LFM Stats And Pure

jee-main 2020 Q52 Locus Identification from Modulus/Argument Equation View

The region represented by $\{ z = x + i y \in C : | z | - \operatorname { Re } ( z ) \leq 1 \}$ is also given by the inequality
(1) $y ^ { 2 } \geq 2 ( x + 1 )$
(2) $y ^ { 2 } \leq 2 \left( x + \frac { 1 } { 2 } \right)$
(3) $y ^ { 2 } \leq \left( x + \frac { 1 } { 2 } \right)$
(4) $y ^ { 2 } \geq x + 1$

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) $\sqrt { 8 }$

jee-main 2020 Q53 Circle Equation and Properties via Complex Number Manipulation View

Let $u = \frac { 2 z + i } { z - k i } , z = x + i y$ and $k > 0$. If the curve represented by Re $( u ) + \operatorname { Im } ( u ) = 1$ intersects the $y$-axis at points P and Q where $\mathrm { PQ } = 5$ then the value of k is
(1) $\frac { 3 } { 2 }$
(2) $\frac { 1 } { 2 }$
(3) 4
(4) 2

jee-main 2021 Q61 Distance and Region Optimization on Loci View

The least value of $| z |$ where $z$ is complex number which satisfies the inequality $e ^ { \left( \frac { ( | z | + 3 ) ( | z | - 1 ) } { | | z | + 1 | } \log _ { \mathrm { e } } 2 \right) } \geq \log _ { \sqrt { 2 } } | 5 \sqrt { 7 } + 9 i | , i = \sqrt { - 1 }$, is equal to :
(1) 3
(2) $\sqrt { 5 }$
(3) 2
(4) 8

jee-main 2021 Q61 Geometric Properties of Triangles/Polygons from Affixes View

Let a complex number be $w = 1 - \sqrt { 3 } i$. Let another complex number $z$ be such that $| z w | = 1$ and $\arg ( z ) - \arg ( w ) = \frac { \pi } { 2 }$. Then the area of the triangle (in sq. units) with vertices origin, $z$ and $w$ is equal to
(1) 4
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 4 }$
(4) 2

jee-main 2021 Q61 Locus Identification from Modulus/Argument Equation View

The equation $\arg \left( \frac { z - 1 } { z + 1 } \right) = \frac { \pi } { 4 }$ represents a circle with:
(1) centre at $( 0,0 )$ and radius $\sqrt { 2 }$
(2) centre at $( 0,1 )$ and radius 2
(3) centre at $( 0 , - 1 )$ and radius $\sqrt { 2 }$
(4) centre at $( 0,1 )$ and radius $\sqrt { 2 }$

jee-main 2021 Q61 Intersection of Loci and Simultaneous Geometric Conditions View

Let $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$ be three sets defined as $S _ { 1 } = \{ z \in \mathbb { C } : | z - 1 | \leq \sqrt { 2 } \}$, $S _ { 2 } = \{ z \in \mathbb { C } : \operatorname { Re } ( ( 1 - i ) z ) \geq 1 \}$ and $S _ { 3 } = \{ z \in \mathbb { C } : \operatorname { Im } ( z ) \leq 1 \}$. Then, the set $S _ { 1 } \cap S _ { 2 } \cap S _ { 3 }$
(1) is a singleton
(2) has exactly two elements
(3) has infinitely many elements
(4) has exactly three elements

jee-main 2021 Q62 Circle Equation and Properties via Complex Number Manipulation View

If the equation $a | z | ^ { 2 } + \overline { \bar { \alpha } z + \alpha \bar { z } } + d = 0$ represents a circle where $a , d$ are real constants then which of the following condition is correct?
(1) $| \alpha | ^ { 2 } - a d \neq 0$
(2) $| \alpha | ^ { 2 } - a d > 0$ and $a \in R - \{ 0 \}$
(3) $| \alpha | ^ { 2 } - a d \geq 0$ and $a \in R$
(4) $\alpha = 0 , a , d \in R ^ { + }$

jee-main 2021 Q63 Distance and Region Optimization on Loci View

If $z$ is a complex number such that $\frac { z - i } { z - 1 }$ is purely imaginary, then the minimum value of $| z - ( 3 + 3i ) |$ is :
(1) $3 \sqrt { 2 }$
(2) $2 \sqrt { 2 }$
(3) $2 \sqrt { 2 } - 1$
(4) $6 \sqrt { 2 }$

jee-main 2021 Q81 Distance and Region Optimization on Loci View

A point $z$ moves in the complex plane such that $\arg \left( \frac { z - 2 } { z + 2 } \right) = \frac { \pi } { 4 }$, then the minimum value of $| z - 9 \sqrt { 2 } - 2 i | ^ { 2 }$ is equal to

jee-main 2021 Q81 Locus Identification from Modulus/Argument Equation View

Let $z _ { 1 }$ and $z _ { 2 }$ be two complex numbers such that $\arg \left( z _ { 1 } - z _ { 2 } \right) = \frac { \pi } { 4 }$ and $z _ { 1 } , z _ { 2 }$ satisfy the equation $| z - 3 | = \operatorname { Re } ( z )$. Then the imaginary part $z _ { 1 } + z _ { 2 }$ is equal to

jee-main 2022 Q61 Circle Equation and Properties via Complex Number Manipulation View

Let a circle $C$ in complex plane pass through the points $z _ { 1 } = 3 + 4 i , z _ { 2 } = 4 + 3 i$ and $z _ { 3 } = 5 i$. If $z \neq z _ { 1 }$ is a point on $C$ such that the line through $z$ and $z _ { 1 }$ is perpendicular to the line through $z _ { 2 }$ and $z _ { 3 }$, then $\arg z$ is equal to
(1) $\tan ^ { - 1 } \frac { 24 } { 7 } - \pi$
(2) $\tan ^ { - 1 } \frac { 2 } { \sqrt { 5 } } - \pi$
(3) $\tan ^ { - 1 } 3 - \pi$
(4) $\tan ^ { - 1 } \frac { 3 } { 4 } - \pi$

jee-main 2022 Q61 Solving Complex Equations with Geometric Interpretation View

The area of the polygon, whose vertices are the non-real roots of the equation $\bar { z } = i z ^ { 2 }$ is
(1) $\frac { 3 \sqrt { 3 } } { 2 }$
(2) $\frac { 3 \sqrt { 3 } } { 4 }$
(3) $\frac { \sqrt { 3 } } { 4 }$
(4) $\frac { \sqrt { 3 } } { 2 }$

jee-main 2022 Q61 Distance and Region Optimization on Loci View

For $z \in \mathbb{C}$ if the minimum value of $(|z - 3\sqrt{2}| + |z - p\sqrt{2}i|)$ is $5\sqrt{2}$, then a value of $p$ is $\_\_\_\_$.
(1) 3
(2) $\frac{7}{2}$
(3) 4
(4) $\frac{9}{2}$

jee-main 2022 Q61 Distance and Region Optimization on Loci View

Let $S_1 = \{z_1 \in \mathbb{C} : |z_1 - 3| = \frac{1}{2}\}$ and $S_2 = \{z_2 \in \mathbb{C} : |z_2 - z_2 + 1| = |z_2 + z_2 - 1|\}$. Then, for $z_1 \in S_1$ and $z_2 \in S_2$, the least value of $|z_2 - z_1|$ is
(1) 0
(2) $\frac{1}{2}$
(3) $\frac{3}{2}$
(4) $\frac{3}{2}$

jee-main 2022 Q61 Intersection of Loci and Simultaneous Geometric Conditions View

Let $A = \left\{ z \in C : \left| \frac { z + 1 } { z - 1 } \right| < 1 \right\}$ and $B = \left\{ z \in C : \arg \left( \frac { z - 1 } { z + 1 } \right) = \frac { 2 \pi } { 3 } \right\}$. Then $A \cap B$ is
(1) a portion of a circle centred at $\left( 0 , - \frac { 1 } { \sqrt { 3 } } \right)$ that lies in the second and third quadrants only
(2) a portion of a circle centred at $\left( 0 , - \frac { 1 } { \sqrt { 3 } } \right)$ that lies in the second quadrant only
(3) an empty set
(4) a portion of a circle of radius $\frac { 2 } { \sqrt { 3 } }$ that lies in the third quadrant only

jee-main 2022 Q61 Intersection of Loci and Simultaneous Geometric Conditions View

The number of points of intersection $| z - ( 4 + 3 i ) | = 2$ and $| z | + | z - 4 | = 6 , z \in C$ is
(1) 1
(2) 2
(3) 3
(4) 4

jee-main 2022 Q61 Geometric Properties of Triangles/Polygons from Affixes View

Let $O$ be the origin and $A$ be the point $z _ { 1 } = 1 + 2i$. If $B$ is the point $z _ { 2 } , \operatorname { Re } \left( z _ { 2 } \right) < 0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true?
(1) $\arg z _ { 2 } = \pi - \tan ^ { - 1 } 3$
(2) $\arg \left( z _ { 1 } - 2 z _ { 2 } \right) = - \tan ^ { - 1 } \frac { 4 } { 3 }$
(3) $\left| z _ { 2 } \right| = \sqrt { 10 }$
(4) $\left| 2 z _ { 1 } - z _ { 2 } \right| = 5$

jee-main 2022 Q61 Distance and Region Optimization on Loci View

Let the minimum value $v _ { 0 }$ of $v = | z | ^ { 2 } + | z - 3 | ^ { 2 } + | z - 6 i | ^ { 2 } , z \in \mathbb { C }$ is attained at $z = z _ { 0 }$. Then $\left| 2 z _ { 0 } ^ { 2 } - \bar { z } _ { 0 } ^ { 3 } + 3 \right| ^ { 2 } + v _ { 0 } ^ { 2 }$ is equal to
(1) 1000
(2) 1024
(3) 1105
(4) 1196

jee-main 2022 Q62 Intersection of Loci and Simultaneous Geometric Conditions View

Let $A = \{ z \in \mathrm { C } : 1 \leqslant | z - ( 1 + i ) | \leqslant 2 \}$ and $B = \{ z \in A : | z - ( 1 - i ) | = 1 \}$. Then, $B$
(1) is an empty set
(2) contains exactly two elements
(3) contains exactly three elements
(4) is an infinite set

jee-main 2022 Q62 Intersection of Loci and Simultaneous Geometric Conditions View

If $z = x + i y$ satisfies $|z - 2| = 0$ and $|z - i| - |z + 5i| = 0$, then
(1) $x + 2 y - 4 = 0$
(2) $x ^ { 2 } + y - 4 = 0$
(3) $x + 2 y + 4 = 0$
(4) $x ^ { 2 } - y + 3 = 0$

jee-main 2022 Q62 Intersection of Loci and Simultaneous Geometric Conditions View

Let $S = \{z = x + iy : |z - 1 + i| \geq |z|, |z| < 2, |z + i| = |z - 1|\}$. Then the set of all values of $x$, for which $w = 2x + iy \in S$ for some $y \in \mathbb{R}$, is
(1) $\left(-\sqrt{2}, \frac{1}{2\sqrt{2}}\right)$
(2) $\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right)$
(3) $\left(-\sqrt{2}, \frac{1}{2}\right)$
(4) $\left(-\frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}\right)$

jee-main 2022 Q62 Intersection of Loci and Simultaneous Geometric Conditions View

For $n \in N$, let $S _ { n } = \left\{ z \in C : \left| z - 3 + 2i \right| = \frac { n } { 4 } \right\}$ and $T _ { n } = \left\{ z \in C : \left| z - 2 + 3i \right| = \frac { 1 } { n } \right\}$. Then the number of elements in the set $\left\{ n \in N : S _ { n } \cap T _ { n } = \phi \right\}$ is
(1) 0
(2) 2
(3) 3
(4) 4

jee-main 2022 Q63 Locus Identification from Modulus/Argument Equation View

Let $z_1$ and $z_2$ be two complex numbers such that $\bar{z}_1 = i\bar{z}_2$ and $\arg\frac{z_1}{\bar{z}_2} = \pi$, then the argument of $z_1$ is
(1) $\arg z_2 = \frac{\pi}{4}$
(2) $\arg z_2 = -\frac{3\pi}{4}$
(3) $\arg z_1 = \frac{\pi}{4}$
(4) $\arg z_1 = -\frac{3\pi}{4}$

jee-main 2022 Q81 Distance and Region Optimization on Loci View

Let $S = \{ z \in \mathbb { C } : | z - 3 | \leq 1$ and $z ( 4 + 3 i ) + \bar { z } ( 4 - 3 i ) \leq 24 \}$. If $\alpha + i \beta$ is the point in $S$ which is closest to $4 i$, then $25 ( \alpha + \beta )$ 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