HSC Science इयत्ता १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

If X is the random variable with distribution function F(x) given by,
F(x) = `{{:(0",", - oo < x < 0),(1/2(x^2 + x)",", 0 ≤ x ≤ 1),(1",", 1 ≤ x < oo):}`
then find the probability density function f(x)

If X is the random variable with distribution function F(x) given by,
F(x) = `{{:(0",", - oo < x < 0),(1/2(x^2 + x)",", 0 ≤ x ≤ 1),(1",", 1 ≤ x < oo):}`
then find P(0.3 ≤ X ≤ 0.6)

Choose the correct alternative:

Let X be random variable with probability density function
`f(x) = {(2/x^3, x ≥ 1),(0, x < 1):}`
Which of the following statement is correct?

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A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is 
`f(x) = {{:(1/l, 0 < x < l),(0, l ≤ x < 2l):}`
The mean and variance of the shorter of the two pieces are respectively

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If the function f(x) = `1/12` for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?

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The random variable X has the probability density function
`f(x) = {{:("a"x + "b", 0 < x < 1),(0, "otherwise"):}`
and E(X) = `7/12`, then a and b are respectively

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If `f(x) = {{:(2x, 0 ≤ x ≤ "a"),(0, "otherwise"):}` is a probability density function of a random variable, then the value of a is

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A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

If the direction cosines of a line are `(1/c, 1/c, 1/c)` then ______.

If z = (ax + b) (cy + d), then find `(∂z)/(∂x)` and `(∂z)/(∂y)`.

Test for consistency and if possible, solve the following systems of equations by rank method:

x – y + 2z = 2, 2x + y + 4z = 7, 4x – y + z = 4

Test for consistency and if possible, solve the following systems of equations by rank method:

3x + y + z = 2, x – 3y + 2z = 1, 7x – y + 4z = 5

Test for consistency and if possible, solve the following systems of equations by rank method:

2x + 2y + z = 5, x – y + z = 1, 3x + y + 2z = 4

Test for consistency and if possible, solve the following systems of equations by rank method:

2x – y + z = 2, 6x – 3y + 3z = 6, 4x – 2y + 2z = 4

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have no solution

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have unique solution

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have infinitely many solution

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have no solution

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have a unique solution

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have an infinite number of solutions