Maths-1999-Set I

(i) All Questions are compulsory
(ii) Question number 1 to 15 are of 2 marks each
(iii) Question number 16 to 25 are of 4 marks each
(iv) Question number 26 to 30 are of 6 marks each

Q1) A coin is tossed 12 times. What is the Probability of getting exactly 8 tails? (Marks 2)
Ans1) When a coin is tossed, we have S = {H, T}
p = P (getting a tail) = 1/2
… q = 1 – p = 1/2
Here n = 12
Now the Probability Distribution
B(n, p) = B(12, 1/2)
i.e. (q + p)12 where q = p = 1/2
… P (X = 8) = 12C8 . q12 – 8 . p8
= 12C4 . q4 p8 = 12×11x10×9 x (1/2)4 + 8
4
= 498 = 495
212 4096

Q2) Three coins are tossed simultaneously. List the sample space for the event. (Marks 2)
Ans2) Sample space S, of tossing 3 coins simultaneously is given by:-
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Q3) Two cards are drawn without replacement from a well shuffled pack of 52 cards. What is the probability that one is red queen and the other is a king of black colour? (Marks 2)
Ans3) We know that there are 2 red queens and two kings of black colour in a pack of 52 cards.
P (drawing a red queen) = 2/52 = 1/26
Since the card is not replaced, after the firs card, we have 51 cards left.
… P (drawing a black king) = 2/51
But these two draws are interchangeable.
… The reqd. probability = 2 x 1/26 x 2/51 = 2/(13 x 51) = 2/663

Q4) Find a unit vector perpendicular to both = 3 + – 2 and = 2 + 3 – (Marks 2)
Ans4) Here, = 3 + – 2 and = 2 + 3 -
=> x =
3 1 -2
2 3 -1

= 5 – + 7
… = Unit vector | to and
= ( x )/[ x ] = (5 – + 7)/(52 + (-1)2 + 72)
= (5 – + 7)/(75) = 1/(53) (5 – + 7)

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