JEE Mathematics Complex Numbers MCQs Set 34

Answer: (c) \(a=1,\ b=0\)

Explanation:
1. \(\dfrac{1-i}{1+i}=-i\) and \((-i)^{100}=1\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (d) cannot be determined

Explanation:
1. The terms alternate \(1,-1,1,-1,\ldots\); the sum is 1 for even \(n\) and 0 for odd \(n\).
2. So its value depends on \( n \) and cannot be fixed.

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (d) \(x=\dfrac5{13},\ y=\dfrac{14}{13}\)

Explanation:
1. \(x+iy=\dfrac{4+i}{2-3i}=\dfrac{(4+i)(2+3i)}{13}=\dfrac{5+14i}{13}\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (c) -2

Explanation:
1. \(z=(p+iq)^3=p^3-3pq^2+i(3p^2q-q^3)\), so \(x=p^3-3pq^2\) and \(y=q^3-3p^2q\).
2. \(\dfrac xp+\dfrac yq=p^2-3q^2+q^2-3p^2=-2(p^2+q^2)\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (b) \(\cos\dfrac\pi2-i\sin\dfrac\pi2\)

Explanation:
1. \((i^{25})^3=i^{75}=i^3=-i\).
2. \(-i=\cos\dfrac\pi2-i\sin\dfrac\pi2\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (d) first quadrant

Explanation:
1. \(\dfrac{z_1}{z_2}=\dfrac{(\sqrt3+i\sqrt3)(\sqrt3-i)}4=\dfrac{(3+\sqrt3)+i(3-\sqrt3)}4\).
2. Both real and imaginary parts are positive.

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (b) \(|z_1+z_2|\le|z_1|+|z_2|\)

Explanation:
1. \(|z_1+z_2|=|2+2i|=2\sqrt2\approx2.83\) and \(|z_1|+|z_2|=\sqrt{10}+\sqrt2\approx4.58\).
2. The triangle inequality gives \( |z1 + z2| \) <= \( |z1| + |z2| \) (here strictly less); the other options fail numerically.

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (c) \(-2\sqrt3+2i\)

Explanation:
1. \(z=4\left(\cos150^\circ+i\sin150^\circ\right)=4\left(-\dfrac{\sqrt3}2+\dfrac i2\right)\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (d) \(\dfrac\pi2\)

Explanation:
1. The two terms are conjugates: \(\dfrac{(3-i)(2-i)}5=1-i\) and \(\dfrac{(3+i)(2+i)}5=1+i\), so \(z=2\).
2. \(zi=2i\) has argument \(\dfrac\pi2\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (d) \(2^5\)

Explanation:
1. \((1+i)^2=2i\), so \((1+i)^8=(2i)^4=16\) and similarly \((1-i)^8=16\).
2. Sum = \( 32 = 2^5 \).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (c) \(\dfrac{-7-26i}{25}\)

Explanation:
1. \(\dfrac{(2+5i)(4+3i)}{25}=\dfrac{-7+26i}{25}\); its conjugate is \(\dfrac{-7-26i}{25}\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (a) \(-\dfrac i2\)

Explanation:
1. \(z^2=2i\) and \(\dfrac1{2i}=-\dfrac i2\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (b) \(\dfrac\pi3\)

Explanation:
1. The denominator is a positive real number, so the argument is that of \(1+\sqrt3i\), i.e. \(\dfrac\pi3\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (c) 16

Explanation:
1. \(1+\dfrac1i=1-i\), so the product is \([(1+i)(1-i)]^4=2^4=16\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (d) \(-2-10i\)

Explanation:
1. \((1+i)^6=(2i)^3=-8i\) and \((1-i)^3=(-2i)(1-i)=-2-2i\).
2. Sum = \( -2 - 10i \).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (c) \(i\)

Explanation:
1. \(i^{4n+1}=i\) and \(i^{4n-1}=-i\), so the value is \(\dfrac{i-(-i)}2=i\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (b) \(-3\sqrt2\)

Explanation:
1. \(i\sqrt3\cdot i\sqrt6=i^2\sqrt{18}=-3\sqrt2\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (d) \(-2^{10}\)

Explanation:
1. \(z=\dfrac{(3+i)(2+i)}5=1+i\).
2. \(z^{20}=((1+i)^2)^{10}=(2i)^{10}=2^{10}i^{10}=-2^{10}\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (a) \(-\dfrac8{25}-\dfrac{31}{25}i\)

Explanation:
1. \(\dfrac{4-5i}{3+4i}=\dfrac{(4-5i)(3-4i)}{25}=\dfrac{12-16i-15i-20}{25}=\dfrac{-8-31i}{25}\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (b) \(\dfrac32i\)

Explanation:
1. \(\sqrt{5+12i}=3+2i\) and \(\sqrt{5-12i}=3-2i\).
2. The fraction is \(\dfrac6{4i}=-\dfrac32i\), whose conjugate is \(\dfrac32i\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.

Answer: (c) \(2^7\)

Explanation:
1. \(|z|=\dfrac{\sqrt{50}}5=\sqrt2\), so \(|z|^{14}=2^7\).

Teacher's Note:
1. Convert the expression to the form \( a \) + ib by multiplying with the conjugate of the denominator.
2. Use modulus and argument for products, quotients and powers; they multiply and add respectively.