Important Instructions to the Examiners:

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1 Summer 0 Examnaton Subject & Code: asc Maths (70) Model Answer Page No: / Important Instructons to the Examners: ) The Answers should be examned by key words and not as word-to-word as gven n the model answer scheme. ) The model answer and the answer wrtten by canddate may vary but the examner may try to assess the understandng level of the canddate. ) The language errors such as grammatcal, spellng errors should not be gven more mportance. (Not applcable for subject Englsh and Communcaton Sklls.) ) Whle assessng fgures, examner may gve credt for prncpal components ndcated n the fgure. The fgures drawn by the canddate and those n the model answer may vary. The examner may gve credt for any equvalent fgure drawn. 5) Credts may be gven step wse for numercal problems. In some cases, the assumed constant values may vary and there may be some dfference n the canddate s Answers and the model answer. 6) In case of some questons credt may be gven by judgment on part of examner of relevant answer based on canddate s understandng. 7) For programmng language papers, credt may be gven to any other program based on equvalent concept.

2 Subject & Code: asc Maths (70) Page No: / ) a) Attempt any TEN of the followng: Solve to fnd the value of x, f 6 x 0 x ( x x) ( x x) x x 0 6x x 60 or 6x 60 x 0 6 x x x 8 + 6x x x b) 8x x 6x x x 60 or 6x 60 x 0 If A 7 and 6, fnd A. 6 9 A A 8 9 A

3 Subject & Code: asc Maths (70) Page No: / ) c) 5 0 If A, 6 0, fnd the matrx A I, where I s dentty matrx. 5 0 A A I A I d) If A,, verfy that T T T A + A + A

4 Subject & Code: asc Maths (70) Page No: / ) T T A by and, T ( A )...( ) ( ) ( ) T T T A + A + ( ) e) T ( A + ) T T A by and, ( ) ( ) T T T T A + A + T ( ) ( ) Resolve nto partal fractons. x A + x x + x x + x ( x) A + + x Put x 0 x + A + 0 A

5 Subject & Code: asc Maths (70) Page No: 5/ ) Put + x 0 x x x + x Note for partal fracton problems: The problems of partal fractons could also be solved by the method of equatng equal power coeffcents. Ths method s also applcable. Gve approprate marks n accordance wth the scheme of markng n the later problems as the soluton by ths method s not dscussed. For the sake of convenence, the soluton of the above problem wth the help of ths method s llustrated hereunder. A + x x + x x + x ( x) A ( x) ( A ) ( A ) x x A + + A x A + and A 0 A + + A 0 A A A + x x + x

6 Subject & Code: asc Maths (70) Page No: 6/ ) f) Wthout usng calculator fnd the value of sn ( 0º ) ( ) sn 0º sn 0º ( ) sn 60º 0º sn 0º sn 0º or 0.5 sn 0º sn 0º sn 70º + 60º sn 90º + 60º + cos 60º or 0.5 ( ) ( + ) sn ( 0º ) sn 0º sn 60º 0º 0.5 or g) Wrte the followng formulae: () sn ( A + ) and () cos ( A ) ) sn A + sn Acos + cos Asn ) cos A cos Acos + sn Asn h) If sn A, fnd sn A. sn A sn A sn A... * Note (*): Due to the use of advance scentfc calculator, wrtng drectly the step (*) s allowed. No marks to be deducted.

7 Subject & Code: asc Maths (70) Page No: 7/ ) Gven that sn A. A sn 0º sn A sn 0º sn 90º ) Evaluate cos 75º cos5º wthout usng calculator. cos 75º cos5º cos 75º + 5º + cos 75º 5º cos 90º + cos 60º 0 + or 0.5 cos 75º cos 0º + 5º cos0º cos 5º sn 0º sn 5º cos5º cos 5º 0º cos 5º cos0º + sn 5º sn 0º + + cos75º cos5º + or 0.5

8 Subject & Code: asc Maths (70) Page No: 8/ ) j) Prove that cos ( x) π cos x. Let cos x cosθ ( x) x cosθ θ cos ( π θ ) ( x) π x cos θ π cos x k) Fnd the slope of a lne passng through ponts (, ) and (, 8). y y 8 + slope m x x + 5 l) Fnd the range and the coeffcent of range for the followng data: 0, 00, 0, 50, 50. Smallest Value S 50, L argest Value L 50 Range L S L S Coeff. of Range L + S or 0.5 ) a) The voltage n an electrc crcut are related by the followng equatons: V + V + V 9, V V + V, V + V V. Fnd V, V and V. Note: As n ths problem the method of soluton s not mentoned/prescrbed and as the problem s to be solved wthn the prescrbed currculum only, the problem can be solved by two dfferent methods: Cramer s Method and Inverse Matrx Method. ut the problem s not supposed to be solved by the method of smultaneous lnear equaton as prescrbed n school algebra.

9 Subject & Code: asc Maths (70) Page No: 9/ ) V + V + V 9 V V + V V + V V D D D D 9 V V V D D D D D D 8 6 V + V + V 9 V V + V V + V V V 9 A, X V, V A + +

10 Subject & Code: asc Maths (70) Page No: 0/ ) The cofactor matrx of A s, C ( A) 0 0 (*) 0 0 adj ( A) 0 0 adj ( A) A A the soluton s, X A V, V, V Note: ) (*) In the matrx C(A), f to elements are wrong (ether n sgn or value), deduct mark, f to 6 elements are wrong, deduct marks, f 7 to 9 are wrong, deduct all the marks. Further, f all the elements n the last.e., adj (A) are correct, then only gve mark.

11 Subject & Code: asc Maths (70) Page No: / ) Note ) To fnd the adj (A), there are varous methods are prescrbed n the MSTE Currculum whch are dscussed hereunder for the sake of convenence for marks dstrbuton. The matrx of mnors s, M ( A) the matrx of cofactors s, 0 C ( A) adj ( A) 0 0 A A A A A A A A A Note: In the above, f to elements are wrong, deduct mark, f to 6 elements are wrong, deduct marks, and f 7 to 9 are wrong, deduct all the marks. Further, f all the elements n the followng matrces C(A) and adj (A) are correct, then only gve the marks.

12 Subject & Code: asc Maths (70) Page No: / ) the matrx of cofactors s, 0 C ( A) adj ( A) 0 0 b) x 5 If A y and y x, y. and f A, fnd Gven A x 5 y y 9 6 x 6 5 y y 9 6 x y + 5 and 6y 6 x and y + c) If 0 A, show that A I. A 0 0 A A I

13 Subject & Code: asc Maths (70) Page No: / ) d) Usng matrx nverson method, solve the equatons: 5x + y, x + y 5. d) 5x + y x + y 5 5 x A, X, y 5 5 A 0 7 C ( A) 5 adj ( A) 5 A adj ( A) A 7 5 thesoluton s, X A x, y Note: To fnd the adj (A), students may follow any of methods as shown n the queston (a). Please gve approprate marks, as per scheme of markng dscussed n the queston (a). x + x + x x + x + Resolve nto partal fractons: x + x + A C + + x x + x + x x + x + x + x + x x + x A C x x + x +

14 Subject & Code: asc Maths (70) Page No: / ) x x x x A x x x x C Put x A A A Put x Put x ( ) ( ) 0 0 ( )( ) C C C x + x x x + x + x x + x + f) Resolve nto partal fractons: ( x + x + ) x + x x + x A x + C + x + x + ( x + )( x + ) A x + C x + x x + x + + x + x x x x A x x C Put x + + A A 6 A

15 Subject & Code: asc Maths (70) Page No: 5/ ) Put x 0 A ( C) A + C C 6 C C Put x A ( C) A + + C x + x 6 7x + + x + x + ( x + )( x + ) Note for Partal Fracton Methods: The above Q. (e) & (f) problems of partal fractons could be solved by the method of equatng equal power coeffcents also. Ths method, llustrated n the soluton of Q. (e), s also applcable. Gve approprate marks n accordance wth the scheme of markng. As ths method s very tedous and complcated, hardly someone use ths method n such cases. So such soluton methods for partal fracton problems are not llustrated heren. ) a) Attempt any four If A, show that A 8A s a scalar matrx A A A A A +

16 Subject & Code: asc Maths (70) Page No: 6/ ) A A s a scalar matrx. A A A A A 8A A 8 A s a scalar matrx.

17 Subject & Code: asc Maths (70) Page No: 7/ ) b) Resolve nto partal fractons: ( )( x + x + ) x Put x y x y A + y + y + y + y + ( x + )( x + ) A y ( y + )( y + ) + y + y + y y + A + y + Put y + A + 0 A Put y y + y + y + y + y + c) x + x + x + ( x + )( x + ) x + Resolve nto partal fractons: x x + x + A C x x x x x A C x + x ( x + ) + + x x x x x x A x x C Put x Put x C C

18 Subject & Code: asc Maths (70) Page No: 8/ ) Put x A + + C A + A + A A A C x x x + x x x + d) tan A tan Prove that tan ( A ) + tan Atan e) tan sn cos ( A ) ( A ) ( A ) sn Acos cos Asn cos Acos + sn Asn sn Acos cos Asn cos Acos cos Acos + sn Asn cos Acos sn Acos cos Asn cos Acos cos Acos cos Acos sn Asn + cos Acos cos Acos tan A tan + tan A tan Prove that sn A + sn A + sn A tan A cos A + cos A + cosa sn A + sn A + sn A sn A + sn A + sn A cos A + cos A + cosa cos A + cosa + cos A sn A cos A + sn A cos Acos A + cos A ( A) ( A) sn A cos + cos A cos + sn A cos A tan A

19 Subject & Code: asc Maths (70) Page No: 9/ ) ( sn A + sn A) + ( sn A + sn A) sn A + sn A + sn A cos A + cos A + cosa cos A + cos A + cos A + cosa A A 5A A sn cos + sn cos A A 5A A cos cos + cos cos A A 5A cos sn sn + A A 5A cos cos + cos A 5A sn + sn A 5A cos + cos A sn A cos A cos Acos tan A f) 9 Prove that tan + tan cot tan cot tan tan tan...(*) tan...(**) Note: Due to advance use of calculator, students may drectly wrte the step (**) after step (*) whch s permssble.

20 Subject & Code: asc Maths (70) Page No: 0/ ) Attempt any four: a) Prove that + + cos θ cosθ ( θ ) + + cos θ + + cos ( θ ) + cos + cos + cos θ ( θ ) + cos ( θ ) cos θ b) cos θ cosθ Prove that cos θ cos θ cosθ cos θ cos θ + θ cosθ cos θ snθ sn θ ( ) ( ) cosθ cos θ snθ snθ cosθ cosθ cos θ sn θ cosθ θ θ θ θ cos cos cos cos cos θ cosθ c) sn 7x + sn x Prove that sn x cos x cot x cos5x cosx 7x + x 7x x sn cos sn 7x + sn x cos5x cosx 5x + x 5x x sn sn sn cos sn cos sn x ( x) ( x) ( x) sn ( x) ( x)

21 Subject & Code: asc Maths (70) Page No: / ) d) ( x + x) cos sn x cos x cos x sn x sn x sn x cos x cos x sn x sn x sn x sn x cot x cos x + sn x sn x cot x cos x Prove that sn 0º sn 0ºsn 60º sn80º 6 sn 0º sn 0º sn 60ºsn80º sn 0º sn 0º sn80º ( sn 0º sn 0º ) sn 80º ( cos 60º cos 0º ) sn80º cos 0º sn 80º sn 80º sn 80º cos 0º sn 80º sn 80º cos 0º sn 80º ( sn00º sn 60º ) + sn 80º sn00º 8 cos90ºsn 0º Note: ) If the above problem s proved, usng the values of sn 0º, sn 0º, sn80º wth the help of calculator, no marks to be gven because under the constrant of the MSTE Currculum, t s expected that such problems are to be solved wthout usng calculator.

22 Subject & Code: asc Maths (70) Page No: / ) Note ) The above problem may also be solved by makng varous combnatons of sne ratos. Consequently the solutons vary n accordance wth the combnatons. Please gve the approprate marks n accordance wth the scheme of markng. For the sake of convenence one of the solutons s llustrated hereunder. e) sn 0ºsn 0º sn 60º sn80º sn 0º sn 0º sn 80º ( sn 0ºsn 80º ) sn 0º ( cos0º cos 0º ) sn 0º ( cos ( 90º + 0º ) cos 0º ) sn 0º ( sn 0º cos 0º ) sn 0º cos 0º sn 0º sn 0º sn 0º cos 0º sn 0º sn 0º cos 0º sn 0º sn 60º sn ( 0º ) + + sn 0º + sn 0º Prove that cos + tan tan 5 5 Let A cos tan 5 5 cos A and tan 5 5

23 Subject & Code: asc Maths (70) Page No: / ) tan A (*) A tan cos + tan tan + tan tan tan tan Note: To evaluate value of tan A, varous methods are used by students, such as usng the relaton between sn A and tan A or frst to fnd sn A usng cos A and fnd tan A etc., nstead of usng Trangle Method as llustrated n the above soluton. As man object s to fnd the value of tan A, please consder these methods also. f) tan + tan + tan π Prove that π + + tan tan tan tan tan π + + tan tan π + π tan tan

24 Subject & Code: asc Maths (70) Page No: / ) + tan + tan + tan ( ) tan + π + tan π π + + tan tan + tan tan π 5) Attempt any four: a) sn A + sn A sn A sn Prove that sn ( A + ) sn ( A ) sn sn A + A [ cos [( A + ) + ( A ) ] cos [( A + ) ( A ) ]] [ cos A cos ] sn sn A + sn A sn ( A + ) sn ( A ) [ sn Acos cos Asn ][ sn Acos cos Asn ] ( sn Acos ) ( cos Asn ) sn + sn Acos cos Asn sn A sn sn A sn sn A sn Asn sn + sn Asn sn A sn b) C + D C D Prove that sn C + sn D sn cos We know that, sn Acos sn A + + sn A Put A + C A D

25 Subject & Code: asc Maths (70) Page No: 5/ 5) C + D C D A and C + D C D sn C + sn D sn cos c) x y Prove that tan x tan y tan + +, f xy <. xy Put tan x A and tan y x tan A and y tan tan A + tan tan ( A + ) tan A tan x + y xy A tan x + y xy + x + y x y xy tan + tan tan d) Prove that f θ s the acute angle between the lnes wth slopes m m m and m, then tanθ. + m m Let θ Angle of nclnaton of L θ Angle of nclnaton of L Slope of L s m tanθ Slope of L s m tanθ

26 Subject & Code: asc Maths (70) Page No: 6/ 5) from fgure, θ θ θ tanθ tan θ θ ( ) ( θ ) tan ( θ ) ( θ ) ( θ ) tan + tan tan m m + m m For angletobe acute, m m tanθ + m m e) Fnd the equaton of the lne whch passes through the pont of ntersecton of the lnes x + y, 5x y 7 and perpendcular to the lne x 5y x + y, 5x y 7 x + y 5 x y 7x x y 0 ( x ) Pont of ntersecton, Slope of lne x 5y s m the equaton s, 5 y y 6 5x + 0 A 5 5 Slope of requred lnes m y y m x x 5x + y 6 0 or 5x + y 6 5

27 Subject & Code: asc Maths (70) Page No: 7/ 5) f) Fnd the length of the perpendcular from (, ) on the lne x 6y 5 0. Gven x 6y 5 0 A, 6, C 5 the length of the perpendcular s, p Ax + y + C A + 6) + ( 6) or Note: If ve sgn s left wth the answer, mark s to be deducted. Attempt any four: + a) Fnd the perpendcular dstance between the parallel lnes 5x y + 0 and 0x y. Gven 5x y + 0 and 0x y 0x y + 0 and 0x y 0 A 0,, C and C p C A C or Note: If the ve value s wrtten by the student (.e., or 0.5, deduct mark. 6 + Gven 5x y + 0 and 0x y 5x y + 0 and 5x y 0

28 Subject & Code: asc Maths (70) Page No: 8/ 6) A 5,, C and C + C C p A or b) Fnd the equaton of straght lne passng through the ponts, 6 8,. ( ) and y y x x y y x x y 6 x y 6 x ( y ) ( x ) y 7 9x 6 9x + y 6 0 or 9x y or x + y 0 or x y + 0 y y 6 slope of lne s m x x 8 + the equaton s, y y m x x y 6 ( x + ) or y + ( x 8) y x or y + x + x + y 0 or x y + 0

29 Subject & Code: asc Maths (70) Page No: 9/ 6) c) Fnd the mean devaton from mean for followng dstrbuton: of Students Class x f f x D x x fd d) fx 50 x 7 N 50 fd M. D. N Fnd the S. D. of followng data: of Students Class x f f x x fx fx fx S. D. N N

30 Subject & Code: asc Maths (70) Page No: 0/ 6) Class x f d f d d f d x A A 5 h 0, d h fd fd S. D. h N N Note: Students may take any another value for A n the above/ below example. So the above table and correspondng values vary accordngly. ut the fnal answer wll be the same. e) The two sets of observatons are gven below: Set I Set II x 8.5 x 8.75 σ 7. σ 8.5 Whch of the two sets s more consstent? σ 7. C. V. ( I ) x 8.5 σ 8.5 C. V. ( II ) x 8.75 C. V. I < C. V. II Set I s more consstent.

31 Subject & Code: asc Maths (70) Page No: / 6) f) Fnd the varance and coeffcent of varance of the followng: Class Interval Frequences 7 5 Class x f f x x fx fx 500 x 5 N 00 fx fx S. D. N N Varance ( S. D. ) S. D. Coeff. of Varance 00 x fx fx Varance N N S. D. Coeff. of Varance 00 x

32 Subject & Code: asc Maths (70) Page No: / 6) Class x f d f d d f d x A A 5, h 0, d h fd x A + h N fd fd S. D. h N N Varance ( S. D. ) S. D. Coeff. of Varance 00 x f d f d Varance h N N 60 0 h

33 Subject & Code: asc Maths (70) Page No: / 6) S. D..69 Coeff. of Varance x Important Note In the soluton of the queston paper, wherever possble all the possble alternatve methods of soluton are gven for the sake of convenence. Stll student may follow a method other than the gven heren. In such case, FIRST SEE whether the method falls wthn the scope of the currculum, and THEN ONLY gve approprate marks n accordance wth the scheme of markng.

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