HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Prove that:

`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`

Solve the following initial value problem:-

\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]

If x^myⁿ = (x + y)^m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]

Apply the given elementary transformation of the following matrix.

A = `[(1,0),(-1,3)]`, R₁↔ R₂

Apply the given elementary transformation of the following matrix.

B = `[(1, -1, 3),(2, 5, 4)]`, R₁→ R₁ – R₂

Apply the given elementary transformation of the following matrix.

A = `[(5,4),(1,3)]`, C₁↔ C₂; B = `[(3,1),(4,5)]` R₁↔ R₂.
What do you observe?

Apply the given elementary transformation of the following matrix.

A = `[(1,2,-1),(0,1,3)]`, 2C₂

B = `[(1,0,2),(2,4,5)]`, −3R₁

Find the addition of the two new matrices.

Apply the given elementary transformation of the following matrix.

A = `[(1,-1,3),(2,1,0),(3,3,1)]`, 3R₃ and then C₃ + 2C₂

Apply the given elementary transformation of the following matrix.

A = `[(1,-1,3),(2,1,0),(3,3,1)]`, 3R₃ and then C₃ + 2C₂

and A = `[(1,-1,3),(2,1,0),(3,3,1)]`, C₃ + 2C₂ and then 3R₃
What do you conclude?

Apply the given elementary transformation of the following matrix.

Use suitable transformation on `[(1,2),(3,4)]` to convert it into an upper triangular matrix.

Apply the given elementary transformation of the following matrix.

Convert `[(1,-1),(2,3)]` into an identity matrix by suitable row transformations.

Apply the given elementary transformation of the following matrix.

Transform `[(1,-1,2),(2,1,3),(3,2,4)]` into an upper triangular matrix by suitable column transformations.

The total cost of 3 T.V. sets and 2 V.C.R.’s is ₹ 35,000. The shopkeeper wants a profit of ₹ 1000 per T.V. set and ₹ 500 per V.C.R. He sells 2 T.V. sets and 1 V.C.R. and gets the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. set and a V.C.R.

If A = `((1,0,0),(2,1,0),(3,3,1))`, then reduce it to I₃ by using column transformations.

If A = `[(2,1,3),(1,0,1),(1,1,1)]`, then reduce it to I₃ by using row transformations.

Check whether the following matrix is invertible or not:

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

Check whether the following matrix is invertible or not:

`((1,1),(1,1))`

Check whether the following matrix is invertible or not:

`((1,2),(3,3))`