Complex Numbers & Quadratic Equations · JEE Main previous year questions

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2022 43 questions

Q12022 · 27 Jun 2022 · Shift 1
The area of the polygon, whose vertices are the non-real roots of the equation $\overset{z}{ˉ} = i z^{2}$ is :
1. A. $\frac{3 3}{4}$
2. B. $\frac{3 3}{2}$
3. C. $\frac{3}{2}$
4. D. $\frac{3}{4}$
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Answer: A
Q52022 · 27 Jun 2022 · Shift 1
The number of distinct real roots of $x^{4} - 4 x + 1 = 0$ is :
1. A.4
2. B.2
3. C.1
4. D.0
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Answer: B
Q222022 · 27 Jun 2022 · Shift 1· Numerical answer
If the sum of all the roots of the equation $e^{2 x} - 11 e^{x} - 45 e^{- x} + \frac{81}{2} = 0$ is $lo g_{e} p$ , then $p$ is equal to ____.
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Answer: 45
Q12022 · 27 Jun 2022 · Shift 2
The number of points of intersection of $∣ z - (4 + 3 i) ∣ = 2$ and $∣ z ∣ + ∣ z - 4∣ = 6$ , $z \in C$ , is :
1. A.0
2. B.1
3. C.2
4. D.3
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Answer: C
Q222022 · 27 Jun 2022 · Shift 2· Numerical answer
Let $α, β$ be the roots of the equation $x^{2} - 4 λ x + 5 = 0$ and $α, γ$ be the roots of the equation $x^{2} - (32 + 23) x + 7 + 3 λ 3 = 0$ , $λ > 0$ . If $β + γ = 32$ , then $(α + 2 β + γ)^{2}$ is equal to ________.
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Answer: 98
Q32022 · 26 Jun 2022 · Shift 1
The ordered pair (a, b), for which the system of linear equations $3 x - 2 y + z = b$ $5 x - 8 y + 9 z = 3$ $2 x + y + a z = - 1$ has no solution, is :
1. A. $(3, \frac{1}{3})$
2. B. $(- 3, \frac{1}{3})$
3. C. $(- 3, - \frac{1}{3})$
4. D. $(3, - \frac{1}{3})$
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Answer: C
Q82022 · 26 Jun 2022 · Shift 1
The area bounded by the curve $y = ∣ x^{2} - 9∣$ and the line $y = 3$ is :
1. A. $4 (23 + 6 - 4)$
2. B. $4 (43 + 6 - 4)$
3. C. $8 (43 + 36 - 9)$
4. D. $8 (43 + 26 - 9)$
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Answer: D
Q212022 · 26 Jun 2022 · Shift 1· Numerical answer
The sum of the cubes of all the roots of the equation $x^{4} - 3 x^{3} - 2 x^{2} + 3 x + 1 = 0$ is ____.
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Answer: 36
Q22022 · 26 Jun 2022 · Shift 2
If the system of equations $α x + y + z = 5$ , $x + 2 y + 3 z = 4$ , $x + 3 y + 5 z = β$ has infinitely many solutions, then the ordered pair $(α, β)$ is equal to :
1. A. $(1, - 3)$
2. B. $(- 1, 3)$
3. C. $(1, 3)$
4. D. $(- 1, - 3)$
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Answer: C
Q242022 · 26 Jun 2022 · Shift 2· Numerical answer
Let $X = 000100010$ , $Y = α I + β X + γ X^{2}$ and $Z = α^{2} I - α β X + (β^{2} - α γ) X^{2}$ , $α, β, γ \in R$ . If $Y^{- 1} = \frac{1}{5} 00 \frac{- 2}{5} \frac{1}{5} 0 \frac{1}{5} \frac{- 2}{5} \frac{1}{5}$ , then $(α - β + γ)^{2}$ is equal to ________.
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Answer: 100
Q152022 · 25 Jun 2022 · Shift 1
Let a circle $C$ in complex plane pass through the points $z_{1} = 3 + 4 i$ , $z_{2} = 4 + 3 i$ and $z_{3} = 5 i$ . If $z (\neq = z_{1})$ is a point on $C$ such that the line through $z$ and $z_{1}$ is perpendicular to the line through $z_{2}$ and $z_{3}$ , then $ar g (z)$ is equal to :
1. A. $tan^{- 1} (\frac{2}{5}) - π$
2. B. $tan^{- 1} (\frac{24}{7}) - π$
3. C. $tan^{- 1} (3) - π$
4. D. $tan^{- 1} (\frac{3}{4}) - π$
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Answer: B
Q22022 · 25 Jun 2022 · Shift 2
Let $a, b \in R$ be such that the equation $a x^{2} - 2 b x + 15 = 0$ has a repeated root $α$ . If $α$ and $β$ are the roots of the equation $x^{2} - 2 b x + 21 = 0$ , then $α^{2} + β^{2}$ is equal to :
1. A. $37$
2. B. $58$
3. C. $68$
4. D. $92$
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Answer: B
Q32022 · 25 Jun 2022 · Shift 2
Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\overset{z}{ˉ}_{1} = i \overset{z}{ˉ}_{2}$ and $ar g (\frac{z _{1}}{z _{2}}) = π$ . Then :
1. A. $ar g z_{2} = \frac{π}{4}$
2. B. $ar g z_{2} = - \frac{3 π}{4}$
3. C. $ar g z_{1} = \frac{π}{4}$
4. D. $ar g z_{1} = - \frac{3 π}{4}$
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Answer: C
Q12022 · 24 Jun 2022 · Shift 2
Let $x * y = x^{2} + y^{3}$ and $(x * 1) * 1 = x * (1 * 1)$ . Then a value of $2 sin^{- 1} (\frac{x ^{4} + x ^{2} - 2}{x ^{4} + x ^{2} + 2})$ is :
1. A. $\frac{π}{4}$
2. B. $\frac{π}{3}$
3. C. $\frac{π}{2}$
4. D. $\frac{π}{6}$
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Answer: B
Q22022 · 24 Jun 2022 · Shift 2
The sum of all the real roots of the equation $(e^{2 x} - 4) (6 e^{2 x} - 5 e^{x} + 1) = 0$ is
1. A. $lo g_{e} 3$
2. B. $- lo g_{e} 3$
3. C. $lo g_{e} 6$
4. D. $- lo g_{e} 6$
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Answer: B
Q32022 · 24 Jun 2022 · Shift 2
Let the system of linear equations $x + y + α z = 2$ $3 x + y + z = 4$ $x + 2 z = 1$ have a unique solution $(x^{*}, y^{*}, z^{*})$ . If $(α, x^{*})$ , $(y^{*}, α)$ and $(x^{*}, - y^{*})$ are collinear points, then the sum of absolute values of all possible values of $α$ is
1. A.4
2. B.3
3. C.2
4. D.1
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Answer: C
Q102022 · 24 Jun 2022 · Shift 2
The number of distinct real roots of the equation $x^{7} - 7 x - 2 = 0$ is
1. A.5
2. B.7
3. C.1
4. D.3
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Answer: D
Q212022 · 24 Jun 2022 · Shift 2· Numerical answer
Let $S = {z \in C : ∣ z - 3∣ \leq 1 and z (4 + 3 i) + \overline{z} (4 - 3 i) \leq 24}$ . If $α + i β$ is the point in S which is closest to $4 i$ , then $25 (α + β)$ is equal to ____.
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Answer: 80