Complex Numbers & Quadratic Equations · JEE Main previous year questions

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2024 38 questions

Q152024 · 31 Jan 2024 · Shift 2
Let $z_{1}$ and $z_{2}$ be two complex numbers such that $z_{1} + z_{2} = 5$ and $z_{1}^{3} + z_{2}^{3} = 20 + 15 i$ Then, $z_{1}^{4} + z_{2}^{4}$ equals -
1. A. $1515$
2. B. $303$
3. C. $253$
4. D.75
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Answer: D
Q52024 · 30 Jan 2024 · Shift 1
The maximum area of a triangle whose one vertex is at $(0, 0)$ and the other two vertices lie on the curve $y = - 2 x^{2} + 54$ at points $(x, y)$ and $(- x, y)$ , where $y > 0$ , is :
1. A.108
2. B.122
3. C.88
4. D.92
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Answer: A
Q162024 · 30 Jan 2024 · Shift 1
If $z = x + i y, x y \neq = 0$ , satisfies the equation $z^{2} + i \overset{z}{ˉ} = 0$ , then $∣ z^{2} ∣$ is equal to :
1. A.9
2. B. $\frac{1}{4}$
3. C.4
4. D.1
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Answer: D
Q172024 · 30 Jan 2024 · Shift 1
If $2 sin^{3} x + sin 2 x cos x + 4 sin x - 4 = 0$ has exactly 3 solutions in the interval $[0, \frac{nπ}{2}], n \in N$ , then the roots of the equation $x^{2} + n x + (n - 3) = 0$ belong to :
1. A. $(0, \infty)$
2. B. $Z$
3. C. $(- \frac{17}{2}, \frac{17}{2})$
4. D. $(- \infty, 0)$
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Answer: D
Q242024 · 30 Jan 2024 · Shift 1· Numerical answer
Let $α, β \in N$ be roots of the equation $x^{2} - 70 x + λ = 0$ , where $\frac{λ}{2}, \frac{λ}{3} \in / N$ . If $λ$ assumes the minimum possible value, then $\frac{( α - 1 + β - 1 ) ( λ + 35 )}{∣ α - β ∣}$ is equal to :
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Answer: 60
Q282024 · 29 Jan 2024 · Shift 1· Numerical answer
Let $α, β$ be the roots of the equation $x^{2} - x + 2 = 0$ with $Im (α) > Im (β)$ . Then $α^{6} + α^{4} + β^{4} - 5 α^{2}$ is equal to _______.
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Answer: 13
Q172024 · 29 Jan 2024 · Shift 2
Let $r$ and $θ$ respectively be the modulus and amplitude of the complex number $z = 2 - i (2 tan \frac{5 π}{8})$ , then $(r, θ)$ is equal to
1. A. $(2 sec \frac{11 π}{8}, \frac{11 π}{8})$
2. B. $(2 sec \frac{3 π}{8}, \frac{3 π}{8})$
3. C. $(2 sec \frac{5 π}{8}, \frac{3 π}{8})$
4. D. $(2 sec \frac{3 π}{8}, \frac{5 π}{8})$
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Answer: B
Q252024 · 29 Jan 2024 · Shift 2· Numerical answer
Let $α, β$ be the roots of the equation $x^{2} - 6 x + 3 = 0$ such that $Im (α) > Im (β)$ . Let $a, b$ be integers not divisible by 3 and $n$ be a natural number such that $\frac{α ^{99}}{β} + α^{98} = 3^{n} (a + ib), i = - 1$ . Then $n + a + b$ is equal to _________.
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Answer: 49
Q262024 · 29 Jan 2024 · Shift 2· Numerical answer
Let for any three distinct consecutive terms $a, b, c$ of an A.P, the lines $a x + b y + c = 0$ be concurrent at the point $P$ and $Q (α, β)$ be a point such that the system of equations $x + y + z = 6$ , $2 x + 5 y + α z = β$ and $x + 2 y + 3 z = 4$ , has infinitely many solutions. Then $(P Q)^{2}$ is equal to _________.
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Answer: 113
Q22024 · 27 Jan 2024 · Shift 1
If $S = {z \in C : ∣ z - i ∣ = ∣ z + i ∣ = ∣ z - 1∣}$ , then, $n (S)$ is :
1. A.1
2. B.2
3. C.3
4. D.0
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Answer: A
Q292024 · 27 Jan 2024 · Shift 1· Numerical answer
If $α$ satisfies the equation $x^{2} + x + 1 = 0$ and $(1 + α)^{7} = A + B α + C α^{2}$ , $A, B, C \geq 0$ , then $5 (3 A - 2 B - C)$ is equal to ________.
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Answer: 5
Q72024 · 27 Jan 2024 · Shift 2
If $α, β$ are the roots of the equation, $x^{2} - x - 1 = 0$ and $S_{n} = 2023 α^{n} + 2024 β^{n}$ , then :
1. A. $2 S_{12} = S_{11} + S_{10}$
2. B. $S_{12} = S_{11} + S_{10}$
3. C. $S_{11} = S_{10} + S_{12}$
4. D. $2 S_{11} = S_{12} + S_{10}$
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Answer: B
Q292024 · 27 Jan 2024 · Shift 2· Numerical answer
Let the complex numbers $α$ and $\frac{1}{α ˉ}$ lie on the circles $∣ z - z_{0} ∣^{2} = 4$ and $∣ z - z_{0} ∣^{2} = 16$ respectively, where $z_{0} = 1 + i$ . Then, the value of $100∣ α ∣^{2}$ is _______.
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Answer: 20