JEE Mathematics Complex Numbers MCQs Set 35

Practice MCQs for JEE Mathematics Complex Numbers

Review structured MCQ sets for JEE Mathematics Complex Numbers. Built according to official JEE guidelines, these downloadable questions support daily revision and core concept reinforcement.

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Multiple Choice Questions

Question 1: The modulus of \(\dfrac{(1+i\sqrt3)(2+2i)}{\sqrt3-i}\) is
(a) \(2\sqrt2\)
(b) 2
(c) 4
(d) \(3\sqrt2\)
Show Answer & Explanation

Answer: (a) \(2\sqrt2\)

Explanation:
1. \(\dfrac{2\cdot2\sqrt2}{2}=2\sqrt2\).

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.

Question 2: The argument of the complex number \(\dfrac i2-\dfrac2i\) is equal to
(a) \(\dfrac\pi4\)
(b) \(\dfrac{3\pi}4\)
(c) \(\dfrac\pi{12}\)
(d) \(\dfrac\pi2\)
Show Answer & Explanation

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

Explanation:
1. \(-\dfrac2i=2i\), so the number is \(\dfrac i2+2i=\dfrac52i\) with 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.

Question 3: The square root of \(7-24i\) is
(a) \(\pm(3-5i)\)
(b) \(\pm(3+4i)\)
(c) \(\pm(4-3i)\)
(d) \(\pm(4+5i)\)
Show Answer & Explanation

Answer: (c) \(\pm(4-3i)\)

Explanation:
1. \((4-3i)^2=16-24i-9=7-24i\).

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.

Question 4: The value of \((1+i)^5\times(1-i)^5\) is
(a) -8
(b) 32
(c) 8i
(d) 8
Show Answer & Explanation

Answer: (b) 32

Explanation:
1. \((1+i)^5(1-i)^5=[(1+i)(1-i)]^5=2^5=32\).

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.

Question 5: If \(z_1=2-i\) and \(z_2=1+i\), then the value of \(\left|\dfrac{z_1+z_2+1}{z_1-z_2+1}\right|\) is
(a) 2
(b) \(\sqrt2\)
(c) \(2i\)
(d) \(\sqrt2i\)
Show Answer & Explanation

Answer: (b) \(\sqrt2\)

Explanation:
1. Numerator \(=4\); denominator \(=2-2i\) with modulus \(2\sqrt2\).
2. \(\dfrac4{2\sqrt2}=\sqrt2\).

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.

Question 6: If \(\dfrac{(1+i)^3}{(1-i)^3}-\dfrac{(1-i)^3}{(1+i)^3}=x+iy\), then
(a) \(x=-2,\ y=0\)
(b) \(x=1,\ y=1\)
(c) \(x=0,\ y=-2\)
(d) \(x=-1,\ y=-1\)
Show Answer & Explanation

Answer: (c) \(x=0,\ y=-2\)

Explanation:
1. \(\dfrac{1+i}{1-i}=i\), so the first term is \(i^3=-i\); the second is \((-i)^3=i\).
2. Difference \(=-i-i=-2i\).

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.

Question 7: If \(\bar z\) is the conjugate of \(z\), which of the following relations is false?
(a) \(\arg z=\arg\bar z\)
(b) \(|z|=|\bar z|\)
(c) \(z\bar z=|z|^2\)
(d) \(\overline{z_1+z_2}=\bar z_1+\bar z_2\)
Show Answer & Explanation

Answer: (a) \(\arg z=\arg\bar z\)

Explanation:
1. \(\arg\bar z=-\arg z\), so the equality of arguments fails for non-real z; the other relations are standard identities.

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.

Question 8: If \(\sqrt{a+ib}=x+iy\), then a possible value of \(\sqrt{a-ib}\) is
(a) \(x^2+y^2\)
(b) \(\sqrt{x^2+y^2}\)
(c) \(x-iy\)
(d) \(x+iy\)
Show Answer & Explanation

Answer: (c) \(x-iy\)

Explanation:
1. Taking conjugates of both sides: \(\overline{\sqrt{a+ib}}=\sqrt{a-ib}=x-iy\).

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.

Question 9: If \(z=i^9+i^{19}=a+bi\) with \(a,b\) real, then \(a\) equals
(a) 1
(b) 0
(c) 2
(d) 3
Show Answer & Explanation

Answer: (b) 0

Explanation:
1. \(i^9=i\), \(i^{19}=i^3=-i\), so \(z=0\) and \(a=0\).

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.

Question 10: The value of \(i^{57}+\dfrac1{i^{25}}\) is
(a) \(2i\)
(b) \(-2i\)
(c) 2
(d) 0
Show Answer & Explanation

Answer: (d) 0

Explanation:
1. \(i^{57}=i\) and \(\dfrac1{i^{25}}=\dfrac1i=-i\), so the sum is 0.

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.

Question 11: If \(\dfrac{c+i}{c-i}=a+ib\), where \(a,b,c\) are real, then \(a^2+b^2\) is equal to
(a) 7
(b) 1
(c) \(c^2\)
(d) \(-c^2\)
Show Answer & Explanation

Answer: (b) 1

Explanation:
1. \(|a+ib|=\dfrac{|c+i|}{|c-i|}=1\), so \(a^2+b^2=1\).

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.

Question 12: If \(x+iy=\dfrac{a+ib}{a-ib}\), then \(x^2+y^2\) equals
(a) 2
(b) 0
(c) 1
(d) 4
Show Answer & Explanation

Answer: (c) 1

Explanation:
1. \(|x+iy|=\dfrac{|a+ib|}{|a-ib|}=1\).

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.

Question 13: If \(z=x+iy\), \(z^{1/3}=a-ib\) and \(\dfrac xa-\dfrac yb=k(a^2-b^2)\), then \(k\) equals
(a) 2
(b) 6
(c) 1
(d) 4
Show Answer & Explanation

Answer: (d) 4

Explanation:
1. \(z=(a-ib)^3\) gives \(x=a^3-3ab^2\) and \(y=b^3-3a^2b\).
2. \(\dfrac xa-\dfrac yb=a^2-3b^2-b^2+3a^2=4(a^2-b^2)\).

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.

Question 14: If \(z=i^{-39}\), then \(z=a+i\). The value of \(a\) is
(a) 1
(b) 2
(c) 3
(d) 0
Show Answer & Explanation

Answer: (d) 0

Explanation:
1. \(i^{-39}=i^{-39+40}=i\), so \(z=0+i\) and \(a=0\).

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.

Question 15: If \((1-i)^n=2^n\), then the value of \(n\) is
(a) 1
(b) 2
(c) 0
(d) None of these
Show Answer & Explanation

Answer: (c) 0

Explanation:
1. Taking moduli: \((\sqrt2)^n=2^n\) so \(n=0\), and indeed \((1-i)^0=1=2^0\).

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.

Question 16: The value of \((1+i)^5(1-i)^5\) is \(2^n\). Then \(n\) equals
(a) 5
(b) 2
(c) 3
(d) 4
Show Answer & Explanation

Answer: (a) 5

Explanation:
1. \([(1+i)(1-i)]^5=2^5\).

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.

Question 17: If \(|z_1+z_2|=|z_1|+|z_2|\) for two non-zero complex numbers, then \(\arg z_1-\arg z_2\) is equal to
(a) \(\dfrac\pi2\)
(b) 0
(c) \(-\pi\)
(d) \(-\dfrac\pi2\)
Show Answer & Explanation

Answer: (b) 0

Explanation:
1. Equality in the triangle inequality holds only when the two vectors point in the same direction, so the arguments are equal.

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.

Question 18: If \(|z-4|<|z-2|\), its solution is given by
(a) \(\text{Re}(z)>0\)
(b) \(\text{Re}(z)<0\)
(c) \(\text{Re}(z)>3\)
(d) \(\text{Re}(z)>2\)
Show Answer & Explanation

Answer: (c) \(\text{Re}(z)>3\)

Explanation:
1. With \(z=x+iy\): \((x-4)^2<(x-2)^2\) gives \(-8x+16<-4x+4\), so \(x>3\).

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.

Question 19: The value of \(\dfrac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}-1\) is
(a) 0
(b) -1
(c) 1
(d) -2
Show Answer & Explanation

Answer: (d) -2

Explanation:
1. Numerator: \(1-1+1-1+1=1\); denominator: \(-1+1-1+1-1=-1\).
2. \(\dfrac1{-1}-1=-2\).

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.

Question 20: The modulus and amplitude of \(\dfrac{1+2i}{1-(1-i)^2}\) are
(a) \(\sqrt2\text{ and }\dfrac\pi6\)
(b) \(1\text{ and }\dfrac\pi3\)
(c) 1 and 0
(d) \(1\text{ and }\dfrac\pi4\)
Show Answer & Explanation

Answer: (c) 1 and 0

Explanation:
1. \((1-i)^2=-2i\), so the denominator is \(1+2i\) and the fraction equals 1.
2. Modulus 1, amplitude 0.

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.

Question 21: If \(|z^2-1|=|z|^2+1\), then \(z\) lies on
(a) the real axis
(b) the imaginary axis
(c) the origin only
(d) None of these
Show Answer & Explanation

Answer: (b) the imaginary axis

Explanation:
1. \(|z^2-1|\le|z^2|+1\) with equality only when \(z^2\) and \(-1\) have the same direction, i.e. \(z^2\le0\).
2. That means z is purely imaginary (including 0).

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.

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Complex Numbers Objective Questions & Solutions for JEE Mathematics

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