HSC Science Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions

Find the value of m so that the function y = e^mx solution of the given differential equation.

y’ + 2y = 0

Find the value of m so that the function y = e^mx solution of the given differential equation.

y” – 5y’ + 6y = 0

The Slope of the tangent to the curve at any point is the reciprocal of four times the ordinate at that point. The curve passes through (2, 5). Find the equation of the curve

Show that y = e^–x + mx + n is a solution of the differential equation `"e"^x(("d"^2y)/("d"x^2)) - 1` = 0

Show that y = `"a"x + "b"/x ≠ 0` is a solution of the differential equation x²yⁿ + xy’ – y = 0

Show that y = ae^–3x + b, where a and b are arbitrary constants, is a solution of the differential equation `("d"^2y)/("d"x^2)  + 3("d"y)/("d"x)` = 0

Show that the differential equation representing the family of curves y² = `2"a"(x + "a"^(2/3))`, where a is a postive parameter, is `(y^2 - 2xy ("d"y)/("d"x))^3 = 8(y ("d"y)/("d"x))^5`

Show that y = a cos bx is a solution of the! differential equation `("d"^2y)/("d"x^2) + "b"^2y` = 0

Choose the correct alternative:

The general solution of the differential equation `("d"y)/("d"x) = y/x` is

Choose the correct alternative:

The solution of the differential equation `2x ("d"y)/("d"x) - y = 3` represents

Choose the correct alternative:

The solution of the differential equation `("d")/("d"x) + 1/sqrt(1 - x^2) = 0` is

For the random variable X with the given probability mass function as below, find the mean and variance.

`f(x) = {{:(1/10, x = 2","  5),(1/5, x = 0","  1","  3","  4):}`

For the random variable X with the given probability mass function as below, find the mean and variance.

`f(x) = {{:((4 - x)/6,  x = 1","  2","  3),(0,  "otherwise"):}`

For the random variable X with the given probability mass function as below, find the mean and variance.

`f(x) = {{:(2(x - 1), 1 < x ≤ 2),(0, "otherwise"):}`

For the random variable X with the given probability mass function as below, find the mean and variance.

`f(x) = {{:(1/2 "e"^(x/2),  "for"  x > 0),(0,  "otherwise"):}`

Two balls are drawn in succession without replacement from an urn containing four red balls and three black balls. Let X be the possible outcomes drawing red balls. Find the probability mass function and mean for X

If µ and σ² are the mean and variance of the discrete random variable X and E(X + 3) = 10 and E(X + 3)² = 116, find µ and σ² 

Four fair coins are tossed once. Find the probability mass function, mean and variance for a number of heads that occurred

A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station. Let X denote the amount of time, in minutes, that the student waits for the train from the time he reaches the train station. It is known that the pdf of X is

`f(x) = {{:(1/30, 0 < x < 30),(0, "elsewhere"):}`
Obtain and interpret the expected value of the random variable X

The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function
`f(x) = {{:(3"e"^(-3x), x > 0),(0, "eleswhere"):}`
Find the expected life of this electronic equipment