WINTER 16 EXAMINATION

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1 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer ject Code: Important Instructions to examiners: ) The answers should be examined 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 examiner 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, examiner may give credit for principal components indicated in the figure. The figures drawn by candidate and model answer may vary. The examiner 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 answer and model answer. 6) In case of some questions credit may be given by judgement on part of examiner 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. 7 Q. wer Attempt any TEN of the following: 0 a) b) Find x, if x x x x x 9x 90 0 x 0 x x If A,, find 3 5, where is the unit matrix of 4 7 B 5 A B I I order two A 3B 5I Page 0/30

2 [[ Q. (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 b) c) = If A, show that is null matrix. A 4 4 A d) Resolve into partial fraction : Let A B x x x x A x Bx Put x0 A, Put x B x x x x x x 0 e) Prove that cos cos cos cos cos cos sin sin cos sin cos cos cos cos cos Page 0/30 0

3 [ Q. (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 f) Find sin, if tan 3 tan or or 3 tan 0 sin sin 60 or sin = or g) Without using calculator, find the value of sin 765 sin 765 sin = sin or sin 8 45 = sin 45 0 = or h) Find the principal value of sec cos. sec cos 3 0 sec30 or sec 6 or Page 03/30

4 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 i) Define compound angle. If A & B are any two angles, then sum or difference i.e A B or A B is compound angle. 0 j) Prove that the lines 3x y 5, and x 3y 6 are perpendicular. slope of 3xy 5 is 3 m slope of x3y 6 is m mm 3 lines are perpendicular. 0 k) l) Find the range & coefficient of range of the following data: 50, 90,0,40,80,00,80. Range L S L S Coefficient of range = = 0.67 L S Find AB if A, 3 B 0 5 AB [ 0 a) Attempt any FOUR of the following Solve the following equations using Cramer's rule x 3y 5, y 3z, z 3x 4 6 Page /30

5 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 a) b) x 3y 5, y 3z, 3x z D D D D x y z Dx 5 x D 5 Dy 5 y D 5 Dz 5 z D If A I 3, obtain the matrix A I A I A 3 I [ Page 05/30

6 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 b) c) A I = = = A I A I cos 0 sin Show that matrix A 0 0 is an orthogonal matrix. sin 0 cos cos 0 sin cos 0 sin T A 0 0 A 0 0 sin 0 cos sin 0 cos cos 0 sin cos 0 sin T AA sin 0 cossin 0 cos cos 0 sin cos sin sin cos = sin cos 0 cos sin sin 0 cos 0 0 = 0 0 I 0 0 A is an orthogonal matrix. Page 06/30

7 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 d) 3 Find the inverse of the matrix ; A 4 5 by adjoint method A A 0 A exists Matrix of minors Matrix of cofactors OR C C C C C , C , C , C , C , 4 3 Matrix of cofactors Page 07/30

8 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 d) 3 Adj. A A Adj. A A A e) Resolve into partial fraction x 3 x 4 x x x x 3 x 4 x 3 4x + 5 x 3 x x 5x x x x x 4 5x A B x x x x 5x x A x B x Put 5 0 A 4 5 A Put x 5 0 B 4 5 B 5 5 5x x x x x A B Page 08/30

9 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 e) f) x x 5 x or x x 4 x x x x x Resolve into partial fraction : x 4 x x 3x A B C x 4x x x 4 x x x A x x B x x C x x Put x 4 A A 9 3 A 7 A 7 Put x 3 B B 5 7 B 4 0 B 7 Put x C C C 9 0 3x x 4 x x x 4 x x Page 09/30

10 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 3 Attempt any FOUR of the following: 6 a) Using matrix inversion method solve the system of equations : x y z 3, 3x y 3z 4, 5x 5y z Let A A A A exists Matrix of minors Matrix of cofactors OR C C C C , C , C , C , C Page 0/30

11 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 3 a) C , Matrix of cofactors Adj. A A Adj. A A = x z = = x y = z x, y, z. y A B b) tan Resolve into partial fraction : tan tan 3 Let tan t t A B t t 3 t t 3 3 t t A t B Page /30

12 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 3 b) c) put t 3 A put t B B 3 A B t 3 t t 3 t t 3 tan 3 tan tan 3 tan tan 3 Resolve into partial fractions : x 3x A Bx C x3 x x3 x x3 x 3 3 x x x A x Bx C Put x A A A 6 Put x 0, A 6 C C C Put x, A 6, C 3 x 3x 6 3B 4 4B 8 B 7 x 3x 6 7x x3 x x3 x Page /30

13 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 3 d) e) Prove that tan tan 3 4 x y tan xtan y tan xy tan tan tan tan tan tan If tan A, tan B, Where 0 A, B, find sin A B 3 4 tan A, A A lies in first quadrant sin A tan B, 4 3 B 4 7 B lies in third quadrant sin B sin A B sin Acos B cos Asin B A B 3, cos A 0 0 4, cos B 7 7 Page 3/30

14 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 3 e) f) [ 4 3 = = or Without using calculator, find the value of : tan 585º. cot 495º cot 405º. tan 495º 0 tan 585º tan º tan 45º cot 495º cot 495º 0 0 cot º tan 45 cot 405º cot º 0 cot 45º tan 495º tan 495º cot 45º 0 tan º tan 585º. cot 495º cot 405º. tan 495º 0 Note: The above example may be proved in different ways by expressing the ratio in many ways e.g., instead of tan 585º tan 690º 45º, one can express it as expressing tan 585º tan 790º 45º and the get the desired value. Further here in this example it is expected that it must be proved without using calculator. If directly calculator is used, no marks to be given Page 4/30

15 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 4 Attempt any FOUR of the following: 6 a) Prove that : sin A B sin Acos B cos Asin B Right Angled Triangle Acute Angle Trigonometric Ratios PM OM OMP MOP = A sin A, cos A OP OP OPQ POQ = B PQ OP sin B, cos B OQ OQ PRQ QPR = A RQ PR sin A, cos A PQ PQ ONQ NOQ = A-B sin( ) QN ON A B, cos( A B) OQ OQ QN RM sin( A B) OQ OQ PM PR OQ PM PR OQ OQ PM OP PR PQ OP OQ PQ OQ sin Acos B cos Asin B. Page 5/30

16 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 4 a) OR Q R 7 P P S S x Consider a standard unit circle Let P,Q,R,S be points such that XOP A, XOQ B, XOR A B From fig. POQ A B POQ XOR P cos A,sin A, Q cos B,sin B Chord PQ Chord RS cos,sin, S,0 R A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B cos cos sin sin cos sin 0 cos cos sin sin cos sin 0 cos A cos B cos Acos B sin A sin B sin Asin B cos cos sin cos cos sin sin cos cos cos sin sin cos Replace B by B in above equation cos Acos B sin Asin B cos A B Consider sin A B cos A B cos A B cos Acos B sin Asin B sin Acos B cos Asin B Page 6/30

17 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 4 b) Prove that cos A cos A cos A cos A A cosa cos A sin Asin A cos Asin cos A cos cos A cos cos A A A A c) If tan x y and tan x y, find i tan x, ii tan y. 3 i tan x tan x y x y tan x y tan x y = tan x y tan x y = 3 3 = ii tan y tan x y x y tan x y tan x y = tan x y tan x y = 3 3 = Page 7/30

18 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 4 d) e) sin A sin A sin 3A sin 4A 5A Prove that tan cos A cos A cos 3A cos 4A L.H.S. sin A sin 4A sin A sin 3A cos A cos 4A cos A cos3a 5A 3A 5A A sin cos sin cos 5A 3A 5A A cos cos cos cos 5A 3A A sin cos cos 5A 3A A cos cos cos 5A sin 5A cos 5A tan R.H.S Prove that cos cos cos Let cos A 5 4 cos A 5 sin A cos A sin A cos 3 cos B B Page 8/30

19 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 4 e) sin B cos B sin B 3 cos A B cos Acos B sin Asin B cos A B A B cos cos cos cos OR 4 Let cos A 5 4 cos A 5 3 tan A 4 3 A tan cos tan 5 4 cos B 3 cos B 3 Page 9/30

20 [[[[[ Q. (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 4 e) 5 tan B 5 B tan 5 cos tan L. H. S. tan tan tan tan tan tan 56 Let tan C tan C cosc C cos C f) Prove that : tan tan tan tan tan tan tan tan Page 0/30

21 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 4 f) tan tan tan tan tan 5 7 tan tan tan tan 4 5 a) Attempt any FOUR of the following : x y Prove tan xtan y tan, xy if xy 0 put tan and tan tan and tan x A y B x A y B tan A tan B x y tan A B OR RHS tan tan Atan B xy x y tan A tan B = tan xy tan Atan B x y xy A B tan =tan tan A AB B x y xy tan x tan y tan = tan x tan y =LHS Page /30

22 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 5 b) c) Prove that sin0 sin 30 sin 50 sin 70 sin0 sin 30 sin 50 sin sin0 sin 50 sin sin0 sin 50 sin cos 40 cos 60 sin cos 40 sin cos 40 sin 70 sin 70 4 cos sin 70 sin sin0 sin 30 sin sin sin sin 70 sin C D C D Prove that sin Csin Dcos sin We know that sin A B sin A B cos A sin B Let A B C A B D A C D C D A C D B C D C D sin Csin D cos sin [ Page /30

23 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 5 d) Show that the distance between two parallel lines ax by C 0 & ax by C 0 is given by d C a C b L : ax by C 0 L : ax by C 0 Let P x, y be any point on the line L ax by C 0 ax by C PM is perpendicular on the line L e) PM d or d C ax by C a C a C b a C b b Find the length of the perpendicular on the line 3x 4y 5 0 from the point 3,4. p ax by c a b Page 3/30

24 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 5 e) f) = = = 5 0 = 5 p = 4 units Find the equation of the line passing through the point the intersection of lines x 3y 3, 5x y 7 0 and perpendicular to the line 3x y 7 0. x3y 3 5x y 7 x3y x3y 7x 34 x 5 y 7 y 3 y 3 Point of intersection, 3 Slope of the line 3x y 7 0. is, a 3 3 m0 b Slope of the required line is, m m equation is, y y m x x y 3 x 3 x 3y3 0 Page 4/30

25 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 6 a) Attempt any FOUR of the following: Find the equation of line passing through the point of intersection of lines x y 0 and x y 9 and a point,5 x y 0 x y 9 6 3x 9 x 3 y 3 Point of intersection 3, 3 equation is, y y y x x y x x y5 x x y 0 OR Point of intersection 3, 3 y y Slope m x x equation is, y y m x x 35 3 y 5 8 x OR y 3 8 x 3 8x y 0 8 b) Find the mean deviation from median of the following distribution: Weight (in gms) of items Page 5/30

26 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 6 b) Class xi f i cf D x Med i i fd i i N cf Median L h f =5 5 5 =8 fd i MD.. N 6.5 i c) Calculate : i) standard deviation ii) Co-efficient of variation from the following data: Rainfall of places Page 6/30

27 (ISO/IEC Certified) WINTER 6 EXAMINATION 6 c) Model wer wer ject Code: 7 Class d i xi a h fidi fidi i) S. D h N N fd i i ii) Mean x=a+ h N 70 = = Co-efficient of variation = 00 x 8.5 = OR Page 7/30

28 6 (ISO/IEC Certified) WINTER 6 EXAMINATION Class Model wer wer ject Code: fx i i 00 Mean x N 00 i) S.D. fx N i i x ii) Co-efficient of variation = 00 x 8.5 = Page 8/30

29 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 6 d) The weights of 00 students are given by the following distribution: Weight above or equal to of students Calculate: i) Mean,ii)Variance of the data using step deviation method No student has weight above 75 kg. Class Class boundries d i xi a h fd i i i) Mean x=a+ h N 3 = = 5.85 fidi fidi ii S. D. h N N =7.44 Variance Page 9/30

30 (ISO/IEC Certified) WINTER 6 EXAMINATION Model wer wer ject Code: 7 6 e) In the two factories A & B engaged in the same industry, the average weekly wages & standard deviation are as follows : Factories Average Wages Standard deviation A B Which factory is more consistent? For factory A CV. 00 x 5.0 = =4.49% For factory B CV. 00 x 4.5 = =5.79% C. V of A < C. V of B Factory A is more consistent 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 30/30

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