Solve the following differential equations:
i) 2 x + y + 3) dy/dx = x + 2 y + 1 .ii) ( x2 + 1) dy/dx + 2 xy = 4 x2.
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Determine the value of the constant a if the function f (x) defined below is continuous at x = 2.
Evaluate the following integrals:
i) 
ii) ∫ x ln | x + 1| dx
The probability that a person over 60 years of age in a certain community drinks alcohol 2/5 is
and the probability that a person over 60 years of age has heart disease is 2/15 . The
probability that a person over 60 years of age drinks alcohol and has heart disease is 1/16 . Are
‘drinking alcohol’ and ‘heart disease’ independent events?
For what values of m will the equation
x 2 − 2x (1 + 3m) + 7 (3 + 2m) = 0
have equal roots?
Solve the following differential equations:
i) 2 x + y + 3) dy/dx = x + 2 y + 1 .ii) ( x2 + 1) dy/dx + 2 xy = 4 x2.
Which of the following statements are true? Give reasons for your answers.
i) The derivative of the quotient of two functions is equal to the quotient of the derivatives of the two functions.
ii) The function of f ( x) = sin x/2 + sinx has a maxima at x = π/2 .
iii) 2x + 3 y + 4z = 0 represents a line in 3-dimenisonal space.
v) In a Binomial distribution if mean is known then standard deviation is also known.
v) If two events are mutually exclusive then they are independent.
A training director wishes to find out whether a particular training programme increases the efficiency of the employees. He plans to take a random sample of employees and record their efficiency rating before the training starts. After the completion of the programme, the efficiency of the same sample of employees is recorded as given in the table below. At 0.05 level of significance do you reject or accept the null hypothesis
H 0 : µd = 0 against H1 : µd > 0
where µd = 0 is the mean of the difference?