The random variable X can take only the values 0, 1, 2, 3. Given that P(X =0) = P(X = 1) = p and P(X = 2) = P(X = 3) such that Σp_ix_i² = 2Σp_ix_i, find the value of p.

Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six.
Find the probability that it is actually a six.
Do you also agree that the value of truthfulness leads to more respect in the society?

Solve the following L.P.P. graphically :
Minimise Z = 5x + 10y
Subject to x + 2y ≤ 120
Constraints x + y ≥ 60
x – 2y ≥ 0
and x, y ≥ 0

Use product to solve the system of equations x + 3z = 9,
–x + 2y – 2z = 4, 2x – 3y + 4z = –3

Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b = a − b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A. 

Consider f : R₊ → [−5, ∞), given by f(x) = 9x² + 6x − 5. Show that f is invertible with f⁻¹(y).

Hence Find
(i) f⁻¹(10)
(ii) y if f⁻¹(y)=43,

where R+ is the set of all non-negative real numbers.

27.

If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum, when the angle between them is π/3.

Solve the differential equation 

Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x³ – 3xy²)dx = (y³ – 3x²y) dy, where C is a parameter.

If  are mutually perpendicular vectors of equal magnitudes, show that the vector  is equally  inclined to Also, find the angle which  with .