x) defined as

f(x) = `{((x^2 -2x-3)/(x+1), x != -1),(k, x != -1):}`

is continous at x = -1

Show that the function f(x) = `{(x^2, x<=1),(1/2, x>1):}` is continuous at x = 1 but not differentiable.