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===================================================================================================Triumphant Institute of Management Education Pvt. Ltd. (T.I.M.E.) HO: 95B, 2nd Floor, Siddamsetty Complex, Secunderabad β 500 003. Tel : 040β27898194/95 Fax : 040β27847334 email : [email protected] website : www.time4education.com Sol.SMM631101/1 ===================================================================================================
SOLUTIONS FOR
BASIC STUDY MATERIAL
Sol.SMM631101
MATHEMATICS
===================================================================================================Triumphant Institute of Management Education Pvt. Ltd. (T.I.M.E.) HO: 95B, 2nd Floor, Siddamsetty Complex, Secunderabad β 500 003. Tel : 040β27898194/95 Fax : 040β27847334 email : [email protected] website : www.time4education.com Sol.SMM631101/2 ===================================================================================================
2nd Floor, 95B, Siddamsetty Complex, Secunderabad β 500 003.
Tel : 040β27898194/95 Fax : 040β27847334 email : [email protected] website : www.time4education.com HINTS/SOLUTIONS for M1101
(Sets)
Classroom Discussion Exercise 1. (c) A = {Ο, x} Subsets of A are Ο {Ο}, {x} and A β P(A) = { Ο , {Ο}, {x}, A} 2. (d) 2m β 2n = 96 = 32 Γ 3 = 25 (4 β 1) = 25 (22 β 20) = 27 β 25 m = 7 and n = 5 3. (c) e x β x for any x β R So A β© B = Ο. 4. (b) A = {x : x β₯ 3}; B = {x : x < 5} Clearly A β© B = {x : x β R, 3 β€ x < 5} 5. (d) X = {3, 5, 7, 9} 6. (d) The curves will intersect when x = 0 β΄ The curves meet at (0, 1) β΄ A β© B = {(0, 1)} 7. (c) The points of intersection are given by y2 = 4 β y = Β± 2 β΄Points of intersection are (1, 2) and (1, β2) 8. (d) A β B = {x / x β A, and x β B} β΄ (A β B) β© B = Ο. 9. (c) n (A) = 76, n (B) = 44, n (A βͺB) = 100 n (A β© B) = n (A) + n (B) β n (A βͺ B) = 76 + 44 β 100 = 20. 10. (a) The required number of subsets = 5C3 = 10 11. (d) All (i) , (ii) and (iii) represent the shaded region 12. (a) A = {7, 14, 21, 28,β¦,.105} B = {4, 8, 12, 16, β¦., 60} A β© B = {28, 56} β΄ n (A β B) = n(A) β 2 = 13. 13. (c) P β© (P βͺ Q)β = P β© (Pβ β© Qβ) = (P β© Pβ) β© (P β© Qβ) = Ο β© (P β© Qβ) = Ο
14. (d) U = {1, 2, 3,β¦β¦β¦β¦, 10} A = {3, 4, 5, 6, 7, 8} B = {2, 3, 5, 7} A β B = {4, 6, 8} (A β B)β = {1, 2, 3, 5 ,7, 9, 10}
C B
A
C ββββ B
B C
A
A ββββ C
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15. (d)
There is no region common to A β C and C β B. β΄ (A β C) β© (C β B) = Ο 16. (a) People who are not teenagers is U β C = Cβ People who have their weights less than 40 kg = U β D = D β β΄ Required set = Cβ β© Dβ 17. (c) Since k and m are relatively prime, the L.C.M.
of k and m is km. β΄β΄β΄β΄ kN β©β©β©β© mN = (mk)N = nN ββββ mk = n 18. (d) n (A) = 300; n (B) = 500; n (A β© B) = 100 n (A βͺ B) = 300 + 500 β 100 = 700 n (Aβ β© Bβ) = n [(A βͺ B)β] = 1000 β n (A βͺ B) = 1000 β
700 = 300
19. (c) Since A β B = (A βͺ B) β (A β© B), therefore A β© B = Ο 20. (c) A βͺ B = A βͺ C and A β© B = A β© C β B = C 21. (a) A β (A βͺ B)β = A 22. (c) )BA(n)B(n)A(n)BA(n β©β+=βͺ
)BA(n)B(n)A(n)BA(n βͺβ+=β©β
)BA(n6465)BA(n βͺβ+=β©β
But 100)BA(n β€βͺ
1006465)BA(n β+β₯β©β
%29)BA(n β₯β©β
)1(............%29x β₯β
and)A(n)BA(n β€β©
)B(n)BA(n β€β©
and%65)BA(n β€β©β
%64)BA(n β€β©
%64)BA(n β€β©β
)2.....(..........%64x β€β Combining (1) and (2) 29% %64x β€β€ 23. (c) Let A, B, C denote the set of all families
reading βThe Hinduβ, βHindustan Timesβ and βThe Chronicleβ respectively.
n (A) = 10000 Γ 400010040 =
n (B) = 10000 Γ 200010020 =
n (C) = 10000 Γ 100010010 =
n (A β© B) = 10000 Γ 500100
5 =
n (B β© C) = 10000 Γ 300100
3 =
n (A β© C) = 10000 Γ 400100
4 =
n (A β© B β© C) = 10000 Γ 200100
2 =
n (Aβ β© Bβ β© Cβ) = 10000 β n (A βͺ B βͺ C) = 10000 β {n (A) +
n (B) + n (C) β [n (A β© B) + n (B β© C) ] + n (A β© C)] +
n (A β© B β© C)} i.e., n (Aβ β© Bβ β© Cβ) =10000 β {7000 β 1200 + 200} = 10000 β 6000 = 4000 = 40%.
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24. (d) A = { 2 } B = { β 2, 1 } A β B = { 2 } B β A = { β 2 , 1 }
[ ] .3)AB()BA(n}1,2,2{)AB()BA(
=ββͺββ΄β=ββͺβ
25. (c) )BA(n β
)BA(n2)B(n)A(n β©β+= (1)
Now n(A β© B) = n(A) β n(A β B) β n(A β© B) = 5 β 4 = 1 Thus (1) gives 12 = 5 + n(B) β 2 β n(B) = 9
Regular Homework Exercise 1. (c) x 2 + 9 = 0 β x 2 = β 9 β x is imaginary. 2. (b) All equal sets are equivalent 3. (c) n(X βͺ Y) = n(X) + n(Y) β n(X β© Y) is maximum
if n(X β© Y) = 0 β΄ maximum value of n(X β© Y) is n(X) + n(Y) = 12 4. (d) n(Aβ©B) = n(A) = 6 if A β B 5. (c) Number of elements of P(A) = Number of subsets of A = 2n(A)
= 25 6. (c) It represents the elements belonging to
exactly two of the sets A, B, C 7. (c) Let n(A) = m, n(B) = n Given 2m β 2n = 32 β 2n(2m β n β 1) = 25 β n = 5 (Ξ 2m β n β 1 is odd) β m = 6 8. (a) A β© (A βͺ B) = (A β© A) βͺ(A β© B) = Aβͺ(A β© B) = A (since A β© B
β A) 9. (c) P{A} = {Ο , {Ο} } β n [P(A)] = 2 10. (a) The number of subsets containing at least one
element = 25 β 1 = 31 11. (c) n(A βͺ B βͺC) = n(A) + n(B ) + n(C) β n(A β© B)
β n(B β© C) β n (C β© A) + n(A β© B β© C).
12. (b) )CA()BA()CB(A βͺβ©βͺ=β©βͺ 13. (b)
)ZX()YX()ZY(X ββͺβ=β©β 14. (a) x β₯ 100 β (20 + 15 + 30) = 35 β minimum value of x = 35 15. (d) A = {5, 6, 7} , B = {1, 3, 5, 7} β A βͺ B = {1, 3, 5, 6, 7} β Aβ β© Bβ = (A βͺ B)β = {2, 4, 8} 16. (b) A β B β (A βͺ B) = B β΄ n(A βͺ B) = n(B) = 7 17. (b) Let A, B and C be the set of basketball players,
cricket players and general athletics players respectively.
n (A) = 21; n (B) = 26; n (C) = 29
n (A β© B) = 14; n (B β© C) = 15; n (A β© C) = 12; n (A β© B β© C) = 8 n (A βͺ B βͺ C) = n (A) + n (B) + n (C) β n (A β© B) β n (B
β© C) β n (A β© C) + n (A
β© B β© C) = 21 + 26 + 29 β 14
β 15 β 12 + 8 = 43. 18. (d) .}..........,24,12,8,4{N4 =
.....},.........24,18,12,6{N6 =
..}..........,36,24,12{NN 64 =β©β΄
= N12 19. (c) Number of students who like at least one juice
=125 + 118 + 117 β (60 + 60) + 20 = 260 β΄ total number of students = 260 + 70 = 330 20. (c) Clearly the shaded region represents B β (A βͺ C)
Assignment Exercise 1. (d) A = {18, 45, 108, β¦β¦β¦β¦} B = {18, 27, 36, 45,β¦β¦β¦.} β΄ B A β
2. (a) n (A β B) + n (A β© B) = n (A). 3. (b)
A B
X
Y Z
A β B
A B Aβ©B
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We observe that A β (A β B) = A β© B. 4. (b) A β B = {5, 6}. So, number of subsets of A β B = 2 2 = 4.
5. (b) We have ex = .........!2
x!1
x1
2
+++ where x is real.
Hence, ex β x for any real x. β΄ A β© B = Ο 6. (c)
n (B β M) = n (Mβ)
= 40 β 30 = 10 7. (a) n (Aββ©Bβ) = n (AβͺB)β (By De Morganβs Law ) = n (U) β n (AβͺB) = n (U) β
[ ])BA(n)B(n)A(n β©β+
= 1000 β (400 + 300 β 100) = 400 8. (c) A β B β Aβ β Bβ β Aβ β© Bβ = Bβ.
9. (b) B only = 900500010018 =Γ
10. (b) Total number of subsets of two sets are 2m and 2n
324822 4nm Γ==ββ΄
024n4m 22322 β==ββ ββ
04nand24m =β=ββ
4nand6m ==β 11. (c) Since )BA(x βͺβ ,
β΄ x β A and x β B Option (c) is wrong. 12. (c) One half of the men belong to club A β 6 belong to club A One third of the men belong to club B β 4 belong to club B One forth of the men belong to both clubs β 3 belong to club A and club B β n (AβͺB)=7 Thus 12 β 7 = 5 belong to neither clubs. 13. (b) n (A) = 4 and n ( B ) = 7 Minimum of n ( 7)B(n)BA ==βͺ when A β B 14. (c) The shaded region represents A. C)BA(C =βͺβ©β΄ . 15. (d) A = BC'BC β=β© The shaded region represents A Ο=β©β΄ BA
A B
3
7
18
C
2
3 3 1
n(A)=6 n(B)=4 U = 12
5 M
B
8 22
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Additional Practice Exercise 1. (d) By definition. 2. (c) )BA(n2)B(n)A(n)BA(n Ξβ+=β
= 250 + 350 β 200 = 400 3. (d) n(A βͺ B) = n(A) + n(B) β n(A β© B) = 100 + 50 β 25 = 125. β΄ n(Aβ β© Bβ) = 75. 4. (c) Obviously the smallest set is A = {1, 2, 5}.
5. (b) BA)BA(A β©=ββ
= )B'A(A βͺβ©
6. (a) Let m be the number of elements in S. Then 9m = 3 Γ 45 β m = 15 Again 5n = 10m = 10 Γ 15 n = 30 7. (b) Let the number of news papers be x
Then the number of subscriptions = 30x
Every body subscribes 6 newspapers
β΄ Number of people = 6
x30
β 6
x30 = 240 β x = 48
8. (d) BA β = )AB()BA( ββͺβ
= )BA()BA( β©ββͺ
)BA(n β = )BA(n)BA(n β©ββͺ
= )BA(n2)B(n)A(n β©β+ 9. (d) All of them are correct 10. (c) A = { x / xβR, β2 < x < 2} B = { x / xβR, 2 β€ |x β 2|} = {x / xβR, xβ(ββ, β2] βͺ[4, β)} A βͺ B = (ββ, 2) βͺ [4, β) = R β {x/x β R, 2 β€ x < 4} 11. (c) {a, b} β P({a, b} ) 12. (d) Number of students who passed with
distinction = 240 Γ 4810020 =
β΄ By the given condition, number of girls who
passed with a distinction = 24248 =
β΄ Percentage of girls who passed with
distinction = 10016024 Γ = 15%.
13. (a)
Number of persons belonging to at least two clubs = 53
14. (b) AβB, Aβ©B and BβA are mutually disjoint. 15. (b) (A β© B) β C = (A β C) β© (B β C) 16. (c) C βD = (A β B) β (B β A)
But there is no region common to both (A β B) and (B β A)
A β B B β A β΄ ( A β B) β (B β A) = (A β B) = C 17. (d) A βͺ B = C βͺ B and A β© B = C β© B β A = C 18. (b) n (m) = 120, n (p) = 90, n (c) = 70, n (m β© p) = 40, n (p β© c) = 30, n (m β© c) = 50, n (mβ β© pβ β© cβ) = 20 n(m βͺ p β© c) = 200 β 20 = 180 ie n(m) + n(p) + n(c) β n (m β© p) β n (m β© c) β n(p β©c) + n (m β© p β© c) = 180 β 120 + 90 + 70 β (40 + 30 + 50) + n(m β© p β© c) =
180 β n (m β© p β© c) = 20 19. (b) Delete both a and s from X. Then the resulting set
has 3 elements. Number of subsets of this set is 8. Now put βaβ to each of these 8 sets. Thus there are 8 subsets of X containing βaβ but not βsβ.
20. (c) )CB(A β©β )CA()BA( ββͺβ=
A
C B
24 12 12
6
11
15
B A
20
C
A B
2 β2 0 4
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21. (b) BABA β©=β is an identity. 22. (c) Aββ (A β© B) = Aβ β Aβ (since A β© B β A) 23. (c) B and A β B are disjoint sets Ο=ββ©β΄ )BA(B
24. (c) The shaded region represents A β B .BA)BA(A β©=βββ΄ 25. (c) Two sets A and B are said to be
equivalent if n (A) = n ( B ) 26. (d) A β (A β© B)C = A β© (A β© B) (Ξ A β B = A β© BC) = A β© B 27. (d) n (m ) = 60 % n ( p) = 65 % n ( m βͺ p) = 100 β 20 = 80% Given that 80% of the students = 16
β΄ Total number of students = 2080
10016 =Γ
β΄ only 3.
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28. (b) A β© B = B β Aβ 29. (c)
The shaded region represents P = (AβB)β Also Q = BA β© and R = BA βͺ
'RQP βͺ=β΄ 30. (c)
HINTS/SOLUTIONS for M1102 (Relation & Functions)
Classroom Discussion Exercise 1. (a) (x, x + y) = (3, 7) β x = 3, y = 4
2. (c) n(B) = ( )
( ) 514
70
An
BAn==
Γ
3. (b) Since (B β© C) = Ο, β΄ (A Γ B) β© (A Γ C) = Ο. 4. (a) (A Γ B) β© (C Γ D) = (A β©C) Γ (B β© D). 5. (d) Since AΓB contains 18 elements, there are 218 relations from A to B 6. (d) Domain = {β1, 3, 4, 6} 7. (d) The relation βis perpendicular toβ is not
reflexive and transitive. A line cannot be
perpendicular to itself and line l1 β₯ to l2 and l2 β₯ l3 does not imply that l1 β₯ l3.
8. (b) Symmetric property is not satisfied. 9. (d) Total number of functions from A to B is (n(B))n(A) β΄ Number of functions on A is 33 = 27 10. (a) In choice (a), 3 it is related to more than one
element and hence is not a function. 11. (d) f1 = {(1, 1), (2, 3), (3, 5), (4, 7), (5, 9)} But 7,
9 β A. f2 = {(1, 5), (2, 4), (2, 5), β¦.}, which is not a
function. f3 = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} But 6
β A. f4 = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)} which
is a function from A to A.
12. (c) ( ) ( )yxfyxf β+
[ ] [ ]yxyxyxyx 3321
3321 +ββββ+ ++=
[ ]x2y2y2x2 333341 ββ +++=
= ( ) ( )[ ]y2y2x2x2 333341 ββ +++
= [ ])y2(f)x2(f21 +
13. (c) The graph shows that for all the positive
as well as negative values of x, y takes only positive
values. Hence the graph represents y = |x|. 14. (d) Rf = {β1, 0, 1}. 15. (a) x β [x] = {x}, the fractional part function
whose range is [0, 1) 16. (a) Range of cosx is [β1, 1] 17. (c) [x] is an integer
β΄ The values of ]x[2
sinΟ are β 1, 0 and 1.
18. (c) [Ο] = [3.14] = 3 and = [β Ο] = β 4 Ξ f(x) = cos 3x β sin4x
β΄β΄β΄β΄ 23
134
sincos3
f +β=ΟβΟ=
Ο
= 2
23 β
19. (b) Since the function is real, we have 49 β x2 > 0 β x β (β7, 7)
20. (c) The domain of ( )( ),
xbax
1
ββ with a < b is
(a, b)
21. (a) Let y = 2x1
x
+β yx2 β x + y = 0
b2 β 4ac β₯ 0 β 1 β 4y2 β₯ 0
β 21
y21 β€β€β
B A
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22. (a) f (x) = |x| + | x+ 4|
Minimum value of f(x) is 4 and it has no maximum.
β΄ Range = [4,β)
23. (d) f (x) + g (x) = 2
ee xx β+β₯ 1 for all real x.
β΄ Range of (f + g) (x) = [1, β) 24. (d) ( ) ( ) ( )yfxfyxf β =+
( ) ( ) ( ) ( ) 23331f1f11f2f =Γ=β =+=
( ) ( ) ( ) 32 3331f2f3f =Γ=β =
( ) ( ) ( ) 43 3331f3f4f =Γ=β = . 25. (a) f (x) = 5x β |x| f (2x) = 10x β2 | x| f (βx) = β 5x β |βx| = β 5x β |x| f (2x)β f (βx) = 10x β 2 |x| + 5x + |x| = 15x β |x| = f(x) + 10x β΄ f(2x) β f(βx) β 10x = f(x)
Regular Homework Exercise
1. (b) n(AΓB) = n(A) . n(B) = 0 2. (a) By definition of equality of ordered pairs, (A Γ B) β© (B Γ A) = Ο. 3. (d) A Γ (B βͺ C) = (A Γ B)βͺ (A Γ C). 4. (a) Co-domain is a superset of range Range = {6, 7, 8, 9} 5. (c) The first coordinate of the ordered pair
should be from B and second coordinate should be from A.
6. (d) Since R is reflexive (a, a)βR β a β A, where
n is the given set having R elements. Since n(A) = n, R having at least n ordered pair. β΄ n(R) β₯ n
7. (a) By definition. 8. (d) By definition.
9. (a) f(x) + 2f x21x3x =
β+ ______(1)
let 1x3x
yβ+= , β
1y3y
xβ+=
β΄
β+=+
β+
1y3y
2)y(f21y3y
f
replacing y by x
( ))1x(3x2
)x(f21x3x
fβ+=+
β+ _____(2)
2 Γ (2) β (1) β x2)1x()3x(4
)x(f3 ββ+=
=
)1x(x2x212x4 2
β+β+
3f(x) = ( )1xx212x6 2
ββ+
β΄ )1x(3
x212x6)x(f
2
ββ+=
10. (a) 2x
1x
x
1x
x
1xf
2
22 +
β=+=
β
β΄ f (z) = z 2 + 2 β f (x) = x 2 + 2. 11. (b) There is only one element in the range of a
constant junction. 12. (d) The possible values of signum function are
Β±1 and 0. Hence range will contain 3 elements.
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13. (b) β1 β€ cos 3x β€ 1 3 β₯ 2 β cosx β₯ 1
β 1x3cos2
131 β€
ββ€
β΄ Range of f (x) =
1,
31
14. (a) f (x) is not defined when 16 β x 2 < 0 β 16 < x 2 or when x < β4 and x > 4 β΄ x β [β4, 4]. 15. (c) By definition of f(x), f(x) is manyβone, into. 16. (b) 0x1 2 >β
( ) 0x11 2 β₯ββ
so 1β 0x11 2 β₯ββ All these hold when 0x1 2 β₯β β x2 β€ 1 β΄ The domain of f is [β1,1]
17. (d) The domain of bxax
ββ when a < b is
( ] ( )ββͺββ ,ba,
18. (c) Since 2x is +ve the least value of 2
2
x2
x
+
is 0
also 1x2
xx2x
2
222 <
+β΄+<
β΄The range is [ )1,0 19. (d) By definition, 4x2 β 0 β x2 β 0β x β 0 β΄ x = R β {0}
20. (c) f (x) =5x
11x
β+β
x β 1 β₯ 0 β x β₯ 1, x β5 β 0 β x β 5 β΄ Domain is x β₯1, x β 5
Assignment Exercise
1. (a) Number of subsets = 60125 22 =Γ 2. (a) A = {2, 4, 8}, B = {5, 6}. 3. (c) ( ) ( )ba2,54,2a +=+ ba24and52a +==+β 2band3a β== . 4. (d) The number of relations from A to B = the number
subsets of A Γ B = 224
5. (b) By definition R is reflexive. 6. (d) By definition. 7. (d) R = {(1, 5), (2, 6), (3, 7), (4, 8), (5, 9), (6, 10)}
8. (d) ( )x1
xxf +=
( )[ ] ( )33 xfxf β
+β+++=3
33
3
x
1x
x
1x3
x3x
+=x1
x3 .
( )
==x1
f3xf3
9. (c) Obviously, the range is {1, β1}.
10. (c) Domain is R and range is [0, β)
11. (a) a x β 0 and a x > 0 β x β R.
12. (c) 0x7 2 β₯β
β x β [ ]7,7β 13. (b) The domain of ( )( )xbaxlog ββ for a < b is (a,
b). log(x β 2) (5 β x) = log(x β 2) + log(5 β x) β x > 2 and x < 5
14. (d) Let y = 9x5x
x2 +β
β ( ) 0y91y5xyx2 =++β
0ac4b2 β₯β (as x is real)
β ( ) ( )( ) 0y9y41y5 2 β₯β+
β β 01y10y11 2 β₯++
01y10y11 2 β€βββ
β y β
β1,
111
β΄ Range of the given function is
β1,
111
15. (d) The range of ( )7x45x3
xfβ+= is same as the
domain of ( )3x45x7
xf 1
β+=β which is given by x
β R; x β 43
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Additional Practice Exercise
1. (c) If A β B, then A Γ C β B Γ C. 2. (a) A β© B = Ο β (A β© B) β© (A βͺ B) = Ο β [(A β© B) β© (A βͺ B)] Γ A = Ο 3. (c) n[(A Γ B) β© (B Γ A)] = 52 = 25 4. (b) A βͺ C = C β΄ (A βͺ C) Γ B contains 4 Γ 2 = 8 elements. 5. (c) n[(A Γ B)β© (B Γ A)] = n[(Aβ©B) Γ (B β© A)] = n [(A β©B) Γ (A β© B)] = n(A β© B) Γ n(A β© B) = 3 Γ 3 = 9. 6. (c) Domain = {x: | x | β€ 4, xβZ} = { β4, β3, β2, β1, 0, 1, 2, 3, 4} 7. (d) Clearly (a, b) β (a, b) and (a, b) β (c, d) β (c, d) β (a, b) Let (a, b) β (c, d) and (c, d) β (e, f) β ad = bc & cf = de
β cfbc
dead =
β af = be β (a, b) β (e, f) β΄ β is transitive also. 8. (c) 2 1 = 2. 9. (d) Number of possible relations is a subset of A
Γ B A Γ B has p q elements β΄ number of subsets for A Γ B = 2pq. 10. (d) By definition, (a, b) β [a, b] 11. (d) R = { } }Nx;x,x β
xx β for all x β N except for x = 1 β΄(x, x) β R β x β N (x, y) β R β y = x β (y, x) β R since x β
y
(x, y) β R and (y, z) β R β y2 = x and z2 = y β x = z4 β (x, z) β R 12. (b) 2mn = 4096 = 212 where n (A) = m and n (B) = n β 26n = 212 β n = 2 β΄ B has 2 elements. 13. (c) n4 = 625 β n = 5 β΄β΄β΄β΄ B has 5 elements. 14. (a) 25.n(B) = 1024 = 210
β n(B) = 2 15. (c) 2mn 16. (c) Since n(B) < n(A), there is no one-one
function from A to B 17. (c) R = {(1,1) (1,2) (1,3) ,(2,1),
(2,2),(2,3),(3,1),(3,2)} 18. (d) By definition of f + g, f β g, Ξ±f and fg, all the
given statements in (a), (b), (c) are true. 19. (b) Either 11 β |x| β₯ 0 and 12 β |x| > 0 or 11 β |x| β€ 0 and 12 β |x| < 0 β either |x| β€ 11 or |x| > 12 β x β (ββ, β12) βͺ [β11, 11] βͺ (12, β)
20. (b) 02x3x;2x3x
x 2
2>+β
+β
(x β 1) (x β 2) > 0 β x β (1, 2) or x β ( ββ, 1) βͺ (2, β)
21. (a) ( )
x1x
x1xx1x
x1x
x2
ββ=
βββ=
ββ
For 1x0x1,0x1
x2
>β<ββ₯
ββ
and x = 0
β΄β΄β΄β΄ x β (1, β) βͺ {0}
22. (b) 2222 baxcosbxsinaba +β€+β€+β
β 2x3sinx3cos2 β€+β€β
222x3sinx3cos0 +β€++β€β 23. (a) Substitute x = β 1 and x = 3, image is (8, 72) 24. (c) f(x) is not defined when sin2 x = 0 β when x = nΟ; n β I.
25. (a) f (x) = 2
2
x1
x
+
2x0 β€ < 1+ x2
β 0 β€ ( ) )1,0[xf1x1
x2
2
ββ<+
β΄ Range is [0, 1) 26. (d) β΄ 10 β x β₯ 0 and 4 + x β₯ 0 10 β₯ x and x β₯ β 4 x β€ 10 and β4 β€ x β 4 β€ x β€ 10 D (f) = β4 β€ x β€ 10
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27. (d) g (x) = log
β+
x1x1
g(x1) + g (x2)
= log
β+
1
1
x1x1 + log
β+
2
2
x1x1
= log ( )
++β+++
2121
2121
xxxx1xxxx1
= log ( )( )
+β++++
2121
2121
xxxx1xxxx1
=
++β
+++
21
21
21
21
xx1xx
1
xx1xx
1log =
++
21
21
xx1xx
g
28. (c) Any polynomial satisfying the equation. f(x) .f(1/x) = f(x) + f(1/x) is of the form
1xn + or -xn +1 f(4) = 65 = 64 + 1 = 43 + 1 β΄ f(x) = x3 +1 f(3) = 33 +1 = 28
29. (d) 555 yx =+
β xy 555 β=
β y= ( )x5 55log β
β΄ 1x055 x <β>β
30. (c) 2101x
x β<+
β 100
11x
x100
1 <
+<β
β 100 > 100x
1x >+
β 100 > 1 + 100x1 >
i.e., β 101 > 99x1 >
991
x101
1 <<β
HINTS/SOLUTIONS for M1103 (Trigonometric Functions)
Classroom Discussion Exercise 1. (a) lr2P +=
ΞΈ= 2r21
A
ΞΈ+= rr2P
2r
A2.rr2P +=
A2r2Pr 2 +=
i.e., 0A2Prr2 2 =+β 2. (d) The ratio of the radii is the ratio of the angles in
radian measures. Hence required ratio is 5: 4. 3. (c) For 45Β°< x < 90Β°, cosx < sinx Hence y < 0
4. (d) tan ΞΈ = 1, cot ΞΈ = 1
==+ 1xiff2x1
xΞ
β΄β΄β΄β΄ tan3 ΞΈ + cot3 ΞΈ = 2 5. (a) m2 β n2 = (m + n) ( m β n) = 4 sinxtanx 6. (a) Let x = sec Ξ± + tan Ξ±
x1β΄ = sec Ξ± β tan Ξ±
x1
x +β΄ = 2 sec Ξ±
= 2a + a21
β΄ x = 2a or a21
7. (c) sinx = cos2x β¦β¦β¦β¦.. (1) cos12x + 3 cos10x + 3 cos8x + cos6x = (cos4x + cos2x)3 = (sin2x + cos2x)3 = 1 by (1).
8. (a) A + B = C2
βΟ
tan (A + B) = tan
βΟC
2
1Btan.AtanCtan.BtanCtan.Atan
Ctan1
CcotBtan.Atan1
BtanAtan
=++β
==β
+
9. (a)
+β=
+β
0
0
00
00
21tan1
21tan1
21sin21cos
21sin21cos
00
00
21tan45tan
21tan45tan
+β=
( )00 2145tan β= 024tan=
r
r
l ΞΈ
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10. (d) log tan 10 + log tan 20 + β¦ + log tan 890 = log [tan 10 . tan 20 β¦ tan 890] = log [tan 10 . tan 20 β¦ tan 450 . cot 440
cot 2 0 . cot 1 0] = log 1 = 0. 11. (b) sin210 + sin220 + cos220 + cos210 = 2
12. (d) 4
B2
CA Ο==+
2
CAΟ=+
tan(A +C) =CtanAtan1CtanAtan
β+
CtanAtan1CtanAtan
2tan
β+=Οβ΄
tanA tanC = 1
But tan B = tan 4
Ο= 1
β΄ tanA tanB tanC = 1 13. (a) tanA β tanB = x
yAtan
1Btan
1 =β
yx
Btan.Atan =β
cot(A β B) = BtanAtanBtan.Atan1
β+
=x
yx
1
+
y1
x1
xyyx +=+=
14. (a) xsin2x2sin
x2cos22β
=( )
xsin2xcosxsin2
xsinxcos22
22
ββ
=( )( )
( )xsinxcosxsin2xsinxcosxsinxcos2
ββ+
=1 + cotx. 15. (c) x2 + y2 = 1 Let x = cos ΞΈ, y = sin ΞΈ β΄β΄β΄β΄ (4 cos3 ΞΈ β 3 cos ΞΈ)2 + (4 sin3 ΞΈ β 3 sin ΞΈ)2 = (cos 3ΞΈ)2 + (β sin 3 ΞΈ)2 = 1
16. (a) sin ΞΈ . cos ΞΈ = 2
2sin ΞΈ
Minimum of sin 2ΞΈ = β1
Minimum value = ( ) .2
11
2
1 β=β
17. (c) ( )Ξ²Ξ±β
Ξ²+Ξ±=Ξ²+Ξ±
tantan1tantan
tan
Ξ²β
Ξ²
=Ξ²
2tan1
2tan2
tan2
34
4
11
21
.2=
β=
β΄β΄β΄β΄ ( ) 3
3
4.
3
11
3
4
3
1
tan =β
+=Ξ²+Ξ±
18. (d) sin ΞΈ [sin 2 60 β sin 2 ΞΈ]
=
ΞΈβΞΈ 2sin43
sin
4
sin4sin3 3 ΞΈβΞΈ= ΞΈ= 3sin41
19. (b) cos20 cos40 cos80
= 20sin2
80cos40cos20cos20sin2
= 20sin4
80cos40cos40sin2
= 20sin8
160sin20sin8
80cos80sin2 = =81
20. (b) ΞΈ+ΞΈ=ΞΈ+ΞΈ 2sin8sin4sin6sin ββββ 2sin5 ΞΈΞΈ=ΞΈΞΈ 3cos5sin2cos
( ) 03coscos5sin =ΞΈβΞΈΞΈ
β sin5 0sin.2sin2. =ΞΈΞΈΞΈ
β sin5 0sinor02sinor0 =ΞΈ=ΞΈ=ΞΈ
β Ο=ΞΈΟ=ΞΈΟ=ΞΈ nor2
nor
5n
21. (d) 02sin3sin2 2 =βΞΈ+ΞΈ
( ) ( ) 02sin1sin2 =+ΞΈβΞΈ
2sinor2
1sin β=ΞΈ=ΞΈ ( not possible )
6
sin21
sinΟ==ΞΈ
β΄β΄β΄β΄ ( )6
1n n Οβ+Ο=ΞΈ .
22. (a) 1sin2
3cos
2
1 =ΞΈ+ΞΈ
1sin3
sincos3
cos =ΞΈΟ+ΞΈΟ
13
cos =
ΟβΞΈ
Ο=ΟβΞΈ n23
3
n2Ο+Ο=ΞΈ
In the interval [0, 2 Ο ], 3Ο=ΞΈ only
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23. (d) tan3x . tan7x = -1
tan7x = x3tan
1β
= β cot3x
=
+Οx3
2tan
x32
nx7 +Ο+Ο=β΄
Ο
+=β΄2
1n2x4
Ο
+=β8
1n2x .
24. (b) The given equation is
( )222 xcosxsin +
xcosxsinxcosxsin2 22 =β
i.e. xcosxsinxcosxsin21 22 =β
i.e. 2
x2sin2
x2sin1
2
=β
i.e. x2sinx2sin2 2 =β
sin22x + sin2x β 2 = 0
sin2x = 2
811 +Β±β =
2
31Β±β= β2, 1
1x2sin =β΄
2
n2x2Ο+Ο=
4
nxΟ+Ο=β΄ ( )1n4
4+Ο= .
25. (d) The given equation is
ΞΈ+ΞΈ 2tantan )2tantan1(3 ΞΈΞΈβ=
i.e; 32tantan1
2tantan =ΞΈΞΈβ
ΞΈ+ΞΈ
ββββ 33tan =ΞΈ
3
n3Ο+Ο=ΞΈ
β΄β΄β΄β΄ ( )1n3993
n +Ο=Ο+Ο=ΞΈ .
Regular Homework Exercise
1. (b) We know that Ξ» = rΞΈ
β 4
.r2
Ο=Ο
β r = 2 cm 2. (c) Since the value of sinΞΈ lies between β1 and 1, we
get sinΞΈ1 = sinΞΈ2 = sinΞΈ3 = β1
β ΞΈ1 = ΞΈ2 = ΞΈ3 = 2
3Ο
β΄ cosΞΈ1 + cosΞΈ2 + cosΞΈ3 = 0
3. (c) sec A + tan A = 23
β sec A β tan A = 32
β΄ 2 sec A = 23
+ 32
β sec A = 1213
cos A = 1312
β sin A = .135
4. (d) sin25 + sin210 +β¦+ sin290 = (sin25 + sin285) + (sin210 + sin280) +β¦.+ sin245
+ sin290
= 8 + 21
+ 1 (since sin285 = sin2(90 β 5) = cos25)
= 921
5. (b) 37
cos6
sinΟ+Ο
= 121
21 =+
6. (c)
ΟβΟ+Ο+Ο8
3cos
83
cos8
cos 222
ΟβΟ+8
cos2
= +
Ο+Ο8
sin8
cos 22
Ο+Ο8
3sin
83
cos 22 = 2
7. (d) cos36Β° = sin(90 β 36)Β° = sin54Β° and cos72Β° = sin(90 β 72) = sin18Β° β΄ sin18Β°.sin54Β° β cos36Β°.cos72Β° = 0 8. (d) Given expression = 3 (sin4 Ξ± + cos4 Ξ±)
β 2 (sin6 Ξ± + cos6 Ξ±) = 3 (1 β 2 sin2 Ξ±
cos2 Ξ±)
β 2(1 β 3 sin2 Ξ± cos2 Ξ±) = 1.
9. (c) Given expression =
Ο+Ο4
3cos
4cos2 44 = 1
10. (b) sin(18 + ΞΈ)cos(72 β ΞΈ) + cos(18 + ΞΈ)sin(72 β ΞΈ) = sin[(18 + ΞΈ) + (72 β ΞΈ)] = sin90 = 1
11. (b) Given expression = +βBsinAsin
BsinAcosBcosAsin
AsinCsinAsinCcosAcosCsin
CsinBsin
CsinBcosCcosBsin
β+
β
= cot B β cotA + cot C β cot B + cot A β cot C = 0
12. (a) Bcot
2
CAsin
2
CAsin2
2
CAsin
2
CAcos2
=
β
+
β
+
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B2CA
Bcot2
CAcot
=+β
=
+β
β A, B, C are in A.P.
13. (d) 34cos
11sin11cos +
= 34cos
79cos11cos +
= 34cos
34cos45cos2 = 2
14. (b) ( )A16cos12222 ++++
= A8cos2222 +++
= A4cos222 ++ = A2cos22 +
= ( )A2cos12 +
= Acos.2.2 = 2 cos A
15. (c) cos4ΞΈ + sin4ΞΈ =1 β ΞΈ2sin21 2 which is maximum
when ΞΈ = 4Ο
or β4Ο
16. (b) 32Ο=ΞΈ β΄ General solution is
32
n2Ο+Ο=ΞΈ .
17. (d) tanx (tan2x β 3) = 0
tanx = 0 or tanx = 3Β±
β΄ x = nΟ or 3
nxΟΒ±Ο=
18. (a) 2sincos =ΞΈ+ΞΈ
1sin2
1cos
2
1 =ΞΈ+ΞΈ
1sin4
sincos4
cos =ΞΈΟ+ΞΈΟ
cos 0cos4
=
ΟβΞΈ
0n24
Β±Ο=ΟβΞΈ
4
n2Ο+Ο=ΞΈ
19. (c) tanx = Β± 1 and cot x = Β± 1
x = 4
nΟ+Ο .
20. (c) Multiplying by cos x and solving for sinx we
get ΞΈ = 10
)1(n n Οβ+Ο .
Assignment Exercise
1. (c) 6264
l ΓΟΓ= = 8Ο.
2. (a) Angle covered in 1second = 270 Γ 2Ο radians
Time taken to turn 6210Ο radians = ΟΟ
5406210
= 11.5 sec.
3. (d) 2
2
2
2
2
2
c
z
b
y
a
x ++
= ΞΈ+ΟΞΈ+ΟΞΈ 22222222 cosrcossinrsinsinr
= r2 sin2 ΞΈ (sin2 Ο + cos2 Ο ) + r2 cos2 ΞΈ
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= r2 (sin2 ΞΈ + cos2 ΞΈ ) = r2
4. (b)
Ο+Ο+
Ο+Ο8
sin8
5sin
8sin
Ο+Ο+8
5sin
=8
5sin
8sin
85
sin8
sinΟβΟβΟ+Ο
= 0
5. (c) cos (x + y) = cosx cosy β sinx siny = 25
1β
6. (a) 1m
BsinAcosBcosAsin
1m
BtanAtan =β=
Applying compendo dividendo, we get
1m1m
BsinAcosBcosAsinBsinAcosBcosAsin
+β=
+β
β ( )( ) 1m
1mBAsinBAsin
+β=
+β
7. (a) 33
BcosAcosBsinAsin
BsinAsinBcosAcos
β++
β+
=
3
2BA
sin2
BAcos2
2BA
cos2
BAcos2
β+
β+
+
3
2BA
sin2
BAsin2
2BA
cos2
BAsin2
β+β
β+
= 3
3
2BA
cot2
BAcot
ββ+β = 0.
8. (d) tan (A+B+C) = 0
β AtanCtanCtanBtanBtanAtan1CtanBtanAtanCtanBtanAtan
ββββ++
= 0
β΄ tanA + tanB + tanC = tanAtanBtanC 9. (d) cosxcos2xcos4xcos8x
= xsin2
12sinxcosxcos2xcos4xcos8x
= xsin2
1sin2xcos2xcos4xcos8x
= xsin2
12
sin4xcos4xcos8x
= xsin2
13
sin8xcos8x
= xsin2
14
sin16x
10. (c) 13tan.2tan =ΞΈΞΈ
13cos2cos3sin2sin =
ΞΈΞΈΞΈΞΈ
i.e, 03sin2sin3cos2cos =ΞΈΞΈβΞΈΞΈ
i.e, cos ( ) 032 =ΞΈ+ΞΈ
2
cos05cosΟ==ΞΈ
ββββ 2
n25ΟΒ±Ο=ΞΈ
β΄β΄β΄β΄ 10
n52 ΟΒ±Ο=ΞΈ .
11. (c) 0coteccos =ΞΈ+ΞΈ
0sincos
sin1 =
ΞΈΞΈ+
ΞΈ
1 + cosΞΈ = 0 β cosΞΈ = β1 β ΞΈ = Ο 12. (d) Since sinx, sin2x, sin3x lies in the interval [-1,1],
the given equation is true only if sinx = 1, sin2x = 1 and sin3x = 1 which is impossible.
13. (b) x is in the third quadrant.
Particular solution is x = 4
5Ο
β΄ General solution is x = 24
5n
Ο+Ο .
14. (b) secΞΈ + tan ΞΈ = 3
secΞΈ β tan ΞΈ = 3
1
Solving, secΞΈ = 21
3
13
+ =
3
2
β΄β΄β΄β΄ ΞΈ = 6
n2Ο+Ο .
15. (d) Dividing by 2, we get ΞΈ = 3Ο
or 6Ο
Additional Practice Exercise
1. (c)
Arc length = r ΞΈ = Ο=ΟΓ 53
15
2. (d) Secant increases in the second quadrant and sine and cosine increases in the fourth quadrant
3. (b) cos(β 765Β°) = cos 765Β° = cos (720 + 45)Β°
= cos 45Β° =2
1
4. (a) tanΞΈ being an increasing function, the value of
tanΞΈ increases as ΞΈ increases 5. (b) secΞΈ + tan ΞΈ = x
1
1
13Ο
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secΞΈ β tan ΞΈ = x1
Solving, secΞΈ = x2
1x2 +
β cosΞΈ = 1x
x22 +
β΄ sinΞΈ = xcos1 2β = 1x
1x2
2
+β
6. (a) )yxcos(22ba 22 β+=+ a2 β b2 = cos 2x + cos 2y + 2 cos (x + y) =
)yxcos(2)yxcos()yxcos(2 ++β+
= 2cos(x + y)[cos(x β y) + 1] = cos(x + y)(a2 + b2)
β cos(x + y) = 22
22
ba
ba
+β
β΄ sin (x + y) = 22 ba
ab2
+.
7. (d) )xsin()x(eccos
1βΟ=
βΟ=sinx
8. (b)
ΟβΟ
ΟβΟΟΟ5
sin.52
sin.52
sin.5
sin
= 5
sin.52
sin.52
sin.5
sinΟΟΟΟ
= 52
sin.5
sin 22 ΟΟ
= sin236 Γ sin272
=
+
β16
521016
5210=
165
16
201002
=β
9. (c) cos x = 2p 2 β 1 = 2 cos 2 200 β 1 β cosx = cos40Β° = cos(360 β 40) = cos320Β° 10. (b) sin(45 + x) β cos(45 β x) = sin45cosx + cos45sinx β [cos45cosx +
sin45sinx]
= 2
1(cosx + sinx β cosx β sinx)
= 0 11. (c) (sinΞΈ + cosΞΈ)2 = 1 + 2sinΞΈ cosΞΈ
β ac
21ab
2
+=
β
β a2 β b2 + 2ac = 0
12. (c)
ΞΈβΟ
ΞΈ+Ο4
sec4
sec
=
ΞΈβΟ
ΞΈ+Ο4
cos4
cos
1
= ΞΈβΟ 22 sin
4cos
1
= ΞΈβ 2sin21
2
= 2sec2ΞΈ 13. (b) sin(Ξ±+ 2Ξ²) = sin(Ξ±+ Ξ²) cosΞ² + cos(Ξ±+ Ξ²)sinΞ² = cosΞ². 14. (c) Since A + B = 225, β΄ cot(A + b) = cot(225) = 1
β BcotAcot1Bcotcot
+β
= 1
β cotAcotB β 1 = cotA + cotB β cotAcotB = 1 + cotA + cotB β 2cotAcotB = (1 + cotA)(1 + cotB) (by adding cotAcotB to both LHS and
RHS)
β 21
Bcot1Bcot
.Acot1
Acot =++
15. (a) cos74
cos.7
2cos.7
ΟΟΟ
=
7sin8
78
sin
Ο
Ο
= 81β
16. (c) 4 sin 230 sin 370 sin 830 = 2 sin 230 (2sin 830 sin 370 ) = 2 sin 230 (β cos 1200 + cos 460) = sin 230 + 2 cos 460 sin 230 = sin 230 + sin 690 β sin 230 = sin 690 = cos 210
17. (c) cosA + cosB =
2C
sin4 2
2
Csin4
2
BAcos.
2
BAcos2 2=
β
+β
2C
sin22
BAcos =
ββ
=
+2C
sin2
BAcoscesin
2C
cos.2
Csin.2.2
2C
cos.2
BAcos2 =
β
Csin22
BAsin
2BA
cos2 =
+
β
Csin2BsinAsin =+β .
18. (d)
8cos
8sin
8tan
Ο
Ο
=Ο
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=
8cos2
8cos
8sin2
2 Ο
ΟΟ
=
2
11
2
1
4cos1
4sin
+
=Ο+
Ο
= 1212
1 β=+
19. (d) 2524
491149
11
tan1
tan12cos
2
2=
+
β=
ΞΈ+ΞΈβ=ΞΈ
sinΟ = 10
1 and cosΟ =
10
3
β΄ 4sinΟ β 2524
2sin2Ο = 2524
56 β
sin3Ο β 2524
sin4Ο = 2sin2Οcos2Ο
= 2524
Ο+Οβ=Ο
Ο+Ο=Ο
2
2
2
tan1
tan12cos
tan1
tan22sin
20. (a) sin2A + sin2B + sin2C = 2sin(A + B)cos(A β B) + sin2C = 2sinCcos(A β B) + 2sinCcosC = 2sinC[cos(A β B) + cosC] =
2sinC
ββ+β2
CBAcos
2CBA
cos2
= 2sinC.2sinB.sinA = 4sinAsinBsinC
21. (a) x2sinx2cos
x2sinx6sin22 β
β =
4x cos2x sinx 4 2cos
= 2sin2x. 22. (a) cos6Ξ± + sin6Ξ± = 1 + msin22Ξ±(cos2Ξ± + sin2Ξ±)3 β
3cos2Ξ±sin2Ξ±(cos2Ξ± + sin2Ξ±) = 1 + msin22Ξ± i.e; 1 β 3sin2Ξ±cos2Ξ± = 1 + msin22Ξ±
β΄ 4
3(sin2Ξ±cos2Ξ±) = msin22Ξ±
β΄ m = β4
3
23. (a) tanA
β
+
+
β
Atan31
Atan3
Atan31
Atan3
=
ββ
Atan31
Atan3Atan
2
2
= A3tanAtan31
AtanAtan32
3
=
ββ
.
24. (a) cos4x = cos2x β΄ 4x = x2n2 Β±Ο β 6x = 2nΟ or 2x = 2nΟ
β x = nΟ or 3
nΟ
β΄ 3n
xΟ= which includes x = nΟ also
25. (d) The given equation is 2sin4 ΞΈ cos3 ΞΈ + sin4 ΞΈ = 0 i.e;sin4 ΞΈ (2 cos3 ΞΈ +1) = 0
i.e; sin4 ΞΈ = 0 or cos 3 ΞΈ = 21β
sin4 ΞΈ = 0 β 0, 2
,4
ΟΟ
cos3 ΞΈ = β2
1 β ΞΈ =
9
4,
9
2 ΟΟ
26. (a) 06cos4cos2cos =ΞΈ+ΞΈβΞΈ
β [ ] 04cos6cos2cos =ΞΈβΞΈ+ΞΈ
04cos2cos4cos2 =ΞΈβΞΈΞΈ
i.e; ( ) 012cos24cos =βΞΈΞΈ
β cos4 ΞΈ=ΞΈ 2cos2or0 =1
3
n22or2
n24ΟΒ±Ο=ΞΈΟΒ±Ο=ΞΈβ΄
6
nor82
n ΟΒ±Ο=ΞΈΟΒ±Ο=ΞΈβ΄ .
27. (c) sin3x + sinxcosx + cos3x = 1 β΄ sin3x + cos3x β (1 β sinxcosx) = 0 (sinx + cosx)(sin2x + cos2x β sinxcosx)
β (1 β sinxcosx) = 0 i.e; (1 β sinxcosx)(sinx + cosx β 1) = 0 i.e; sinxcosx = 1 (1) or sinx + cosx = 1 (2) (1) implies sin2x = 2 which is not possible From (2)
2
1xcos
2
1xsin
2
1 =+
2
1
4xcos =
Οβ
4
n24
xΟΒ±Ο=Οββ΄
2
n2orn2xΟ+ΟΟ=β΄ .
28. (b) Solving, cos x = 21
or cos x = 23
cosx = 2
1 β x =
35
,3
ΟΟ
cosx = 2
3 is not possible
29. (d) Solving we get
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tanx = 3
1β x =
6n
Ο+Ο .
30. (d) Both 2Ο
and 4Ο
does not satisfy the given
equation. Hence choice (d).
HINTS/SOLUTIONS for M1104 (Straight Lines)
Classroom Discussion Exercise 1. (d) Let the third vertex be (x, y). Then
23
12x =++ and 2
321y =+β
5y,3x ==β΄ (3, 5) is the third vertex. 2. (a) Given (x β a1)
2 + (y β b1)2 = (x β a2)
2 + (y β b2)2
β x2 + a12 β 2a1x + y2 + b1
2 β 2b1y = x2 + a2
2 β 2a2x + y2 + b22 β
2b2y β 2(a1 β a2)x + 2(b1 β b2)y + a2
2 + b2
2 β a1
2 β b12 = 0
β (a1 β a2)x + (b1 β b2)y
+ ( ) 0baba21 2
121
22
22 =ββ+
3. (a) Since y = mx + c passes through (3, β5) and
(2, 4), therefore β5 = 3m + c and 4 = 2m + c Solving m = β9, c = 22
4. (c) A(1, -2) and Q (2, -1) are points on QS
β΄ 21
2y121x
+β+=
ββ
β x β y β 3 = 0
5. (a) Required equation is 2qy
px =+
β 3y
2x + = 2
β 3x + 2y β 12 = 0 6. (d) By the intercept form, equation is 5x + 7y + 35 = 0 7. (c) 3x + 4y β 5 β k(x + 2y β 3) = 0 β x(3 β k) + y(4 β 2k) + (3k β 5) = 0 This is parallel to xβaxis. β Slope is zero
i.e; k24k3
ββ
= 0 β k = 3
8. (a) If the required ratio is K:1, the point of division is
given by
++
+β
1K
1K8,
1K
1K7This being a point on
5yx2 =+ , gives 17
6K =
Thus the required ratio is 6 : 17 is 6 : 17. 9. (b) k2 + 13k + 5 + k2 + 1 + 14 = 0 β 2k2 + 13k + 20 = 0
β k = 2
5β, β4
10. (a) The midpoint of the diagonal is (3, 2) which lie on
the line y = 2x + a Hence the value of a is β 4. 11. (b) The given line is
( ) ( ) 0b4a5y3x2by2xa =+ββ++
i.e., a(x + 2y β 5) + b(2x β 3y + 4) = 0 This represents the family of lines through the
point of intersection of x + 2y β 5 = 0 and 2x β 3y + 4 = 0
Solving these two equations we get x = 1, y = 2. 12. (b) Equation of AB is y β 1 = β1 (x β 4) i.e., x + y β 5 = 0. B is the point of intersection of this line with
y = 3x.
β΄ B is
415
,45
β΄ AB2 = 22
14
1545
4
β+
β = 22
411
411
+
2
411
2
Γ=
β΄ AB = 4
211
13. (c) Point of intersection of 4x + 3y = 1 and y = x + 5 is
(β2, 3) Substituting this in 5y + bx + 3 = 0 we get, b = 9.
14. (d) 2
cab
+=
0cy2
caax =+
++β΄
ie ( ) ( ) 02ycyx2a =+++ This represents a family of lines passing through
the point of intersection of the lines 2x + y = 0 and y + 2 = 0.
A (4, 1)
135Β°
y = 3x
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15. (c) The equation of the required line is of the form 2x + y + 6 + Ξ» (x β 2y+3) = 0. Since this passes through the origin, we get Ξ» = β 2.
Substituting, the required line is y = 0. 16. (a) a 1 a 2 + b 1 b 2 = 0 β΄β΄β΄β΄ ΞΈ = 900 17. (d) The lines 3x - 4y + 14 = 0 and 4x + 3y β 23 = 0 are perpendicular β΄ the orthocenter is the point of intersection of
these lines. Solving we get x = 2, y = 5. 18. (c) For parallel lines slopes are same. Hence k = 4.
19. (d) Dividing throughout with 211 =+ , we get
2
6y
2
1x
2
1 =β
β cos Ξ± = 2
1; sin Ξ± =
2
1β
β Ξ± = 315 0 = 4
7Ο
20. (d) The points (x1, y1) and (x2, y2) are on the same
side of the line ax + by + c = 0 if cbyaxcbyax
22
11
++++
is
positive
781
7+ββ
< 0
(β7, 0) is a point on the line
761
7++
> 0
β΄ (0, 0) and (1, 3) are on the same side of x + 2y + 7 = 0
21. (c) 13c1394
c0302=β=
+
+Γ+Γ
22. (a) The image (x2, y2) of the point (x1, y1) in the line
ax + by + c = 0 is given by
22111211
ba
)cbyax(2b
yya
xx
+++β=β=β
( )
( ) ( ).4,1y,x91
724323
8y1
3x
22
22
ββ=β+
β+β=β=ββ
23. (c) Distance = 22 43
205
+
+ = 5
24. (b) ( ) 2222 512
2y5x12
43
7y4x3
+
β+β=β+
+β
13
2y5x125
7y4x3 β+β=+β
39x β 52y + 91 = β60x β 25y + 10 99x β 27y + 81 = 0 11x β 3y + 9 = 0. 25. (c) Let the origin be shifted to (h ,k). Then the new equation is
( ) ( ) ( ) 02hx8ky4ky 2 =β+++++
Coefficient of y = 0 β 2k+ 4 = 0 β k = -2
Constant term = 0 β 02h8k4k2 =β++
β 4 β 8 + 8h β 2 = 0 β 4
3h =
Regular Homework Exercise
1. (c)
+++β3
903,
3
512 = (2, 4)
2. (a) Let the point be P(x, y) (x + a)2 + y2 + (x β a)2 + y2 = (2a)2 β x2 + y2 = 2a2 3. (c) Slope of AB is 1 β΄AC is vertical where C is the new position of B
2222AB 22 =+=
β΄ The required point is ( )22,2
4. (c) Given slope = 43 and c = β2
Equation of the line is y = mx + c
β y = 43
x β 2
β 3x β 4y β 8 = 0. 5. (c) Required equation is 1byax =+
6. (c) Let the line be by
ax + = 1
It passes through (1, 3) β 1b3
a1 =+
(1, 3) is the midpoint
β 2a
= 1 and 2b
= 3
β a = 2 and b = 6
β΄ Required line is 6y
2x + = 1
β 3x + y β 6 = 0
7. (a) Since a, b, c are in A.P, therefore b = 2
ca +
β΄ ax + by + c = 0 β 2a
(2x + y) + 2c
(y + 2) = 0,
which always passed through (1, β2)
450
A (2, 0)
B (4, 2) C
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D
B
A
C
x + y β 6 = 0
x - 3y β 2 = 0
5x - 3y + 2 = 0
8. (c) Since the lines are concurrent,
312
32a
143
βββ = 0
β a = 5 9. (a) Equation of the line is (4x + 3y β 7) + Ξ» (8x + 5y β 1) = 0 (4 + 8Ξ») x + (3 + 5Ξ») y β 7 β Ξ» = 0
Slope = 23ββ
23
5384 β=
Ξ»+Ξ»+β
8 + 16Ξ» = 9 + 15Ξ» i.e., Ξ» = 1. β΄ Equation of the line is (12x + 8y β 8 = 0) i.e., 3x + 2y β 2 = 0.
10. (d) 0
bac
acb
cba
=
i.e., a3 + b3 + c3 β 3abc = 0 11. (d) Equation of a line parallel to 4x + 10y β 8 = 0 is
0Ky10x4 =++ . It passes through ( )2,2 β .
Hence 12K0K21024 =β=+βΓ+Γ 01210y4xisequationrequired =++β΄ 12. (a) 3m1 =
1m2 =
21
12
mm1
mmtan
+β
=Ξ±β΄ 21
312 =+
=
113
m,32
m 43 ==
31132
1
113
32
tan
ΓΓ+
β=Ξ² =
31
633922 =
+β
( )61
1
31
21
tanβ
+=Ξ²+Ξ±β΄ =1
4Ο=Ξ²+Ξ±β΄ .
13. (d)
By solving the equations 5x β 3y + 2 = 0 and
x + y β 6 = 0 we get the point A (2, 4) Altitude AD is 3x + y + k = 0 It passes through (2, 4) β k = β10 β΄ equation of the altitude is 3x + y = 10.
14. (b) Any line perpendicular to the given line is x β 2y + k = 0. Since it passes through (1, 2), we get k = 3
Hence the required line is x β 2y + 3 = 0.
15. (b) Let the required ratio be Ξ» : 1.
Then P is
+Ξ»+Ξ»
+Ξ»+Ξ»β
134
,12
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P lies on x + y + 1 = 0
β΄ Ξ» = β23
16. (d) With the given sides, the vertices are A(0, 0),
B(β1, 3), C(2, β1) Equation of the line through (β1, 3) and
perpendicular to x + 2y = 0 is 2x β y + 5 = 0 β¦β¦..(1) Equation of the line through (0, 0) and perpendicular to 4x + 3y = 5 is 3x β 4y = 0β¦β¦β¦β¦(2)
Point of intersection of (1) and (2) is the orthocentre
β΄ Orthocentre is (β4, β3) 17. (a) The lines 3x β 4y β 2 = 0 and 4x + 3y β 11 = 0 are
perpendicular β΄ The orthocentre is the point of intersection of
these two lines. 3x β 4y β 2 = 0 4x + 3y β 11 = 0
169
1338
y644
x+
=+β
=+
β x = 2, y = 1.
18. (a) If (x, y) is the image of (x1, y1) in the line ax + by + c = 0, then
( )
221111
ba
cbyaxb
yya
xx
+++β=β=β
( )
169520
45y
30x
+βββ=
ββ=β
β x = 3, y = 1 19. (c) If P (h, k) is the image of A (4,β13) in the lines
5x + y + 6 = 0, then
113k
54h +=β
1125
613202 β=
++βΓβ=
14k,1h β=β=β΄ .
20. (b) 103
16a
125
2=
+
β
β 103
16a2
252
=+
β
β 516a2 =+
β a2 + 16 = 25 β a2 = 9 β΄ a = Β± 3.
Assignment Exercise
1. (c) [ ] [ ]22 )ab(y)ba(x ββ++β
= [ ] [ ]22 )ba(y)ba(x +β+ββ
bx β ay = 0 2. (b) P (h, k) lies on 3x + 2y β 13 = 0 β 3h + 2k β 13 = 0 Q (k, h) lies on 4x β y β 5 = 0 β 4k β h β 5 = 0 Solving them, we get h = 3, k = 2 β΄ Equation of PQ is y β 2 = β1 (x β 3) β x + y β 5 = 0
3. (b) The equation of the line is 1by
ax =+
where 2a = 3b
This passes through (3, β1) β 1b1
a3 =β
1a23
a3 =ββ β
23
a =
β΄ b = 1
β΄ Required line 1y
23x =+
β 2x + 3 y β 3 = 0.
4. (b) ax + by + 13 = 0 passes through (2, 5) and (β3, β 0)
β 013b5a2 =++ and 013ba3 =+ββ
Solving, a = 6, b = β 5
5. (d) c = 2
ba +
β΄ ax + by + c = 0 β ,021
yb21
xa =
++
+
which always passes through
ββ21
,21
6. (a) The equations of the sides are
Area = ( )
1221
2121
baba)dd(cc
βββ
= 28912 =
βΓ
7. (c) 0
bac
acb
cba
=
i.e; a3 + b3 + c3 β 3abc = 0
β ( )( ) 0bcacabcbacba 222 =βββ++++
β a + b + c = 0 (since a β b β c, β΄ a2 + b2 + c2 β ab β bc β ca β 0) 8. (b) Solving the first and the second equations, the
point of intersection is (-1, 1). This lies on the third line. Substituting, we get k = 4
9. (c) 4x + 3y β 7 = 0 8x + 5y β 1 = 0 Solving, the point of intersection is (-8, 13).
β΄ The required equation is ( )8x23
13y +β=β
β 2y β 26 = β 3x β 24 β 3x + 2y β 2 = 0. 10. (b) x β y = 0 or x = y This line is equally inclined to the x and y axes β΄ The angle it makes with y = 0 (x β axis) is 450. 11. (c) The midpoint of (1, 1) and (3, 5) is (2, 3). Slope of the line joining (1, 1) and (3, 5) is 2
β΄ Slope of its perpendicular is 21β
β΄ Equation of the right bisector is
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y β 3 = ( )2x21 ββ
β x + 2y β 8 = 0 12. (b) The lines x = 7 and y = 5 are perpendicular β΄ the circumcentre lies on the third line 5x + 7y β 35 = 0. 13. (b) The required line can be taken as
kya
secx
btan =ΞΈ+ΞΈ
It passes through ( )ΞΈΞΈ tanb,seca
ka
sectanbb
tanseca =ΞΈΞΈ+ΞΈΞΈβ΄
ΞΈΞΈ+=β΄ tansecab
bak
22
Equation becomes
ΞΈΞΈ+=ΞΈ+ΞΈ tansecab
basec
ay
tanbx 22
ie; 22 batanby
secax +=
ΞΈ+
ΞΈ
ie; 22 bacotbycosax +=ΞΈ+ΞΈ . 14. (a) Slopes of the three lines are
m1 = β 1, m2 = β 3, m3 = 31β
Angle between the lines x + y = 0 and 3x + y β 4 = 0 is given by
Ξ± =
++ββ
3131
tan 1 =
β
21
tan 1
Angle between the lines x + y = 0 and x + 3y β 6 = 0 is given by
Ξ² =
+
+ββ
31
1
131
tan 1 =
β
21
tan 1
Angle between the lines 3x + y β 4 = 0 and x + 3y β 6 = 0 is given by
Ξ³ =
β=
+
+βββ
34
tan113
13
tan 11
β΄ The triangle is isosceles with the line x + y = 0 as the base.
15. (a) The lines are parallel.
Distance between the lines is 22 125
2
1725
+
β
26
33=
5233
radius =β΄
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Additional Practice Exercise
1. (c) Area = ba1ac1cb1
21
= cbba0cabc0
cb1
21
ββββ
= [ ])ca()ba()cb()bc(21 βββββ
=
[ ])bcabaca(bcbcbc21 222 +βββ+ββ
= [ ]abcabccba21 222 βββ++β
=
[ ]ab2ca2bc2c2b2a241 222 βββ++β
= [ ]222 )ba()ac()cb(41 β+β+ββ
2. (a) Required area is 4
49ab2c2
= .
3. (d) Area of triangle formed by the straight line
0cbyax =++ with the coordinate axes is
ab2c2
If a, c, b are in G.P; abc 2 =
β Area of the triangle is 21
units.
4. (a) Solving the equations (0, -1) is the point of
intersection. 5. (d)
Orthocenter is O (0, 0) and circumcenter C is the
midpoint of AB. ie;
=23
,2C
49
4OC +=β΄ =4
25=
25
= 2.5
6. (d) Solving 4x + 5y = 0 and 11x + 7y = 9 we get
β34
,35
solving 7x + 2y = 0 and 11x + 7y = 9 we
get
β34
,35
midpoint is
21
,21
.
The other diagonal should pass through (0, 0) and
21
,21
. Hence its equation is y = x
Aliter:
The given sides pass through the origin, while the given diagonal does not. Hence, the required
diagonal must pass through the origin. Thus, the only correct choice is (d).
7. (b) Consider P(a, b), Q(a, c), R(d, c)
Slope of PQ = 0
bc β
Slope of RQ = ad
0β
β΄ PQ is perpendicular to QR β΄ Q(a, c) is the orthocentre. 8. (b) Equation of any line through ( )2,3 is
( )1...m3x2y =
ββ
Now, slope of the line 21
is3y2x =β
12m
1m2
2
m1
2
1m
45tan Β±=+β
β
+
β=
31
mor3mβ==β΄
Thus from (1) the required equations are 09y3x,07yx3 =β+=ββ
9. (a) Any line perpendicular to 03yx3 =β+ is
0Ky3x =+β .
It passes through (2, 2) β k = 4. β΄ equation of the line is 04y3x =+β Putting x = 0, we get the intercept.
y = 3
4
10. (c) The sides are given by 5x + 2y + 5 = 0,
5x + 2y β 5 = 0, 5x β 2y + 5 = 0, 5x β 2y β 5 = 0. These lines form a rhombus.
Area = units.sq525
552 =Γ
ΓΓ
11. (d) Slope of AC = 12
34
ββ+
=37β
β΄ slope of BD = 7
3
Midpoint of AC is
β21
,,2
1which is (the midpoint
of BD also) β΄ equation of BD is
+=β21
x73
21
y
ie; 6x β 14y + 10 = 0. ie; 3x β 7y + 5 = 0
12. (c) The angle ΞΈ is given by 1
23
1
21
3tan =
+
+β=ΞΈ
O A(4, 0)
B(0, 3)
23
,2C
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β ΞΈ = 4Ο
13. (b) The triangle is right angled. So the circumcentre is the midpoint of the hypotenuse, Hence circumcentre is (3, 3). 14. (a) Equation of the line is 4x + 2y + c = 0
y-intercept = 2cβ = 5
β c = -10 β΄ equation is 4x + 2y β 10 = 0 i.e., 2x + y β 5 = 0.
15. (a) Any line perpendicular to 1sinby
cosax =ΞΈ+ΞΈ is
kcosay
sinbx =ΞΈβΞΈ
This passes through ( a cosΞΈ, b sinΞΈ)
β΄ k = ab
ba 22 βsinΞΈ cosΞΈ
β΄ required line is
ΞΈΞΈβ=ΞΈβΞΈ cossinab
bacos
a4
sinbx 22
β ax secΞΈ β by cosec ΞΈ = a2 β b2
16. (a) Slope of PR = 25
Slope of PQ = m
tan ΞΈ = 1 1
2m5
1
m2
5
=+
β
m25
1m25 +=β or
2m5
1m25 +=+β
m = 73
of 37β
When m = 7
3, PQ is ( )3x
7
34y β=β
i.e., 7y β 28 = 3x β 9 β 3x β 7y + 19 = 0. 17. (c) The equation in normal form is
a60siny60cosx =+ ΞΏΞΏ
a2
y32x =+β
a2y3x =+β 18. (b) 1 + 1 β 4 < 0
041a3a2 2 <ββ+ββ΄
i.e. 08a2a2 <β+
ie; (a + 4) (a β 2) < 0 β΄ a lies in the interval (β 4, 2)
19. (b) ΞΈ+ΞΈ
=22 eccossec
ap
ΞΈ+
ΞΈ
=
22
22
sin
1
cos
1a
p
ΞΈΞΈ= 222 cossina
β ΞΈ= 2sinap4 222 (1)
Also ΞΈ+ΞΈ
ΞΈ=22 sincos
2cosap2
ΞΈ= 2cosap4 222 (2)
(1) + (2) β 22 ap8 =
β 8a
p2
2 = .
20. (d) b
siny
a
cosx ΞΈ+ΞΈ = 1 β bxcosΞΈ + aysinΞΈ β ab = 0
Let Ξ± = 22 ba β β΄ Point is (Ξ±, 0) and (βΞ±, 0)
β΄ perpendicular distance from (Ξ±, 0) is
P1 = ΞΈ+ΞΈ
βΞΈΞ±2222 cosbsina
abcosb and
P2 = ΞΈ+ΞΈ
βΞΈΞ±β2222 cosbsina
abcosb
β΄ P1P2 = ΞΈ+ΞΈΞΈΞ±β
2222
22222
cosbsina
cosbba
= ( )
ΞΈ+ΞΈΞΈββ
2222
222222
cosbsina
cosbabba
= [ ]
ΞΈ+ΞΈΞΈ+ΞΈβ
2222
222222
cosbsina
cosbcosaab
= ( )
ΞΈ+ΞΈΞΈ+ΞΈ
2222
22222
cosbsina
cosbsinab = b2
21. (b) p = ΞΈ+ΞΈ 22 eccossec
a
β p2 =
ΞΈ+
ΞΈ 22
2
sin
1
cos
1a
= a2 sin2 ΞΈ cos2 ΞΈ
β 4p2 = a2 sin2 2ΞΈ Similarly q2 = a2 cos2 2ΞΈ β΄ 4p2 + q2 = a2. 22. (c) The lines are 6x + 8y β 10 = 0 and 6x + 8y β 45 = 0 Distance between them
= 5.310
35
6436
4510
ba
cc
22
12 ==+
+β=
+
β
23. (d) Using
+
++β=
β=
β22
1111
ba
cbyax2
b
yy
a
xx
1
8y
1
6x
ββ=β
( ) ( )6,8y,x2
862 =β
ββ=
24. (c) The side of the square is equal to the distance between the parallel lines which is equal to
P (3, 4)
R (1, -1)
Q
S
O
ΞΈ
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= 2019
86
257
22=
+
+
β΄ diameter of the circle is 20
219
β΄ area =
2
202219
4
ΓΟ
=800361Ο
25. (a) Distance between the lines = 25
86
629
22=
+
ββ
26. (a) Let the image be (h, k). Then
( )
191143
21
4k3
1h+
β+ββ=β=+
β h = 5 , k = 6 27. (b) Line joining (β1, 2) and (5, 4) is x β 3y + 7 = 0. If (h, k) is the foot of the perpendicular, then
( )
917031
130y
11h
++Γββ=
ββ=β
β h = 5
12k,
51 =
28. (b) 2x + y = 7 passes through (2, 3) but it is parallel to
the given lines, so it will not make any intercepts with them. Point of intersection of x β 2 = 0 with 2x + y β 3 = 0 and 2x + y β 5 = 0 are (2, β1) and (2, 1) respectively. β΄ The intercepts made by x β 2 = 0 with them is 2 unit. x β 2y + 4 = 0 passes through (2, 3) but perpendicular to the given parallel lines
β΄ The required intercept is the distance between
the parallel lines is 5
2
14
53 =+ββ β
β 2
Finally, 2x + 3y β 4 = 0 does not pass through (2, 3)
β΄ choice (b)
29. (b) Locus of P(x, y) such that PA = PB is the required equation
( ) ( ) ( ) ( )2222 3y2x1y1x β+β=β+ββ΄
011y4x2 =β+β 30. (a) The other bisector is perpendicular to x + y β 2 = 0 Hence the equation is x β y + k = 0 This passes through (1, 1) β k = 0 β΄ Equation of the other bisector is x β y = 0
HINTS/SOLUTIONS for M1105 (Complex numbers and Quadratic Equations)
Classroom Discussion Exercise 1. (a) 5i.
2. (d) Let ibai247 +=β β 7 β 24i = a2 β b2 + 2iab. Equating the real and imaginary parts, we get
a2β b2 = 7 and ab = β12.
Now a2 + b2 = ( ) 22222 ba4ba +β
= 25 β 2a2 = 32 β a = Β± 4 When a = 4, b = β3 When a = β4, b = 3.
β΄ ( )i34i247 βΒ±=β 3. (d) 5i5 + 4i4 β 3i3 + 2i2 β i = 5i + 4 + 3i β 2 β I = 2 + 7i
4. (d) tanβ1
+
β
rs
tanqp 1
=
β
+β
rs
pq
1
rs
pq
tan 1 =
β+β
qsprpsqr
tan 1
= ( )4
n1tan 1 Ο+Ο=β
5. (d) (x + 1 + i) (x + 1 β i) (x β 1 + i) (x β 1 β i) =(x2 + 2x + 1 + 1) (x2 β 2x + 1 + 1) =[ (x2 + 2) + 2x ] [(x2 + 2) β2x] =(x2 + 2)2 β 4x2 = x4 + 4. 6. (c) The given equation is (1 + i) (3 β i) x β 2i (3 β i) +
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(2 β 3i) (3 + i) y + i (3 + i) = i (3 + i) (3 β i) β (4 + 2i) x β 6i β 2 + (9 β 7i) y + 3i β 1 = 10i β (4x + 9y β 3) + (2x β 7y) = 13i Equating real and imaginary parts, 4x + 9y β 3 = 0 2x β 7y = 13 Solving, we get x = 3, y = -1.
7. (c) 3423175675
5473322115
iiiii
iiiii
β+ββ+β+β
( )32211554732
5473322115
iiiiii
iiiii
+β++ββ+β+β
= 1i
12
=β
Aliter : Zkwhere
ii
1i
ii
1i
3k4
2k4
1k4
k4
β
β=β=
==
+
+
+
Substituting for each term in the numerator and denominator, we get the answer.
8. (b) (5 + 2i) 2 = 21 + 20i β΄ Conjugate of (5 + 2i)2 is 21 β 20i.
9. (b) iyxyx 22 +=+
= 1i2i2 =
β+
10. (d) Complex conjugate of Ο+Οcosi
2sin
ΟβΟ= cosi2
sin
2
sini2cosΟ+Ο=
11. (c) ( ) ( ) ( )
( )i1i1i23i2
β+ββ
= i1
i1i23i2
β+ββ
=
.652
2135=
12. (a) Given 40 Γ (x 2 + y 2) Γ 25 Γ 10 = 2500
β 41
yx 22 =+ .
13. (d) The given expression is
Ο+Ο
Ο+Ο
Ο+Ο
3sini
3cos
6sini
6cos
4sini
4cos
= sini364
cos +
ΟβΟ+Ο
ΟβΟ+Ο364
= 12
sini12
cosΟ+Ο
14. (d) =β+
1z1z
2
2
k3ik1
k3ik3
β++
β++
= ( )( ) 22
22
k3k1
k3k3
β++β++
=
3k24k612 =
++
15. (a) AB = BC = CD = DA = 13 Also AC2 = | 17 β 7 i | = 338 AB2 + BC2 = 169 + 169 = 338 = AC2
β΄β ABC is right angle β΄ABCD is a square.
16. (b) ( ) iba3i25 3940+=+
β iba3i25 3940
+=+
β 223940 ba.33 +=
β 3ba 22 =+ β a2 + b2 =9. 17. (b) z = sinΞΈ + icosΞΈ
ΞΈβΟ+
ΞΈβΟ=2
sini2
cos
β΄ z2 = cos (Ο β 2ΞΈ) + i sin (Ο β 2ΞΈ)
and 2z
1= cos(Ο β 2ΞΈ) β isin (Ο β 2ΞΈ)
( )ΞΈβΟ=+β΄ 2cos2z
1z
22
18. (a) The point P can be obtained by rotating the
complex number i3 + in anticlockwise and clockwise direction. Hence P can be either
( ) ( )i3iori3i +β+ 19. (d) ΞΈ+ΞΈ= sinicosxLet
ΞΈβΞΈ= sinicosx1
ΞΈ+ΞΈ= 6sini6cosx6
ΞΈβΞΈ= 6sini6cosx
16
ΞΈ=β 6sini2x
1x
66
20. (a) Putting x = 23β in the given equation,
2 x 49
- 23
p β 6 = 0
β p = β1
21. (c) =Ξ± 10
i3i3
1 β=+
10
i3
+=Ξ²
106=Ξ²+Ξ± and
101
Ξ±Ξ² =
Required equation is 0101
x1062x =+β
ie; 10x2 β 6x + 1 = 0 β΄ a = 10, b = β6 and c = 1
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22. (d) Since Ξ±, Ξ² satisfy x3 β 1 = 0 Ξ±3 = 1 and Ξ²3 = 1, Ξ±2 + Ξ± + 1 = 0 Ξ²2 + Ξ² + 1 = 0 β΄ (Ξ±2 β Ξ²2) + (Ξ± β Ξ²) = 0 Ξ± + Ξ² + 1 = 0, β Ξ Ξ± β Ξ² β Ξ± + Ξ² = β1 β΄ (Ξ± + Ξ²)27 + (Ξ±100 + Ξ²100)27 = Ξ± + Ξ² = β1 = (Ξ± + Ξ²)27 + (Ξ± + Ξ²)27 Ξ Ξ±100 = Ξ±, Ξ²100 =
Ξ² = (β1)27 + (β1)27 = β2
23. (a) 01x2x4 2 =β+
β΄41
- and21 =Ξ²Ξ±β=Ξ²+Ξ±
Since Ξ± is a root, therefore Ξ±+Ξ± 24 2 β 1 = 0
β 24Ξ± = 1 β 2 Ξ±
β 34Ξ± = 22 Ξ±βΞ±
= 2
)21( Ξ±ββΞ±
= 2
14 βΞ±
21
2614
34 3 βΞ±β=Ξ±ββΞ±=Ξ±βΞ± = Ξ².
24. (a) Ξ±
β=+Ξ±β=+Ξ±+Ξ± rqp 0rqp 2 and
Ξ²
β=+Ξ²β=+Ξ²+Ξ² rqp 0rqp 2
β΄ rrqpqp
Ξ±Ξ²βΞ±Ξ²β=+Ξ²
Ξ±++Ξ±
Ξ²
pr
xr2β=
=p2β
25. (d) |z1 + z2| β₯β₯β₯β₯ |z1| + |z2|.
Regular Homework Exercise
1. (b) Let z = i2i2
+β
z = i2i2
β+
= ( )
( ) ( )i2i2i2 2
+β+
5
i43 += .
2. (a) Let ibai6011 +=+β
β β11 + 60i = a2 β b2 + 2iab β a2 β b2 = β11 and ab = 30
β΄ a2 + b2 = ( ) 22222 ba4ba +β = 61
β 2a2 = 50 β a = Β± 5 When a = 5, b = 6 When a = β5, b = β6
β΄ ( )i65i6011 +Β±=+β .
3. (c) ( )( )( )( )i2i2
i23i2i23i2
+β+β=
ββ
2
2
i4
i36i2i4
βββ+= i
51
58 +β= .
4. (a) 3x + i(4x β 6y) = 2 β i Equating the real and imaginary parts, we get
3x = 2 and 4x β 6y = β1.
β΄ x = 32
Thus 4x β 6y = β1 β y = 1811
5. ( ) ( )22i484i484 ββ++β = ( ) ( )
+β22 i4842
= β 64
6. (d) Modulus is a non-negative real number.
7. (c) 2
zz.z = = 130.
8. (c) 0z0z0zz2 =β=β=
9. (b) Complex conjugate of ΟβΟcosi
2sin
Ο+Ο= cosi2
sin
2
sini2cosΟβΟ=
10. (d) We cannot compare two complex numbers since there is no ordering in the set of complex numbers. [Note that the comparisons in (a), (b) and (c) are in R, the set of real numbers]
11. (b) Argument = 1243Ο=ΟβΟ
12. (b) 2.5.10 β¦β¦ ( 1 + n2) = |1 + i| |1 + 2i| β¦.|1+ ni| = x2 + y2
13. (c) xx5858 = which is true only
when x = 0. Thus number of solutions = 1. 14. (d) The given complex number has modulus 1 and
argument
ΞΈβΟ2
.
15. (b) [ ]20
1010
2i3
3iyx3
+=+
Now 2020
6sini
6cos
2i
23
Ο+Ο=
+
620
sini6
20cos
Ο+Ο=
21
xβ=β and
23
yβ=
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21
431
yx 22 β=β=β
16. (d) Given that 91
1k5 2
=β
β k2 = 2 Now, discriminant = 4k2β 4(5k2 β 1) = 8 β 4 (10 β 1) < 0 β΄ The roots are imaginary. 17. (c) Interchanging the constant term and the
coefficient of x we get the required equation as qx2 β px + 1 = 0
18. (c) If 3i2 + is a root then 3i2 β also a root.
Sum of the roots = 22 Product of the roots = 5
β΄β΄β΄β΄ Required equation is 05x22x2 =+β
19. (c) x2 + x + 1 = 01x,1x1x3
β βββ
β΄Ξ±, Ξ² are the roots of x2 + x + 1 = 0
β 01
13
=βΞ±βΞ±
Ξ± β 1 β Ξ±3 = 1
11011 3
3
=Ξ²ββ Ξ²β=βΞ²βΞ²
Ξ±400 = (Ξ±399. Ξ±) = Ξ±
Ξ²400 = (Ξ²399 . Ξ²) = Ξ² Again Ξ±2 + Ξ± + 1 = 0 and Ξ²2 + Ξ² + 1 = 0 β Ξ± + Ξ² = β1 β΄ Ξ±400 + Ξ²400 = Ξ± + Ξ² = β1 20. (c) We have
( )
23
i21
41i323
313i
3i
3i2
+=β+=+
+=+β
+
= cos3
sini3
Ο+Ο
3
200sini
3200
cos3i
3ii200
Ο+Ο=
+β+
Similarly 3
200sini
3200
cos3i
3i200
ΟβΟ=
+β
β΄
200200
3i
3i
3i
3i
+β+
β+
121
23
200cos2 β=βΓ=Ο=
= (βΟ2)200 + Ο200 = Ο400 + Ο200 = β1
Assignment Exercise
1. (b) i 6 + i7 + i8 + i9 = β1 β i + 1 + i = 0.
2. (c) i21i21
+β
= ( )
5i21 2β
β΄ Square root = ( )i215
1 βΒ± .
3. (b) ( )
( ) ( ) i11
1i21i1i1
i1i1i1 2
=+
β+=β+
+=β+
which lies on
yβaxis. 4. (d) β3 + ix2 y & x2 + y β 4i are conjugates β x2 + y = β3, x2 y = β 4. Solving, we get x = Β± 1, y = β4
5. (a) (cos t + i sin t) (cos t β i sin t) = cos 2 t + sin 2 t =
1. 6. (d) |z1| + |z2| = |1 + 2i| + |2 + 3i|
= 9441 +++ = 135 +
7. (a) Modulus, r = 213 =+
tan ΞΈΞΈΞΈΞΈ =3
1ββββ ΞΈΞΈΞΈΞΈ =
6Ο
+=+
2i
23
2i3
= 2
Ο+Ο6
sini6
cos
8. (a) Equating modulus on both sides, we get
ibaix1ix1 β=
+β
β 1 = 22 ba + β 1ba 22 =+ 9. (a) Let z = x + iy.
β΄ ( ) ( )2222 1yx1yx1iziz ++=β+β=
+β
β y = 0, which represents the x - axis
10. (c) arg ( )3
3i1Ο=+
arg ( )6
i3Ο=+
β΄663i3
3i1arg
Ο=ΟβΟ=
+
+
11. (d)
+=+ i
23
21
2i31 =
Ο+Ο3
sini3
cos2
β΄ ( )
Ο+Ο=+3
5sini
35
cos2i31 55
and ( )
ΟβΟ=β35
sini3
5cos23i1 55
Thus, ( ) ( )553i13i1 β++ = 26 cos
35Ο
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= 25 = 32
12. (c) We have 2121 zzzz +β€+
β΄β΄β΄β΄ 2z2z +β€+
β€ 4 ( )4z β€Ξ
13. (b) Sitaβs equation is (x - 2) (x - 5) = 0 i.e. x2 β 7x + 10 = 0 Mamataβs equation is (x + 6) (x + 1) = 0 i.e. x2 + 7x + 6 = 0 β΄ Correct equation is x2 β 7x + 6 = 0 β x = 1 or 6. 14. (b) Sum of the roots = β 6i β the other root = β3 β 10i Product of the roots = k
β k = 31 β 42i
15. (c) ( )1003i1iyx β=+
= ( )1003i1+β
=2100
100
23
i21
+β
=2100
+β
23
i21
Ο=Ο=
+ 100
100
23
i1cesin
β΄ x = β299, y = 299 3
Additional Practice Exercise
1. (c) The co-ordinates of the vertices of the triang le
are ( ) ( ) ( )37,3,7,,7,3,,3,7 +ββ
Area = ( )( ))yy(x)yy(xyyx21
213132321 β+β+β
= ( )( ) ( )[ 3373377721 β+++ββ
( )( )]7337 +β+
= [ ] 553721217721 =β=β++βββ
2. (b) =β+
i31i32 ( )( )
( )( )i31i31i31i32
+β++
= 10
i97 +β
β΄β΄β΄β΄ Im 109
i31i32 =
β+
3. (b) i32i21
+β
= ( ) ( )( ) ( )i32i32
i32i21
β+ββ
= 13
i7
13
4
94
6i72 ββ=+
ββ.
4. (d) i 4 + i8 = 1 + 1 = 2.
5. (a) ( )( )iba
ibaz
β+=
( ) ( )( ) ( )ibaiba
ibaiba
+β++= =
22
22
ba
biba2a
+β+
Im z = 22 ba
ab2
+.
6. (c) z = ( ) ( )i1i1
i1
i1
1
β+β=
+ =
2
i1β
Re z = 2
1.
7. (c) The point 4i lies on positive y β axis.
8. (c) Multiplicative inverse of a + ib = iba
1
+
= 22 ba
iba
+β
9. (a) ( ) ( )2222 5yx5yx1i5z
i5z ++=β+β=+β
β y = 0
10. (a) ( ) ( ) 01izi1izz2 =+++ (Ξ i2 = β1)
( )( ) 01iziz2 =++βi1
zoriz2 β=β=β
In both cases |z| = 1
11. (c) The given points form a rectangle.
12. (a) (b + ia)5 = ( )[ ]5ibai β
= ( ) ( )Ξ²βΞ±=β iiibai 55 = Ξ²+Ξ±i .
13. (b) ( ) qipyix 31
+=+
( )3iqpyix +=+β = p3 β 3pq2 + i (3p2 q β q3) β x = p3 β 3pq2 and y = 3p2 q β q3
2222 qp3q3pqy
px β+β=+β΄
= 4 (p2 β q2).
14. (a) Let iyxi125 +=β
β΄ 5 β 12i = x2 β y2 + 2ixy β x2 β y2 = 5 and xy = β 6
Now x2 + y2 = ( ) 22222 yx4yx +β
(β5, 2) (5, 2)
(5, β2) (β5, β2)
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= 13 β΄ 2x2 = 18 β x = Β± 3 When x = 3, y = β2 When x = β3, y = 2
β΄ ( )i23i125 βΒ±=β
15. (b) 1z7 β=
28917586 zzz ++
( ) ( ) ( ) 24172572127 zzzzz β ++β =
( ) ( ) ( ) 24125212 z.11z1 β+β+β β=
1z1z 22 β=ββ= .
16. (c) x + 2 = 7i
β x2 + 4 + 4x = β49
β x2 + 4x + 53 = 0
β΄ x3 + 4x2 + 53x + 5 = 5
17. (a) ( ) 2ttit1iyx 2 +++β=+
)t1(x β= 2tty 2 ++=
( ) ( ) 2x1x12tty 222 +β+ββ++=
4x3x2 +β=
β+
β=49
423
x2
β
ββ
47
23
x
47y
2
2
= 1, a hyperbola
18. (d) 212
1
2
1 zzzz
zz ββ =
19. (b) We have 6Z9
Z =+
β΄ |Z| = Z9
Z9
Z β+
β€ Z9
Z9
Z ++
β€ |Z|
96 +
β |Z|2 β 6 |Z| + 9 β€ 18
(|Z| β 3)2 β€ 18 β |Z| β 3 β€ 18
β |Z| β€ 3 + 18
So maximum value of |Z| is 3 + 18
20. (d) 1 +
Ο+Ο=3
sini3
cos23i
β΄ ( ) 33
3sini
3cos83i1
Ο+Ο=+
= 8 (cos Ο + i sin Ο) = β 8.
21. (c) ( )
i2
i1i1i1 2
=+=β+
β΄arg
β+
i1i1
= 2Ο
22. (d) ( ) ( )
β=ββzz
argzargzarg
= arg (-1) = Ο.
23. (a) 2iz1iz =β+β
β 2iziiz 2 =β++
β ( ) 2izizi =β++
β 2iziz =β++ , which represents a line
segment [Note: [|z β z1| + |z β z2| = k represent (i) a straight line if |z1 β z2| = k (ii) an ellipse if |z1 β z2| > k ]
24. (c) ( ) ( ) ( )4192i413i41322
β=β++
=β 64.
25. (b) 1 β sin Ξ± + i cos Ξ±
=
Ξ±βΟ+
Ξ±βΟβ2
sini2
cos1
=
Ξ±βΟ
Ξ±βΟ+
Ξ±βΟ24
cos24
sin2i24
sin2 2
=
Ξ±βΟ+
Ξ±βΟ
Ξ±βΟ24
cosi24
sin24
sin2
=
Ξ±+Ο+
Ξ±+Ο
Ξ±βΟ24
sini24
cos24
sin2
β argument = 24Ξ±+Ο
.
26. (d) |Ο| = 1
β 1
2i3
z
z =β
β 1
2i3
z
|z| =β
β |z| = i23
z β
β x2 + y2 = x2 + 2
23
y
β (where z = x + iy)
β y = 43
, a straight line.
27. (b) Let z = x + iy
( )( )i54z
i23zzzzz
2
1
+β+β=
ββ
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( ) ( )( ) ( )5yi4x
2yi3xβ+ββ+β=
( ) ( )[ ] ( ) ( )[ ]
( ) ( )22 5y4x
5yi4x2yi3x
β+βββββ+β=
( )( ) ( )( )
( ) ( )22 5y4x
5y2y4x3x
β+βββ+ββ=
( )( ) ( )( )
( ) ( )22 5y4x
5y3x4x2yi
β+ββββββ+
( ) ( )22
22
5y4x
22y7yx7x
β+β+β+β=
( ) ( )22 5y4x
7yx3i
β+βββ+
arg 4zz
zz
2
1 Ο=
ββ
122y7yx7x
7yx322
=+β+β
βββ
β x2 β 10x + y2 β 6y + 29 = 0 β (x β 5)2 + (y β 3)2 = 5 β |(x β 5) + i (y β 3)|2 = 5
β | (x + iy) β (5 + 3i) |2 = 5
β | z β (5 + 3i) | = 5
28. (b) In a parallelogram diagonals bisect each other.
β΄β΄β΄β΄ 2
zz2
zz 4231 +=+
β z1 + z3 = z2 + z4 29. (a) In a square, lengths of the diagonals are equal. β΄β΄β΄β΄ |z1 β z3| = |z2 β z4|
30. (b)
Ο+Ο=+6
sini6
cos2i3
6sini
6cos
2i3 Ο+Ο=+
β 6x
sini6x
cos2
i3x
Ο+Ο=
+
So from the given equation, we get
16
xsini
6
xcos =
Ο+
Ο
β cos6xΟ
= 1 and sin6xΟ
= 0
sin6xΟ
= 0 β 6xΟ
= nΟ; n β Ξ
β x = 6n
Now cos6xΟ
= 1 β cosnΟ = 1
β n is a multiple
of 2
β x = 12,
24,β¦β¦β¦
HINTS/SOLUTIONS for M1106 (PMI, Sequence and Series)
Classroom Discussion Exercise
1. (b) t1 = 252; tn = 798; d = 7
n = 1d
tt 1n +β = 17
252798 +β = 79
2. (b) a = 5 , d = 5
t20 = a + 19d = 5 +19 5 = 520
3. (c) cb
1,
ac1
,ba
1+++
are in A.P
βba
1
ac
1
ac
1
cb
1
+β
+=
+β
+
β bacb
cbba
+β=
+β
β 2222 cbba β=β
β 222 b2ca =+
β 222 c,b,a are in A.P
4. (b) a + 17d = 108 and a + 107d = 18 β d = β1 and a = 125 β΄ 126th term = 125 + (126 β 1) x β 1 = 0
5. (a) 1bc
)cb(a ++ , 1ca
)ac(b ++ , 1ab
)ba(c ++
are in A.P.
bc
cabcab ++ ,ca
cabcab ++ ,ab
cabcab ++ are in
A.P.
P.Ainareab1
,ca1
,bc1
β abc
bc1 , abc
ca1 , abc
ab1 are in A.P.
β a, b, c are in A.P. 6. (d) x2 = x1 + d d = 12 xx β = ( ) ( )1212 xxxx β+
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β d
xx
xx
1 12
12
β=
+
Similarly, d
xx
xx
1 23
23
β=
+
d
xx
xx
1 34
34
β=
+
d
xx
xx
1 1nn
n1n
β
β
β=
+
n1n3221 xx
1.......
xx
1
xx
1
+++
++
+β΄
β
= d
xx 1n β
But x n = x1 + (n ββββ 1)d
(n ββββ 1)d = xn β x1 = ( ) ( )1n1n xxxx β+
ββββ
+β=
β
1n
1n
xx
1nd
xx
n1n3221 xx
1.....
xx
1
xx
1
+++
++
+β΄
β
=
1n xx
)1n(
+β
7. (d) a = 1; tn = 101; d = 2; n = 51
S = [ ]1011251 + = 2601
210251 =Γ
8. (b) 2nd term = S2 β S1 = 30 β 9 = 21 9. (b) a + 2d = 5 a + 6d = 3 (a + 2d) + 6 β a = β3 and d = 4 β΄ S32 = 16 [β 6 + 31 x 4] = 1888 10. (c) a + (n β 1) d = 164
( )[ ] n5n3d1na2.2n 2 +=β+
a + 164 = 3n + 5 Now, a = 3 Γ 1 + 5 Γ 1 = 8 Thus 3n + 5 = 86 β n = 27
11. (d) [ ]2
)1m(m1x)1m(2
2m
S1+=β+=
S2 = ( )[ ] ( )2
1m3m31m4
2m +=Γβ+
[ ])1p2)(1m(p22m
Sp ββ+=
= [ ]1mp2pm2p22m +ββ+
= [ ]1)1p2(m2m +β
β΄β΄β΄β΄2
mS.....SSS p321 =+++ [ ]{ }1)1p2(m +ββ
= [ ]pp.m2m 2 +
)1mp(2
mp +=
12. (d) 2b = a + c β΄ 72b = 7a + c β (7b)2 = 7a. 7c
13. (d) ar4 = 2 a . ar . ar2 β¦ ar8 = a9 . r36 = (ar4)9 = 29 = 512 14. (b) Let the numbers be a, ar, ar2. Given a + ar + ar2 = 21 β a ( r + r + r2) = 21
β a . 21r1r1 3
=β
β
(1) Also given a2 + (ar)2 + (ar2)2 = 189 β a2 (1 + r2 + r4) = 189
β a2 189r1
r1 6
=β
β (2)
Solving these two equations, we get a and r. Aliter: The only numbers in the given choices
whose sum is 21 are 3, 6, 12 15. (d) 434 214 34 21
digitsndigitsn
2 )5...555()3...333( +
= (3 + 30 + 300 + β¦β¦..+ n terms )2 + (5 + 50 + 500 + β¦β¦β¦+ n terms )
= ( )( )
( )110
1105
110
1103 n2n
ββ+
ββ
= ( )
+ββ
59
11039
110 nn
= ( ) ( )
271410110 nn +β
16. (a) Let the terms be ra , a,ar.
a3 =21
a81 =β
( )16
21ar
r
aaraa
r
a =
Γ+Γ+
Γ
1621
arar
a 222
=++
1621
r1
r141 =
++
4
21
r
1r1 =++
( ) r211rr4 2 =++
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04r17r4 2 =+β
41
or4r =
Numbers are 81
,21
,2or2,21
,81 .
17. (b) Given arn β 1 = arn + arn + 1
β rrrrr nnn
β +=
r2 + r β 1 = 0
β r = 2
51Β±β
But r is positive
β΄ r = 018sin22
15 =β.
18. (c) (2k + 3) β (k + 2) = (4k β 1) β (2k + 3) β k = 5
19. (b) Given ( ) 21
nn
1n1n
abba
ba =++ ++
21
n21
21
21
n1n1n b.ab.aba
++++ +=+β
( ) ( )babbaa 2
1n
2
1n
β=ββ++
β 1ba 2
1n
=
+
or 21
n021
nβ=β=+
. 20. (d) b 2 = ac gives x = β 4 so that the numbers
are β 4, β 6, β 9
β΄ 4th term = 227β
21. (d) a = 12; ar5 = 384
r5 = 3212384 =
β΄ r = 2
22. (c) n th term nn
n
2
11
2
12 β=β=
Sum to n terms = ββ
βn
1n
n
21
n
=
β
β
β
21
1
21
121
n
n
= n21n β+β .
23. (d) ( ) ( )( )6
1n21nn71
21nn ++Γ=+
β 2n + 1 = 21 or n = 10
24. (d) tn = 2
1nn
n +=β
β΄ Sn = ( )2
1nβ +
=
( )
2
n2
1nn ++
= ( )4
3nn +
25. (d) (12 β 22) + (32 β 42) + β¦ + (492 β 502) + 512
= β1(3 + 7 + 11 + β¦ + 99) + 512 = 1326
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Regular Homework Exercise
1. (d) 7 Γ t7 = 11 Γ t11 7 [a + 6d] = 11 [a + 10d] 7a + 42d = 11a + 110d 4a + 68d = 0 or a + 17d = 0 β΄ t`18 = 0 2. (b) Tn = a+(n-1)d 60th term = 3+59 Γ 5 = 298 3. (a) a = 20; d = 1; tn = 99
S = 2n (a + tn)
n = 11
2099 +β = 80
β΄ S = ( ) =+ 99202
80 40 Γ 119 = 4760
4. (a) Tn = a + (nβ1) d = 4 β 3i + (n β 1) (i β 2) = 6 β 2n + i (n β 4) Equating the imaginary part equal to
zero, we get n = 4. β΄ The 4th term of the sequence is purely real
. 4th term = T4
= β 2.
5. (c) n = 9, 61
dβ= ;
2
1a =
( )23
61
1921
229
S9β=
ββ+Γ=
6. (d) 6
2561
)1n(61 =β+ β n = 25
7. (a) a = 11, d = 2, a n = 99, an = 11 + (n β 1) 2 99 = 9 + 2n or n = 45. S = 2475. 8. (c) Let the numbers be a-d, a, a+d a = 5 5(5 β d) + 5(5 + d) + (52 β d2) = 71 d2 = 75 β 71 = 4 i.e. d = Β± 2 The numbers are 3, 5, 7 9. (d) a = 1,x = t n = a + (n β 1) d where d = 5
β 5
4xn
+=
β΄ ( )[ ] 148d1na22n =β+ gives
( ) 148x110
4x =+
+
β x = β 41, 36
But, the given A.P is increasing, and hence x = 36.
10. (d) 4k
4
4k2
4k
β=
+β Solving k = 16
11. (d) a =4; r = 3 ; tn = 36 Γ 34 arn β 1 = 36 Γ 34
( ) 61n3434 Γ=Γ
β
62
1n
33 =β
β 62
1n =β β n = 13
12. (d) a = 3, arn-1=192, Sn = 381 rn-1 = 64 i.e. rn = 64r
3811r
)1r(3 n
=β
β
i.e. rn β 1 = 127 (r-1) 64r β 1 = 127r-127 r = 2 2n-1 = 64 = 26 i.e. n = 7 13. (b) ar2 = 32 product of 1st 5 terms a.ar.ar2.ar3.ar4 = a5.r10 = (ar2) 5 = (32)5.
14. (d) 43
a = , r=2, nth term = 384
384)2(43 1n =β i.e 2n-1 = 384 x
34
= 29 n = 10
Sn = =β
β
12
)12(43 10
43069
1023x43 =
15. (b) x2 β 5x + 6 = 0 (x β 3).(x β 2) = 0 x = 3 or 2 β΄ roots are 3 and 2. Ξ± = 3 and Ξ² = 2 G.M. = 632 =Γ=Ξ±Ξ² .
16. (c) There are now n + 2 terms in the A.P.
2 + 2d = 21 (7 β 2d). Solving, d =
21 .
Also, 2 + (n + 1)d = 7. Solving, n = 9. 17. (d) T1 = 2; T2 = 3 = 2+12 T3 = 7 = 2+12+22, --------- Tn = 2+12+22+--------+(n-1)2
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= 6
)1n2(n)1n(2
ββ+
= 6
12)1n2)(1n(n +ββ
18. (d) ( ) ( ) ( )nn.....2211 222 ++++++
β β= =
+=n
1i
n
1i
2 ii = ( )( ) ( )2
1nn6
1n21nn ++++
= ( )( ) ( )6
1nn31n21nn ++++
= ( ) ( )6
2n21nn ++ = ( )( )3
2n1nn ++ .
19. (c) 552
)1n(nn...321 =+=++++
( )
+=++++4
1nnn....321
233333
. = 55 Γ 55 = 3025
20. (b) n .1 + (n +1)2 +(n +2)3 +-----+ ( )n1nn β+ = [ n +2n +3n +---+n.n]+1Γ 2+2Γ3+---+(nβ1)n = n (1+1+3+----+n) + ( )n1n ββ
= ( )nn
2
1nnn 2 βββ+
+
= n ( ) ( )( ) ( )2
1nn
6
1n21nn
2
1nn +β+++
+
= (nβ1) ( ) ( )3
1n2
2
1nn
2
1nn +Γ++
+
= ( )
++β+3
1n23n3
2
1nn
= ( )( )6
2n51nn β+ .
Assignment Exercise
1. (d) b1
c1
a1
b1 β=β
bc
cbab
ba β=β
ac
abbc
bacb ==
ββ
2. (d) 2b = a + c
b2 β ac = ac2
ca2
β
+
4 (b2 β ac) = (a β c)2 3. (d) Ξ± , Ξ± + d, Ξ± + 2d be the angles Ξ± + Ξ± + d + Ξ± + 2d = 180 3Ξ± + 3d = 1800 Given Ξ± + 2d = 2Ξ± 2d = Ξ±
d = 2Ξ±
3Ξ± + 2
3Ξ± =1800
2
3Ξ± = 600
Ξ± = 400 β΄ largest angle is 800 4. (d) Sn = 16200, a = 100, d=20
[ ] 1620020)1n(2002n =β+
n(10+n-1) = 1620 n2 + 9n β 1620 = 0 (n + 45) (n-36) = 0 n = -45 or 36
5. (d) a + (p β 1) d = q and a + (p + q β 1) d = 0 a + ( p β 1) d + qd = 0 q + qd = 0 β d = β1 a + (p + q β 1) d = 0 a + (q β 1) d + pd = 0 β΄ a + (q β 1)d β p β 1 = 0 a + (q β 1) d = p 6. (d) 6a + 69d = 267 β 2a + 23d = 89
( ) 10688912d23a22
24ni
24
1
=Γ=+=β
7. (c) G1 = {3}, n(G1) = 1 β Ist term of G1 = t1 of
A.P. G2 = {7, 11}, n (G2) = 2 β Ist term of G2 = t2
of A.P G3 = {15, 19, 23, 29} n (G3) = 4 = 22 β΄ Ist term of G3 = ( ) 15P.Aoft 22
=
G4 = {31, β¦.}, n(G4) = 8 = 23 β΄ Ist term of G4 = ( ) APoft 32
β΄ n(G8) = 28β1 = 27 β΄ Ist term of G8 = ( ) P.Aoft 72
is t128 = 3 + (127) 4 = 511
8. (d) ( )[ ]
[ ] 3419410
111n182n
=Γ+
β+
n[11n +7] = (60)(84) This has no positive integral solution.
Thus n does not exist.
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9. (b) nth term of the first A.P. = β4 + (n β 1)7 = 7n β 11 nth term of the second A.P. = 61+(n-1)2 = 2n+59 β΄ 7n-11 = 2n + 59 5n = 70 i.e. n = 14 10. (a) ββ+++ 777777
= [ ]terms n to99999997 ββ+++
= [ ]ββββ+β+β+β )11000()1100()110(97
= ( )[ ]n10......101097 n2 β+++
= ( )10n910817 1n ββ+ .
11. (a) 51n 3243)3.(3 ==β β n=10 12. (a) ar 2 = 8 a . ar . ar 2 . ar 3 . ar 4 = a 5 . r 10 = (ar 2) 5 = 8 5
13. (a) Let the numbers be ra , a, ar
6a216ar,a,ra =β=
22
r3636r
36 ++ =364
91r99r
9 22
=++
09r82r9 24 =+β
i.e. 0)9r()1r9( 22 =ββ
r = 3 or 31 (+ve Nos.)
The numbers are 2, 6, 18
14. (a) c, a, b, d are in A.P. a β c = b β a = d β b
2
cbca
β=ββ
15. (d) 1 + 8 + 27 + 64 + β¦. n terms
= ( )
4
1nnn...........321
223333 +=++++ .
Additional Practice Exercise
1. (d) It is only a sequence. 2. (c) p[a+(p-1)d] = q[a+(q-1)d] a(p-q) + [(p2-q2) β (p-q)]d = 0 i.e. a+(p+q-1)d = 0, Ξ p β q (p + q)th term = 0 3. (c) Let there be 2n terms in the A.P β΄ a + (a + 2d) +β¦β¦ + [a + (2n β 2) d] = 72
β ( ) 72]d21na2[2
n =β+
β n [a + nd β d] = 72 (1) (a + d) + (a + 3d) + β¦β¦ + (a+(2nβ1)d) =
90
]d2)1n()da(2[2n β++ = 90
n [a+ (n β 1)d + d] = 90 i.e., n(a + (n β 1)d) + nd = 90 i.e., 72 + nd = 90, using (1) β΄ nd = 18 Moreover, [a+ (2n β 1)d] β a = 30 β΄ (2n β 1) d = 30 2nd β d = 30 2 Γ 18 β d = 30 β΄ d = 6 β΄ n = 3
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4. (c) a+(m-1)d = n1 and a+(n-1) =
m1
Solving, d = mn1 and a=
mn1
thmn term = a+(mn-1)d
= 1min
1mnmn1 =β+
5. (b) [ ]
[ ]D)1n(A22n
d)1n(a22
n
1n51n3
β+
β+=
+β
D
21n
A
d2
1na
β+
β+=
Since we are looking for the ratio of
the 4 th term, 32
1n =β i.e. n = 7
β΄ Required ratio
= 95
3620
17x517x3 ==
+β
.
6. (c) A, B, C are in A.P β B = 60
2
3CsinBsin
2
3cb =β=
β΄ sinC =
75A45C2
160sin
3
2 =β΄=β=
7. (d) tp = 9p + 2 t1 = 9 + 2 = 11 tn = 9n + 2
β΄ ( ) ( )2n9112n
tt2n
S n1n ++=+= = [ ]13n92n +
8. (c) Tn = a . rn-1 (G.P)
8th term = 3.729128
32
7
=
9. (a) 26r.a 24 = and 36.2r.a 511 =
737 )6(3.6.2r ==β΄
i.e. 6r = and 2a =
β΄ 3rd term = a . r2 = 26 10. (c) q β p, r β q, p are in G.P β (r β q)2 = p (q β p) (1) Since p, q, r are in A.P, r β q = q β p = d, the common difference. β΄ (1) β d2 = p.d β d = p, Ξ d β 0 β΄ q = p + d = 2p r = p+2d = 3p
β΄ p : q : r = 1 : 2 : 3. 11. (d) r > 1 a + arnβ1 = 516 2048arar 2n =Γ β
β 1n2ra β = 2048
β 02048a516a2 =+β β a = 512 or 4
β ( ) 22
1n
4
2048or
512
2048r =β
β 12816
2048r1r 1n ==β΄> β and a = 4
Also given that ( )( )1r
1ra n
ββ =1020
β ( )1r
1r128a
ββ = 1020
β ( )1020
1r
1r1284 =β
β
β 128r β 1 = 127r = 254 β r = 2. rnβ1 = 128 β 2nβ1 = 27 β n β 1 = 7 β n = 8.
12. (a) Let ar,a,r
a be the numbers in G.P.
β΄β΄β΄β΄ .P.areinAar,a2,r
a
β΄ ( ) 01r4rrr1
aa22 2 =+ββ
+=
32r Β±=β Given G.P is increasing.
32r +=β΄ .
13. (d) a2 = 49
1
21
2
14
3=Γ
a = 71
14. (d) ar8 = 256 ar6 = 64 r2 = 4 r = Β± 2 Since 9th term is > 7th term, r = +2 15. (b) t n = 2n(2n + 2) and n = 20 gives t 20 = 1680 16. (c) ar 9 = 9 β¦β¦β¦.. (1) ar 3 = 4 β¦β¦β¦β¦(2)
Dividing (1) by (2); r 6 = 49 or r 3
= Β± 23
β a = Β± 38 β t 7 = a r 6 = Β± 6
49
38 Β±=Γ .
17. (c) arp + q β1. arp β q β 1 = m n
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β΄ a2. r2p β 2 = m n (arp β 1)2 = m n arp β 1 = mn 18. (b) ar4 = x; ar7 = y; ar10 = z i.e. x.z = a.r4.a.r10 = (a.r7)2 = y2 19. (c) 2p = a + b β (1) 2q = b + c β (2)
from (1) and (2); p + q = 2
cb2
ba +++
= 2
cb2a ++
and 4pq = (a + b) (b + c) = ab + b2 + ac + bc = ab + 2b2 + bc = b (a + 2b + c)
2pq = ( )cb2a2b ++
β=+
bqp
pq2 p, b, q are in H.P.
20. (c) r1
aS
β=β
2
21
1
1S1 =
β= , 3
23
x2
31
1
2S2 ==
β=
ββββ==β
= 434
x3
41
1
3S3
1pp
1pxp
1p1
1
pSp +=+=
+β
=
p321 SSSS ββββ+++β΄
=2+3+4+------+p+(p+1)
= 2
22p3p1
2)2p()1p( 2 β++=β++
= )3p(p21 +
21. (a) ar 2 = 6
a . ar . ar 2 . ar 3 . ar 4 = a 5 . r 10 = (ar 2) 5 = 6 5 22. (d) a, ar, ar2, -----, arnβ1,---------
β΄ =β=
1000
1nn2a ar +ar3 +-------1000 terms
β ar1
=Ξ±β
Ξ± ( 1+r2 +-----+r1998) ---(1)
β=
β
1000
1n1n2a = a +ar2 +------1000 terms
β =Ξ±+
Ξ±1
a ( 1+r2 +-----+r1998)----(2)
(1) Γ· (2) β r11 =
Ξ±βΞ±+
β 1 +Ξ± = rβrΞ± β Ξ± +rΞ± = rβ1
β Ξ± = 1r
1r
+β .
23. (c) xy22
yx =+
12
xy2
yx =+
13
xy2yx
xy2yx=
β+
++ β
( )( ) 1
3
yx
yx2
2
=β
+
β 13
yx
yx=
β
+ β
13
13
y
x
β+=
β 32
32yx
β+=
24. (a) From the data given, we have a, b, b + 3, b
+ 6 are the numbers so that a = b + 6 β΄ b + 6, b, b + 3, b + 6 are the numbers β΄ b2 = (b + 6) ( b + 3) b2 = b2 + 9b + 18 β b = β 2 β΄ numbers are 4, β 2, 1, 4 25. (c) a, x, b are in A.P i.e. 2x = a+b a, y, z, b are in GP y2 = az and z2 = by y3+z3 = yz (a+b) = 2xyz
26. (d) Ξ± + Ξ² = a
bβ Ξ± Ξ² =
a
c
2b = a + c β 2 a
c1
a
b +=
β΄ 2β (Ξ± + Ξ²) = 1 + Ξ± Ξ²
is 1 + Ξ± Ξ² + 2Ξ± + 2 Ξ² = 0
1 + 2 Ξ± = βΞ² ( Ξ± + 2)
Ξ²2 β
+Ξ±Ξ±+2
21
27. (a) 4, a1, a2, ------a7, 52 are in A.P. β΄ 52 = 4+ (9β1) d 48 = 8d 6 = d β΄ a6 β a5 = d = 6 a1 + a7 = 4 + 52 = 56 (a2 + a6 = a3 + a5β¦..)
28. (c) S = 361002
20192
=
Γ
29. (a) S = sum of squares of first 20 numbers
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Sn = ( ) ( )6
1n21nn ++
S20 = 6
412120 ΓΓ = 2870
30. (b) Ξ± + Ξ² = abβ Ξ±Ξ² =
ac
ab11
22β=
Ξ²+
Ξ±
β΄ ( ) a
b2
22β=
Ξ±Ξ²Ξ²+Ξ±
i.e., ( )
( ) ab2
2
2
β=Ξ±Ξ²
Ξ±Ξ²βΞ²+Ξ±
i.e., 2
2
2
2
a
cab
ac
2a
b β=β
ac2
a
bc
a
b3
2
2
2
=+
Multiply by bca2
β΄β΄β΄β΄ ba
2ac
cb =+
β΄ ac
andba
,cb
are in A.P
β΄ bc
,ab
,ca
are in H.P.
HINTS/SOLUTIONS for M1107 (Limits & Derivatives)
Classroom Discussion Exercise
1. (a) [ ]
1x1x
lim1x +
+ββ
=[ ]
1h11h1
lim0h +β
+ββ
=
[ ]h2h2
lim0h β
ββ
= 21
h21
lim0h
=ββ
2. (c) 3xx3
lim3x
|3x|lim
3x3x ββ
=ββ
ββ ββ= β1
3. (d) ( )xflim1x ββ
= 53x2lim1x
=+β
( ) ( ) 61x3limxflim1x1x
=+ββ +
β ( ) ( )xflimxflim1x1x +ββ
β
β ( )xflim1xβ
does not exist
4. (a) 7xlim2x
14x5xlim
2x
2
2x+=
ββ+
ββ= 9
5. (c) x31x
2xlim
2x ββββ
β
=( ) ( )
βββ
ββββ x31x
x31x21
lim2x
= ( )x31xlim21
2xβ+β
β= 1
6. (d) LHL = ( ) ( )
]x[x5x4x
lim5x β
ββββ
( ) ( )4x
5x4xlim
5x βββ
β
= ( ) 05xlim5x
=ββ
RHL = ( ) ( )5x
5x4xlim
5x βββ
+β
= ( ) 14xlim5x
=ββ
LHL β RHL β΄ Limit does not exist.
7. (a)
β
β=
ββ
βββ
25
x
25
xlim
52x2
5x2lim
nn
25x1nn
nnn
25x
1n
25
nβ
=
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8. (b)
22sin
lim
180x180
x2lim
limx
x2limlim
00x0x=
ΞΈΞΈ=
Ο
Ο
=βΞΈβΒ°
Β°
β
9. (a)
4x
cos
2xlim
2x Οβ
β
=
Ο+Οβ
4t
2cos
tlim
0t(put xβ2 = t)
=
4t
sin
tlim
0t Οββ =
Οβ4
10. (d) ( ) ( )
xx10sinx10sin
lim0x
ββ+β
=x
xsin10cos2lim
0xβ
= 2 cos10
11. (a) ( )( )2
2
x x
2/xsinsinlim
βΟΟ
Οβ, put t = Ο β x
= 2
2
0t t
2t
2sinsin
lim
βΟΟ
β
= ( )
2
2
0t t
2/tcossinlim
Οβ
=( )
2
2
0t t
2/tsinsinlim
ΟβΟβ
=( )
2
2
0t t
2/tsinsinlim
Οβ
=
( )( ) 22
22
0t t2/tsin
2/tsin2/tsinsinlim
ΟΟΓΟ
β=
4Ο
12. (c) Put ΞΈ = x β 3Ο
when x β 0,3
βΞΈΟ
β΄
Ο+ΞΈβ
ΞΈβΞΈ
3cos21
sin2lim
0
= ΞΈ+ΞΈβ
ΞΈβΞΈ sin3cos1
sin2lim
0
=
2cos
2sin32
2sin2
2cos
2sin4
lim20 ΞΈΞΈ+ΞΈ
ΞΈΞΈ
βΞΈ
=
ΞΈ+ΞΈ
ΞΈ
βΞΈ
2cos3
2sin
2cos2
lim0
= 3
2
13. (a) [ ] 1x,0x1For β=<<β
[ ][ ]
[ ] 010sin
xx1sinLt
0x =β
=+β΄ ββ
14. (b) (2x+3)2 = 4x2 + 12x + 9
β fβ(x) = 8x + 12
15. (b) y = x+x1
2x
11
dxdy β=
dxdy
= 0 at x = 1
16. (d) f(x) = sin (Ο+x)
β f(x) = βsinx
β fβ(x) = βcosx β fβ
Ο3
= 2
1β
17. (a) LHL =( ) ( )
h
3fh3flim
0h
β+β
=
h
0|3h3|lim
0h
ββ+β
=1
RHL=( ) ( )
h
3fh3flim
0h βββ
β =
h
0|3h3|lim
0h ββββ
β= β1
β fβ(3) does not exist
18. (a)
βββ=
ββ
ββ 2xx95
lim2x
)x(f)2(flim
2
2x2x
=
β+β
+ββ 2
2
2xx95)2x(
x95lim
=
β+β
β+β 22x
x95)2x(
)2x()2x(lim
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= 5
2
55
4 =+
.
19. (a) ( )1xsecxcosdxdy
xcos 222 +=
= 1 + cos2 x = 1 + (1 β sin2x)
= 2 β sin2x.
20. (b) ( ) ( )
2xxf22xf
lim2x β
ββ
= ( ) ( ) ( ) ( )( )
2xxf22f22f22xf
lim2x β
β+ββ
=( ) ( ) ( ) ( )( )
ββ+β
β 2xxf2f22f2x
lim2x
= f(2) β ( ) ( )
ββ
β 2x2fxf
2lim2x
= f(2) β2 fβ(2)
21. (d) fβ(x) =
xsinx
dxd
= xsin
xcosxxsin2
β
= cosecxβ x cotx cosecx
22. (a) f(x) = xsin1
xcos2
+
= xsin1xsin1 2
+β
= 1β sinx
β fβ(x) = βcosx
β fβ 02
=
Ο
23. (c) f(x) = ( )
xsin1xcos3 2 β
= β3 sinx
fβ(x) = β3 cosx
24. (b) xy = x+y
β y = 1x
xβ
β ( )
( )21x
x1xdxdy
βββ=
= ( )21x
1
ββ
25. (c) xy = c2 β y = x
c2
β 2
2
x
cdxdy β=
Regular Homework Exercise
1. (c) 21
11111
1x1xx
lim37
1x=
ββ+ββ=
β++
ββ
=
21
h21
lim0n
=ββ
2. (a) ( ) 31xlimxflim 2
2x2x=β=
ββ +
3. (c) xx
xlim
0x ββ
= 1x
1lim
0x ββ
= β1
4. (c) 21x
1xlim
2
2
1x β+
ββ
= 21xlim 2
1x++
β = 22
5. (d) RHL = 4h4
4h4
0h
lim
β+β+
β
1h
h0h
lim=
β=
LHL 4h4
4h4
0h
lim
ββββ
β=
=h
h
0h
lim
ββ
β
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= 1h
h0h
limβ=
ββ
β RHL β LHL β the limit does not exist.
6. (c) xbb|x|
limbx β
βββ
= 0xxbbx
limbx
>βββ
ββΞ
= β1
7. (a) ( ) Ξ»+=ββ
6xflim2x
( ) 4xflim2x
=+β
β Ξ» = β2
8. (d) 216xx
8xlim
22
8x βββ
β
=2/32/3
22
8x 8x
8x8x8x
limββΓ
βββ
β
= β16Γ2/18
23
1
Γ =
328β
9. (a) ( )t
tsinlim
xxsin
lim0tx
βΟ=βΟ βΟβ
(put Οβx = t)
= 1t
tsinlim
0t=
β
10. (a) ( )xx2sinxsin2
limx βΟ
βΟβ
= ( ) ( )
tt2sintsin2
lim0t
βΟββΟβ
(Put t = Ο βx)
= t
t2sintsin2lim
0t
+β
= t
t2sinlim
ttsin2
lim0x0t ββ
+
= 4
11. (d) 5cos2x
xsin5cos2lim
0x=β
β.
12. (d) ( )2x x
2x2cos1lim
+Ο++
Οββ
= ( )( )2x x
1xcos2lim
+Ο+
Οββ
= ( )( )
20x t
1tcos2lim
+Οββ
= ( )
20x t
tcos12lim
ββ
=
44t
2t
sin22lim
2
2
0xΓ
Γβ
= 2
1
13. (c) 1
xxsin
xcoslim
0x=
β
14. (d) ( ) 5xflim
1x=
+β so (a) is true
( ) 7xflim2x
=ββ
( ) 7xflim2x
=+β
β 2x
limβ
exists. So (b) is true
The domain of the function is [1, 3]
so that (d) is also true
15. (d) f(x) = x
2
fβ(x) = 2x
2β
β fβ(β1) = β2
16. (a) f(x) = sina cosx + cosa sinx
β fβ(x) = βsina sinx + cosa cosx
= cos(x+a)
β fβ(βa) = 1
17. (d) Let f(x) = [x]
Left fβ(0) = ( ) ( )
h
0fh0flim
0h βββ
β
β Left fβ(0) = h1
lim0h β
β=β
β the left limit does not exist
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β fβ(0) here does not exist
18. (d) f(x) = tan2x, fβ(x) = 2 tanx . sec2x
( )
hatanhatan
lim22
0h
β+β
= ( ) ( )
hafhaf
lim0h
β+β
= fβ(a)
= 2 tan a sec2a
19. (c) Left Limit = h
)0(f)h0(flim
0h βββ
β
=
ββ+
β h0h1
lim3
0h= β
β΄β΄β΄β΄ fβ(0) does not exist
β΄β΄β΄β΄ f(x) is not differentiable at x = 0
20. (d) f(x) = sin 2x = 2 sinx cosx
fβ(x) = 2(cosx cosx β sinx sinx)
= 2(cos2x β sin2x)= 2 cos2x
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Assignment Exercise
1. (a) 32
39
1x
3xlim
2
3x=β=
ββ
β
2. (d) 412x
4xlim
4x β+β
β
= 412x
1612xlim
4x β+β+
β =
412xlim4x
++β
= 8
3. (d) 0x
limβ
( ) ( )
x
ax1ax1 ββ+
=( ) ( )
( )
β++ββ+
β ax1ax1x
ax1ax10x
lim22
2
a20x
lim
β= = a
4. (c) LHL = ] 0xx2 = = 0,
RHL= ] 0xx2 = = 0
β lim = 0.
5. (d) x
xlim
0xβ does not exist
6. (c) ( )
x11x
lim8
0x
β+β
( )
( ) 11x11x
lim88
11x β+β+
β+= 8
7. (a) 20x x
x5cosx3coslim
ββ
=
20x x
xsinx4sin2limβ
= 8
8. (d) 0x
limβ 3x
xcos.xsin2xsin2 β
= ( )
30x x
xcos1xsin2lim
ββ
= 3
2
0x x2
xsin2.xsin2
limβ
=
2
0x x2x
sin.
xxsin
4lim
β
= 4.1. 121
2
=
9. (c)
βΟ
β xcosxsin1
lim
2x
= 020
xsin1xcos
lim
2n
==
+Οβ
10. (a) f(x) = x9 β93
fβ(x) = 9x8
β fβ (1) = 9
11. (d) Standard result.
12. (b) 0100sin
1x
xsinlim
2
2
2
0x=
β=
ββ
13. (a) f(x) = 2cosx+1
β fβ(x) = β2 sinx
14. (a) 3. 23
x.4
xsin22
2
β 2
xxsin
= 23
1.23 2 =
15. (c) 1x
lim
β
( )
++
++
β
β+
23x
23xx
1x
23x
2
22
= ( )( )
++β
β
23x1x
1x
2
2
23x
1x2 ++
+= =2
1
4
2 =
Additional Practice Exercise
1. (d) xa
x2lim
ax +β is = 1
2. (c) 2222
2x22x
lim2x
==β
+β
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3. (a) [ ]xxlim1x
βββ
is
= [ ]xlimxlim1x1x ββ ββ
β
= 1 β 0 = 1
4. (b) [ ] 1xlim4x
+ββ
β 4.
5. (d) By definition
6. (b) ( )
( )x3x33x
lim
31
x1
3xlim
3x3x ββ=
β
βββ
= β9
7. (b) 1xsin
xlimecxcosxlim
0x0x==
ββ
8. (b) x
xsin2lim
xx2cos1
lim0x0x ββ
=β= 2
9. (b) ( )x2cos1x
xcos1lim
20x +β
β
=
x2cos11
limx
2x
sin2lim
0x2
2
0x +Γ
ββ
= 241
4x
2x
sin2lim
2
2
0xΓΓ
β
= 41
10. (a) 30x x
xsin3x3sinlim
ββ
β =
3
3
0x x
xsin4lim
ββ
β
= 4
11. (d) x
x8sinx5tanlim
0x
ββ
= β3
12. (c) ( )
6xx
2xsinlim
22x β+β
β
=( )
( )( )3x2x2xsin
lim2x +β
ββ
= 51
13. (a) x3sinx5x3x5tan
lim0x β
ββ
13535
x
x3sin5
3x
x5tan
lim0x
=ββ=
β
β
β
Since mxmxsin
lim0x
=β
and
mxmxtan
lim0x
=β
14. (b) xcos1
xsinxlim
0x Ξ»ββ=
2x
sin2
xsinxlim
20x Ξ»β
= 2
1β Ξ»
= 2
15. (c) xtan1
1xsin2lim
4x +
+Οββ
= ( )( )xtan1xsin21
xsin21lim
2
4x +β
βΟββ
= ( )( )xtan1xsin21
x2coslim
4x +βΟβ
β
= ( )
( )( )xtan1xsin21
xtan1
xtan1
lim2
2
4x +β
+β
Οββ
=
( )( )xsin21xtan1
xtan1lim
2
4x β+
βΟβ
β
=
( )
ββ+
ββ
2
1.2111
11 =
21
2.22 =
16. (b) 2x
.x
xsin
x
x
x2
xsin =
β 1.0 = 0 as x β 0
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17. (c) 111
x18
x18
sinlim
1
xsin
xlim
0x
0
0
0x==
Ο
Ο=
β
β
18. (b) 0
limβΞΈ ΞΈΞΈ
ΞΈΞΈ.3.2
sin.3sin2 . 3 = 3.
19. (a) 1x
limβ
( )( )x
2cos
x1x1Ο
β+=
1x
lim
β( )( )
( )2
2
x12
sin
x1x1Ο
Ο
βΟβ+
1x
limβ
( )
( )( ) 1xas
42.22x1.
x12
sin
x12 β
Ο=
Οβ
Ο+
βΟ
βΟ
20. (b) 1x1x
lim2
1x ββ
+β
( ) ( )
( )1x1x1x
lim1x β
β++β
ββ+=
0h1x
h1x
( ) ( )
( )1h11h11h1
lim0h β+
β+++β
= 2.
21. (a)
ΞΈβΞΈΞΈβΞΈ
=βΞΈ 2cos1
2cos2sin
2sinlimitlim
0
= ( )ΞΈβ
ΞΈβΞΈΞΈ
βΞΈ 2cos12cos
12cos2sin
lim0
= 010
2cos2sin
lim0
=β=ΞΈΞΈβ
βΞΈ
22. (d)
+Οβ 0
0x2cos1
x3coslim
2x
= x2sin2
x3sinlim
2x β
βΟ
β (LβHospitalsβ rule)
x2sinx3sin
Lim23
2x
Οβ=
ββ=
=Οβ=Ο0
22sin,1
23
sinΞ
23. (b)
2
limΟ
βΞΈ ΞΈβΟΞΈ
2
cot
11eccos
lim2
2
=β
ΞΈβΟ
βΞΈ.
24. (c) ( )xtan
nxlimxcotnxlim
nxnx Οβ=Οβ
ββ
( ) Ο=
ΟΟβ
1
xsec
1lim
2nx
25. (a) ( ) ( )( ) ( ) 3
1323
1
31
3231
xx8x8
xx8x8
++β+
β+β+
= 3
1323
1
31
3231
8xx
128x
12
8xx
128x
12
++β
+
β+β
+
(Expand using binomial series and neglect higher powers)
=
++β+
β+β+
24xx
124x
1
24xx
124x
1
32
32
=
24x
24x
24x
24x
24x
24x
32
32
ββ
+β=
2
2
xx1
xx1
ββ+β
β΄ Limit = 1.
26. (d) ( ) ( )
( )
ββ+++β
=+
β 00
x12x2nnx1n1 1nn2
Lt1x
( ) ( )( )
( ) ( )( )
( ) ( )[ ]
( ).
21nn
21n2n1nn
21n2nn1nn
2x1n2nnx1nn1
2
n1n2Lt
1x
+=
+β++=
++++β=
++++β=β
β
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27. (b) f(x) =
( )1x
n1x1x
xn
β
βββ
= ( ) ( )( )2
n
1x
1xn1xx
ββββ
= ( )2
1n
1x
nxnxx
β+ββ+
Use Lβ Hospitalβs rule two times, Required limit
= ( )
( )1x21nx1n
limn
1x βββ+
β
= ( )
2x1nn
lim1n
1x
β
β
+
= ( )2
1nn + .
28. (d)
0xlimβ
x log sin x =
0x
limβ
20x
x
1
xcos.xsin
1
lim
x
1xsinlog
β=
β
0xsec
x2lim
xtanx
lim20x
2
0x=β=
β=ββ
By L Hospitalsβ rule.
29. (d) f(x) = x2cos1+
= 2 cosx
β fβ(x) = β 2 sinx
β fβ
Οβ2
= + 2
30. (b) f(x) = x2sin1+
= ( ) 2/122 xcossin2xsinxcos ++
= (cosx + sinx)
β fβ(x) = βsinx+cosx
β fβ(0) = 1