Short Answer TypeThe 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 Σpixi2 = 2Σpixi, 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
Long Answer TypeUse 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.Â
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.
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.
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.
Given equation is
(x3 – 3xy2)dx = (y3 – 3x2y) dy
which is a homogeneous equation.
Therefore substituting y = vx
We have,
Integrating the equation both sides, 
thus substituting the values of I1 and I2, 
If 
 are mutually perpendicular vectors of equal magnitudes, show that the vector 
 is equally 
 inclined to Also, find the angle which 
 with 
.
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