HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

Use quantifiers to convert the given open sentence defined on N into a true statement

Y + 4 > 6

The negation of p ^ (q → r) is ______.

Without using truth table prove that (p ∧ q) ∨ (∼ p ∧ q) v (p∧ ∼ q) ≡ p ∨ q

Simplify the given circuit by writing its logical expression. Also, write your conclusion.

Construct the truth table for the statement pattern:

[(p → q) ∧ q] → p

Apply the given elementary transformation of the following matrix.

A = `[(1,0),(-1,3)]`, R₁↔ R₂

If A = `[("x",0,0),(0,"y",0),(0,0,"z")]` is a non-singular matrix, then find A⁻¹ by using elementary row transformations. Hence, find the inverse of `[(2,0,0),(0,1,0),(0,0,-1)]`

Find the inverse of A = `[("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)]` by elementary row transformations.

Express the following equations in matrix form and solve them by the method of reduction:

x − y + z = 1, 2x − y = 1, 3x + 3y − 4z = 2

If A = `[(-2, 4),(-1, 2)]` then find A² 

Find the matrix X such that AX = I where A = `[(6, 17),(1, 3)]`

Find the inverse of A = `[(2, -3, 3),(2, 2, 3),(3, -2, 2)]` by using elementary row transformations.

Solve the following system of equations by the method of inversion.

x – y + z = 4, 2x + y – 3z = 0, x + y + z = 2

Solve the following system of equations by the method of reduction:

x + y + z = 6, y + 3z = 11, x + z = 2y.

In Δ ABC with the usual notations prove that `(a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2`

In any ΔABC if  a² , b² , c² are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.

In a Δ ABC, with usual notations prove that:` (a -bcos C) /(b -a cos C )= cos B/ cos A`

In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a +   c - b)

In any ΔABC, with usual notations, prove that b² = c² + a² – 2ca cos B.

In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.

(A) `1/5`

(B) `sqrt(1/5)`

(C) `4/5`

(D) `2/5`