LFM Stats And Pure

jee-main 2016 Q62 Powers of i or Complex Number Integer Powers View

Let $z = 1 + a i$, be a complex number, $a > 0$, such that $z ^ { 3 }$ is a real number. Then, the sum $1 + z + z ^ { 2 } + \ldots + z ^ { 11 }$ is equal to :
(1) $1365 \sqrt { 3 } i$
(2) $- 1365 \sqrt { 3 } i$
(3) $- 1250 \sqrt { 3 } i$
(4) $1250 \sqrt { 3 } i$

jee-main 2016 Q67 Systems of Equations via Real and Imaginary Part Matching View

A value of $\theta$ for which $\frac{2+3i\sin\theta}{1-2i\sin\theta}$ is purely imaginary, is: (1) $\frac{\pi}{3}$ (2) $\frac{\pi}{6}$ (3) $\sin^{-1}\left(\frac{\sqrt{3}}{4}\right)$ (4) $\sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$

jee-main 2017 Q62 Roots of Unity and Cyclotomic Expressions View

Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$. If
$$\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^2 - 1 & \omega^2 \\ 1 & \omega^2 & \omega^7 \end{vmatrix} = 3k$$
Then $k$ can be equal to:
(1) $-z$
(2) $\frac{1}{z}$
(3) $-1$
(4) $1$

jee-main 2018 Q62 Roots of Unity and Cyclotomic Expressions View

If $\alpha , \beta \in C$ are the distinct roots of the equation $x ^ { 2 } - x + 1 = 0$, then $\alpha ^ { 101 } + \beta ^ { 107 }$ is equal to
(1) 2
(2) - 1
(3) 0
(4) 1

jee-main 2018 Q62 Identifying Real/Imaginary Parts or Components View

The set of all $\alpha \in R$, for which $w = \frac { 1 + ( 1 - 8 \alpha ) z } { 1 - z }$ is a purely imaginary number, for all $z \in C$ satisfying $| z | = 1$ and $\operatorname { Re } z \neq 1$, is
(1) $\{ 0 \}$
(2) an empty set
(3) $\left\{ 0 , \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right\}$
(4) equal to $R$

jee-main 2018 Q63 Trigonometric/Polar Form and De Moivre's Theorem View

The least positive integer $n$ for which $\left( \frac { 1 + i \sqrt { 3 } } { 1 - i \sqrt { 3 } } \right) ^ { n } = 1$ is
(1) 2
(2) 5
(3) 6
(4) 3

jee-main 2018 Q63 Identifying Real/Imaginary Parts or Components View

The set of all $\alpha \in R$, for which $w = \frac { 1 + ( 1 - 8 \alpha ) z } { 1 - z }$ is a purely imaginary number, for all $z \in C$ satisfying $| z | = 1$ and $\operatorname { Re } ( z ) \neq 1$, is :
(1) $\{ 0 \}$
(2) $\left\{ 0 , \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right\}$
(3) equal to $R$
(4) an empty set

jee-main 2019 Q62 Trigonometric/Polar Form and De Moivre's Theorem View

Let $z = \left( \frac { \sqrt { 3 } } { 2 } + \frac { i } { 2 } \right) ^ { 5 } + \left( \frac { \sqrt { 3 } } { 2 } - \frac { i } { 2 } \right) ^ { 5 }$. If $R ( z )$ and $I ( z )$ respectively denote the real and imaginary parts of $z$, then
(1) $I ( z ) = 0$
(2) $R ( z ) < 0$ and $I ( z ) > 0$
(3) $R ( z ) > 0$ and $I ( z ) > 0$
(4) $R ( z ) = - 3$

jee-main 2019 Q62 Systems of Equations via Real and Imaginary Part Matching View

Let $z \in C$ with $\operatorname { Im } ( z ) = 10$ and it satisfies $\frac { 2 z - n } { 2 z + n } = 2 i - 1$ for some natural number $n$. Then
(1) $n = 20$ and $\operatorname { Re } ( z ) = 10$
(2) $n = 40$ and $\operatorname { Re } ( z ) = 10$
(3) $n = 20$ and $\operatorname { Re } ( z ) = - 10$
(4) $n = 40$ and $\operatorname { Re } ( z ) = - 10$

jee-main 2019 Q62 Modulus Computation View

Let $z _ { 1 }$ and $z _ { 2 }$ be any two non-zero complex numbers such that $3 \left| z _ { 1 } \right| = 4 \left| z _ { 2 } \right|$. If $z = \frac { 3 z _ { 1 } } { 2 z _ { 2 } } + \frac { 2 z _ { 2 } } { 3 z _ { 1 } }$ then maximum value of $| z |$ is
(1) $\frac { 7 } { 2 }$
(2) $\frac { 9 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 1 } { 2 } \sqrt { \frac { 17 } { 2 } }$

jee-main 2019 Q62 Modulus Computation View

Let $z _ { 1 }$ and $z _ { 2 }$ be two complex numbers satisfying $\left| z _ { 1 } \right| = 9$ and $\left| z _ { 2 } - 3 - 4 i \right| = 4$. Then the minimum value of $\left| z _ { 1 } - z _ { 2 } \right|$ is :
(1) 2
(2) $\sqrt { 2 }$
(3) 0
(4) 1

jee-main 2019 Q62 Identifying Real/Imaginary Parts or Components View

Let $z \in C$ be such that $| z | < 1$. If $\omega = \frac { 5 + 3 z } { 5 ( 1 - z ) }$, then:
(1) $5 R e ( \omega ) > 1$
(2) $5 \operatorname { Im } ( \omega ) < 1$
(3) $5 R e ( \omega ) > 4$
(4) $4 \operatorname { Im } ( \omega ) > 5$

jee-main 2019 Q63 Roots of Unity and Cyclotomic Expressions View

Let $z_0$ be a root of quadratic equation, $x^2 + x + 1 = 0$. If $z = 3 + 6iz_0^{81} - 3iz_0^{93}$, then $\arg(z)$ is equal to:
(1) 0
(2) $\frac{\pi}{4}$
(3) $\frac{\pi}{6}$
(4) $\frac{\pi}{3}$

jee-main 2019 Q63 Powers of i or Complex Number Integer Powers View

If $z = \frac { \sqrt { 3 } } { 2 } + \frac { i } { 2 }$ $( i = \sqrt { - 1 } )$, then $1 + i z + z ^ { 5 } + i z ^ { 8 }$ is equal to:
(1) - 1
(2) 1
(3) 0
(4) $- 1 + 2 i ^ { 9 }$

jee-main 2020 Q52 Roots of Unity and Cyclotomic Expressions View

Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$. If $a = (1 + \alpha)\sum_{k=0}^{100}\alpha^{2k}$ and $b = \sum_{k=0}^{100}\alpha^{3k}$, then $a$ and $b$ are the roots of the quadratic equation.
(1) $x^{2} + 101x + 100 = 0$
(2) $x^{2} - 102x + 101 = 0$
(3) $x^{2} - 101x + 100 = 0$
(4) $x^{2} + 102x + 101 = 0$

jee-main 2020 Q52 Trigonometric/Polar Form and De Moivre's Theorem View

The value of $\left( \frac{1 + \sin\frac{2\pi}{9} + i\cos\frac{2\pi}{9}}{1 + \sin\frac{2\pi}{9} - i\cos\frac{2\pi}{9}} \right)^{3}$ is
(1) $\frac{1}{2}(1 - i\sqrt{3})$
(2) $\frac{1}{2}(\sqrt{3} - i)$
(3) $-\frac{1}{2}(\sqrt{3} - i)$
(4) $-\frac{1}{2}(1 - i\sqrt{3})$

jee-main 2020 Q52 Identifying Real/Imaginary Parts or Components View

The imaginary part of $( 3 + 2 \sqrt { - 54 } ) ^ { \frac { 1 } { 2 } } - ( 3 - 2 \sqrt { - 54 } ) ^ { \frac { 1 } { 2 } }$, can be
(1) $- \sqrt { 6 }$
(2) $- 2 \sqrt { 6 }$
(3) 6
(4) $\sqrt { 6 }$

jee-main 2020 Q52 Roots of Unity and Cyclotomic Expressions View

If $a$ and $b$ are real numbers such that $( 2 + \alpha ) ^ { 4 } = a + b \alpha$, where $\alpha = \frac { - 1 + i \sqrt { 3 } } { 2 }$, then $a + b$ is equal to:
(1) 9
(2) 24
(3) 33
(4) 57

jee-main 2020 Q52 Trigonometric/Polar Form and De Moivre's Theorem View

The value of $\left(\frac{-1+i\sqrt{3}}{1-i}\right)^{30}$ is:
(1) $6^5$
(2) $2^{15}\mathrm{i}$
(3) $-2^{15}$
(4) $-2^{15}\mathrm{i}$

jee-main 2020 Q52 Systems of Equations via Real and Imaginary Part Matching View

Let $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ be a non-zero complex number such that $\mathrm{z}^{2}=\mathrm{i}|\mathrm{z}|^{2}$, where $\mathrm{i}=\sqrt{-1}$, then z lies on the:
(1) line, $y=-x$
(2) imaginary axis
(3) line, $y=x$
(4) real axis

jee-main 2021 Q61 Powers of i or Complex Number Integer Powers View

If $\alpha$ and $\beta$ are the distinct roots of the equation $x ^ { 2 } + ( 3 ) ^ { \frac { 1 } { 4 } } x + 3 ^ { \frac { 1 } { 2 } } = 0$, then the value of $\alpha ^ { 96 } \left( \alpha ^ { 12 } - 1 \right) + \beta ^ { 96 } \left( \beta ^ { 12 } - 1 \right)$ is equal to:
(1) $56 \times 3 ^ { 25 }$
(2) $56 \times 3 ^ { 24 }$
(3) $52 \times 3 ^ { 24 }$
(4) $28 \times 3 ^ { 25 }$

jee-main 2021 Q61 Solving Equations for Unknown Complex Numbers View

Let $n$ denote the number of solutions of the equation $z ^ { 2 } + 3 \bar { z } = 0$, where $z$ is a complex number. Then the value of $\sum _ { k = 0 } ^ { \infty } \frac { 1 } { n ^ { k } }$ is equal to
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 3 } { 2 }$
(4) 2

jee-main 2021 Q62 Geometric Interpretation and Triangle/Shape Properties View

The area of the triangle with vertices $P ( z ) , Q ( i z )$ and $R ( z + i z )$ is
(1) 1
(2) $\frac { 1 } { 2 } | z | ^ { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 2 } | z + i z | ^ { 2 }$

jee-main 2021 Q62 Systems of Equations via Real and Imaginary Part Matching View

If $\alpha , \beta \in R$ are such that $1 - 2 i$ (here $i ^ { 2 } = - 1$ ) is a root of $z ^ { 2 } + \alpha z + \beta = 0$, then ( $\alpha - \beta$ ) is equal to:
(1) - 7
(2) 7
(3) - 3
(4) 3

jee-main 2021 Q81 Solving Equations for Unknown Complex Numbers View

Let $z$ and $w$ be two complex numbers such that $w = z \bar { z } - 2 z + 2 , \left| \frac { z + i } { z - 3 i } \right| = 1$ and $\operatorname { Re } ( w )$ has minimum value. Then, the minimum value of $n \in N$ for which $w ^ { n }$ is real, is equal to $\_\_\_\_$.

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LFM Stats And Pure

Completing the square and sketching 61

Inequalities 181

Discriminant and conditions for roots 65

Solving quadratics and applications 131

Curve Sketching 228

Factor & Remainder Theorem 61

Polynomial Division & Manipulation 35

Binomial Theorem (positive integer n) 197

Function Transformations 28

Composite & Inverse Functions 211

Partial Fractions 12

Generalised Binomial Theorem 10

Complex Numbers Arithmetic 207

Complex Numbers Argand & Loci 159

Permutations & Arrangements 255

Combinations & Selection 237

Measures of Location and Spread 181

Probability Definitions 212

Tree Diagrams 18

Principle of Inclusion/Exclusion 33

Independent Events 35

Conditional Probability 167

Discrete Probability Distributions 176

Uniform Distribution 3

Binomial Distribution 101

Geometric Distribution 8

Hypergeometric Distribution 9

Negative Binomial Distribution 1

Modelling and Hypothesis Testing 22

Hypothesis test of binomial distributions 2

Data representation 21

Continuous Probability Distributions and Random Variables 322

Continuous Uniform Random Variables 3

Geometric Probability 16

Normal Distribution 87

Approximating Binomial to Normal Distribution 4

Sign Change & Interval Methods 26

Fixed Point Iteration 9

Newton-Raphson method 6