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

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 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]

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.