JEE Mathematics Complex Numbers MCQs Set 36

Mathematics Objective Questions and Answers: Complex Numbers

Access targeted multiple-choice questions for Complex Numbers designed to align with the latest JEE academic syllabus for JEE Mathematics. These objective practice sets help students evaluate their conceptual understanding and improve exam readiness.

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Navigate directly to the 50 objective questions for Complex Numbers using the digital viewer below. Each practice set includes verified answer keys, allowing students to instantly cross-check their work and identify areas requiring further revision.

Multiple Choice Questions

Question 1: If \(z=2+i\), then \((z-1)(\bar z-5)+(\bar z-1)(z-5)\) is equal to
(a) 2
(b) 7
(c) -1
(d) -4
Show Answer & Explanation

Answer: (d) -4

Explanation:
1. The expression is \(2\,\text{Re}[(z-1)(\bar z-5)]\); \((1+i)(-3-i)=-2-4i\).
2. So the value is \(2(-2)=-4\).

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 square root of \(i\) is
(a) \(\pm\dfrac1{\sqrt2}(-1+i)\)
(b) \(\pm\dfrac1{\sqrt2}(1+i)\)
(c) \(\pm\dfrac1{\sqrt2}(1-i)\)
(d) None of these
Show Answer & Explanation

Answer: (b) \(\pm\dfrac1{\sqrt2}(1+i)\)

Explanation:
1. \(\left(\dfrac{1+i}{\sqrt2}\right)^2=\dfrac{2i}2=i\).

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.

Question 3: The complex number \(z\) which satisfies \(\left|\dfrac{i+z}{i-z}\right|=1\) lies on
(a) \(\text{the circle }x^2+y^2=1\)
(b) the imaginary axis
(c) the real axis
(d) \(\text{the line }x+y=1\)
Show Answer & Explanation

Answer: (c) the real axis

Explanation:
1. \(|z+i|=|z-i|\) means \(z\) is equidistant from \(i\) and \(-i\), so it lies on the perpendicular bisector, the real axis.

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 equation \(|z_1+z_2|=|z_1|+|z_2|\) is possible if
(a) \(z_2=\bar z_1\)
(b) \(z_2=\dfrac1{z_1}\)
(c) \(|z_1|=|z_2|\)
(d) \(\arg z_1=\arg z_2\)
Show Answer & Explanation

Answer: (d) \(\arg z_1=\arg z_2\)

Explanation:
1. Equality in the triangle inequality needs the two numbers in the same direction, so equal arguments.

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: The modulus of the complex number \(z\) such that \(|z+3-i|=1\) and \(\arg z=\pi\) is equal to
(a) 2
(b) 9
(c) 4
(d) 3
Show Answer & Explanation

Answer: (d) 3

Explanation:
1. \(\arg z=\pi\) means \(z=-t\), \(t>0\); \(|(3-t)-i|=1\) gives \((3-t)^2=0\).
2. So \(t=3\) and \(|z|=3\).

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 6: If \(Z=\dfrac{i-1}{\cos\frac\pi3+i\sin\frac\pi3}\), then the polar form of \(Z\) is
(a) \(\sqrt2\left(\cos\dfrac{5\pi}{12}-i\sin\dfrac{5\pi}{12}\right)\)
(b) \(\sqrt2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)\)
(c) \(\sqrt2\left(\cos\dfrac{5\pi}{12}+i\sin\dfrac{5\pi}{12}\right)\)
(d) \(\sqrt2\left(\cos\dfrac\pi4-i\sin\dfrac\pi4\right)\)
Show Answer & Explanation

Answer: (c) \(\sqrt2\left(\cos\dfrac{5\pi}{12}+i\sin\dfrac{5\pi}{12}\right)\)

Explanation:
1. \(i-1=\sqrt2\left(\cos\dfrac{3\pi}4+i\sin\dfrac{3\pi}4\right)\).
2. Dividing: argument \(\dfrac{3\pi}4-\dfrac\pi3=\dfrac{5\pi}{12}\).

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 \((x-iy)(3+5i)\) is the conjugate of \(-6-24i\), then \(x\) and \(y\) are
(a) \(x=3,\ y=-3\)
(b) \(x=3,\ y=3\)
(c) \(x=-3,\ y=3\)
(d) \(x=-3,\ y=-3\)
Show Answer & Explanation

Answer: (a) \(x=3,\ y=-3\)

Explanation:
1. The conjugate is \(-6+24i\). \((x-iy)(3+5i)=(3x+5y)+i(5x-3y)\).
2. \(3x+5y=-6\) and \(5x-3y=24\) give \(x=3,\ y=-3\).

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 8: If \(x+iy=\sqrt{\dfrac{a+ib}{c+id}}\), then \((x^2+y^2)^2\) equals
(a) \(\dfrac{a+b}{c+d}\)
(b) \(\dfrac{a^2+b^2}{c^2+d^2}\)
(c) \(\dfrac{c^2+d^2}{a^2+b^2}\)
(d) \(\left(\dfrac{a^2+b^2}{c^2+d^2}\right)^2\)
Show Answer & Explanation

Answer: (b) \(\dfrac{a^2+b^2}{c^2+d^2}\)

Explanation:
1. \(|x+iy|^2=\sqrt{\dfrac{a^2+b^2}{c^2+d^2}}\), so squaring gives \((x^2+y^2)^2=\dfrac{a^2+b^2}{c^2+d^2}\).

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 9: If \(|z+4|\le3\), then the maximum value of \(|z+1|\) is
(a) 0
(b) 4
(c) 6
(d) 10
Show Answer & Explanation

Answer: (c) 6

Explanation:
1. \(|z+1|=|(z+4)-3|\le|z+4|+3\le6\), attained at \(z=-7\).

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.

Question 10: The value of \(\dfrac{(\cos\theta+i\sin\theta)^4}{(\cos\theta-i\sin\theta)^3}\) is
(a) \(\cos5\theta+i\sin5\theta\)
(b) \(\cos4\theta+i\sin4\theta\)
(c) \(\cos\theta+i\sin\theta\)
(d) \(\cos7\theta+i\sin7\theta\)
Show Answer & Explanation

Answer: (d) \(\cos7\theta+i\sin7\theta\)

Explanation:
1. The numerator is \(\text{cis}4\theta\); the denominator is \(\text{cis}(-3\theta)\).
2. The quotient is \(\text{cis}7\theta\).

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: Consider: I. The additive inverse of (1 - i) is -1 + i. II. If z1 and z2 are two complex numbers, then z1 - z2 is the sum of z1 and the additive inverse of z2. III. The simplest form of (5 + sqrt2 i)/(1 - sqrt2 i) is 1 + 2 sqrt2 i. Which are correct?
(a) I, II and III
(b) Only I and II
(c) Only II and III
(d) None of these
Show Answer & Explanation

Answer: (a) I, II and III

Explanation:
1. I and II are definitions.
2. III: \( (5 + sqrt2 i)(1 + sqrt2 i)/3 = (3 + 6 sqrt2 i)/3 = 1 + 2 sqrt2 i \).

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.

Question 12: Consider: I. If z1 and z2 are complex numbers with |z1 + z2| = |z1| + |z2|, then arg z1 - arg z2 = 0. II. The roots of \( x^2 + 3x + 5 = 0 \) are (-3 +- i sqrt11)/2. Which statements are correct?
(a) Both I and II
(b) Only I
(c) Only II
(d) Neither I nor II
Show Answer & Explanation

Answer: (a) Both I and II

Explanation:
1. I: equality in the triangle inequality means the same argument.
2. II: \( x = (-3 \pm \sqrt{9 - 20})/2 = (-3 \pm i sqrt11)/2 \).

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 13: Consider: I. The value of \( x^3 + 7x^2 - x + 16 \) when x = 1 + 2i is -17 + 24i. II. If \( iz^3 + z^2 - z + i = 0 \) then |z| = 1. Which statements are correct?
(a) Both I and II
(b) Only I
(c) Only II
(d) Neither I nor II
Show Answer & Explanation

Answer: (a) Both I and II

Explanation:
1. I: \( (1 + 2i)^2 = -3 + 4i, (1 + 2i)^3 = -11 - 2i \), so the value is \( -11 - 2i - 21 + 28i - 1 - 2i + 16 = -17 + 24i \).
2. II: the roots of the polynomial all have modulus 1, so \( |z| = 1 \).

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 14: Match the complex numbers with their multiplicative inverses: A. 4 - 3i, B. sqrt5 + 3i, C. -i with 1. sqrt5/14 - (3/14)i, 2. 4/25 + (3/25)i, 3. 0 + i. The correct pairing is
(a) A-2, B-1, C-3
(b) A-1, B-2, C-3
(c) A-1, B-3, C-2
(d) A-2, B-3, C-1
Show Answer & Explanation

Answer: (a) A-2, B-1, C-3

Explanation:
1. A: \( 1/(4 - 3i) = (4 + 3i)/25 \).
2. \( B \): \( 1/(sqrt5 + 3i) = (sqrt5 - 3i)/14 \).
3. \( C \): \( 1/(-i) = i \).

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.

Question 15: Simplify the numbers in column I and match with column II: A. \( (1 - i)^4, B \). \( (1/3 + 3i)^3, C \). \( (-2 - i/3)^3 \) with 1. -(22/3 + (107/27) i), 2. -4 + 0i, 3. -242/27 - 26i. The correct pairing is
(a) A-2, B-3, C-1
(b) A-1, B-2, C-3
(c) A-2, B-1, C-3
(d) A-3, B-1, C-2
Show Answer & Explanation

Answer: (a) A-2, B-3, C-1

Explanation:
1. A: \( (1 - i)^2 = -2i \), so \( (1 - i)^4 = -4 \).
2. \( B \): \( (1/3 + 3i)^3 = 1/27 + 3(1/9)(3i) + 3(1/3)(-9) - 27i = -242/27 - 26i \).
3. \( C \): \( (-2 - i/3)^3 = -(22/3 + (107/27) i) \).

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.

Question 16: Match: A. \( i^{-1}, B \). \( i^{-2}, C \). \( i^{-3}, D \). \( i^4 \) with 1. -1, 2. -i, 3. i, 4. 1. The correct pairing is
(a) A-2, B-1, C-3, D-4
(b) A-1, B-2, C-3, D-4
(c) A-2, B-1, C-4, D-3
(d) A-3, B-4, C-1, D-2
Show Answer & Explanation

Answer: (a) A-2, B-1, C-3, D-4

Explanation:
1. \( i^{-1} = -i, i^{-2} = -1, i^{-3} = i \) and \( i^4 = 1 \).

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 17: Match the complex numbers with their polar forms: A. 1 - i, B. -1 + i, C. -1 - i, D. sqrt3 + i with 1. sqrt2[cos(-3pi/4) + i sin(-3pi/4)], 2. 2[cos(pi/6) + i sin(pi/6)], 3. sqrt2[cos(-pi/4) + i sin(-pi/4)], 4. sqrt2[cos(3pi/4) + i sin(3pi/4)]. The correct pairing is
(a) A-3, B-4, C-1, D-2
(b) A-3, B-1, C-4, D-2
(c) A-4, B-3, C-1, D-2
(d) A-1, B-4, C-3, D-2
Show Answer & Explanation

Answer: (a) A-3, B-4, C-1, D-2

Explanation:
1. \( 1 - i \) is in the fourth quadrant (arg \( -\pi/4 \)); \( -1 + i \) in the second \( (3pi/4) \); \( -1 - i \) in the third \( (-3pi/4) \); \( sqrt3 + i \) has modulus 2 and argument \( \pi/6 \).

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: Assertion: if z1 and z2 are complex numbers and \( |z1 + z2|^2 = |z1|^2 + |z2|^2 \), then z1/z2 is purely imaginary. Reason: if z is purely imaginary, then z + conj z = 0. Which is correct?
(a) Both are true and Reason explains Assertion
(b) Assertion is true, Reason is false
(c) Assertion is false, Reason is true
(d) Both are true but Reason does not explain Assertion
Show Answer & Explanation

Answer: (d) Both are true but Reason does not explain Assertion

Explanation:
1. The condition gives Re(z1 conj z2) = 0, so \( z1/z2 \) is purely imaginary: the assertion is true.
2. The reason is a true fact but it is the converse property, so it does not explain the assertion.

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: Assertion: if P and Q are the points in the plane representing z1 and z2, then the distance |PQ| = |z2 - z1|. Reason: the locus of the point P(z) satisfying |z - (2 + 3i)| = 4 is a straight line. Which is correct?
(a) Both are true and Reason explains Assertion
(b) Both are true but Reason does not explain Assertion
(c) Assertion is true, Reason is false
(d) Assertion is false, Reason is true
Show Answer & Explanation

Answer: (c) Assertion is true, Reason is false

Explanation:
1. The distance formula for points as complex numbers is correct.
2. \( |z - (2 + 3i)| = 4 \) describes a circle with centre \( 2 + 3i \) and radius 4, not a line.

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: Assertion: the equation \( ix^2 \) - 3ix + 2i = 0 has non-real roots. Reason: if a, b, c are real and \( b^2 \) - 4ac is greater than or equal to 0, the roots of \( ax^2 \) + bx + c = 0 are real, and if \( b^2 \) - 4ac is negative the roots are non-real. Which is correct?
(a) Both are true and Reason explains Assertion
(b) Assertion is false, Reason is true
(c) Both are true but Reason does not explain Assertion
(d) Assertion is true, Reason is false
Show Answer & Explanation

Answer: (b) Assertion is false, Reason is true

Explanation:
1. Dividing by \( i \) gives \( x^2 - 3x + 2 = 0 \) with roots 1 and 2, which are real, so the assertion is false.
2. The reason holds for real coefficients.

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.

Free study material for Complex Numbers

Multiple Choice Questions (MCQs) for JEE Mathematics Complex Numbers

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