HSC Arts इयत्ता १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions

Evaluate the following:

`int_0^1 (sin(3tan^-1 x)tan^-1 x)/(1 + x^2)  "d"x`

Evaluate the following:

`int_0^(1/sqrt(2)) ("e"^(sin^-1x) sin^-1 x)/sqrt(1 - x^2)  "d"x`

Evaluate the following:

`int_0^(pi/2) x^2 cos 2x  "d"x`

Choose the correct alternative:

If `f(x) = int_0^x "t" cos  "t"  "dt"`, then `("d"f)/("d"x)` =

Find the differential equation of the family of all non-vertical lines in a plane

Find the differential equation of the family of all non-horizontal lines in a plane 

Form the differential equation of all straight lines touching the circle x² + y² = r²

Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis

Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis

Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin

Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be ^–8x, where A and B are arbitrary constants

Find the differential equation of the curve represented by xy = ae^x + be^–x + x²

Choose the correct alternative:

The slope at any point of a curve y = f(x) is given by `("d"y)/("d"x) - 3x^2` and it passes through (-1, 1). Then the equation of the curve is

The probability density function of X is given by
`f(x) = {{:(kx"e"^(-2x),  "for"  x > 0),(0,  "for"  x ≤ 0):}`
Find the value of k

The probability density function of X is `f(x) = {(x, 0 < x < 1),(2 - x, 1 ≤ x ≤ 2),(0, "otherwise"):}`
Find P(0.2 ≤ X < 0.6)

The probability density function of X is `f(x) = {(x, 0 < x < 1),(2 - x, 1 ≤ x ≤ 2),(0, "otherwise"):}`
Find P(1.2 ≤ X < 1.8)

The probability density function of X is `f(x) = {(x, 0 < x < 1),(2 - x, 1 ≤ x ≤ 2),(0, "otherwise"):}`
Find P(0.5 ≤ X < 1.5)

Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function
`f(x) = {{:(k, 200 ≤ x ≤ 600),(0, "otherwise"):}`
Find the value of k

Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function
`f(x) = {{:(k, 200 ≤ x ≤ 600),(0, "otherwise"):}`
Find the distribution function

Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function
`f(x) = {{:(k, 200 ≤ x ≤ 600),(0, "otherwise"):}`
Find the probability that daily sales will fall between 300 litres and 500 litres?