Page No 13.3:
Question 1:
Evaluate the following:
(i) i457
(ii) i528
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Answer:
Page No 13.31:
Question 1:
Express the following complex numbers in the standard form a + i b:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
Answer:
Page No 13.31:
Question 2:
Find the real values of x and y, if
(i)
(ii)
(iii)
(iv)
Answer:
Page No 13.31:
Question 3:
Find the conjugates of the following complex numbers:
(i) 4 − 5 i
(ii)
(iii)
(iv)
(v)
(vi)
Answer:
Page No 13.32:
Question 4:
Find the multiplicative inverse of the following complex numbers:
(i) 1 − i
(ii)
(iii) 4 − 3i
(iv)
Answer:
Page No 13.32:
Question 5:
If
Answer:
Page No 13.32:
Question 6:
If find
(i) Re
(ii) Im
Answer:
Page No 13.32:
Question 7:
Find the modulus of
Answer:
Page No 13.32:
Question 8:
If , prove that x2 + y2 = 1
Answer:
Page No 13.32:
Question 9:
Find the least positive integral value of n for which is real.
Answer:
Page No 13.32:
Question 10:
Find the real values of θ for which the complex number is purely real.
Answer:
Page No 13.32:
Question 11:
Find the smallest positive integer value of m for which is a real number.
Answer:
Page No 13.32:
Question 12:
If , find (x, y).
Answer:
Also,
It is given that,
Thus, (x, y) = (0, −2).
Page No 13.32:
Question 13:
If , find x + y.
Answer:
It is given that,
Thus, x + y = .
Page No 13.32:
Question 14:
If , find (a, b).
Answer:
It is given that,
Thus, (a, b) = (1, 0).
Page No 13.32:
Question 15:
If , find the value of .
Answer:
Thus, .
Page No 13.32:
Question 16:
Evaluate the following:
(i)
(ii)
(iii)
(iv)
(v)
Answer:
Page No 13.32:
Question 17:
For a positive integer n, find the value of .
Answer:
Thus, the value of is 2n.
Page No 13.33:
Question 18:
If , then show that .
Answer:
Hence, .
Page No 13.33:
Question 19:
Solve the system of equations .
Answer:
Let .
Then ,
and
According to the question,
Thus, .
Page No 13.33:
Question 20:
If is purely imaginary number (), find the value of .
Answer:
Let .
Then,
If is purely imaginary number, then
Thus, the value of is 1.
Page No 13.33:
Question 21:
If z1 is a complex number other than −1 such that and , then show that the real parts of z2 is zero.
Answer:
Let .
Then,
Now,
Thus, the real parts of z2 is zero.
Page No 13.33:
Question 22:
If , find z.
Answer:
Let .
Then,
Thus,
Page No 13.33:
Question 23:
Solve the equation .
Answer:
Let .
Then,
Thus,
Page No 13.33:
Question 24:
What is the smallest positive integer n for which ?
Answer:
Thus, the smallest positive integer n for which is 2.
Page No 13.33:
Question 25:
If z1, z2, z3 are complex numbers such that , then find the value of .
Answer:
Thus, the value of is 1.
Page No 13.33:
Question 26:
Find the number of solutions of
Answer:
Let .
Then,
For
Thus, there are infinitely many solutions of the form .
Page No 13.39:
Question 1:
Find the square root of the following complex numbers:
(i) −5 + 12i
(ii) −7 − 24i
(iii) 1 − i
(iv) −8 − 6i
(v) 8 −15i
(vi)
(vii)
(viii) 4i
(ix) −i
Answer:
Page No 13.4:
Question 2:
Show that 1 + i10 + i20 + i30 is a real number.
Answer:
Page No 13.4:
Question 3:
Find the values of the following expressions:
(i) i49 + i68 + i89 + i110
(ii) i30 + i80 + i120
(iii) i + i2 + i3 + i4
(iv) i5 + i10 + i15
(v)
(vi) 1+ i2 + i4 + i6 + i8 + ... + i20
(vii) (1 + i)6 + (1 − i)3
Answer:
(vii) (1 + i)6 + (1 − i)3
= [(1 + i)2]3 + (1 − i)3
= [12 + i2 + 2i]3 + (13 − i3 + 3i2 − 3i)
= [1 − 1 + 2i]3 + (1 + i − 3 − 3i) [∵ i2 = −1, i3 = −i]
= (2i)3 + (−2 − 2i)
= 8i3 − 2 − 2i
= −8i − 2 − 2i [∵ i3 = −i]
= −10i − 2
Page No 13.57:
Question 1:
Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
(i) 1 + i
(ii)
(iii) 1 − i
(iv)
(v)
(vi)
(vii)
(viii)
Answer:
Page No 13.57:
Question 2:
Write (i25)3 in polar form.
Answer:
Let .
Then, .
Let θ be the argument of z and α be the acute angle given by .
Then,
Clearly, z lies in fourth quadrant. So, arg(z) = .
∴ the polar form of z is .
Thus, the polar form of (i25)3 is .
Page No 13.57:
Question 3:
Express the following complex in the form r(cos θ + i sin θ):
(i) 1 + i tan α
(ii) tan α − i
(iii) 1 − sin α + i cos α
(iv)
Answer:
Page No 13.57:
Question 4:
If z1 and z2 are two complex numbers such that and arg(z1) + arg(z2) = , then show that .
Answer:
Let θ1 be the arg(z1) and θ2 be the arg(z2).
It is given that and arg(z1) + arg(z2) = .
Since, z1 is a complex number.
Hence, .
Page No 13.57:
Question 5:
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, prove that .
Answer:
Given that z1, z2 and z3, z4 are two pairs of conjugate complex numbers.
Then,
and
Hence, .
Page No 13.58:
Question 6:
Express in polar form.
Answer:
Page No 13.62:
Question 1:
Write the values of the square root of i.
Answer:
Page No 13.62:
Question 2:
Write the values of the square root of −i.
Answer:
Page No 13.62:
Question 3:
If x + iy = , then write the value of (x2 + y2)2.
Answer:
Page No 13.62:
Question 4:
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of .
Answer:
Page No 13.62:
Question 5:
If n is any positive integer, write the value of .
Answer:
Page No 13.62:
Question 6:
Write the value of .
Answer:
Page No 13.62:
Question 7:
Write 1 − i in polar form.
Answer:
Page No 13.62:
Question 8:
Write −1 + i in polar form
Answer:
Page No 13.62:
Question 9:
Write the argument of −i.
Answer:
Page No 13.62:
Question 10:
Write the least positive integral value of n for which is real.
Answer:
Page No 13.62:
Question 11:
Find the principal argument of .
Answer:
Page No 13.62:
Question 12:
Find z, if
Answer:
We know that,
Thus, .
Page No 13.62:
Question 13:
If , then find the locus of z.
Answer:
Hence, the locus of z is real axis.
Page No 13.63:
Question 14:
If , find the value of .
Answer:
Hence, .
Page No 13.63:
Question 15:
Write the value of .
Answer:
Hence, .
Page No 13.63:
Question 16:
Write the sum of the series upto 1000 terms.
Answer:
We know that,
Thus, the sum of the series upto 1000 terms is 0.
Page No 13.63:
Question 17:
Write the value of .
Answer:
Let z be a complex number with argument θ.
Then,
⇒ argument of is −θ.
Thus, .
Page No 13.63:
Question 18:
If , then find the greatest and least values of .
Answer:
Hence, the greatest and least values of is 6 and 0.
Page No 13.63:
Question 19:
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of .
Answer:
Hence, .
Page No 13.63:
Question 20:
Write the conjugate of .
Answer:
∴ Conjugate of
Hence, Conjugate of is .
Page No 13.63:
Question 21:
If n ∈ , then find the value of .
Answer:
Thus, the value of is 0.
Page No 13.63:
Question 22:
Find the real value of a for which is real.
Answer:
Since, is real.
Hence, the real value of a for which is real is −2.
Page No 13.63:
Question 23:
If , find z.
Answer:
We know that,
Hence, .
Page No 13.63:
Question 24:
Write the argument of .
Disclaimer: There is a misprinting in the question. It should be instead of .
Answer:
Let the argument of be α. Then,
Let the argument of be β. Then,
Let the argument of be γ. Then,
∴ The argument of
Hence, the argument of .
Page No 13.63:
Question 1:
The value of is
(a) 2
(b) 0
(c) 1
(d) i
Answer:
(b) 0
(1+ i) (1 + i2) (1 + i3) (1 + i4)
= (1+ i) (1 1) (1 i) (1 + 1) (i2 = 1, i3 = i and i4 = 1)
= (1 + i) (0) (1 i) (2)
= 0
Page No 13.63:
Question 2:
If is a real number and 0 < θ < 2π, then θ =
(a) π
(b)
(c)
(d)
Answer:
(a) π
Given:
is a real number
On rationalising, we get,
For the above term to be real, the imaginary part has to be zero.
For this to be zero,
sin = 0
= 0,
But
Hence,
Page No 13.63:
Question 3:
If is equal to
(a)
(b)
(c)
(d)
(e)
Answer:
(c) a2 + b2
(1 + i)(1 + 2i)(1 + 3i) ......(1 + ni) = a + ib
Taking modulus on both the sides, we get:
Squaring on both the sides, we get:
2
Page No 13.63:
Question 4:
If then possible value of is
(a)
(b)
(c) x + iy
(d) x − iy
(e)
Answer:
(d) x iy
Page No 13.64:
Question 5:
If , then
(a)
(b)
(c)
(d)
Answer:
(d)
Page No 13.64:
Question 6:
The polar form of (i25)3 is
(a)
(b) cos π + i sin π
(c) cos π − i sin π
(d)
Answer:
(d)
(i25)3 = (i)75
= (i)418+ 3
= (i)3
= i ( i4 = 1)
Modulus, r =
Polar form = r (cos + i sin )
= cos+i sin
= cos i sin
Page No 13.64:
Question 7:
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
(a) 1
(b) −1
(c) i
(d) 0
Answer:
(d) 0
Page No 13.64:
Question 8:
If , then the value of arg (z) is
(a) π
(b)
(c)
(d)
Answer:
(c)
z =
Rationalising z, we get,
Page No 13.64:
Question 9:
If a = cos θ + i sin θ, then
(a)
(b) cot θ
(c)
(d)
Answer:
(c)
Page No 13.64:
Question 10:
If (1 + i) (1 + 2i) (1 + 3i) .... (1 + ni) = a + ib, then 2.5.10.17.......(1+n2)=
(a) a − ib
(b) a2 − b2
(c) a2 + b2
(d) none of these
Answer:
(c) a2 + b2
(1 + i)(1 + 2i)(1 + 3i) ......(1 + ni) = a + ib
Taking modulus on both the sides, we get,
Squaring on both the sides, we get:
2×5×10×.....(1 + n2) = a2 + b2
Page No 13.64:
Question 11:
If is equal to
(a)
(b)
(c)
(d) none of these
Answer:
(a)
Taking modulus on both the sides, we get:
Page No 13.64:
Question 12:
The principal value of the amplitude of (1 + i) is
(a)
(b)
(c)
(d) π
Answer:
(a)
Let z = (1+i)
Therefore, arg (z) =
Page No 13.64:
Question 13:
The least positive integer n such that is a positive integer, is
(a) 16
(b) 8
(c) 4
(d) 2
Answer:
Page No 13.64:
Question 14:
If z is a non-zero complex number, then is equal to
(a)
(b)
(c)
(d) none of these
Answer:
(a)
Page No 13.64:
Question 15:
If a = 1 + i, then a2 equals
(a) 1 − i
(b) 2i
(c) (1 + i) (1 − i)
(d) i − 1.
Answer:
(b) 2i
a = 1 + i
On squaring both the sides, we get,
a2 = (1 + i)2
a2 = 1 + i2 + 2i
a2 = 11 + 2i ( i2 = 1)
a2 = 2i
Page No 13.64:
Question 16:
If (x + iy)1/3 = a + ib, then
(a) 0
(b) 1
(c) −1
(d) none of these
Answer:
(d) none of these
Page No 13.64:
Question 17:
is equal to
(a)
(b)
(c)
(d) none of these.
Answer:
(b)
Page No 13.65:
Question 18:
The argument of is
(a) 60°
(b) 120°
(c) 210°
(d) 240°
Answer:
(d) 240°
Page No 13.65:
Question 19:
If , then z4 equals
(a) 1
(b) −1
(c) 0
(d) none of these
Answer:
(a) 1
Rationalising the denominator:
Page No 13.65:
Question 20:
If , then arg (z) equal
(a) 0
(b)
(c) π
(d) none of these.
Answer:
(a) 0
Page No 13.65:
Question 21:
(a)
(b)
(c)
(d) none of these
Answer:
(a)
Page No 13.65:
Question 22:
(a) 1
(b)
(c)
(d) none of these
Answer:
(b)
Page No 13.65:
Question 23:
(a)
(b)
(c)
(d)
Answer:
(c)
Page No 13.65:
Question 24:
If , then x2 + y2 =
(a) 0
(b) 1
(c) 100
(d) none of these
Answer:
(c) 100
Page No 13.65:
Question 25:
If , then Re (z) =
(a) 0
(b)
(c)
(d)
Answer:
(b)
Page No 13.65:
Question 26:
If then y =
(a) 9/85
(b) −9/85
(c) 53/85
(d) none of these
Answer:
(c)
Page No 13.65:
Question 27:
If , then =
(a) 1
(b) −1
(c) 0
(d) none of these
Answer:
(a) 1
Page No 13.65:
Question 28:
If θ is the amplitude of , than tan θ =
(a)
(b)
(c)
(d) none of these
Answer:
(b)
Page No 13.65:
Question 29:
If , then
(a)
(b)
(c) amp (z) =
(d) amp (z) =
Answer:
(d) amp (z) =
Page No 13.65:
Question 30:
The amplitude of is equal to
(a) 0
(b)
(c)
(d) π
Answer:
(c)
Page No 13.66:
Question 31:
The argument of is
(a)
(b)
(c)
(d)
Answer:
(a)
Page No 13.66:
Question 32:
The amplitude of is
(a)
(b)
(c)
(d)
Answer:
(c)
Page No 13.66:
Question 33:
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
(a)
(b)
(c) 1
(d)
Answer:
(a)
Page No 13.66:
Question 34:
equals
(a) i
(b) −1
(c) −i
(d) 4
Answer:
(c) i
Page No 13.66:
Question 35:
The value of is
(a) −1
(b) −2
(c) −3
(d) −4
Answer:
(b) 2
Page No 13.66:
Question 36:
The value of is
(a) 8
(b) 4
(c) −8
(d) −4
Answer:
(c) 8
Page No 13.66:
Question 37:
If lies in third quadrant, then also lies in third quadrant if
(a)
(b)
(c)
(d)
Answer:
Since, lies in third quadrant.
Now,
Since, also lies in third quadrant.
From (1) and (2),
Hence, the correct option is (c).
Page No 13.66:
Question 38:
If , where , then is
(a)
(b)
(c)
(d) none of these
Answer:
Since ,
Hence, the correct answer is option (a).
Page No 13.66:
Question 39:
A real value of x satisfies the equation
(a) 1
(b) −1
(c) 2
(d) −2
Answer:
Hence, the correct option is (a).
Page No 13.66:
Question 40:
The complex number z which satisfies the condition lies on
(a) circle x2 + y2 = 1
(b) the x−axis
(c) the y−axis
(d) the line x + y = 1
Answer:
Hence, the correct option is (b).
Page No 13.66:
Question 41:
If z is a complex number, then
(a)
(b)
(c)
(d)
Answer:
It is obvious that, for any complex number z,
Hence, the correct option is (b).
Page No 13.66:
Question 42:
Which of the following is correct for any two complex numbers z1 and z2?
(a)
(b)
(c)
(d)
Answer:
Since, it is known that
,
and
Hence, the correct option is (a).
Page No 13.66:
Question 43:
If the complex number satisfies the condition , then z lies on
(a) x−axis
(b) circle with centre (−1, 0) and radius 1
(c) y−axis
(d) none of these
Answer:
Hence, the correct option is (b).
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