HSC Science (Computer Science) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

Solve the differential equation

`y (dy)/(dx) + x` = 0

Form the differential equation of all lines which makes intercept 3 on x-axis.

The solution of the differential equation `dx/dt = (xlogx)/t` is ______.

Find the particular solution of the differential equation `dy/dx` = e^2y cos x, when x = `π/6`, 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 ______.

A random variable X has the following probability distribution:

then E(X)=....................

From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain -

(a) exactly 1 defective bulb.

(b) at least 1 defective bulb.

The time (in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable taking values between 25 and 35 minutes with p.d.f 

`f(x) = {{:(1/10",", 25 ≤ x ≤ 35),(0",", "otherwise"):}`

What is the probability that preparation time exceeds 33 minutes? Also, find the c.d.f. of X.

Probability distribution of X is given by

Find P(X ≥ 2) and obtain cumulative distribution function of X

Find the probability distribution of number of heads in two tosses of a coin.

From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)¹⁹ = 0⋅1348]

A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).

For the following probability density function (p. d. f) of X, find P(X < 1) and P(|x| < 1) 

`f(x) = x^2/18, -3 < x < 3`

            = 0,             otherwise

If X ∼ N (4,25), then find P(x ≤ 4)

The p.m.f. of a random variable X is
`"P"(x) = 1/5` , for x = I, 2, 3, 4, 5 
        = 0 , otherwise.
Find E(X).