Mathematics T 2

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1. The variables x and y , x 0 , satisfy the differential equation Using the substitution , show that the differential equation may reduced to . . Hence find the particular solution for the given differential equation given [ 10 ] 2. (a) Show that, if tan  2 t sin 2  (b) When, tan  2 4t (1  t 2 ) 1  2t 2  t 4 [3]  1 4 Deduce that, sin 2  cos   cos 2  sin  480  sin   cos  289 4 [4] (c) If , cos  2   1 , find the value of tan . 3 4 [3] 3. The position v
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  1.   The variables  x and  y ,  x > 0 , satisfy the differential equation       .Using the substitution  , show that the differential equation may reduced to     . Hence find the particular solution for the given differential equationgiven     [ 10 ]2.   (a) Show that, if tan t   2     422 21)1(4 2sin t t t t      (b) When, tan 412     Deduce that, 2894804cossinsincoscossin 22          (c)   If , cos 312      , find the value of tan 4   .[ 3 ][ 4 ][ 3 ]3.   The position vectors of the points  A, B, C  and  D , relative to srcin, are  x i   + 2  j ,  5 i + 10  j , i   +  y  j   and -2 i   + 4  j   respectively.  M  is the mid-point of   AB . If   DM   is perpendicular to  AB ,  (a)   Find  x .(b)   If  CB is parallel to  DM  , find  y and  APM   .[ 6 ][ 6 ]  4.   P, Q, M  and  N  are the mid points of   AB, AC, BE  an CE  respectively.   Prove that, (a)    BC   2PQ(b)   PQ MN     [ 5 ][ 2 ]5.   In the given figure, KBHL is tangents to the circle  ABCE  and parallel to  AED. ACL  and  BCD are straight lines.  AB is parallel to CH  .Prove that(a)    DCE  ACB   (b)    LB LH  LC   2  [ 3 ][ 5 ]   6.   During the checking of the printing of a Mathematics book, the number of misprintsper page has a Poisson distribution with mean 3.(a)   Calculate the probability that a randomly chosen page has a perfectprinting. Give your answer correct to two decimal places.(b)   If the whole book of 200 pages is checked. By using a suitableapproximation , calculate the probability that at least 3 pages has nomisprint.7.   X and Y are independent , normally distributed, random variables with thefollowing parameters.: X  N (100 , 25) and Y  N (80 , 20)Calculate the following probabilities.a) P (2X  –  Y > 110)b) Find the value of  a if P (X + Y < a ) = 0.2085[ 2][ 4 ][ 3 ][ 4 ]8.   The probability that Nasir take tuition for mathematics is 0.4. If he takes tuition, theprobability that she will pass the mathematics paper is 0.8. If he does not taketuition, the probability that he will pass the mathematics paper is 0.3.(a)   Find the probability that Nasir passes the mathematics paper.(b)   Find the probability that Nasir takes tuition if he does not pass themathematics paper.[ 3 ][ 3 ]9.   Random variable X takes values -2, -1, 0, 1, 2 with probabilities  p , q , 2  p , 2 q , 2  p  respectively.(a) Express q in terms of   p .(b) Find, in terms of   p , the expected value of X.(c) If X 1 , X 2 and X 3 are three independent observations of X. Find E (Y)where Y = X 1 + X 2 + X 3 , in terms of   p .(d) If   p =  , find P (X 1 + X 2 = 3).[ 2 ][ 2 ][ 2 ][ 4 ]  10. The cumulative distribution function for a continuous random variable X isgiven byF (  x ) = {  (  )  (  )   Find(a) Find the value of  k  .(b) P ( 2 < X < 3 )(c) the probability density function for X.(d) the mean of X.[3][2][2][3]11. The incomes and the mode of transportation to work of 200 workers in a certainfactory are shown in the following table.Income Transportation(number of workers)Bus CarMore than RM 1000 100 25Less than RM 1000 60 15A worker is selected from the group. Find the probability that(a)   If the worker income’s is at least RM 1000 ,find the probability that the worker  drove a car to work (b)   the worker income’s is less than RM 1000, given that the worker went to work  by bus.[3][3]
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