EXERCISE - A
SIMPLIFY:
1) i²⁸. 1
2) i²⁵³ -1
3) i⁻¹³ -i
4) i⁹ + 1/i⁵ 0
5) i¹³⁵. -i
6) i¹⁹ - i
7) i⁴⁵⁷. i
8) i⁵²⁸ 1
9) 1/i⁵⁸. -1
10) 1/i⁹⁹⁹. i
11) 6i⁵⁴+5i³⁷+5i¹⁹ +16i⁶⁸. 10
12) i³⁷ + 1/i⁶⁷ 2i
13) (i⁴¹ + 1/i²⁵⁷)⁹ 0
14) (i⁷⁷ + i⁷⁰ + i⁸⁷ + i⁴¹⁴)³. -8
15) i³⁰ + i⁴⁰ + i⁶⁰ 1
16) i⁴⁹ + ⁶⁸ + i⁸⁹+ i¹¹⁰. 2i
17) i⁴⁹ + i⁶⁸ + i⁸⁹ + i¹¹⁰ 2i
18) i³⁰ + i⁸⁰ + i¹²⁰ 1
19) i+ i² + i³ + i⁴ 0
20) i⁵ + i¹⁰ + i¹⁵ - 1
21) {i¹⁹ + 1/i²⁵}² - 4
22) {i¹⁷ - 1/i)³⁴}² 2i
23) {i⁸ + (1/I)²⁵}³ 0
24) (i⁵⁹² + i⁵⁹⁰+ i⁵⁸⁸+ i⁵⁸⁶ +⁵⁸⁴)/(i⁵⁸²+ i⁵⁸⁰ + i⁵⁷⁸+ i⁵⁷⁶ +⁵⁷⁴) -1
25) 1 + i² + i⁴ + i⁶ + i⁸ +.......+ i²⁰ 1
26) Prove iⁿ + iⁿ⁺¹+ iⁿ⁺² + iⁿ⁺³= 0, for all n ∈ N.
27) (1+i³)(1+1/i)³(i⁴+1/i⁴). 4 - 4i
28) Show that 1 + i¹⁰ + i²⁰ + i³⁰ is real number.
EXERCISE - B
Simplify
1) √(-144) 12i
2) √(-4) ×√((-9/4). - 3
3) √(-25)+ 3√(-4) +2√(-9) 17i
4) (4√-25 - 3√-81)x (9√-8 +8√-18) 294√2
5) 6i⁵⁴ + 5i³⁷ + 5i¹⁹ + 16i⁶⁸. 10
6) (1+i³)(1+1/i)³(i⁴+1/i⁴). -8
7){(1+i)(3+i)}/(3- i)+{(1-i)(3-i)}/(3+i). 2/5
8) (-√-1)⁴ⁿ⁺³. i
EXERCISE-C
Express in the form A+ iB
1) (-5i)(i/8) 5/8 + 0i
2) (-i)(2i)(-i/8)³. 0+ i/216
3) (5i)(-3i/5) 3+ 0i
4) i⁹ + i¹⁹. 0+0i
5) 1/i³⁹ 0+ i
6) (1- i)⁴ -4+0i
7) 3(7+7i)+ i(7+7i). 14+ 28i
8) ((1-i)- (-1+ 6i). 2 - 7i
9) (1/5 + 2i/5) - (4+ 5i/2). -19/5 - 21i/10
10) {(1/3 + 7i/3)+(4+i/3)} - (-4/3 + i). 17/3 + 7i/3.
11) (1/3 + 3i)³ -242/27 - 26i
12) (-2 - i/3) -22/3 - 107i/27
13) 1/(3 - 4i) 3/25 + 4i/25
14) (5+4i)/(4+5i) 40/41- 9i/41
15) (1+i)²/(3-i) 1/5 + 3i/5
16) {(3-2i)(2+3i)}/{(1+2i)(2-i)} 63/25 - 16i/25
17) 1/(-2 +√-3). -2/7 -√3/7i
18) (2- √-25)/(1-√-16) 22/17+ 13i/17
19) (3-√-16)/(1-√-9) 3/2+ i/2
20) 1/(2+i)² 3/25 - 4i/25
21) (3-4i)/{(4-2i)(1+i)}. -11/125+ 2i/125
22) (1-2i)⁻³. --11/125 -2i/125
EXERCISE - D
Find Conjugate:
1) 3 - 4i 3+4i
2) 1/(3+5i) 1/34(3+5i)
3) 1/(1+i) 1/2(1+i)
4) (1-i)/(1+i). 0 - i
5) (1+i)²/(3-i). -1/5 - 3i/5
6) (2+3i)²/(2-i) -22/5 - 19i/5
7) (3-i)²/(1+i) 1 + 7i
8) {(1+i)(2+i)}/(3+i). 3/5- 4i/5
9) (2+i)³/(2+3i) 37/13 - 16i/13
10) (x+iy)/(x-iy) (x²-y²)/(x²+y²) - 2xyi/(x²+ y²)
11) (2-i)/(1-2i)³. -26/125 - 7i/125
EXERCISE - E
Find Modulus:
1) 2 - 3i √13
2) (3-2i)²/(-2+5i). √4901/29
3) (1+2i)/(1- 2i) - (1-2i)/(1+2i)
4) (2+i)/{4i+(1+i)²}.
5) -2√3+ 2√(2i). 2√5
6) (4-3i)(√3 +3i). √354
7) (1-i)/(3-4i). √2/5
8) (1+i)³/(2-i)². 2√2/5
9) (√3- i√2)/(√2- i√3). 1
EXERCISE - F
Find Multiplicative Inverse:
1) 3- 2i
2) - i
3) √(7) + 4i
4) (2+i)/(3-i)
EXERCISE-G
Find Amplitude of:
1) 4 0
2) 2 - 2i -π/4
3) (1+i)(√3 +i). 5π/2
4) (1-i)/(1+i) -π/2
5) (√3 +i)/(-1-i√3).. √(5/13)
6) i³/(1+i). Sec(¢/2)
EXERCISE - H
Find real values of x and y if:
1) if - 3 +ix²y and x² + y + 4i are conjugate of each other. ±1, ±4
2) if (x- iy)(3+5i)are conjugate of - 6-24i. 3, -3
3) (x- iy)(2-3i)= 4. 5/13,14/13
4) (3x- 2i)(2+i)²= 10(1+i) 14/15,1/5
5) (1+ i)(x +iy)= 2 - 5i. -3/2, -7/2
6) {(1+ i)(x-2i)}/(3+i) + {(2-3i)y +1}/(3-i) =i. 3, -1
7) If (x+y- i) and {5+i(2x-y)} are Conjugate to each other. 2, 3
8) If (x²+ 5 +iy) and (3y+ 2xi) are Conjugate to each other. -5, 10
9) 2 + (x +iy)= (3-i). 1, -1
10) x + 4iy = ix + y +3. 4, 1
11) (1-i)x +(1+i)y= 1 - 3i. 2, -1
12) (x+ iy)(2-3i)= 4+ i. 5/13, 14/13
13) {(x-1)/(3+i)} + {(y-1)/(3-i)}=i. -4,6
14) (1+i)y²+ (6+i)= (2+ i)x. 5, ±2
EXERCISE - I
Represent the following in the polar form :
1) 4. 4(coso + i sin0),4,0
2) -2. 2(cosπ + i sinπ), 2,π
3) 2i. 2(cosπ/2 + i sinπ/2),2,π/2
4) (1+ i√3). 2(cosπ/3+i sinπ/3), 2,π/3
5) -√3+ i. 2(cos5π/6 + i sin5π/6), 2, 5π/6
6) - 1+ i. √2(cos3π/4 +i sin3π/4), √2, 3π/4
7) (2+ 6√3i)/(5+√3 i). 2(cosπ/3+ i sinπ/3), 2, π/3
8) (1-3i)/(1+2i). √2(cos5π/4 + i sin5π/4)
EXERCISE- J
SOLVE:
1) x²+2= 0. ± √2
2) x² + 5= 0. ±√5 i
3) 2x²+1= 0. ± 1/√2
4) x² + x+1= 0. (-1± i√3)/2
5) x² - x+2 = 0. (1± √7i)/2
6) x² + 2x+2 = 0. -1± i
7) 2x² - 4x+ 3= 0. 1 ± i/√2
8) 9x² + 10x+3= 0. (-5±√2 i)/9
9) 25x² - 30x+ 11= 0. (3± √2 i)/5
10) 17x² +1= 8x. (4±i)/17
11) x² + 3ix+10= 0. (2i, -5i)
12) 2x² + 3ix+2= 0. i/2, -2i
13) x² + x/√2+1= 0. (-1±√7i)/2√2
EXERCISE- J
Find the Square roots of :
1) i ±(1+i)/√2
2) -5 +12i ±(2+3i)
3) - 2i. ±(1-i)
4) - 7 - 24i ±(3-4i)
5) 4 + 6√-5. ±(3+i√5)
5) 2i/(3-4i) ±(1+3i)/5
7) x² - 1 +2ix. ±(x+i)
8) 3a +i√(a⁴-7a²+1). ±√{√(a²+3a+1) + i√(a²-3a+1)}
9) 2ab - i(a² - b²). ±1/√2{(a+b)-i(a-b)}.
EXERCISE - K
Find the Value of:
1) If x= 4 + 3i, y= 4 - 3i find (x²+ xy+y²)/(x² - xy +y²). -39/11
2) If a= 3+2i & b= 3 - 2i then
I) a²+ab+b² II) a³+b³ .
3) Find the least positive integral value of m for which {(1+i)/(1-i)}ᵐ=1
EXERCISE - L
Prove:
1) If (x+iy)³= (u+iv) then (u/x + v/y) = 4(x² - y²).
2) If (a+ib)/(c+id) = (p +iq) then
I) (p-q)=(a-ib)/(c-id).
II) (p² +q²)=(a²+ b²)/(c² +d²).
3) If (a+ ib)= √{(1+i)/(1-i)} then (a²+b²)= 1.
4) If (a+ib)(c+id)(e+if)(g+ih)=A+iB then (a²+b²)(c²+d²)(e²+f²)(g²+h²)= (A² + B²).
5) If ³√(x+ iy)= (a+ ib), then (x/a + y/b) = 4(a² - b²).
6) If (x+iy)= {(a+ib)/(a-ib), then x²+ y²= 1.
EXERCISE - M
If w be an imaginary cube root of unity, then show that..
1) (3+3w+5w²)⁶= (3+5w+3w²)⁶=64
2) (1+w)(1+w²)(1+w⁴)⁶(1+w⁸)⁶=1
3) (1- w+w²)(1 - w²+w⁴)(1-w⁴-w⁸)=8
4) (x+yw+zw²)²+ (xw+yw²+z)² + (xw²+ y + zw)²= 0
5) (1-w)(1-w²)(1-w⁵)(1-w¹⁰)=9
6) (1- w+w²)⁵+(1+w-w²)⁵=32
7) (1-w+w²)(1-w²+w⁴)- (1+w³)²= 0
8) (x+y)²+(xw+yw²)+ (xw²+yw)²= 6xy
9) (1-w+w²)(1-w²+w⁴)(1-w⁴+w⁸).... to 2n factors= 2²ⁿ
10) (a+w+w²)(a-w²-w⁴)(a+w⁴+w⁸)(a-w⁸- w¹⁶) ....2n factors = (a² -1)ⁿ
11) If x= a+b, y= aw+bw², z= aw²+bw Prove that xyz= a³ + b³.
12) If x= a+b, y= a+b¢², z= a+b¢², where 1, ¢, ¢² are the cube roots of unity, Prove that x³+ y³+ z³=3(a³ + b³)
13) Simplify: a+bw+cw²)/(b+ cw+aw²)+ (a+bw+cw²)/(c+aw+bw²)
14) w/9 (1- w(1-w²)(1-w⁴)(1--w⁸) + w{(c+aw+bw²)/(b + cw+ aw²)}= -1.
15) (aw+b+w²)/(bw²+a+w)= w
16) (1+w-w²)(2- w+w²) = 4
17) (1- 2w+w²)(1- 2w²+ w) = 9
18) (1 - w -w²)³ - (1+ w -w²)³= 16.
19) (a+w+w²)(a+ w²-w⁴)(a+w⁴+w⁸) ....... to n factors = (a-1)ⁿ
EXERCISE - N
Continue....
11) Express as a+ib
1/(1- cosx +2i sinx)
Mg. A- R.1
1) Value √(-4) x √(-9).
a) -1 b) -2 c) -4 d) -6
2) i⁹ +1/i⁵.
a) 0 b) 1 c) -1 d) i
3) (1+i)²+ (1-i)².
a) 0 b) 1 c) -1 d) i
4) (1+i)¹²+ (1-i)¹².
a) 1 b) 2 c) 8 d) 128 e) -128
5) {(1+i)/(1-i)}² + {(1-i)/(1+i)}².
a) 0 b) 1 c) 2 d) -2 e) -1
6) Simplify:
20/(√3- √-2) + 30/(3√-2 - 2√3) - 14/(2√3- √-2).
a) 0 b) 1 c) -1 d) none
7) if {(1+ i)/(1- i)}³ - {(1- i)/(1+ i)}³= p + iq, then p, q is
a) 0,0 b) 1,2 c) 0,-2 d) 1,-2
8) If x= 2+ 3i and y= 2 - 3i, find the value of (x³- y³)/(x³+ y³).
a) 9 b) 9i c) -9i d) -9i/46
9)
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