LFM Stats And Pure

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 Q61 Roots of Unity and Cyclotomic Properties View

Let $\omega$ be a complex number such that $2 \omega + 1 = z$ where $z = \sqrt { - 3 }$. If $$\left| \begin{array} { c c c } { 1 } & { 1 } & { 1 } \\ { 1 } & { - \omega ^ { 2 } - 1 } & { \omega ^ { 2 } } \\ { 1 } & { \omega ^ { 2 } } & { \omega ^ { 7 } } \end{array} \right| = 3 k$$ then $k$ is equal to:
(1) $z$
(2) $- z$
(3) $- 1$
(4) 1

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 2017 Q89 Roots of Unity and Cyclotomic Properties View

The value of $\sum _ { k = 1 } ^ { 10 } \left( \sin \frac { 2 k \pi } { 11 } + i \cos \frac { 2 k \pi } { 11 } \right)$ is:
(1) 1
(2) $- 1$
(3) $- i$
(4) $i$

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 Algebraic Conditions for Geometric Properties (Real, Imaginary, Collinear) View

If $\frac { z - \alpha } { z + \alpha } ( \alpha \in R )$ is a purely imaginary number and $| z | = 2$, then a value of $\alpha$ is :
(1) 1
(2) $\frac { 1 } { 2 }$
(3) $\sqrt { 2 }$
(4) 2

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 Vieta's formulas: compute symmetric functions of roots View

If $\alpha$ and $\beta$ be the roots of the equation $x^2 - 2x + 2 = 0$, then the least value of $n$ for which $\frac{\alpha^n}{\beta} = 1$ is
(1) 5
(2) 4
(3) 2
(4) 3

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 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 2019 Q76 Inequalities and Estimates for Complex Expressions View

The greatest value of $c \in R$ for which the system of linear equations $x - cy - cz = 0$, $cx - y + cz = 0$, $cx + cy - z = 0$ has a non-trivial solution, is
(1) $-1$
(2) $2$
(3) $\frac{1}{2}$
(4) $0$

jee-main 2020 Q51 Solving Complex Equations with Geometric Interpretation View

If the equation $x ^ { 2 } + b x + 45 = 0 , b \in R$ has conjugate complex roots and they satisfy $| z + 1 | = 2 \sqrt { 10 }$, then
(1) $b ^ { 2 } - b = 30$
(2) $b ^ { 2 } + b = 72$
(3) $b ^ { 2 } - b = 42$
(4) $b ^ { 2 } + b = 12$

jee-main 2020 Q52 Algebraic Conditions for Geometric Properties (Real, Imaginary, Collinear) View

If $\frac { 3 + i \sin \theta } { 4 - i \cos \theta } , \theta \in [ 0,2 \pi ]$, is a real number, then an argument of $\sin \theta + i \cos \theta$ is
(1) $\pi - \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$
(2) $\pi - \tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)$
(3) $- \tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)$
(4) $\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$

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 2020 Q53 Distance and Region Optimization on Loci View

If $z$ is a complex number satisfying $| \operatorname { Re } ( z ) | + | \operatorname { Im } ( z ) | = 4$, then $| z |$ cannot be
(1) $\sqrt { \frac { 17 } { 2 } }$
(2) $\sqrt { 10 }$
(3) $\sqrt { 7 }$
(4) $\sqrt { 8 }$

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

Solving quadratics and applications 106

Curve Sketching 295

Factor & Remainder Theorem 90

Polynomial Division & Manipulation 44

Binomial Theorem (positive integer n) 244

Function Transformations 65

Composite & Inverse Functions 249

Partial Fractions 22

Generalised Binomial Theorem 16

Generalised Binomial Theorem and Partial Fractions 3

Complex Numbers Arithmetic 283

Complex Numbers Argand & Loci 135

Permutations & Arrangements 276

Combinations & Selection 268

Measures of Location and Spread 233

Probability Definitions 413

Tree Diagrams 5

Principle of Inclusion/Exclusion 23

Independent Events 51

Conditional Probability 127

Discrete Probability Distributions 210

Binomial Distribution 107

Geometric Distribution 11

Hypergeometric Distribution 10

Negative Binomial Distribution 2

Modelling and Hypothesis Testing 38

Hypothesis test of binomial distributions 12

Data representation 29

Continuous Probability Distributions and Random Variables 30

Continuous Uniform Random Variables 15

Geometric Probability 28

Normal Distribution 96

Approximating Binomial to Normal Distribution 7

Sign Change & Interval Methods 74

Fixed Point Iteration 12