ANSWERS EXERCISE 1.1 EXERCISE 1.2

Transcription

1 ANSWERS EXERCISE.. (i), (iv), (v), (vi), (vii) and (viii) are sets.. (i) (ii) (iii) (vi) (v) (vi). (i) A = {,,, 0,,,, 4, 5, 6 } (ii) B = {,,, 4, 5} (iii) C = {7, 6, 5, 44, 5, 6, 7, 80} (iv) D = {,, 5} (v) E = {T, R, I, G, O, N, M, E, Y} (vi) F = {B, E, T, R,} 4. (i) { : = n and n 4 } (ii) { : = n and n 5 } (iii) { : = 5 n and n 4 } (iv) { : is an even natural number} (v) { : = n and n 0 } 5. (i) A = {,,, 5,... } (ii) B = {0,,,, 4 } (iii) C = {,, 0,, } (iv) D = { L, O, Y, A } (v) E = { February, April, June, September, November } (vi) F = {b, c, d, f, g, h, j } 6. (i) (c) (ii) (a) (iii) (d) (iv) (b) EXERCISE.. (i), (iii), (iv). (i) Finite (ii) Infinite (iii) Finite (iv) Infinite (v) Finite. (i) Infinite (ii) Finite (iii) Infinite (iv) Finite (v) Infinite 4. (i) Yes (ii) No (iii) Yes (iv) No 5. (i) No (ii) Yes 6. B= D, E = G EXERCISE.. (i) (ii) (iii) (iv) (v) (vi) (vii). (i) False (ii) True (iii) False (iv) True (v) False (vi) True. (i), (v), (vii), (viii), (i), (i) 4. (i) φ { a }, (ii) φ, { a }, { b } { a, b } (iii) φ, { }, { }, { }, {, }, {, }, {, } {,, } (iv) φ (i) ( 4, 6] (ii) (, 0) (iii) [ 0, 7 ) (iv) [, 4 ] 7. (i) { : R, < < 0 } (ii) { : R, 6 } (iii) { : R, 6 < } (iv) { R : < 5 } 9. (iii)

2 44 MATHEMATICS EXERCISE.4. (i) X Y = {,,, 5 } (ii) A B = { a, b, c, e, i, o, u } (iii) A B = { : =,, 4, 5 or a multiple of } (iv) A B = { : < < 0, N} (v) A B = {,, }. Yes, A B = { a, b, c }. B 4. (i) {,,, 4, 5, 6 } (ii) {,,, 4, 5, 6, 7,8 } (iii) {, 4, 5, 6, 7, 8 } (iv) {, 4, 5, 6, 7, 8, 9, 0) (v) {,,, 4, 5, 6, 7, 8 } (vi) (,,, 4, 5, 6, 7, 8, 9, 0} (vii) {, 4, 5, 6, 7, 8, 9, 0 ) 5. (i) X Y = {, } (ii) A B = { a } (iii) { } 6. (i) { 7, 9, } (ii) {, } (iii) φ (iv) { } (v) φ (vi) { 7, 9, } (vii) φ (viii) { 7, 9, } (i) {7, 9, } () { 7, 9,, 5 } 7. (i) B (ii) C (iii) D (iv) φ (v) { } {vi){ : is an odd prime number } 8. (iii) 9. (i) {, 6, 9, 5, 8, } (ii) {, 9, 5, 8, } (iii) {, 6, 9,, 8, } (iv) {4, 8, 6, 0 ) (v) {, 4, 8, 0, 4, 6 } (vi) { 5, 0, 0 } (vii) {0 ) (viii) { 4, 8,, 6 } (i) {, 6, 0, 4} () { 5, 0, 5 } (i) {, 4, 6, 8,, 4, 6} (ii) {5, 5, 0} 0. (i) { a, c } (ii) {f, g } (iii) { b, d }. Set of irrational numbers. (i) F (ii) F (iii) T (iv) T EXERCISE.5. (i) { 5, 6, 7, 8, 9} (ii) {,, 5, 7, 9 } (iii) {7, 8, 9 } (iv) { 5, 7, 9 ) (v) {,,, 4 } (vi) {,, 4, 5, 6, 7, 9 }. (i) { d, e, f, g, h} (ii) { a, b, c, h } (iii) { b, d, f, h } (iv) { b, c, d, e ). (i) { : is an odd natural number } (ii) { : is an even natural number } (iii) { : N and is not a multiple of } (iv) { : is a positive composit number and = ]

3 ANSWERS 45 (v) { : is a positive integer which is not divisible by or not divisible by 5} (vi) { : N and is not a perfect square } (vii) { : N and is not a perfect cube } (viii) { : N and = } (i) { : N and = } () { : N and < 7 } (i) { : N and > 9 } 6. is the set of all equilateral triangles. 7. (i) U (ii) A (iii) φ (iv) φ EXERCISE , Miscellaneous Eercise on Chapter. A B, A C, B C, D A, D B, D C. (i) False (ii) False (iii) True (iv) False (v) False (vi) True 7. False. We may take A = {, }, B = {, }, C = {, } , 0 6. EXERCISE.. = and y =. The number of elements in A B is 9.. G H = {(7, 5), (7, 4), (7, ), (8, 5), (8, 4), (8, )} H G = {(5, 7), (5, 8), (4, 7), (4, 8), (, 7), (, 8)} 4. (i) False P Q = {(m, n) (m, m) (n, n), (n, m)} (ii) False A B is a non empty set of ordered pairs (, y) such that A and y B (iii) True 5. A A = {(, ), (, ), (, ), (, )} A A A = {(,, ), (,, ), (,, ), (,, ), (,, ), (,, ), (,, ), (,, )} 6. A = {a, b}, B = {, y} 8. A B = {(, ), (, 4), (, ), (, 4)} A B will have 4 = 6 subsets. 9. A = {, y, z} and B = {,}

4 46 MATHEMATICS 0. A = {, 0, }, remaining elements of A A are (, ), (, ), (0, ), (0, 0), (, ), (, 0), (, ) EXERCISE.. R = {(, ), (, 6), (, 9), (4, )} Domain of R = {,,, 4} Range of R = {, 6, 9, } Co domain of R = {,,..., 4}. R = {(, 6), (, 7), (, 8)} Domain of R = {,, } Range of R = {6, 7, 8}. R = {(, 4), (, 6), (, 9), (, 4), (, 6), (5, 4), (5, 6)} 4. (i) R = {(, y) : y = for = 5, 6, 7} (ii) R = {(5,), (6,4), (7,5)}. Domain of R = {5, 6, 7}, Range of R = {, 4, 5} 5. (i) R = {(, ), (,), (, ), (, 4), (, 6), ( 4), (, 6), (, ), (4, 4), (6, 6), (, ), (, 6)} (ii) Domain of R = {,,, 4, 6} (iii) Range of R = {,,, 4, 6} 6. Domian of R = {0,,,, 4, 5,} 7. R = {(, 8), (, 7), (5, 5), (7, 4)} Range of R = {5, 6, 7, 8, 9, 0} 8. No. of relations from A into B = 6 9. Domain of R = Z Range of R = Z EXERCISE.. (i) yes, Domain = {, 5, 8,, 4, 7}, Range = {} (ii) yes, Domain = (, 4, 6, 8, 0,, 4}, Range = {,,, 4, 5, 6, 7} (iii) No.. (i) Domain = R, Range = (, 0] (ii) Domain of Function = { : } (iii) Range of Function = { : 0 }. (i) f (0) = 5 (ii) f (7) = 9 (iii) f ( ) = 4. (i) t (0) = (ii) t (8) = 4 (iii) t ( 0) = 4 (iv) (i) Range = (, ) (ii) Range = [, ) (iii) Range = R

5 Miscellaneous Eercise on Chapter ANSWERS Domain of function is set of real numbers ecept 6 and. 4. Domain = [, ), Range = [0, ) 5. Domain = R, Range = non-negative real numbers 6. Range = Any positive real number such that 0 < 7. (f + g) = 8. a =, b = 9. (i) No (ii) No (iii) No (f g) = + 4 f + =, g 0. (i) Yes, (ii) No. No. Range of f = {, 5,, } EXERCISE. 5π 9π 4π. (i) (ii) (iii) (iv) 6π (i) 9 0 (ii) (iii) 00 (iv) 0. π π 7. (i) 5 (ii) 5 (iii) EXERCISE : sin =, cosec =, sec =, tan =, cot = cosec =, cos =, sec =, tan =, cot = sin =, cosec =, cos =, sec =, tan = sin =, cosec =, cos =, tan =, cot = 5

6 48 MATHEMATICS sin =, cosec =, cos =, sec =, cot = EXERCISE (i) + (ii) EXERCISE π 4π π,,n π +, n Z. 5π π 5π,,n π ±, n Z nπ = or = nπ, n Z 6. n 7π π = nπ + ( ) or(n+ ), n Z 6 nπ nπ π =, or +, n Z 9. 8 π 5π π,, n π ±, n Z 7π π n 7π,,n π + ( ), n Z π π = (n + ), or nπ ±, n Z 4 nπ π =, or n π ±,n Z Miscellaneous Eercise on Chapter ,, 5 5 6,, ,,

7 ANSWERS 49 EXERCISE i i 5. 7 i 6. 9 i = i i i i i 4 4. i 4. 7 i EXERCISE 5.. π,. 5π,. 6 π π cos sin + i π π cos + i sin π π cos + i sin (cos π + i sin π) 7. π π cos + i sin π π cos + i sin EXERCISE 5.. ± i. ± 7 4 i. ± i 4. ± 7i 5. ± i 6. ± 7 i 7. ± 7 i 8. ± 4i ± ( 4 ) 9. i 0. ± 7 i

8 440 MATHEMATICS Miscellaneous Eercise on Chapter 5. i i (i) π π cos + i sin 4 4, (ii) π π cos + i sin ± i 7. ± i ± 7 i ± i (i) (ii) 0 5 5, π., 4. =, y = EXERCISE 6.. (i) {,,, 4} (ii) {...,,, 0,,,,4,}. (i) No Solution (ii) {... 4, }. (i) {...,, 0, } (ii) (, ) 4. (i) {, 0,,,,...} (ii) (, ) 5. (, ) 6. (, ) 7. (, ] 8. (, 4] 9. (, 6) 0. (, 6). (, ]. (, 0]. (4, ) 4. (, ] 5. (4, ) 6. (, ] 7. <, 8., 9. >, 0. <, 7. More than or equal to 5. Greater than or equal to 8. (5,7), (7,9) 4. (6,8), (8,0), (0,) 5. 9 cm 6. Greater than or equal to 8 but less than or equal to

9 ANSWERS 44 EXERCISE

10 44 MATHEMATICS EXERCISE 6...

11 ANSWERS

12 444 MATHEMATICS

13 ANSWERS Miscellaneous Eercise on Chapter 6. [, ]. (0, ]. [ 4, ] 4. (, ) , 6., 7. ( 5, 5) 8. (, 7) 9. (5, ) 0. [ 7, ]. Between 0 C and 5 C. More than 0 litres but less than 80 litres.. More than 56.5 litres but less than 900 litres. 4. Atleast 9.6 but more than 6.8. EXERCISE 7.. (i) 5, (ii)

14 446 MATHEMATICS EXERCISE 7.. (i) 400, (ii) 8. 0, No (i) 0, (ii) 50 EXERCISE , (i), (ii) (i) 60, (ii) 70, (iii) (i) 84400, (ii) 4900, (iii) EXERCISE (i) 5, (ii) Miscellaneous Eercise on Chapter (i) 504, (ii) 588, (iii) C 48 C C 7 + C EXERCISE (.) 0000 > (a b + ab ); ( ), 98

15 ANSWERS 447 EXERCISE 8. r r r ( ) 6 4. ( ) r 4 r r Cr..y y ; m = 4 9 Cr..y y 5 0. n = 7; r = Miscellaneous Eercise on Chapter 8. a = ; b = 5; n = 6. n = 7, 4. a = a 8 + a 6 0a 4 4a n = a 5 + 7a 4 6a + 7a 4 54a 5 + 7a 6 EXERCISE 9.., 8, 5, 4, ,,,,., 4, 8, 6 and ,,, and 5. 5, 5, 65, 5, ,,, and 7. 65, ,, 5, 07, ; ,,,, ;

16 448 MATHEMATICS.,,, 0, ; ( ) ,,, and 5 EXERCISE or n ( 5n + 7) 8. q , 4, 7, 0 and Rs EXERCISE , n (a) th, (b) th, (c) 9 th 6. ± 7. (. ) 0 7 n + 8. ( ). ( ) ;; ( ) 7 7 ( a) + a n 0. n ( ) r = or ; Terms are,, or,, n ,,,... or 4, 8, 6,, 64,.. 0 n n rr., 6,, and 7 8. ( ) 7. n = 0. 0, 480, 0 ( n ). Rs 500 (.) = 0 EXERCISE 9.4 n n+ +.. ( )( n ) ( + ) ( + ) ( + ) n n n n 4

17 ANSWERS 449 n. ( n )( n 5 n ) n (n + ) (n + ) ( + ) ( n + n+ 4 ) n n n 9. ( n )( n n ) ( ) 6 n n + ( + ) ( n+ ) n n ( n+ )( n ) n Miscellaneous Eercise on Chapter 9. 5, 8, ; 6 9. ± 0. 8, 6, (i) ( 0 n n n ), (ii) ( 0 n ) n n. ( n + n+ 5) 5. ( n + 9n+ ) 7. Rs Rs Rs Rs 7000; Rs days 4 EXERCISE 0.. square unit.. (0, a), (0, a) and ( a, 0) or (0, a), (0, a), and ( a, 0). (i) y y, (ii) , = and, or and, or and, or and 4. 5., 04.5 Crores

18 450 MATHEMATICS EXERCISE 0.. y = 0 and = 0. y + 0 = 0. y = m 4. ( ) ( ) y 4( ) + = 5. + y + 6 = 0 6. y+ = y + = y = y + 8 = y + 0 = 0. ( + n) + ( + n)y = n +. + y = 5. + y 6 = 0, + y 6 = y = 0and + y+ = y + 85 = 0. 9 L = C litres. 9. k + hy = kh ( ). (i) y= + 0,, 0; (ii) 7 7 y. (i) + = 46,, ; (ii) 4 6 (iii) EXERCISE y= +,, ; (iii) y = 0 + 0, 0, 0 y + =,, ; y =, intercept with y-ais = and no intercept with -ais.. (i) cos 0 + y sin 0 = 4, 4, 0 (ii) cos 90 + y sin 90 =,, 90 ; (iii) cos 5 + y sin 5 =,, units 5. (, 0) and (8, 0) 6. (i) 65 units, (ii) 7 p+ r l units. 7. 4y + 8 = 0 8. y + 7 = 9. 0 and ( + ) + ( ) y= 8 + ( ) ( ) or + + y = + 8

19 ANSWERS y = , m =,c= 7. y =, Miscellaneous Eercise on Chapter 0. (a), (b) ±, (c) 6 or.. y = 6, + y= π 6, 8 0,, 0, 5. sin ( φ θ) φ θ 6. sin 5 = 7. y + 8 = 0 8. k square units y = 7, + y = 9. + y = 6 4. : Slope of the line is zero i.e. line is parallel to - ais 7. =, y =. 8. (, 4) units 8 ± y + = , y = 05 EXERCISE.. + y 4y = 0. + y + 4 6y = y 6 8y + = y y = y + a + by + b = 0 6. c( 5, ), r = 6 7. c(, 4), r = c(4, 5), r = 5 9. c ( 4, 0) ; r = y 6 8y + 5 = 0. + y 7 + 5y 4 = 0. + y + 4 = 0 & + y + = 0

20 45 MATHEMATICS. + y a by = y 4 4y = 5 5. Inside the circle; since the distance of the point to the centre of the circle is less than the radius of the circle. EXERCISE.. F (, 0), ais - - ais, directri =, length of the Latus rectum =. F (0, ), ais - y - ais, directri y =, length of the Latus rectum = 6. F (, 0), ais - - ais, directri =, length of the Latus rectum = 8 4. F (0, 4), ais - y - ais, directri y = 4, length of the Latus rectum = 6 5. F ( 5, 0) ais - - ais, directri = 5, length of the Latus rectum = F (0, ), ais - y - ais, directri y =, length of the Latus rectum = y = 4 8. = y 9. y = 0. y = 8. y = 9. = 5y EXERCISE.. F (± 0,0); V (± 6, 0); Major ais = ; Minor ais = 8, e = 0 6, Latus rectum = 6. F (0, ± ); V (0, ± 5); Major ais = 0; Minor ais = 4, e = 5 ; Latus rectum = 8 5. F (± 7, 0); V (± 4, 0); Major ais = 8; Minor ais = 6, e = 7 4 ; Latus rectum = 9

21 ANSWERS F (0, ± 75 ); V (0,± 0); Major ais = 0; Minor ais = 0, e = Latus rectum = 5 5. F (±,0); V (± 7, 0); Major ais =4 ; Minor ais =, e = Latus rectum = 7 7 ; 7 ; 6. F (0, ±0 ); V (0,± 0); Major ais =40 ; Minor ais = 0, e = Latus rectum = 0 ; 7. F (0, ± 4 ); V (0,± 6); Major ais = ; Minor ais = 4, e = ; Latus rectum = 4 8. F( 0,± 5) ; V (0,± 4); Major ais = 8 ; Minor ais =, e = 5 4 ; Latus rectum = 9. F (± 5,0); V (±, 0); Major ais = 6 ; Minor ais = 4, e = 5 ; Latus rectum = 8 0. y + =. 5 9 y + = y + = 6 0. y + = y + = 5. 5 y + = y + = y + = y + = 5 9

22 454 MATHEMATICS 9. y + = y = 5 or 0 40 EXERCISE.4 y + = 5. Foci (± 5, 0), Vertices (± 4, 0); e = 4 5 ; Latus rectum = 9. Foci (0 ± 6), Vertices (0, ± ); e = ; Latus rectum = 8. Foci (0, ± ), Vertices (0, ± ); e = ; Latus rectum = 9 4. Foci (± 0, 0), Vertices (± 6, 0); e = 5 ; Latus rectum 64 = 5. Foci (0,± 4 5 ), Vertices (0,± 6 4 ); e = 5 ; Latus rectum 4 5 = 6. Foci (0, ± 65 ), Vertices (0, ± 4); e = 65 4 ; Latus rectum 49 = 7. y = y = y = y =. 6 9 y = y = 5 0. y = y = y = 5 5 Miscellaneous Eercise on Chapter. Focus is at the mid-point of the given diameter... m (appro.). 9. m (appro.) 4..56m (appro.) 5. y + = 6. 8 sq units y + = a

23 ANSWERS 455 EXERCISE.. y and z - coordinate are zero. y - coordinate is zero. I, IV, VIII, V, VI, II, III, VII 4. (i) XY - plane (ii) (, y, 0) (iii) Eight regions EXERCISE.. (i) 5 (ii) 4 (iii) 6 (iv) 5 4. z = y + 5z 5 = 0 EXERCISE.. (i) 4 7,,, (ii) ( 8, 7, ). : : 5. (6, 4, ), (8, 0, ) Miscellaneous Eercise on Chapter. (,, 8). 7 4, 7. a =, b = 4. (0,, 0) and (0, 6, 0) 5. (4,, 6) 6. k 09 + y + z 7y+ z= EXERCISE. 6, c = π π b a b 4. a b 5. 4 π π

24 456 MATHEMATICS a + b , 6 4. Limit does not eist at = 5. Limit does not eist at = 0 6. Limit does not eist at = a = 0, b = 4 9. lim a f () = 0 and lim f () = (a a ) (a a )... (a a ) a 0. lim a f () eists for all a 0... For lim f () to eists, we need m = n; lim 0 f () eists for any integral value of m and n. EXERCISE (i) (ii) (iii) n n n n 6. n + a( n ) + a ( n ) a (iv) ( ) 7. (i) a b (ii) 4a( a b) + (iii) a b ( b) 8. n n n n n an + a ( a) (i) (ii) (iii) ( ) 6 (v) (vi) ( ) ( ) + ( ) (iv) sin. (i) cos (ii) sec tan (iii) 5sec tan 4sin (iv) cosec cot (v) cosec 5 cosec cot (vi) 5cos + 6sin (vii) sec 7sec tan

25 ANSWERS 457. (i) (ii) Miscellaneous Eercise on Chapter (iii) cos ( + ) (vi) π sin 8. qr. + ps 4. c (a+b) (c + d) + a (c + d) 5. ad bc ( c + d ) 6. 0,, ( ) 7. ( a b) + ( a + b + c) 8. ap bp + ar bq ( p + + r) 9. ap + bp + bq ar ( a + b) 4a b 0. + sin 5. n b. na( a + ) n m. ( a b) ( c d ) mc( a b) na( c d ) cos (+a) 5. cosec cosec cot 6. + sin 7. ( sin cos ) 8. sec tan ( sec + ) 9. n sin n cos 0. bc cos + ad sin + bd cos α. cos ( c+ d cos ). ( 5 cos + sin + 0 sin cos ). sin sin + cos 4. qsin( a + sin ) + ( p+ qcos )( a+ cos ) 5. tan ( + cos ) + ( tan )( sin ) cos + 8 cos + 8 sin 5sin ( + 7cos )

26 458 MATHEMATICS 7. π cos 4 sin cos sin ( ) 9. ( + sec )( sec ) + ( tan ).( + sec tan ) 8. + tan sec ( + tan) 0. sin ncos n sin + EXERCISE 4.. (i) This sentence is always false because the maimum number of days in a month is. Therefore, it is a statement. (ii) This is not a statement because for some people mathematics can be easy and for some others it can be difficult. (iii) This sentence is always true because the sum is and it is greater than 0. Therefore, it is a statement. (iv) This sentence is sometimes true and sometimes not true. For eample the square of is even number and the square of is an odd number. Therefore, it is not a statement. (v) This sentence is sometimes true and sometimes false. For eample, squares and rhombus have equal length whereas rectangles and trapezium have unequal length. Therefore, it is not a statement. (vi) It is an order and therefore, is not a statement. (vii) This sentence is false as the product is ( 8). Therefore, it is a statement. (viii) This sentence is always true and therefore, it is a statement. (i) It is not clear from the contet which day is referred and therefore, it is not a statement. () This is a true statement because all real numbers can be written in the form a + i 0.. The three eamples can be: (i) Everyone in this room is bold. This is not a statement because from the contet it is not clear which room is reffered here and the term bold is not precisely defined. (ii) She is an engineering student. This is also not a statement because who she is. (iii) cos θ is always greater than /. Unless, we know what θ is, we cannot say whether the sentence is true or not.

27 ANSWERS 459 EXERCISES 4.. (i) Chennai is not the capital of Tamil Nadu. (ii) is a comple number. (iii) All triangles are equilateral tringles. (iv) The number is not greater than 7. (v) Every natural number is not an integer.. (i) The negation of the first statement is the number is a rational number. which is the same as the second statement This is because when a number is not irrational, it is a rational. Therefore, the given pairs are negations of each other. (ii) The negation of the first statement is is an irrational number which is the same as the second statement. Therefore, the pairs are negations of each other.. (i) Number is prime; number is odd (True). (ii) All integers are positive; all integers are negative (False). (iii) 00 is divisible by,00 is divisible by and 00 is divisible by 5 (False). EXERCISE 4.. (i) And. The component statements are: All rational numbers are real. All real numbers are not comple. (ii) Or. The component statements are: Square of an integer is positive. Square of an integer is negative. (iii) And. the component statements are: The sand heats up quickily in the sun. The sand does not cool down fast at night. (iv) And. The component statements are: = is a root of the equation 0 = 0 = is a root of the equation 0 = 0. (i) There eists. The negation is There does not eist a number which is equal to its square. (ii) For every. The negation is There eists a real number such that is not less than +. (iii) There eists. The negation is There eists a state in India which does not have a capital.

28 460 MATHEMATICS. No. The negation of the statement in (i) is There eists real number and y for which + y y +, instead of the statement given in (ii). 4. (i) Eclusive (ii) Inclusive (iii) Eclusive EXERCISE 4.4. (i) A natural number is odd implies that its square is odd. (ii) A natural number is odd only if its square is odd. (iii) For a natural number to be odd it is necessary that its square is odd. (iv) For the square of a natural number to be odd, it is sufficient that the number is odd (v) If the square of a natural number is not odd, then the natural number is not odd.. (i) The contrapositive is If a number is not odd, then is not a prime number. The converse is If a number in odd, then it is a prime number. (ii) The contrapositive is If two lines intersect in the same plane, then they are not parallel The converse is If two lines do not interesect in the same plane, then they are parallel (iii) The contrapositive is If something is not at low temperature, then it is not cold The converse is If something is at low temperature, then it is cold (iv) The contrapositive is If you know how to reason deductively, then you can comprehend geometry. The converse is If you do not know how to reason deductively, then you can not comprehend geometry. (v) This statement can be written as If is an even number, then is divisible by 4. The contrapositive is, If is not divisible by 4, then is not an even number. The converse is, If is divisible by 4, then is an even number.. (i) If you get a job, then your credentials are good. (ii) If the banana tree stays warm for a month, then it will bloom.

29 ANSWERS 46 (iii) If diagonals of a quadrilateral bisect each other, then it is a parallelogram. (iv) If you get A + in the class, then you do all the eercises in the book. 4. a (i) Contrapositive (ii) Converse b (i) Contrapositive (ii) Converse EXERCISE (i) False. By definition of the chord, it should intersect the circle in two points. (ii) False. This can be shown by giving a counter eample. A chord which is not a dimaeter gives the counter eample. (iii) True. In the equation of an ellipse if we put a = b, then it is a circle (Direct Method) (iv) True, by the rule of inequality (v) False. Since is a prime number, therefore is irrational. Miscellaneous Eercise on Chapter 4. (i) There eists a positive real number such that is not positive. (ii) There eists a cat which does not scratch. (iii) There eists a real number such that neither > nor <. (iv) There does not eist a number such that 0 < <.. (i) The statement can be written as If a positive integer is prime, then it has no divisors other than and itself. The converse of the statement is If a positive integer has no divisors other than and itself, then it is a prime. The contrapositive of the statement is If positive integer has divisors other than and itself then it is not prime. (ii) The given statement can be written as If it is a sunny day, then I go to a beach. The converse of the statement is If I go to beach, then it is a sunny day. The contrapositive is If I do not go to a beach, then it is not a sunny day. (iii) The converse is If you feel thirsty, then it is hot outside. The contrapositive is If you do not feel thirsty, then it is not hot outside.

30 46 MATHEMATICS. (i) If there is log on to the server, then you have a password. (ii) If it rains, then there is traffic jam. (iii) If you can access the website, then you pay a subscription fee. 4. (i) You watch television if and only if your mind in free. (ii) You get an A grade if and only you do all the homework regularly. (iii) A quadrilateral is equiangular if and only if it is a rectangle. 5. The compound statement with And is 5 is a multiple of 5 and 8 This is a false statement. The compound statement with Or is 5 is a multiple of 5 or 8 This is true statement. 7. Same as Q in Eercise 4.4 EXERCISE EXERCISE 5.. 9, 9.5. n+ n,. 6.5, , , , , , 9. 9, 05.5, , 4.5 EXERCISE 5.. B. Y. (i) B, (ii) B 4. A 5. Weight Miscellaneous Eercise on Chapter 5. 4, 8. 6, 8. 4, 5. (i) 0.,.99 (ii) 0., Highest Chemistry and lowest Mathematics 7. 0,.06

31 ANSWERS 46 EXERCISE 6.. {HHH, HHT, HTH, THH, TTH, HTT, THT, TTT}. {(, y) :, y =,,,4,5,6} or {(,), (,), (,),..., (,6), (,), (,),..., (,6),..., (6, ), (6, ),..., (6,6)}. {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT} 4. {H, H, H, H4, H5, H6, T, T, T, T4, T5, T6} 5. {H, H, H, H4, H5, H6, T} 6. {XB, XB, XG, XG, YB, YG, YG 4, YG 5 } 7. {R, R, R, R4, R5, R6, W, W, W, W4, W5, W6, B, B, B, B4, B5, B6} 8. (i) {BB, BG, GB, GG} (ii) {0,, } 9. {RW, WR, WW} 0. [HH, HT, T, T, T, T4, T5, T6}. {DDD, DDN, DND, NDD, DNN, NDN, NND, NNN}. {T, H, H, H5, H, H, H, H4, H5, H6, H4, H4, H4, H44, H45, H46, H6, H6, H6, H64, H65, H66}. {(,), (,), (,4), (,), (,), (,4), (,), (,), (,4), (4,), (4,), (4,)} 4. {HH, HT, TH, TT, H, T, HH, HT, TH, TT, 4H, 4T, 5HH, 5HT, 5TH, 5TT, 6H, 6T} 5. {TR, TR, TB, TB, TB, H, H, H, H4, H5, H6} 6. {6, (,6), (,6), (,6), (4,6), (5,6), (,,6), (,,6),..., (,5,6), (,,6). (,,6),..., (,5,6),..., (5,,6), (5,,6),... } EXERCISE 6.. No.. (i) {,,, 4, 5, 6} (ii) φ (iii) {, 6} (iv) {,, } (v) {6} (vi) {, 4, 5, 6}, A B = {,,, 4, 5, 6}, A B = φ, B C = {, 6}, E F = {6}, D E = φ, A C = {,,4,5}, D E = {,,}, E F = φ, F = {, }. A = {(,6), (4,5), (5, 4), (6,), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)} B = {(,), (,), (, ), (4,), (5,), (6,), (,), (,), (,4), (,5), (,6)} C ={(,6), (6,), (5, 4), (4,5), (6,6)} A and B, B and C are mutually eclusive. 4. (i) A and B; A and C; B and C; C and D (ii) A and C (iii) B and D 5. (i) Getting at least two heads, and getting at least two tails (ii) Getting no heads, getting eactly one head and getting at least two heads

32 464 MATHEMATICS (iii) (iv) (v) Getting at most two tails, and getting eactly two tails Getting eactly one head and getting eactly two heads Getting eactly one tail, getting eactly two tails, and getting eactly three tails Note There may be other events also as answer to the above question. 6. A = {(, ), (,), (,), (,4), (,5), (,6), (4,), (4,), (4,), (4,4), (4,5), (4,6), (6,), (6,), (6,), (6,4), (6,5), (6,6)} B = {(, ), (,), (,), (,4), (,5), (,6), (,), (,), (,), (,4), (,5), (,6), (5,), (5,), (5,), (5,4), (5,5), (5,6)} C = {(, ), (,), (,), (,4), (,), (,), (,), (,), (,), (4,)} (i) A = {(,), (,), (,), (,4), (,5), (,6), (,), (,), (,), (,4), (,5), (,6), (5,), (5,), (5,), (5,4), (5,5), (5,6)} = B (ii) B = {(,), (,), (,), (,4), (,5), (,6), (4,), (4,), (4,), (4,4), (4,5), (4,6), (6,), (6,), (6,), (6,4), (6,5), (6,6)} = A (iii) A B = {(,), (,), (,), (,4), (,5), (,6), (,), (,), (,), (,4), (,5), (,6), (5,), (5,), (5,), (5,4), (5,5), (5,6), (,), (,), (,), (,5), (,6), (4,), (4,), (4,), (4,4), (4,5), (4,6), (6,), (6,), (6,), (6,4), (6,5), (6,6)} = S (iv) A B = φ (v) A C = {(,4), (,5), (,6), (4,), (4,), (4,4), (4,5), (4,6), (6,), (6,), (6,), (6,4), (6,5), (6,6)} (vi) B C = {(,), (,), (,), (,4), (,5), (,6), (,), (,), (,), (,), (,), (,), (,4), (,5), (,6), (4,), (5,), (5,), (5,), (5,4), (5,5), (5,6)} (vii) B C = {(,), (,), (,), (,4), (,), (,)} (viii) A B C = {(,4), (,5), (,6), (4,), (4,), (4,4), (4,5), (4,6), (6,), (6,), (6,), (6,4), (6,5), (6,6)} 7. (i) True (ii) True (iii) True (iv) False (v) False (vi) False EXERCISE 6.. (a) Yes (b) Yes (c) No (d) No (e) No.. (i) (ii) (iii) 6 (iv) 0 (v) (i) (ii) (a) 5 (b) 5 (c) (i) (ii)

33 ANSWERS Rs 4.00 gain, Rs.50 gain, Re.00 loss, Rs.50 loss, Rs 6.00 loss. P ( Winning Rs 4.00) P (Losing Rs.50) =, P(Winning Rs.50) 6 =, P (Losing Rs 6.00) 4 =. 6 =, P (Losing Re..00) 4 = 8 8. (i) 8 (ii) 8 (iii) (iv) 7 8 (v) 8 (vi) 8 (vii) 8 (viii) 8 (i) (i) (ii) (i) No, because P(A B) must be less than or equal to P(A) and P(B), (ii) Yes. 7 (i) (ii) 0.5 (iii) (i) (ii) No 7. (i) 0.58 (ii) 0.5 (iii) (i) (ii) (iii) Miscellaneous Eercise on Chapter (i) 0 60 C C C5 (ii) 60. C 5 C. C 5 C (i) (ii) (iii) (a) (b) 4. (a) C (b) C 7. (i) 0.88 (ii) 0. (iii) 0.9 (iv) (i) (ii) (c) C C 0 0