Complex Numbers & Quadratic Equations · JEE Main previous year questions

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2023 35 questions

Q82023 · 13 Apr 2023 · Shift 2
Let $α, β$ be the roots of the equation $x^{2} - 2 x + 2 = 0$ . Then $α^{14} + β^{14}$ is equal to
1. A. $- 64$
2. B. $- 642$
3. C. $- 1282$
4. D. $- 128$
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Answer: D
Q102023 · 13 Apr 2023 · Shift 2
If the system of equations $2 x + y - z = 5$ $2 x - 5 y + λ z = μ$ $x + 2 y - 5 z = 7$ has infinitely many solutions, then $(λ + μ)^{2} + (λ - μ)^{2}$ is equal to
1. A. $916$
2. B. $912$
3. C. $920$
4. D. $904$
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Answer: A
Q82023 · 12 Apr 2023 · Shift 1
Let $α, β$ be the roots of the quadratic equation $x^{2} + 6 x + 3 = 0$ . Then $\frac{α ^{23} + β ^{23} + α ^{14} + β ^{14}}{α ^{15} + β ^{15} + α ^{10} + β ^{10}}$ is equal to :
1. A. $72$
2. B. $9$
3. C. $729$
4. D. $81$
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Answer: D
Q42023 · 11 Apr 2023 · Shift 1
Let $w_{1}$ be the point obtained by the rotation of $z_{1} = 5 + 4 i$ about the origin through a right angle in the anticlockwise direction, and $w_{2}$ be the point obtained by the rotation of $z_{2} = 3 + 5 i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_{1} - w_{2}$ is equal to :
1. A. $- π + tan^{- 1} \frac{8}{9}$
2. B. $- π + tan^{- 1} \frac{33}{5}$
3. C. $π - tan^{- 1} \frac{8}{9}$
4. D. $π - tan^{- 1} \frac{33}{5}$
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Answer: C
Q142023 · 11 Apr 2023 · Shift 1
The number of triplets $(x, y, z)$ , where $x, y, z$ are distinct non negative integers satisfying $x + y + z = 15$ , is :
1. A.136
2. B.80
3. C.92
4. D.114
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Answer: D
Q272023 · 11 Apr 2023 · Shift 1· Numerical answer
If $a$ and $b$ are the roots of the equation $x^{2} - 7 x - 1 = 0$ , then the value of $\frac{a ^{21} + b ^{21} + a ^{17} + b ^{17}}{a ^{19} + b ^{19}}$ is equal to _________.
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Answer: 51
Q82023 · 11 Apr 2023 · Shift 2
$x + 1 x x x x + λ x x x x + λ^{2} = \frac{9}{8} (103 x + 81)$ , then $λ, \frac{λ}{3}$ are the roots of the equation :
1. A. $4 x^{2} + 24 x - 27 = 0$
2. B. $4 x^{2} - 24 x + 27 = 0$
3. C. $4 x^{2} - 24 x - 27 = 0$
4. D. $4 x^{2} + 24 x + 27 = 0$
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Answer: B
Q112023 · 10 Apr 2023 · Shift 1
For the system of linear equations $2 x - y + 3 z = 5$ $3 x + 2 y - z = 7$ $4 x + 5 y + α z = β,$ which of the following is NOT correct?
1. A.The system has infinitely many solutions for $α = - 6$ and $β = 9$
2. B.The system has a unique solution for $α \neq = - 5$ and $β = 8$
3. C.The system is inconsistent for $α = - 5$ and $β = 8$
4. D.The system has infinitely many solutions for $α = - 5$ and $β = 9$
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Answer: A
Q152023 · 10 Apr 2023 · Shift 1
Let the complex number $z = x + i y$ be such that $\frac{2 z - 3 i}{2 z + i}$ is purely imaginary. If $x + y^{2} = 0$ , then $y^{4} + y^{2} - y$ is equal to :
1. A. $\frac{4}{3}$
2. B. $\frac{3}{2}$
3. C. $\frac{3}{4}$
4. D. $\frac{2}{3}$
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Answer: C
Q112023 · 10 Apr 2023 · Shift 2
Let A be the point $(1, 2)$ and B be any point on the curve $x^{2} + y^{2} = 16$ . If the centre of the locus of the point P, which divides the line segment AB in the ratio $3 : 2$ is the point $C (α, β)$ , then the length of the line segment AC is :
1. A. $\frac{3 5}{5}$
2. B. $\frac{6 5}{5}$
3. C. $\frac{2 5}{5}$
4. D. $\frac{4 5}{5}$
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Answer: A
Q292023 · 10 Apr 2023 · Shift 2· Numerical answer
Let the quadratic curve passing through the point $(- 1, 0)$ and touching the line $y = x$ at $(1, 1)$ be $y = f (x)$ . Then the x-intercept of the normal to the curve at the point $(α, α + 1)$ in the first quadrant is _________.
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Answer: 11
Q122023 · 8 Apr 2023 · Shift 1
Let $α, β, γ$ be the three roots of the equation $x^{3} + b x + c = 0$ . If $β γ = 1 = - α$ , then $b^{3} + 2 c^{3} - 3 α^{3} - 6 β^{3} - 8 γ^{3}$ is equal to :
1. A. $21$
2. B. $19$
3. C. $\frac{169}{8}$
4. D. $\frac{155}{8}$
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Answer: B
Q152023 · 8 Apr 2023 · Shift 1
If for $z = α + i β, ∣ z + 2∣ = z + 4 (1 + i)$ , then $α + β$ and $α β$ are the roots of the equation :
1. A. $x^{2} + 2 x - 3 = 0$
2. B. $x^{2} + 3 x - 4 = 0$
3. C. $x^{2} + x - 12 = 0$
4. D. $x^{2} + 7 x + 12 = 0$
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Answer: D
Q272023 · 8 Apr 2023 · Shift 1· Numerical answer
Let $[t]$ denote the greatest integer $\leq t$ . If the constant term in the expansion of $(3 x^{2} - \frac{1}{2 x ^{5}})^{7}$ is $α$ , then $[α]$ is equal to ___________.
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Answer: 1275
Q72023 · 8 Apr 2023 · Shift 2
The absolute difference of the coefficients of $x^{10}$ and $x^{7}$ in the expansion of $(2 x^{2} + \frac{1}{2 x})^{11}$ is equal to :
1. A. $1 1^{3} - 11$
2. B. $1 3^{3} - 13$
3. C. $1 2^{3} - 12$
4. D. $1 0^{3} - 10$
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Answer: C
Q232023 · 8 Apr 2023 · Shift 2· Numerical answer
Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^{2} - 12 x + [x] + 31 = 0$ and $x^{2} - 5∣ x + 2∣ - 4 = 0$ respectively, where $[x]$ denotes the greatest integer $\leq x$ . Then $m^{2} + mn + n^{2}$ is equal to ___________.
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Answer: 9
Q112023 · 6 Apr 2023 · Shift 1
The sum of all the roots of the equation $∣ x^{2} - 8 x + 15∣ - 2 x + 7 = 0$ is :
1. A. $11 + 3$
2. B. $9 + 3$
3. C. $9 - 3$
4. D. $11 - 3$
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Answer: B
Q122023 · 6 Apr 2023 · Shift 2
For the system of equations $x + y + z = 6$ $x + 2 y + α z = 10$ $x + 3 y + 5 z = β,$ which one of the following is NOT true?
1. A.System has a unique solution for $α = 3, β \neq = 14$ .
2. B.System has infinitely many solutions for $α = 3, β = 14$ .
3. C.System has no solution for $α = 3, β = 24$ .
4. D.System has a unique solution for $α = - 3, β = 14$ .
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Answer: A
Q52023 · 1 Feb 2023 · Shift 1
Let $S$ denote the set of all real values of $λ$ such that the system of equations $λ x + y + z = 1$ $x + λ y + z = 1$ $x + y + λ z = 1$ is inconsistent, then $\sum_{λ \in S} (∣ λ ∣^{2} + ∣ λ ∣)$ is equal to
1. A. $12$
2. B. $2$
3. C. $4$
4. D. $6$
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Answer: D
Q92023 · 1 Feb 2023 · Shift 1
Let $f (x) = 2 x + tan^{- 1} x$ and $g (x) = lo g_{e} (1 + x^{2} + x), x \in [0, 3]$ . Then
1. A.there exists $\overset{x}{ˉ} \in [0, 3]$ such that $f^{'} (\overset{x}{ˉ}) < g^{'} (\overset{x}{ˉ})$
2. B.there exist $0 < x_{1} < x_{2} < 3$ such that $f (x) < g (x), \forall x \in (x_{1}, x_{2})$
3. C. $min f^{'} (x) = 1 + max g^{'} (x)$
4. D. $max f (x) > max g (x)$
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Answer: D
Q122023 · 1 Feb 2023 · Shift 1
If the orthocentre of the triangle, whose vertices are $(1, 2), (2, 3)$ and $(3, 1)$ is $(α, β)$ , then the quadratic equation whose roots are $α + 4 β$ and $4 α + β$ , is :
1. A. $x^{2} - 20 x + 99 = 0$
2. B. $x^{2} - 22 x + 120 = 0$
3. C. $x^{2} - 19 x + 90 = 0$
4. D. $x^{2} - 18 x + 80 = 0$
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Answer: A
Q162023 · 1 Feb 2023 · Shift 1
The combined equation of the two lines $a x + b y + c = 0$ and $a^{'} x + b^{'} y + c^{'} = 0$ can be written as $(a x + b y + c) (a^{'} x + b^{'} y + c^{'}) = 0.$ The equation of the angle bisectors of the lines represented by the equation $2 x^{2} + x y - 3 y^{2} = 0$ is :
1. A. $3 x^{2} + x y - 2 y^{2} = 0$
2. B. $x^{2} - y^{2} - 10 x y = 0$
3. C. $x^{2} - y^{2} + 10 x y = 0$
4. D. $3 x^{2} + 5 x y + 2 y^{2} = 0$
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Answer: B
Q12023 · 31 Jan 2023 · Shift 1
The number of real roots of the equation $x^{2} - 4 x + 3 + x^{2} - 9 = 4 x^{2} - 14 x + 6$ , is :
1. A.0
2. B.1
3. C.3
4. D.2
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Answer: B
Q82023 · 31 Jan 2023 · Shift 2
The complex number $z = \frac{i - 1}{c o s \frac{π}{3} + i s i n \frac{π}{3}}$ is equal to :
1. A. $cos \frac{π}{12} - i sin \frac{π}{12}$
2. B. $2 (cos \frac{π}{12} + i sin \frac{π}{12})$
3. C. $2 i (cos \frac{5 π}{12} - i sin \frac{5 π}{12})$
4. D. $2 (cos \frac{5 π}{12} + i sin \frac{5 π}{12})$
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Answer: D
Q32023 · 30 Jan 2023 · Shift 1
Let the system of linear equations $x + y + k z = 2$ $2 x + 3 y - z = 1$ $3 x + 4 y + 2 z = k$ have infinitely many solutions. Then the system $(k + 1) x + (2 k - 1) y = 7$ $(2 k + 1) x + (k + 5) y = 10$ has :
1. A.unique solution satisfying $x - y = 1$
2. B.infinitely many solutions
3. C.no solution
4. D.unique solution satisfying $x + y = 1$
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Answer: D