HSC Arts (Marathi Medium) 12th Standard Board Exam [इयत्ता १२ वी] - Maharashtra State Board Important Questions for Mathematics and Statistics

Solve the differential equation:

(1 + y²) dx = (tan⁻¹ y − x) dy

The dual of statement t ∨ (p ∨ q) is ______.

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

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

Prove the following:

`tan^-1(1/2) + tan^-1(1/3) = pi/(4)`

In ΔABC, prove the following:

`(cos A)/a + (cos B)/b + (cos C)/c = (a^2 + b^2 + c^2)/(2abc)`

Find the angle between the line `bar r = (hat i + 2hat j + hat k) + lambda(hat i + hat j + hat k)` and the plane `bar r *(2hat i + hat j + hat k) = 8`.

Find the volume of the parallelopiped whose vertices are A (3, 2, −1), B (−2, 2, −3) C (3, 5, −2) and D (−2, 5, 4). 

Find the shortest distance between the lines `barr = (4hati - hatj) + λ(hati + 2hatj - 3hatk)` and `barr = (hati - hatj -2hatk) + μ(hati + 4hatj - 5hatk)`

If x = f(t) and y = g(t) are differentiable functions of t, so that y is function of x and `(dx)/dt ≠ 0` then prove that `dy/(dx) = (dy/dt)/((dx)/dt)`. Hence find `dy/(dx)`, if x = at², y = 2at.

Find the approximate value of tan⁻¹ (1.002).
[Given: π = 3.1416]

The principle solutions of the equation cos θ = `1/2` are ______.

Find the area of the region bounded by the curve y = x², and the lines x = 1, x = 2, and y = 0.

A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.

Solve:

`1 + (dy)/(dx) = cosec (x + y)`; put x + y = u.

The slope of the tangent to the curve x = sin θ and y = cos 2θ at θ = `π/6` is ______.

Prove the following:

`tan^-1(1/2) + tan^-1(1/3) = pi/(4)`

In ΔABC, prove the following:

`(cos A)/a + (cos B)/b + (cos C)/c = (a^2 + b^2 + c^2)/(2abc)`

Find the volume of the parallelopiped whose vertices are A (3, 2, −1), B (−2, 2, −3) C (3, 5, −2) and D (−2, 5, 4). 

Find the approximate value of tan⁻¹ (1.002).
[Given: π = 3.1416]