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 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
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.
Given, * is a binary operation on Q − {1} defined by a*b=a−b+ab
Commutativity:
For any a, b∈A,
we have a*b=a−b+ab and b*a=b−a+ba
Since, a−b+ab≠b−a+ab
∴a*b≠b*a
So, * is not commutative on A.
Associativity:
Let a, b, c∈A(a*b)*c=(a−b+ab)*c
⇒(a*b)*c=(a−b+ab)−c+(a−b+ab)c
⇒(a*b)*c=a−b+ab−c+ac−bc+abc
a*(b*c)=a*(b−c+bc)
⇒a*(b*c)=a−(b−c+bc)+a(b−c+bc)
⇒a*(b*c)=a−b+c−bc+ab−ac+abc
⇒(a*b)*c≠a*(b*c)
So, * is not associative on A.
Identity Element
Let e be the identity element in A, then
a*e=a=e*a ∀a∈Q−{1}
⇒a−e+ae=a
⇒(a−1)e=0
⇒e=0 (As a≠1)
So, 0 is the identity element in A.
Inverse of an Element
Let a be an arbitrary element of A and b be the inverse of a. Then,
a*b=e=b*a
⇒a*b=e
⇒a−b+ab=0 [∵e=0]
⇒a=b(1−a)
⇒b=a/1−a
Since b∈Q−1
So, every element of A is invertible.
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.
If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to Also, find the angle which with .