Complex Numbers & Quadratic Equations · JEE Main previous year questions

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2022 43 questions

Q32022 · 29 Jul 2022 · Shift 2
If the system of equations $x + y + z = 6$ $2 x + 5 y + α z = β$ $x + 2 y + 3 z = 14$ has infinitely many solutions, then $α + β$ is equal to
1. A.8
2. B.36
3. C.44
4. D.48
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Answer: C
Q122022 · 29 Jul 2022 · Shift 2
Let $S = {z = x + i y : ∣ z - 1 + i ∣ \geq ∣ z ∣, ∣ z ∣ < 2, ∣ z + i ∣ = ∣ z - 1∣}$ . Then the set of all values of $x$ , for which $w = 2 x + i y \in S$ for some $y \in R$ , is :
1. A. $(- 2, \frac{1}{2 2}]$
2. B. $(- \frac{1}{2}, \frac{1}{4}]$
3. C. $(- 2, \frac{1}{2}]$
4. D. $(- \frac{1}{2}, \frac{1}{2 2}]$
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Answer: B
Q212022 · 29 Jul 2022 · Shift 2· Numerical answer
Let $α, β (α > β)$ be the roots of the quadratic equation $x^{2} - x - 4 = 0$ . If $P_{n} = α^{n} - β^{n}, n \in N$ , then $\frac{P _{15} P _{16} - P _{14} P _{16} - P _{15}^{2} + P _{14} P _{15}}{P _{13} P _{14}}$ is equal to __________.
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Answer: 16
Q302022 · 29 Jul 2022 · Shift 2· Numerical answer
If the tangent to the curve $y = x^{3} - x^{2} + x$ at the point $(a, b)$ is also tangent to the curve $y = 5 x^{2} + 2 x - 25$ at the point $(2, - 1)$ , then $∣2 a + 9 b ∣$ is equal to ____.
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Answer: 195
Q122022 · 28 Jul 2022 · Shift 1
Let $S_{1} = {z_{1} \in C : ∣ z_{1} - 3∣ = \frac{1}{2}}$ and $S_{2} = {z_{2} \in C : ∣ z_{2} - ∣ z_{2} + 1∣∣ = ∣ z_{2} + ∣ z_{2} - 1∣∣}$ . Then, for $z_{1} \in S_{1}$ and $z_{2} \in S_{2}$ , the least value of $∣ z_{2} - z_{1} ∣$ is :
1. A.0
2. B. $\frac{1}{2}$
3. C. $\frac{3}{2}$
4. D. $\frac{5}{2}$
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Answer: C
Q12022 · 28 Jul 2022 · Shift 2
Let $S = {x \in [- 6, 3] - {- 2, 2} : \frac{∣ x + 3∣ - 1}{∣ x ∣ - 2} \geq 0}$ and $T = {x \in Z : x^{2} - 7∣ x ∣ + 9 \leq 0}$ . Then the number of elements in $S \cap T$ is :
1. A.7
2. B.5
3. C.4
4. D.3
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Answer: D
Q22022 · 28 Jul 2022 · Shift 2
Let $α, β$ be the roots of the equation $x^{2} - 2 x + 6 = 0$ and $\frac{1}{α ^{2}} + 1$ , $\frac{1}{β ^{2}} + 1$ be the roots of the equation $x^{2} + a x + b = 0$ . Then the roots of the equation $x^{2} - (a + b - 2) x + (a + b + 2) = 0$ are :
1. A.non-real complex numbers
2. B.real and both negative
3. C.real and both positive
4. D.real and exactly one of them is positive
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Answer: B
Q232022 · 28 Jul 2022 · Shift 2· Numerical answer
Let $z = a + ib, b \neq = 0$ be complex numbers satisfying $z^{2} = \overset{z}{ˉ} \cdot 2^{1 - ∣ z ∣}$ . Then the least value of $n \in N$ , such that $z^{n} = (z + 1)^{n}$ , is equal to _________.
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Answer: 6
Q32022 · 27 Jul 2022 · Shift 1
Let the minimum value $v_{0}$ of $v = ∣ z ∣^{2} + ∣ z - 3 ∣^{2} + ∣ z - 6 i ∣^{2}, z \in C$ is attained at $z = z_{0}$ . Then $∣2 z_{0}^{2} - z_{0}^{3} + 3 ∣^{2} + v_{0}^{2}$ is equal to :
1. A.1000
2. B.1024
3. C.1105
4. D.1196
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Answer: A
Q92022 · 27 Jul 2022 · Shift 1
The area of the smaller region enclosed by the curves $y^{2} = 8 x + 4$ and $x^{2} + y^{2} + 43 x - 4 = 0$ is equal to
1. A. $\frac{1}{3} (2 - 123 + 8 π)$
2. B. $\frac{1}{3} (2 - 123 + 6 π)$
3. C. $\frac{1}{3} (4 - 123 + 8 π)$
4. D. $\frac{1}{3} (4 - 123 + 6 π)$
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Answer: C
Q252022 · 27 Jul 2022 · Shift 1· Numerical answer
Let $f (x) = 2 x^{2} - x - 1$ and $S = {n \in Z : ∣ f (n) ∣ \leq 800}$ . Then, the value of $\sum_{n \in S} f (n)$ is equal to _________.
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Answer: 10620
Q22022 · 27 Jul 2022 · Shift 2
Let S be the set of all $(α, β), π < α, β < 2 π$ , for which the complex number $\frac{1 - i s i n α}{1 + 2 i s i n α}$ is purely imaginary and $\frac{1 + i c o s β}{1 - 2 i c o s β}$ is purely real. Let $Z_{α β} = sin 2 α + i cos 2 β, (α, β) \in S$ . Then $\sum_{(α, β) \in S} (i Z_{α β} + \frac{1}{i Z ˉ _{α β}})$ is equal to :
1. A.3
2. B. $3 i$
3. C.1
4. D. $2 - i$
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Answer: C
Q32022 · 27 Jul 2022 · Shift 2
If $α, β$ are the roots of the equation $x^{2} - (5 + 3^{l o g_{3} 5} - 5^{l o g_{5} 3}) x + 3 (3^{(l o g_{3} 5)^{\frac{1}{3}}} - 5^{(l o g_{5} 3)^{\frac{2}{3}}} - 1) = 0,$ then the equation, whose roots are $α + \frac{1}{β}$ and $β + \frac{1}{α}$ , is :
1. A. $3 x^{2} - 20 x - 12 = 0$
2. B. $3 x^{2} - 10 x - 4 = 0$
3. C. $3 x^{2} - 10 x + 2 = 0$
4. D. $3 x^{2} - 20 x + 16 = 0$
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Answer: B
Q112022 · 26 Jul 2022 · Shift 1
The length of the perpendicular from the point $(1, - 2, 5)$ on the line passing through $(1, 2, 4)$ and parallel to the line $x + y - z = 0 = x - 2 y + 3 z - 5$ is :
1. A. $\frac{21}{2}$
2. B. $\frac{9}{2}$
3. C. $\frac{73}{2}$
4. D.1
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Answer: A
Q212022 · 26 Jul 2022 · Shift 1· Numerical answer
If for some $p, q, r \in R$ , not all have same sign, one of the roots of the equation $(p^{2} + q^{2}) x^{2} - 2 q (p + r) x + q^{2} + r^{2} = 0$ is also a root of the equation $x^{2} + 2 x - 8 = 0$ , then $\frac{q ^{2} + r ^{2}}{p ^{2}}$ is equal to ____ .
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Answer: 272
Q232022 · 26 Jul 2022 · Shift 1· Numerical answer
The number of distinct real roots of the equation $x^{5} (x^{3} - x^{2} - x + 1) + x (3 x^{3} - 4 x^{2} - 2 x + 4) - 1 = 0$ is ____ .
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Answer: 3
Q12022 · 25 Jul 2022 · Shift 2
For $z \in C$ if the minimum value of $(∣ z - 32 ∣ + ∣ z - p 2 i ∣)$ is $52$ , then a value of $p$ is ___________.
1. A.3
2. B. $\frac{7}{2}$
3. C.4
4. D. $\frac{9}{2}$
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Answer: C
Q22022 · 25 Jul 2022 · Shift 2
The number of real values of $λ$ , such that the system of linear equations $2 x - 3 y + 5 z = 9$ $x + 3 y - z = - 18$ $3 x - y + (λ^{2} - ∣ λ ∣) z = 16$ has no solutions, is
1. A.0
2. B.1
3. C.2
4. D.4
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Answer: C
Q222022 · 25 Jul 2022 · Shift 2· Numerical answer
Let $f (x)$ be a quadratic polynomial with leading coefficient 1 such that $f (0) = p, p \neq = 0$ , and $f (1) = \frac{1}{3}$ . If the equations $f (x) = 0$ and $f \circ f \circ f \circ f (x) = 0$ have a common real root, then $f (- 3)$ is equal to ___________.
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Answer: 25
Q282022 · 25 Jul 2022 · Shift 2· Numerical answer
Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve $4 x^{3} - 3 x y^{2} + 6 x^{2} - 5 x y - 8 y^{2} + 9 x + 14 = 0$ at the point $(- 2, 3)$ be A. Then 8A is equal to ___________.
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Answer: 170
Q22022 · 30 Jun 2022 · Shift 1
The real part of the complex number $\frac{( 1 + 2 i ) ^{8} \cdot ( 1 - 2 i ) ^{2}}{( 3 + 2 i ) \cdot ( 4 - 6 i )}$ is equal to :
1. A. $\frac{500}{13}$
2. B. $\frac{110}{13}$
3. C. $\frac{55}{6}$
4. D. $\frac{550}{13}$
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Answer: D
Q32022 · 30 Jun 2022 · Shift 1
Let S be the set of all integral values of $α$ for which the sum of squares of two real roots of the quadratic equation $3 x^{2} + (α - 6) x + (α + 3) = 0$ is minimum. Then S :
1. A.is an empty set
2. B.is a singleton
3. C.contains exactly two elements
4. D.contains more than two elements
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Answer: A
Q22022 · 28 Jun 2022 · Shift 2
Let $f (x)$ be a quadratic polynomial such that $f (- 2) + f (3) = 0$ . If one of the roots of $f (x) = 0$ is $- 1$ , then the sum of the roots of $f (x) = 0$ is equal to :
1. A. $\frac{11}{3}$
2. B. $\frac{7}{3}$
3. C. $\frac{13}{3}$
4. D. $\frac{14}{3}$
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Answer: A
Q282022 · 28 Jun 2022 · Shift 2· Numerical answer
Sum of squares of modulus of all the complex numbers $z$ satisfying $\overset{z}{ˉ} = i z^{2} + z^{2} - z$ is equal to ________.
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Answer: 2
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