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Learn/ 10th Mathematics/ Unit 1 /Numericals

Unit 1: Complex Numbers — Exercise 1 2

10th Class Mathematics · Unit 1: Complex Numbers

1.2.1.Simplify and write in the form a+bi: (i) (2+5i)+(3-2i) (ii) (16-3i)+(9+2i) (iii) (9-2i)-(7-3i) (iv) (11+9i)-(9-7i) (v) (3+4i)(2-3i) (vi) (5-2i)(3-4i) (vii) (3-5i)÷(2-4i) (viii) (5+2i)÷(6-3i)
Given
(i) (2+5i)+(3-2i)
= (2+3)+(5i-2i)
Result
(i) 5+(5-2)i
Given
(ii) (16-3i)+(9+2i)
= (16+9)+(-3i+2i)
Result
(ii) 25-i
Given
(iii) (9-2i)-(7-3i)
= (9-7)+(-2i+3i)
Result
(iii) 2+i
Given
(iv) (11+9i)-(9-7i)
= (11-9)+(9i+7i)
Result
(iv) 2+16i
Given
(v) (3+4i)(2-3i)
= 6-9i+8i-12i2 = 6-i+12
Result
(v) 18-i
Given
(vi) (5-2i)(3-4i)
= 15-20i-6i+8i2 = 15-26i-8
Result
(vi) 7-26i
Given
(vii) (3-5i)div(2-4i)
= 3-5i2-4i × 2+4i2+4i = 6+12i-10i-20i24+16 = 26+2i20
Result
(vii) 13+i10
Given
(viii) (5+2i)div(6-3i)
= 5+2i6-3i × 6+3i6+3i = 30+15i+12i+6i236+9 = 24+27i45
Result
(viii) 8+9i15
1.2.2.Write the additive inverse for each complex number: (i) 3+2i (ii) 4-3i (iii) 5-7i (iv) -2/3 + 5/4 i
Given
(i) 3+2i
Result
(i) -(3+2i) = -3-2i
Given
(ii) 4-3i
Result
(ii) -(4-3i) = -4+3i
Given
(iii) 5-7i
Result
(iii) -(5-7i) = -5+7i
Given
(iv) -frac23+frac54 i
Result
(iv) -left(-frac23+frac54 iright) = frac23-frac54 i
1.2.3.Find the multiplicative inverse for each complex number: (i) 4+5i (ii) 6+2i (iii) 7-3i (iv) √5-4i
Given
(i) 4+5i
14+5i × 4-5i4-5i = 4-5i42+52
Result
(i) 4-5i41
Given
(ii) 6+2i
6-2i62+22 = 6-2i40
Result
(ii) 3-i20
Given
(iii) 7-3i
7+3i72+32
Result
(iii) 7+3i58
Given
(iv) sqrt5-4i
sqrt5+4i(sqrt5)2+42
Result
(iv) sqrt5+4i21
1.2.4.If z1 = 2+5i, z2 = 1-3i and z3 = 2+i, then verify that: (i) z1+z2 = z2+z1 (ii) z1z2 = z2z1 (iii) (z1+z2)+z3 = z1+(z2+z3) (iv) (z1z2)z3 = z1(z2z3) (v) z1+(-z1) = (-z1)+z1
Given
(i) z1+z2 = z2+z1
L.H.S = (2+5i)+(1-3i) = 3+2i; quad R.H.S = (1-3i)+(2+5i) = 3+2i
Result
(i) L.H.S=R.H.S=3+2i
Given
(ii) z1z2 = z2z1
L.H.S = (2+5i)(1-3i) = 2-6i+5i-15i2 = 17-i; quad R.H.S = (1-3i)(2+5i) = 17-i
Result
(ii) L.H.S=R.H.S=17-i
Given
(iii) (z1+z2)+z3 = z1+(z2+z3)
L.H.S = (3+2i)+(2+i) = 5+3i; quad R.H.S = (2+5i)+(3-2i) = 5+3i
Result
(iii) L.H.S=R.H.S=5+3i
Given
(iv) (z1z2)z3 = z1(z2z3)
L.H.S = (17-i)(2+i) = 34+17i-2i-i2 = 35+15i
z2z3 = (1-3i)(2+i) = 5-5i; quad R.H.S = (2+5i)(5-5i) = 35+15i
Result
(iv) L.H.S=R.H.S=35+15i
Given
(v) z1+(-z1) = (-z1)+z1
L.H.S = (2+5i)+(-2-5i) = 0; quad R.H.S = (-2-5i)+(2+5i) = 0
Result
(v) L.H.S=R.H.S=0
1.2.5.If (1+i)^2 / (2-i) = x + yi, then find the values of x and y.
Given
(1+i)22-i = x+yi
(1+i)2 = 1+2i+i2 = 2i
Working
2i2-i × 2+i2+i = 2i(2+i)4-i2 = 4i+2i25 = 4i-25
= -frac25+frac45 i
Result
x=-frac25, y=frac45
1.2.6.If (2x+iy)(1-i) = 4+2i, then find the values of x and y.
Given
(2x+iy)(1-i) = 4+2i
= 2x-2xi+iy-i2y = (2x+y)+(-2x+y)i
Working
2x+y=4 quad (i); qquad -2x+y=2 quad (ii)
Adding (i) and (ii): 2y=6 ⇒ y=3; quad from (i): 2x=1 ⇒ x=frac12
Result
x=frac12, y=3
1.2.7.Find the values of a and b, if (a+bi)(1+3i) = -8+11i.
Given
(a+bi)(1+3i) = -8+11i
= a+3ai+bi+3bi2 = (a-3b)+(3a+b)i
Working
a-3b=-8 quad (i); qquad 3a+b=11 quad (ii)
b=11-3a ⇒ a-3(11-3a)=-8 ⇒ 10a=25 ⇒ a=frac52
b=11-3left(frac52right) = 22-152
Result
a=frac52, b=frac72