SUMMER 8 EXAMINATION Important Instructions to eaminers: ) The answers should be eamined by key words and not as word-to-word as given in the model answer scheme. ) The model answer and the answer written by candidate may vary but the eaminer may try to assess the understanding level of the candidate. 3) The language errors such as grammatical, spelling errors should not be given more Importance (Not applicable for subject English and Communication Skills. 4) While assessing figures, eaminer may give credit for principal components indicated in the figure. The figures drawn by candidate and model answer may vary. The eaminer may give credit for any equivalent figure drawn. 5) Credits may be given step wise for numerical problems. In some cases, the assumed constant values may vary and there may be some difference in the candidate s answers and model answer. 6) In case of some questions credit may be given by judgement on part of eaminer of relevant answer based on candidate s understanding. 7) For programming language papers, credit may be given to any other program based on equivalent concept. N.. Attempt any TEN of the following: 0 a) b) 0 0 Find, if 3 0 4 0 0 3 0 4 4 0 0 0 ----------------------------------------------------------------------------------------------------------------- 4 5 If A, B, find A B. 3 4 3 4 5 A B 3 4 3 4 4 5 = 6 8 3 6 9 = 5 5 ----------------------------------------------------------------------------------------------------------------- 0 0 Page of 30
SUMMER 8 EXAMINATION N.. c) d) If A show that A is null matri. A 4 4 4 AA 4 4 8 8 4 4 0 0 0 0 A is null matri 3 5 If A, = verify that 0 B 3 AB BA 3 5 AB 0 3 35 6 0 = 0 4 0 6 = 4 3 5 BA 3 0 3 4 5 0 = 9 4 5 0 5 3 5 AB BA 0 0 e) Resolve into partial fraction 4 A B 4 A B 4 0 Page of 30
SUMMER 8 EXAMINATION N.. e) f) g) h) Put 0 A4 Put B 3 4 4 3 Define Allied angle. If the sum or difference of the measures of two angles is either zero or is an integral 0 multiple of 90,i.e, n. where n I, called as Allied angles. Prove that sin sin cos sin sin sin cos cos sin sincos 0 0 If sin80 +sin 50 =sin cos, find, 0 0 0 0 i) 0 0 sin80 +sin 50 =sin 0 0 0 0 0 0 sin80 +sin 50 cos sin sin sin cos 80 & 50 65 & 5 0 0 OR sincos 80 50 80 50 sin cos sincos sin 65cos5 sin cos 65 & 5 Prove that sin sin Let sin sin sin sin 0 Page 3 of 30
SUMMER 8 EXAMINATION N.. i) sin sin sin sin OR sin sin put sin sin sin sin sin sin sin sin j) k) 0 0 Evaluate cos75.cos5 without using calculator. cos 75.cos5 cos 75 5 cos 75 5 0 0 0 0 0 0 cos90 cos60 0 0 0 Prove that the lines 3 y 6 0 and 3y 0 are perpendicular to each other. 0 0 L : 3 y 6 0 L : 3y 0 Page 4 of 30
SUMMER 8 EXAMINATION N... k) l) a) 3 3 m m 3 3 consider mm.. 3 Lines are perpendicular to each other. Find the coefficient of range of the following distribution. 0,00,30,50,50 L S coefficient of range L S 50 50 50 50 00 or 0.5 00 Attempt any FOUR of the following : Solve the following equations by using Cramer's rule 0 6 3 y z 4, 3y z 7, y 3z 6 3 D 3 3 9 6 3 30 3 D 4 7 3 4 9 6 7 8 30 6 3 D 30 D 30 3 4 D y 7 3 6 4 6 7 30 6 3 30 y D y D 30 Page 5 of 30
SUMMER 8 EXAMINATION N.. a) b) D z 3 4 3 7 3 8 7 7 4 3 60 6 Dz 60 z D 30 0 0 T T T If A 3 0, B = 0 verify that AB B A 4 5 0 0 3 0 0 AB 3 0 0 4 5 0 0 3 4 0 0 0 0 3 = 3 0 0 0 0 0 0 6 4 0 0 0 5 0 0 0 0 5 3 AB 3 6 4 5 0 5 3 4 T AB 5 3 6 0 0 3 4 T T B A 0 0 5 0 0 3 0 4 0 3 0 0 4 0 0 = 0 0 0 0 5 0 0 0 3 0 0 6 0 0 0 5 3 4 5 3 6 0 AB T T T B A ------------------------------------------------------------------------------------------------------------------------------------ Page 6 of 30
SUMMER 8 EXAMINATION N.. c) 0 If A 4 3 4 prove that A I 3 3 4 0 A 4 3 4 3 3 4 0 0 A AA 4 3 4 4 3 4 3 3 4 3 3 4 0 4 3 0 3 3 0 4 4 = 0 4 9 4 6 0 3 9 3 6 0 0 0 0 I 0 0 d) 4 4 4 4 4 4 A 4 4 4 4 4 4 4 4 A AA 4 4 4 4 4 4 4 4 4 6 6 8 8 6 8 6 8 = 8 8 6 6 4 6 6 8 8 8 6 8 6 8 8 6 6 4 36 3 3 3 36 3 3 3 36 If A 4 4, show that A 8 Ais a scalar matri Page 7 of 30
SUMMER 8 EXAMINATION N.. d) e) 4 4 6 3 3 8A 8 4 4 3 6 3 4 4 3 3 6 36 3 3 6 3 3 0 0 0 A 8A 3 36 3 3 6 3 0 0 0 3 3 36 3 3 6 0 0 0 A 8 A is a scalar matri Resolve into partial fraction Let 3 3 3 A B C 3 A B C Put A A4 3 A 4 Put C C 3 5 5 C Put 0 3 A B C 5 3 B 4 B 5 3 4 B 4 Page 8 of 30
SUMMER 8 EXAMINATION N.. e) f) 5 3 4 4 3 Resolve into partial fraction 4 Let 3 A B C 4 4 3 A B 4 C 4 Put 4 A 3 4 4 4 A 9 3 A 7 A 7 Put 3 B 4 5 9 3 B 5 7 B 4 0 B 7 Put C C 3 4 3 3 C 9 Page 9 of 30
SUMMER 8 EXAMINATION N.. 3. f) 0 3 7 7 9 4 4 Attempt any FOUR of the following : 6 a) Using matri inversion method, solve the following equations. 3 y z 3, 3y z 3, y z 4 3 Let A 3 A 3 3 4 3 A 8 3 A 3 3 3 3 3 Matri of minors 3 3 3 3 3 7 3 5 5 7 3 7 Matri of cofactors 3 5 5 7 OR C 3 3, C 3 Page 0 of 30
SUMMER 8 EXAMINATION N. 3. a) C C C C 3 3 3 3 33 3 4 3 7, C 4 3 3 3 3, C 6 5 3 6 5, C 3 4 7 3 3 9, 3 3 7 Matri of cofactors 3 5 5 7 3 5 Adj. A 3 7 7 5 A Adj. A A A 3 5 3 7 8 7 5 X A B 3 5 3 y 3 7 3 8 z 7 5 4 3 9 0 y 9 3 8 8 z 5 44 8 y 6 8 z 8 Page of 30
SUMMER 8 EXAMINATION N. 3. a) b) y z, y, z Resolve into partial fraction : 3 3 A B C A B C c) Put 3 3A A Put 0 A C C C Put A B C A B C B 3 B B B 4 Resolve into partial fraction : Page of 30
SUMMER 8 EXAMINATION N. 3. c) d) 4 4 + + 4 A B Let B A put A A put B B Prove that sin A B.sin A B cos B cos A 4 A B A B sin A.cos B cos A.sin Bsin A.cos B cos A.sin B LHS sin.sin A B A B = sin.cos cos.sin = sin A cos B cos Asin B Page 3 of 30
SUMMER 8 EXAMINATION 3. 4. N. d) e) f) a) A B A B LHS cos cos cos cos cos B cos B.cos A cos A cos B.cos A cos RHS Bcos A 0 0 0 0 0 0 Prove that tan 70 tan 50 tan 0 tan 70.tan 50.tan 0 consider tan 70 tan 50 0 0 0 0 0 0 0 tan 70 = tan 50 0.tan 0 0 0 0 0 0 0 0 0 0 0 tan 70 tan 70.tan 50.tan 0 ta tan 50 tan 0 tan 70 tan 50.tan 0 tan 50 tan 0 n 50 tan 0 0 0 tan 70 tan 50 tan 0 tan 70.tan 50.tan 0 0 0 0 0 0 0 9 Prove that tan tan cot 7 3 LHS tan tan 7 3 tan 7 3 7 3 0 tan 90 tan 9 9 cot RHS Attempt any FOUR of the following: Prove that cos A cos A LHS cos A cos A A 6 Page 4 of 30
SUMMER 8 EXAMINATION N. 4. a) b) c) LHS cos Acos A sin Asin A cos Asin cos A cos A A cos A RHS 3 8 77 If tan y and tan y, show that tan 4 5 36 LHS tan y y y tan y y y tan tan = tan tan 3 8 = 4 5 3 8 4 5 77 36 RHS In any ABC, prove that tan A tan B tan C tan A.tan B.tan C In any ABC A B C 0 80 or 0 A B 80 C 0 A B C tan tan 80 tan A tan B tan C tan A.tan B tan A tan B tan C tan A.tan B tan A tan B tan C tan A.tan B.tan C tan A tan B tan C tan A.tan B.tan C Page 5 of 30
SUMMER 8 EXAMINATION N. 4. d) cos A cos 4A cos6a Prove that cos A tan 3 A.sin A cos A cos3a cos5a cos A cos 4A cos6a LHS cos A cos3a cos5a cos A cos6a cos 4A cos A cos5a cos3a A 6A A 6A.cos.cos cos 4A = A 5A A 5A.cos.cos cos3a cos 4 A.cosA cos 4A = cos3 A.cos A cos3 A A A cos 4Acos cos3acos cos3a A cos3a cos 3 A.cos A sin 3 A.sin A cos3a cos3a cos A tan 3 A.sin A RHS e) 4 33 5 3 65 4 Let cos A 5 4 cos A 5 sin A cos A Prove that cos cos cos without using calculator. 6 5 9 5 3 sin A 5 Page 6 of 30
SUMMER 8 EXAMINATION N. 4. e) cos B 3 cos B 3 sin B cos 44 69 B 5 69 5 sin B 3 cos A B cos Acos B sin Asin B 4 3 5 5 3 5 3 48 5 65 65 33 cos A B 65 33 A B cos 65 4 33 cos cos cos 5 3 65 4 Let cos A 5 4 cos A 5 3 tan A 4 3 A tan 4 cos 4 3 tan 5 4 OR Page 7 of 30
SUMMER 8 EXAMINATION N. 4. e) f) cos B 3 cos B 3 5 tan B 5 B tan 5 cos tan 3 3 5 L. H. S. tan tan 4 3 5 tan 4 35 4 56 tan 33 56 Let tan C 33 56 tan C 33 56 65 33 cos C 65 33 33 C cos 65 4 33 cos cos cos 5 3 65 Prove that tan tan 3 4 LHS tan tan 3 Page 8 of 30
SUMMER 8 EXAMINATION N. 4. 5. f) tan tan tan 3 3 5 6 6 4 Attempt any FOUR of the following: 6 a) C D C D Prove that sin Csin Dsin cos We know that, sin( A B) sin( A B) sin Acos B Put C D A C D B A B C A B D and C D C D sin C sin D sin cos b) sin sin 5 sin 9 sin3 Prove that cot 4 cos cos 5 cos 9 cos3 sin sin 5 sin 9 sin3 LHS cos cos 5 cos 9 cos3 Page 9 of 30
SUMMER 8 EXAMINATION N. 5. b) c) LHS sin sin 9 sin 5 sin3 cos cos 9 cos 5 cos3 9 9 5 3 5 3.sin.cos.sin.cos = 9 9 5 3 3 5.sin.sin.sin.sin sin 5.cos4 sin 9.cos4 = sin 5.sin 4 sin 9.sin 4 cos4sin 5 sin 9 = sin 4 sin 5 sin 9 cos 4 = sin 4 cot 4 RHS - - - y Prove that tan tan y tan if 0, y 0 and y. y - - Let tan A & tan y B tan A y tan B d) Consider tan tan A tan B AB tan Atan B y y y y - A B tan - - - y tan tan y tan y Find the distance between two parallel line 3 y 7 0 and 3 y 6 0 L :3 y 7 0 & L :3 y 6 0 c 7 & c 6 Page 0 of 30
SUMMER 8 EXAMINATION N. 5. d) p c c c c OR p a b a b e) 6 7 7 6 = = 3 3 9 9 = = 0 0 9 = OR.846 0 Find the acute angle between the lines 3 4y 40 and 4 3y 40 f) For 34y 40 a 3 3 slope m = b 4 4 For 43y 40 slope m a 4 b 3 m m tan mm 3 4 4 3 3 4 4 3 = = tan 0 = or 90 Find the equation of a line passing through,5 and the point of intersection of y0 and y9. y 0, y 9 Page of 30
SUMMER 8 EXAMINATION N. 5. 6. f) a) y 0 y 9 3 9 3 y 3 y y point of intersection 3, 3, and given point =,5, Equation of line is y y y y y 3 3 5 3 3 y3 3 8 3 8 3 y y3 8 4 8 y Attempt any FOUR of the following: If m and m are the slope of two lines then prove that angle between two m m lines is tan mm 6 Let = Angle of inclination of L = Angle of inclination of L Page of 30
SUMMER 8 EXAMINATION N. 6. a) Slope of L is m tan Slope of L is m tan from figure b) tan tan is acute tan tan tan tan m m m m m m tan m m m m tan m m Find the equation of a line passing through the poing of intersection of lines y 5 0 and 3y 0 and parallel to the line 34y 0. y 5 3 3y 0 3 6y 5 + 6y 0 5 35 7 7 y 5 y y Point of intersection 7, Page 3 of 30
SUMMER 8 EXAMINATION N. 6. b) c) Slope of the line 34y 0 is, a 3 m b 4 Slope of the required line is, 3 m m 4 equation of line is, y y m 3 y 7 4 3 4y 5 0 The runs scored by two batsman A and B in 5 one day matches are are given below. A 48 50 39 46 37 B 50 5 60 55 53 Who is more consistent? Why? For Batsman A i d i i d i 48 4 6 50 6 36 39-5 5 46 4 37-7 49 i 0 i 0 Mean, 44 N 5 di 30 SD.. 5.099 OR N 5 di 30 Page 4 of 30
SUMMER 8 EXAMINATION N. 6. c) For Batsman A i i 48 3 50 500 39 5 46 6 37 369 i 0 i 0 Mean, 44 N 5 i 980 S D N 5 i 980.. 44 5.099 For Batsman B i d i i d i 50-4 6 5-4 60 6 36 55 53 - i 70 70 Mean, 54 N 5 i d i 58 SD.. 3.406 N 5 d i 58 OR For Batsman B Page 5 of 30
SUMMER 8 EXAMINATION N. 6. c) d) i 70 Mean, 54 N 5 S D N 5 i 4638.. 54 3.406 For Batsman A C. V. A 00 5.099 = 00 44 =.589% For Batsman B C. V. B 00 3.406 = 00 54 =6.307% C. V. B < C. V. A i i i 50 500 5 7 60 3600 55 305 53 809 70 Batsman B is more consistent. i 4638 Calculate mean and standard deviation of the following frequency distribution. Class 0-0 0-0 0-30 30-40 40-50 Frequency 4 3 7 5 Page 6 of 30
SUMMER 8 EXAMINATION N. 6. d) Class i f i f i i i f i i 0-0 5 4 70 5 350 0-0 5 3 345 5 575 0-30 5 7 675 65 6875 30-40 35 735 5 575 40-50 45 5 675 05 30375 f i i 500 Mean 5 N 00 f i i S.D. N OR 78500 00.649 5 00 500 78500 Class 0-0 5 4 - -8 4 56 0-0 5 3 - -3 3 0-30 5 7 0 0 0 0 30-40 35 40-50 45 5 30 4 60 00 00 60 fd i i 0 Mean A h 5 0 5 N 00 fidi fidi S. D. h N N 60 0 0 00 00.649 Page 7 of 30
SUMMER 8 EXAMINATION N. 6. e) f) Find the mean deviation from mean of the following distribution. Marks 0-0 0-0 0-30 30-40 40-50 of students 5 8 5 6 06 Marks i i fi i 0-0 5 5 5-0 0-0 5 8 0-96 0-30 5 5 375-30 30-40 35 6 560 8 8 8 40-50 45 06 70 8 8 08 Mean M.D. 350 50 7 47 50 9.44 f i f i i i f i f i 50 350 47 Find variance and the coefficient of variance for the following distribution. Class-Interval 0-0 0-30 30-40 40-50 50-60 60-70 Frequency 4 6 0 8 9 3 Page 8 of 30
SUMMER 8 EXAMINATION N. 6. f) Class 0-0 5 4 60 5 900 0-30 5 6 50 65 3750 30-40 35 0 350 5 50 40-50 45 8 80 05 36450 50-60 55 9 495 305 75 60-70 65 3 95 45 675 50 060 9350 mean f i i 060 N 50 4. S.D..94 f N i i 9350 50 Variance.94 67.44 4. 865 697.44 67.56 S.D. C.V. 00 Mean. 94 00 3.4 4. OR Page 9 of 30
SUMMER 8 EXAMINATION N. 6. f) Class 0-0 5 4 - -8 4 6 0-30 5 6 - -6 6 30-40 35 0 0 0 0 0 40-50 45 8 8 8 50-60 55 9 8 4 36 60-70 65 3 3 9 9 7 50 3 03 fd i i 3 Mean A h 35 0 4. N 50 fidi fidi S. D. h N N 03 3 0 50 50.94 Variance.94 67.44 S.D..94 C.V. 00 00 3.4 Mean 4. Important Note In the solution of the question paper, wherever possible all the possible alternative methods of solution are given for the sake of convenience. Still student may follow a method other than the given herein. In such case, first see whether the method falls within the scope of the curriculum, and then only give appropriate marks in accordance with the scheme of marking. Page 30 of 30