(a) Define Error and their types in quantitative chemical analysis. 

Define Binomial, Poisson and Normal distribution with appropriate equation and name their terms. 

A bag contains 8 red balls and 5 black balls. Two balls are drawn one by one without replacement. Find the probability that both balls are red. 

 A number is chosen at random from the first 40 natural numbers. Calculate the probability that the selected is divisible by 5 or 7. 

 (a) Two coins are tossed simultaneously then find probability of getting at least one head.

Find A⁻¹, where 

(a) Find eigenvectors for the matrix 

Solve the following system of equations using Cramer's rule. 

x + y - z = 6

3x - 2y + z = -5

x + 3y - 2z = 14

Verify that the following matrix A is orthogonal 

 (a) If A =  then show that 

(i) 1/2 (A + A') is symmetric, and

(ii) 1/2 (A - A') is skew symmetric.

Solve the following differential equation: 

(1/y²) (dy/dt) = 1 - e⁻³ᵗ

(c) Show that the equation 

(y - 2x³)dx = x(1 - xy)dy

becomes exact on multiplication by x⁻² and solve it.

Using the ideal gas equation, estimate the change in the pressure of 1.0 mol of an ideal gas at 0°C when its volume is changed from 22.414 L to 21.414 L. 

 (a) Find the differential equation whose solution is given by 

y = eˣ(A cos x + B sin x)

where, A and B are arbitrary constants.

∫₁³ x²eˣ³ dx

 ∫((1+ln x)³)/x dx

 ∫(2eˣ - 3√x)dx

 (d²y/dx²)¹ᐟ⁵ = k[1+(dy/dx)²]⁵ᐟ²

 (dy/dx)³ = √(1+(dy/dx)²)

√(d³y/dx³) = dy/dx + x⁴

(dy/dx)² = (7x)/(4y²)