Complex Numbers & Quadratic Equations · JEE Main previous year questions

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2020 28 questions

Q12020 · 6 Sept 2020 · Shift 1
The region represented by ${z = x + i y \in C : ∣ z ∣ - Re (z) \leq 1}$ is also given by the inequality :
1. A. $y^{2} \leq 2 (x + \frac{1}{2})$
2. B. $y^{2} \leq x + \frac{1}{2}$
3. C. $y^{2} \geq 2 (x + 1)$
4. D. $y^{2} \geq x + 1$
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Answer: A
Q62020 · 6 Sept 2020 · Shift 1
The values of $λ$ and $μ$ for which the system of linear equations $x + y + z = 2$ $x + 2 y + 3 z = 5$ $x + 3 y + λ z = μ$ has infinitely many solutions are, respectively:
1. A.6 and 8
2. B.5 and 8
3. C.5 and 7
4. D.4 and 9
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Answer: B
Q142020 · 6 Sept 2020 · Shift 1
If $α$ and $β$ be two roots of the equation $x^{2} - 64 x + 256 = 0$ . Then the value of $(\frac{α ^{3}}{β ^{5}})^{1/8} + (\frac{β ^{3}}{α ^{5}})^{1/8}$ is :
1. A.1
2. B.3
3. C.2
4. D.4
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Answer: C
Q22020 · 6 Sept 2020 · Shift 2
If $α$ and $β$ are the roots of the equation $2 x (2 x + 1) = 1$ , then $β$ is equal to :
1. A. $- 2 α (α + 1)$
2. B. $2 α (α + 1)$
3. C. $2 α^{2}$
4. D. $2 α (α - 1)$
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Answer: A
Q112020 · 6 Sept 2020 · Shift 2
Let $z = x + i y$ be a non-zero complex number such that $z^{2} = i ∣ z ∣^{2}$ , where $i = - 1$ , then z lies on the :
1. A.line, $y = - x$
2. B.real axis
3. C.line, $y = x$
4. D.imaginary axis
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Answer: C
Q122020 · 6 Sept 2020 · Shift 2
The area (in sq. units) of the region enclosed by the curves $y = x^{2} - 1$ and $y = 1 - x^{2}$ is equal to :
1. A. $\frac{8}{3}$
2. B. $\frac{4}{3}$
3. C. $\frac{7}{2}$
4. D. $\frac{16}{3}$
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Answer: A
Q212020 · 6 Sept 2020 · Shift 2· Numerical answer
The sum of distinct values of $λ$ for which the system of equations $(λ - 1) x + (3 λ + 1) y + 2 λ z = 0$ $(λ - 1) x + (4 λ - 2) y + (λ + 3) z = 0$ $2 x + (3 λ + 1) y + 3 (λ - 1) z = 0$ has non-zero solutions, is ______ .
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Answer: 3
Q132020 · 3 Sept 2020 · Shift 1
If $Δ = x - 2 2 x - 3 3 x - 5 2 x - 3 3 x - 4 5 x - 8 3 x - 4 4 x - 5 10 x - 17 = A x^{3} + B x^{2} + C x + D$ , then $B + C$ is equal to :
1. A.-1
2. B.-3
3. C.9
4. D.1
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Answer: B
Q162020 · 3 Sept 2020 · Shift 1
If $α$ and $β$ are the roots of the equation $x^{2} + p x + 2 = 0$ and $\frac{1}{α}$ and $\frac{1}{β}$ are the roots of the equation $2 x^{2} + 2 q x + 1 = 0$ , then $(α - \frac{1}{α}) (β - \frac{1}{β}) (α + \frac{1}{β}) (β + \frac{1}{α})$ is equal to :
1. A. $\frac{9}{4} (9 - q^{2})$
2. B. $\frac{9}{4} (9 + q^{2})$
3. C. $\frac{9}{4} (9 - p^{2})$
4. D. $\frac{9}{4} (9 + p^{2})$
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Answer: C
Q82020 · 3 Sept 2020 · Shift 2
The set of all real values of $λ$ for which the quadratic equations, $(λ^{2} + 1) x^{2} - 4 λ x + 2 = 0$ always have exactly one root in the interval $(0, 1)$ is :
1. A. $(- 3, - 1)$
2. B. $(2, 4]$
3. C. $(0, 2)$
4. D. $(1, 3]$
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Answer: D
Q102020 · 3 Sept 2020 · Shift 2
If $z_{1}, z_{2}$ are complex numbers such that $Re (z_{1}) = ∣ z_{1} - 1∣$ , $Re (z_{2}) = ∣ z_{2} - 1∣$ , and $ar g (z_{1} - z_{2}) = \frac{π}{6}$ , then $Im (z_{1} + z_{2})$ is equal to :
1. A. $\frac{3}{2}$
2. B. $\frac{1}{3}$
3. C. $\frac{2}{3}$
4. D. $23$
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Answer: D
Q222020 · 3 Sept 2020 · Shift 2· Numerical answer
Let S be the set of all integer solutions, (x, y, z), of the system of equations $x - 2 y + 5 z = 0$ $- 2 x + 4 y + z = 0$ $- 7 x + 14 y + 9 z = 0$ such that $15 \leq x^{2} + y^{2} + z^{2} \leq 150$ . Then, the number of elements in the set S is equal to _______.
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Answer: 8
Q92020 · 2 Sept 2020 · Shift 1
Let $α$ and $β$ be the roots of the equation $5 x^{2} + 6 x - 2 = 0$ . If $S_{n} = α^{n} + β^{n}, n = 1, 2, 3 \dots$ , then :
1. A. $5 S_{6} + 6 S_{5} = 2 S_{4}$
2. B. $5 S_{6} + 6 S_{5} + 2 S_{4} = 0$
3. C. $6 S_{6} + 5 S_{5} + 2 S_{4} = 0$
4. D. $6 S_{6} + 5 S_{5} = 2 S_{4}$
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Answer: A
Q42020 · 2 Sept 2020 · Shift 2
Let f(x) be a quadratic polynomial such that f(–1) + f(2) = 0. If one of the roots of f(x) = 0 is 3, then its other root lies in :
1. A.(–3, –1)
2. B.(1, 3)
3. C.(–1, 0)
4. D.(0, 1)
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Answer: C
Q32020 · 9 Jan 2020 · Shift 1
Let $z$ be complex number such that $\frac{z - i}{z + 2 i} = 1 and ∣ z ∣ = \frac{5}{2} .$ Then the value of $∣ z + 3 i ∣$ is :
1. A. $23$
2. B. $10$
3. C. $\frac{15}{4}$
4. D. $\frac{7}{2}$
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Answer: D
Q72020 · 9 Jan 2020 · Shift 1
The number of real roots of the equation, $e^{4 x} + e^{3 x} - 4 e^{2 x} + e^{x} + 1 = 0$ is :
1. A.1
2. B.2
3. C.3
4. D.4
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Answer: A
Q42020 · 9 Jan 2020 · Shift 2
Let $a, b \in R, a \neq = 0$ be such that the equation, $a x^{2} - 2 b x + 5 = 0$ has a repeated root $α$ , which is also a root of the equation, $x^{2} - 2 b x - 10 = 0$ . If $β$ is the other root of this equation, then $α^{2} + β^{2}$ is equal to :
1. A.28
2. B.24
3. C.26
4. D.25
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Answer: D
Q132020 · 9 Jan 2020 · Shift 2
If z be a complex number satisfying $∣ Re (z) ∣ + ∣ Im (z) ∣ = 4$ , then $∣ z ∣$ cannot be :
1. A. $10$
2. B. $7$
3. C. $\frac{17}{2}$
4. D. $8$
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Answer: B
Q232020 · 9 Jan 2020 · Shift 2· Numerical answer
If the curves, $x^{2} - 6 x + y^{2} + 8 = 0$ and $x^{2} - 8 y + y^{2} + 16 - k = 0$ , ( $k > 0$ ) touch each other at a point, then the largest value of k is _______.
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Answer: 36
Q82020 · 8 Jan 2020 · Shift 1
If the equation, $x^{2} + b x + 45 = 0$ ( $b \in R$ ) has conjugate complex roots and they satisfy $∣ z + 1∣ = 210$ , then :
1. A. $b^{2} - b = 42$
2. B. $b^{2} + b = 12$
3. C. $b^{2} + b = 72$
4. D. $b^{2} - b = 30$
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Answer: D
Q222020 · 8 Jan 2020 · Shift 1· Numerical answer
The least positive value of 'a' for which the equation $2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a$ has real roots is
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Answer: 8
Q242020 · 8 Jan 2020 · Shift 1· Numerical answer
Let the normal at a point P on the curve $y^{2} - 3 x^{2} + y + 10 = 0$ intersect the y-axis at $(0, \frac{3}{2})$ . If m is the slope of the tangent at P to the curve, then $∣ m ∣$ is equal to
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Answer: 4
Q142020 · 8 Jan 2020 · Shift 2
Let $α = \frac{- 1 + i 3}{2}$ . If $a = (1 + α) \sum_{k = 0}^{100} α^{2 k}$ and $b = \sum_{k = 0}^{100} α^{3 k}$ , then a and b are the roots of the quadratic equation :
1. A. $x^{2} + 101 x + 100 = 0$
2. B. $x^{2} + 102 x + 101 = 0$
3. C. $x^{2} - 102 x + 101 = 0$
4. D. $x^{2} - 101 x + 100 = 0$
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Answer: C
Q152020 · 8 Jan 2020 · Shift 2
The length of the perpendicular from the origin, on the normal to the curve, $x^{2} + 2 x y - 3 y^{2} = 0$ at the point (2,2) is
1. A. $2$
2. B. $42$
3. C.2
4. D. $22$
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Answer: D
Q32020 · 7 Jan 2020 · Shift 1
Let $α$ and $β$ be two real roots of the equation $(k + 1) tan^{2} x - 2 \cdot λ tan x = (1 - k)$ , where $k (\neq = - 1)$ and $λ$ are real numbers. If $tan^{2} (α + β) = 50$ , then a value of $λ$ is:
1. A. $52$
2. B.10
3. C.5
4. D. $102$
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Answer: B