ANSWERS EXERCISE 7.1 EXERCISE C. 1 sin 3 x 3. 1 e cos 2x 1 ( ) ax+ b. ax bx. x + C. + x

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1 5 MATHEMATIS ANSWERS EXERISE.. cos. ( ) a+ b a 5. sin. cos e e e a b + + c + + e log sin + e cos+ +. tan + sec + tan + 0. tan sec +.. A EXERISE. log ( + ) +.. cos (cos ) + 5. (log ) +. log + log + cos( a+ b ) + a ( a+ b ) + a 5 ( + ) ( + ) + 5

2 ANSWERS 59.. ( + ) + ( ) + + ( ) log +. 7 ( ) + ( ) (+ ). (log ) m m + 5. log 9 e e e tan + log( e + ) + e 0... log ( e e ) + +. tan(7 ) +. log sin + cos + 5. tan( ) + (sin ) + ( tan ) + sin + (sin ) +. +sin +. (logsin ) + 0. log (+cos ). log cos + sin +. +cos + log cos sin +. tan + 5. ( log ) + + ( + log ) + cos(tan ) +. D B

3 590 MATHEMATIS... EXERISE sin ( + 0) +. sin sin sin cos(+ ) + cos (+ ) + 5. cos cos cos + cos 7+ cos + cos cos + sin sin +. tan + tan + 0. sin+ sin+. + sin + sin+. sin +. (sin + cosα) +. cos +sin 5. sec sec + tan + + sec cosec +. tan + log tan + tan + 0. log cos + sin + cos( a) +.. log sin( a b) cos( b). A. B + EXERISE. tan +. log + + +

4 ANSWERS 59. log sin tan + + log + log + +. log tan + tan log + + a + log tan +. sin log a+ b log + ( a)( b) + sin log log + + tan log sin log log 5 + log + +

5 59 MATHEMATIS log B 5. B EXERISE 5. ( + ) log + +. log + +. log 5log + log +. log log + log + 5. log + log log log ( + ) + tan + log ( ) 5 log + log log log + log + + log + + log + + log +. log + log ( + ) + tan + + log log + + log log + 5. log + n n + n + tan tan + log tan + + sin log sin + + log + +

6 ANSWERS e log +. log + e. B. A EXERISE. cos + sin +.. e ( + ) log + ( ) sin + +. cos ( ) + 0. ( ) sin + sin + cos+ sin + 9 log + log + 9 tan + tan cos + +. tan + log cos + tan log( + ) +. + log + e 9 sin + (log ) log + + e + +. e tan + e ( ). e 5 (sin cos ) +. tan log ( + ) +. A. B e

7 59 MATHEMATIS EXERISE sin +. sin + + (+) log (+) + + log sin (+) log ( ) + + sin log log A. D EXERISE. ( ) b a e e 5+ e EXERISE 9... log e (e )

8 ANSWERS 595 log. log log log. log + tan (e ) log log log D. EXERISE 0. log.. log. ( + ) log 7 0. B. e ( e ) D EXERISE ( n+ )( n+ ). log 5 0. log.

9 59 MATHEMATIS log. a MISELLANEOUS EXERISE ON HAPTER 7. log +. ( ) ( ) + a + b + ( a b). ( a ) +. a log( + ) + log + + log ( + 9) + tan + sin alog sin ( a) + cosa+. sin sin sin +. cos ( + b) log sin ( a b) cos( + a) +. sin ( ) +. + e log + + e. tan tan + 5. cos + log( + ) + [ f ( a+ b)] an ( +) n+ +. sinα sin ( + α) sin + ( ) sin + +

10 ANSWERS cos + +. e tan +. log + + log cos +. + log e. ( ) sin 0. log9 0.. ( ) A. B. D. B e e EXERISE a. () A. B

11 59 MATHEMATIS. EXERISE. 9 + sin B B Miscellaneous Eercise on hapter. (i) 7 (ii) a m ab ( ) ( ) sin + D. B EXERISE. Order ; Degree not defined. Order ; Degree. Order ; Degree. Order ; Degree not defined 5. Order ; Degree Order ; Degree Order ; Degree. Order ; Degree Order ; Degree 0. Order ; Degree. D. A EXERISE. D. D

12 ANSWERS 599 EXERISE. y = 0. y y + (y )² y y = 0. y y y = 0. y y + y = 0 5. y y + y = 0 yy + = y y y = 0. yy + (y )² yy = 0 yy + (y )² yy = 0 0. (² 9) (y )² + ² = 0. B. EXERISE. y= tan +. y = sin ( + ). y = + Ae. tan tan y = 5. y = log (e + e ) + tan y= + + y = e c. + y = y = sin tan y = ( e ). y= log ( ) ( ) tan y y = log. cos = a. y = sec 5. y = e ( sin cos ) y + = log ( (y + ) ) y =. ( + ) = y + (t + 7) 0. 9%. Rs. log log 0. A EXERISE 5. y =. = log + ( y) e y

13 00 MATHEMATIS. 5. tan y = log( + y ) + + y log = log + y. + y = y+ + y = y cos y =. y y cos = sin y cy = log 0. ye + =. log ( + y ) + tan y = log + y. y + = y. cot = log e y. cos = log e 5. y= ( 0, e) log D y EXERISE. y = 5 (sin cos ) + e. y = e + e. y = +. y (sec + tan ) = sec + tan + 5. y = (tan ) + e tan ylog = (+ log ) +. y = (log ) + y=(+ ) log sin + ( + ) y= cot + sin 0. ( + y + ) = e y. y = + y. = y + y

14 ANSWERS 0. y = cos cos. y ( + ) = tan 5. y = sin sin + y + = e y = e. D Miscellaneous Eercise on hapter 9. (i) Order ; Degree (ii) Order ; Degree (iii) Order ; Degree not defined. y y = 5. ( + yy )² = ( y) y ( + (y ) ) sin y + sin =. cos y = sec tan y + tan (e ) = y 0. e = y+. log y = + y+. ye = ( + ). ysin = (sin 0) EXERISE 0. + y= log, +. In the adjoining figure, the vector OP represents the required displacement.

15 0 MATHEMATIS. (i) scalar (ii) vector (iii) scalar (iv) scalar (v) scalar (vi) vector. (i) scalar (ii) scalar (iii) vector (iv) vector (v) scalar. (i) Vectors a and b are coinitial (ii) Vectors b and d are equal (iii) Vectors a and c are collinear but not equal 5. (i) True (ii) False (iii) False (iv) False. a =, b =, c = EXERISE 0.. An infinite number of possible answers.. An infinite number of possible answers.. =, y = 5. 7 and ; 7iˆand ˆj ˆj kˆ i ˆ + ˆ j+ k ˆ. 0.. i ˆ + ˆ j+ k ˆ 0 i ˆ ˆ j+ k ˆ ,, 5. (i) i ˆ + k ˆ,, i ˆ + ˆ j+ k ˆ (ii) iˆ + kˆ iˆ+ ˆj+ kˆ. () (D) EXERISE 0... cos a =, b =, a + ab. 5b

16 ANSWERS 0. Vector b can be any vector.. Take any two non-zero perpendicular vectors a and b 5. cos (D) EXERISE 0. ± i ˆ ˆ j k ˆ. 5. 7, Either a = 0or b = 0. No; take any two nonzero collinear vectors ;,, (B). () Miscellaneous Eercise on hapter 0. ˆ i + ˆj., y y, z z; ( ) + ( y y) + ( z z) 5. iˆ+ ˆj. No; take a, b and c to represent the sides of a triangle. 5. ± 0 0 iˆ+ ˆj iˆ ˆj+ kˆ. : a + 5 b 0. ( ˆ ˆ ˆ); 5 7 i j+ k. (0 iˆ 5 ˆj+ 70 kˆ ). λ = (B) (D). () (B)

17 0 MATHEMATIS EXERISE ,,. ±, ±, ±. 9,, 5,, ;,, ;,, EXERISE.. r = iˆ + ˆj + kˆ + λ ( iˆ + ˆj kˆ ), where λ is a real number 5. r = iˆ ˆj + kˆ + λ ( iˆ+ ˆj kˆ) and cartesian form is y + z = = + y z + 5 = = 5 r = (5iˆ ˆj+ kˆ) +λ ( iˆ+ 7 ˆj + kˆ). Vector equation of the line: r = λ (5iˆ ˆj + kˆ ); y z artesian equation of the line: = = 5 Vector equation of the line: r = iˆ ˆj 5 kˆ+λ( kˆ) artesian equation of the line: 0. (i) θ =. (i) θ =. 9 cos cos 9 70 p =. 9 9 y + z + 5 = = 0 0 (ii) θ = (ii) θ = cos 5 cos 5. 9

18 ANSWERS 05 EXERISE.. (a) 0, 0, ; (b) (c) 5,, ; (d) 0,, 0;,, ; iˆ 5ˆj kˆ. r + = (a) + y z = (b) + y z = (c) (s t) + ( t) y + (s + t) z = 5. (a),, (b) 5 0,, 5 5 (c),, (d) 0,, (a) [ r ( iˆ kˆ )] ( iˆ+ ˆj k ˆ) = 0; + y z = (b) [ r ( iˆ + ˆj + kˆ ) ] ( iˆ ˆj + k ˆ) = 0; y + z + = 0 (a) The points are collinear. There will be infinite number of planes passing through the given points. (b) + y z = 5 5, 5, 5. y = 7 5y + z = 0 r iˆ+ ˆj + kˆ = 5. z + = 0 0. ( ). 5 cos 7. (a) cos 5 (b) The planes are perpendicular (c) The planes are parallel (d) The planes are parallel (e) 5 o. (a) (b) (c) (d)

19 0 MATHEMATIS Miscellaneous Eercise on hapter. 90. y z = = cos k = 7 r = iˆ + ˆj + kˆ + λ ( iˆ + ˆj 5 kˆ). + y + z = a + b + c ,,. 7,0,. (,, 7). 7 y + z + 5 = 0. p = or 7 5. y z + = 0 + y z = 0 + 5y + 50 z = 0. r = iˆ+ ˆj + kˆ+λ( iˆ+ 5 ˆj + kˆ) 0. r = iˆ+ ˆj kˆ+λ (iˆ+ ˆj + kˆ). D. B. Maimum Z = at (0, ). Minimum Z = at (, 0). Maimum Z = 5 9 at 0 5, 9 9. Minimum Z = 7 at, 5. Maimum Z = at (, ) EXERISE. Minimum Z = at all the points on the line segment joining the points (, 0) and (0, ). Minimum Z = 00 at (0, 0); Maimum Z = 00 at all the points on the line segment joining the points (0, 0) and (0, 0).

20 ANSWERS 07. Minimum Z = 00 at all the points on the line segment joining the points (0, 50) and (0, 0); Maimum Z = 00 at (0, 00) Z has no maimum value 0. No feasible region, hence no maimum value of Z. EXERISE.. Minimum cost = Rs 0 at all points lying on segment joining,0 and,.. Maimum number of cakes = 0 of kind one and 0 cakes of another kind.. (i) tennis rackets and cricket bats (ii) Maimum profit = Rs 00. packages of nuts and packages of bolts; Maimum profit = Rs packages of screws A and 0 packages of screws B; Maimum profit = Rs 0 Pedestal lamps and wooden shades; Maimum profit = Rs Souvenir of types A and 0 of Souvenir of type B; Maimum profit = Rs units of desktop model and 50 units of portable model; Maimum profit = Rs Minimise Z = + y subject to + y 0, + y 00, 0 and y 0, where and y denote the number of units of food F and food F respectively; Minimum cost = Rs kg of fertiliser F and 0 kg of fertiliser F ; Minimum cost = Rs 000. (D) Miscellaneous Eercise on hapter. 0 packets of food P and 5 packets of food Q; Maimum amount of vitamin A = 5 units.. bags of brand P and bags of brand Q; Minimum cost of the miture = Rs 950. Least cost of the miture is Rs ( kg of Food X and kg of food Y).

21 0 MATHEMATIS 5. 0 tickets of eecutive class and 0 tickets of economy class; Maimum profit = Rs 000. From A : 0,50, 0 units; From B: 50,0,0 units to D, E and F respectively and minimum cost = Rs 50 From A: 500, 000 and 500 litres; From B: 000, 0, 0 litres to D, E and F respectively; Minimum cost = Rs bags of brand P and 00 bags of brand Q; Minimum amount of nitrogen = 70 kg. 0 bags of brand P and 50 bags of brand Q; Maimum amount of nitrogen = 595 kg dolls of type A and 00 dolls of type B; Maimum profit = Rs 000 EXERISE.. P( E F ), P( F E) = =. PA B ( ) = 5. (i) 0. (ii) 0. (iii) (i) (ii) 5 (i) (ii) 7 (i) (ii) 0.. (i). (i). 5, (iii) (iii) 7 0. (a), (b) 9 (ii) (ii),. (iii) 5 9, 5. 0 D

22 ANSWERS 09 EXERISE A and B are independent 5. A and B are not independent E and F are not independent (i) p = (ii) p = 0 5. (i) 0. (ii) 0.5 (iii) 0. (iv) A and B are not independent. (i) 0. (ii) 0. (iii) 0.7 (iv) (i) 0 0, (ii), (iii). (i), (ii) 5. (i), (ii) (a) 5, (b), (c) D. B EXERISE A EXERISE.. (ii), (iii) and (iv). X = 0,, ; yes. X =,,, 0. (i) X 0 P(X) (ii) X 0 P(X)

23 0 MATHEMATIS (iii) X 0 P(X) 5. (i) X 0 P(X) 9 9 (ii) X 0 P(X) 5 9 X 0 P(X) X 0. (i) (iv) P(X) 9 k = 0 P(0 < X < ) = 0 k = (b) (ii) 5 5 P(X < ) = (iii) 0 (a) P(X < ) =, P(X ) =, P(X ) = Var(X) = 5., S.D =.5. X P(X) P(X > ) = 00 Mean = 5, Var(X) =.7 and S.D(X) =.9 5. E(X) = 0.7 and Var (X) = 0. B D 5

24 ANSWERS EXERISE.5. (i) (ii) 7 (iii).. (i) (ii) (iii) 0 5. (i) (0.95) 5 (ii) (0.95). (iii) (0.95). (iv) (0.95) (a) (b) (c) A (i) (ii) 0. (i). 0 Miscellaneous Eercise on hapter (ii). 0 0 r 0 r r r = 7 (0.9) (0.) 5. (i) 5 (ii) 7 5 (iii) 5 (iv) 5

25 MATHEMATIS ,, n (i) 0.5 (ii) 0.05 A. B