MATHEMATICS (Three hours and a quarter)

Transcription

1 MTHEMTIS (Three hours and a quarter) (The first 5 minutes of the eamination are for reading the paper onl. andidates must NOT start writing during this time). Total marks: nswer Question from Section and 0 questions from Section. ll working, including rough work, should e done on the same sheet adjacent to the rest of the answers. The intended marks for questions or parts of questions are given in rackets [ ]. Mathematical formulae are given at the end of this question paper. The use of calculator (F-8)/(F-00) is allowed SETION (nswer LL questions) irections: Read the following questions carefull. For each question there are four alternatives,,, and. hoose the correct alternative and write it in our answer sheet. Question [5 0] i) How man 6 digits numers can e formed taking the digits,4,6,8,0 and 9? ii) What is the angle etween the pair of straight lines represented 7+ 0? HSE/0/04 This ooklet contains pages Page of

2 iii) Find the value of sin tan. iv) sin π 0 is ( + cos) v) If a paraola has the origin as its focus and the line as the directri, then the verte of the paraola is at (,0) (0,) (,0) (0,) HSE/0/04 Page of

3 vi) Find the value of k so that the following equations are consistent k vii) d If sin, then is cos+ sin cos+ sin sin+ cos sin+ cos viii) Find the coordinate points which is two- thirds from (, 4,) to (,,4) (,,) (,,) (,,) (,, ) i) What is the standard deviation of the data, 4, 47, 4, 9, 6,, 5, and 44 (in Kg)? 8.6Kg 7.4Kg 8.4Kg 7.5Kg HSE/0/04 Page of

4 ) Find the square root of + 4i. ± ( + i) ± ( + i) ± ( i) ± ( i) i) What is the value of , if the matri 5 ii) is 6+ tan cot 4 4 cot 4 tan 4 4 HSE/0/04 Page 4 of

5 iii) In a single throw of two dice, the proailit that neither a doule nor a total of 0 will appear is, iv) d If sin θand acos θ, then is a a a v) The distance etween the parallel planes + 8z+ 0and4+ 5 z+ 6 0is 5 HSE/0/04 Page 5 of

6 Section (70 marks) nswer an 0 Questions. ll questions in this section have equal marks. Unless otherwise stated, ou ma round our answers to two decimal places. Question a) Integrate. [] ) Find the equations of lines represented the equation Question [4] a) Prove that tan + cot cos [] ) Out of 5 computers in a computer la, 5 are defective. In how man was can the five computers e selected? How man of these selections will include at least one defective computer? [4] Question 4 a) Find the equation of the plane which is perpendicular to the plane + 6z+ 8 0and passes through the intersection of the planes + + z 4 0an z [] ) ag contains lack alls and 7 red alls. all is taken out and not replaced. This is repeated twice. What is the proailit that neither is lack? [4] Question 5 a) The equations of two regression lines in a correlation analsis are + 6and6+. student otains the mean value X 7andY 4and the value of the correlation coefficient is 0.5. o ou agree with him? If not suggest our result. [] ) Find the derivative of the function log sin with respect to. [4] HSE/0/04 Page 6 of

7 Question 6 d a) If sin ( cos ), find. [] ) Solve the following sstem of equations using the matri method: [4] z 8 + z 5 Question 7 d a) Find,if [] 5 π ) Solve: sin + cosec 5 4 [4] Question 8 a) Find the vertices, foci and eccentricit of the conic given elow: ) alculate the area enclosed the curve ( ) Question 9, the ais and ordinates are 0 and. If this area is rotated through four right angles aout ais, find the volume of the solid so formed. [4] a) Find the verte and focus of the paraola [] ) Evaluate the integral as limit of sum ( + ). [4] HSE/0/04 Page 7 of

8 Question 0 a) alculate the Karl Pearson s correlation coefficient etween X and Y for the following data: [4] X Y ) Find the equation of the ellipse whose major ais is 8 and distance etween foci is 4. [] Question a) onvert + i into polar form. [4] i ) alculate the mean deviation aout the mean of the data., 6,, 4, 6, 9,. lso calculate the coefficient of Mean eviation aout the mean. [] Question a) Find the general solution of the differential equation d + sin. π lso find the solution when and 0. [] ) Find all the values of 4 i using e Moivre s Theorem. lso find the Question continued product of all values. [4] a) alculate the acute angle etween two lines whose direction ratios are (,,0) and (,,). [] ) Using properties of determinants, find the value of [4] + HSE/0/04 Page 8 of

9 Question 4 a) Find the eccentricit, distance etween the foci and equation of the directrices of the ellipse [] ) closed right circular clinder has a volume of 56 cuic units. What should e the radius of the ase so that the total surface area ma e minimum? [4] HSE/0/04 Page 9 of

10 FORMULE Trigonometr sin cos tan sin ± sin sin ± ( m ) cos ± cos cos ± m tan ± tan tan, < tan tan sin cos + + cosec sin sec cos cot tan omplenumers r a + tanθ θ tan a a If z r( cosθ isinθ) + then ( cosn isinn ) n n z r θ+ θ n n kπ + θ kπ + θ z r cos + isin, n n k 0,,,,... n o-ordinategeometr + + z z m + m m + m mz + mz (,, z),, m + m m + m m + m a+ + cz 0and a+ + cz 0 z c c ca ca a a ngleetween twoplanes, cosθ ± aa + + cc a + + c a + + c a + + cz+ d distanceof a point froma plane ± a + + c m + m m + m (, ), m + m m + m ngle etween thelinesa + h+ 0, tanθ h a a+ equationofisec tor, a h hf g gh af points of intersection,, a h a h HSE/0/04 Page 0 of

11 lgera ( ) a a+ a a± a ± a+ ± + + a Inthe quadratic equation a c 0, n n p ij r r n! ( n r)! n! r! n r! ( ) i + j M I ij adj det, z, z ( n ) n( n ) 4ac ( n ) n( n )( n ) ( ) n n ( n ) LULUS n n, n,, cf(), cf () d du dv If u±v,then ± d dv du If uv,then u +v du dv v -u u d If, then v v d d du du du uv u v v. a n f lim h f( a rh) h 0 + r 0 d p + p Q, IF. e, generalsolution, IF. ( QIF. ) + c V π a Volumeofone π 4 Volume of Sphere π π a r h r h Volume of linder r h Sreaofone. πrl+ πr SreaofSphere. 4πr S. rea of linder π rh+ π r ata and Proailit X f or X f n HSE/0/04 Page of

12 i N Median L+ c f i or n n n f f f f n + n X n + n Mean eviation ov ( ) f n + n + nd + nd n + n ( X, Y) ( X X)( Y Y) n ( )( ) n ( ) ( ) n ( ) n ( ) r r f ( )( ) n ( X Y) cov, Y Y X X r X X ( X Y) cov, X X Y Y r Y Y r na + a + ( ) ( ) r P( ) P + P P( ) P + P P P ( ) ( ) ( ) P P ( ) P P 6 d nn. r, orrection factor m m r± YX XY r r n n n n ( ) ( ) HSE/0/04 Page of