Q24 of 29 Page 1

Using properties of integral, evaluate

OR


Find:


We know,



Let


Now,













⇒ I = π


OR


Dividing numerator and denominator by cos3x




Put tanx = t, then sec2xdx = dt



By partial fractions


Let



⇒ t = At2 – At + A + Bt + Bt2 + Ct + C


⇒ t = t2(A + B) + (-A + B + C) t + A + C


On comparing, A + B = 0


-A + B + C = 1, A + C = 0


On solving, ,


∴





Where, u = t2 – t + 1 and du = 2t – 1





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22

The random variable X has a probability distribution P(X) of the following form, where k is some number:


(i) Find the value of k


(ii) Find P(X <2)


(iii) Find P(X ≤ 2)


(iv) FindP(X ≥ 2)


 23

A bag contains (2n +1) coins. It is known that ‘n’ of these coins have a head on both its sides whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is find the value of ‘n’.

 25

Does the following trigonometric equation have any solutions? If Yes, obtain the solution(s):


OR


Determine whether the operation * define below on Q is binary operation or not.


a * b = ab+1


If yes, check the commutative and the associative properties. Also check the existence of identity element and the inverse of all elements in.


 26

Find the value of x, y and z, if A =satisfies A’ = A1

OR


Verify: A(adjA) = (adjA)A = |A|I for matrix A =