StudyBeacon

Home  ›  Mathematics  ›  Complex numbers Revision Complex Numbers — Complete JEE Mains 2026 Revision Capsule Compact, exam-focused capsule for complex numbers. Learn the algebra, geometry, identities, De Moivre’s rules, roots of unity, loci in Argand plane, and typical JEE question patterns — all formulas in MathJax for crisp rendering. 🔁 View Complete Revision Series Cartesian $z=a+ib$, $\Re(z)=a$, $\Im(z)=b$, $|z|=\sqrt{a^2+b^2}$ Polar / Euler $z=r(\cos\theta+i\sin\theta)=re^{i\theta}$ De Moivre & Roots $z^n = r^n e^{in\theta}$, $n$th roots given by $r^{1/n}e^{i(\theta+2k\pi)/n}$ 1. Basics & Notation Complex number: $z=a+ib$ where $a,b\in\mathbb{R}$ and $i^2=-1$. Conjugate: $\bar z = a-ib$. Modulus: $|z|=\sqrt{a^2+b^2}$. Argument: $\arg z = \theta$ where $z$ corr...

Home  ›  Mathematics  ›  P and C Revision   Permutation & Combination — Complete JEE Mains 2026 Revision (Full) The complete P&C handbook for JEE: every formula, every method, worked mini-examples, JEE traps, and a practice set. Keep this as the single-page cheat-sheet for counting problems. 🔁 View Complete Revision Series Permutation (ordered) $^nP_r = \\dfrac{n!}{(n-r)!}$ , $n!$ , $n^r$ Combination (unordered) $^nC_r = \\dfrac{n!}{r!(n-r)!}$ , $^nC_r = ^nC_{n-r}$ Stars & Bars non-neg sols: $\\binom{n+r-1}{r-1}$ ; positive sols: $\\binom{n-1}{r-1}$ 1 — Counting Principles & Factorial Addition rule: for mutually exclusive choices: add. Multiplication rule: for sequential independent choices: multiply. Factorial: $n! = n(n-1)(n-2)\cdots1$ and $0!=1$. ...

Q. Two vertical poles \(AB = 15\text{ m}\) and \(CD = 10\text{ m}\) stand on level ground with feet \(A\) and \(C\) on the ground. Lines \(BC\) and \(AD\) intersect at point \(P\). What is the height of \(P\) (in meters) above the line \(AC\)? (JEE Mains 2022) 1. \( \frac{10}{3} \) 2. \( 6 \) 3. \( 5 \) 4. \( \frac{20}{3} \) Show Answer Correct Answer: Option 4 — \( \frac{20}{3} \) MathJax Solution Place points so that the ground line \(AC\) is horizontal. Heights: \(AB = 15\), \(CD = 10\). Because \(AD\) and \(BC\) are straight segments between tops and bottoms of the poles, the height varies linearly along each segment. Parameterize the two segments: On \(AD\): \(A = (0,0)\), \(D = (0,15)\). So any point \(P\) ...

Q. The angle of elevation of a cloud \(C\) from a point \(P\), 200 m above a still lake, is \(30^\circ\). The angle of depression of the image of the cloud in the lake from the same point \(P\) is \(60^\circ\). Then \(PC\) (in meters) is: (JEE Mains 2020) 1. \(100\) 2. \(400\sqrt{3}\) 3. \(200\sqrt{3}\) 4. \(400\) Show Answer Correct Answer: Option 3 — \(200\sqrt{3}\) Mathematical Solution (Using MathJax) Let the cloud be at height \(h\) above the lake surface. The point \(P\) is 200 m above the lake. Let the horizontal distance from the foot of the perpendicular to the cloud be \(x\). Angle of elevation to cloud: \[ \tan 30^\circ = \frac{h - 200}{x} \quad\Rightarrow\quad \frac{1}{\sqrt{3}} = \frac{h-200}{x} \quad\Rightarrow\quad h - 200 = ...

Q. A ray of light coming from the point \(P(2,\;2\sqrt{3})\) is incident at an angle \(30^\circ\) on the vertical line \(x=1\) at point \(A\). The ray reflects from the line \(x=1\) and meets the x-axis at point \(B\). The line \(AB\) passes through which point? (JEE Mains 2020 ) 1. \( (4,\; -\frac{\sqrt{3}}{2}) \) 2. \( (3,\; -\sqrt{3}) \) 3. \( (4,\; -\sqrt{3}) \) 4. \( (3,\; -\frac{1}{\sqrt{3}}) \) Tip: Choose an option and then press Show Answer to reveal reasoning and see the interactive diagram. Show Answer Correct Answer: Option 2 — \( (3,\; -\sqrt{3}) \) Mathematical Derivation (MathJax) Given: \(P(2,\,2\sqrt{3})\). Mirror: vertical line \(x=1\). The ray is incident at angle \(30^\circ\) on the line \(x=1\) — i.e. the angle between the ray and the vertical line is \(30^\circ\). ...

Complete Calculus Revision for JEE Mains and Advanced Welcome to this detailed calculus revision guide! This post covers all key topics required for JEE Mains and Advanced, from Relations and Functions to Differential Equations. Each section includes formulas, examples, and illustrations to ensure conceptual clarity. 1. Relations and Functions This chapter forms the basis of calculus. Key topics include: Topic Explanation Domain and Range For a function f(x), the domain is the set of all valid inputs, and the range is the set of outputs. Inverse Functions If f(x) is bijective, then its inverse f^{-1}(x) satisfies f(f^{-1}(x)) = x. Illustration: Domain, Range, and Inverse Functions Example: For f(x) = x^2, the domain is all real numbers (-\infty, \infty), and the range is [0, \infty) because the output is alw...

 Q. If the shortest distance of the parabola y² = 4x from the centre of the circle x² + y²– 4x- 16y + 64 = 0 is d, then d² is equal to:   ( JEE mains 2024) A. 16 B.  24 C. 20 D. 36 Show Answer Ans:- (c) Equation of normal to parabola  y = mx-2m-m³  this normal passing through center of circle (2, 8)  8=2m-2m -m³  m =-2  So point P on parabola (am², -2am) = (4, 4)  And  C = (2, 8)  PC = √4+16=√20  d² = 20 Sequence and series

 Q. The number of common terms in  the progressions 4, 9, 14, 19, , up to 25th term and 3, 6, 9, 12, up to 37th term is:    ( JEE mains 2024) A. 9 B. 5 C. 7 D. 8 Show Answer Ans:- (c) 4, 9, 14, 19, ...., up to 25th term T₂5=4+(25-1)54+120=124 3, 6, 9, 12, up to 37th term T37=3+(37-1)3=3+108= 111 Common difference of Ist series d₁ = 5  Common difference of IInd series d₂ = 3  First common term = 9, and their common difference = 15 (LCM of d, and d₂)  then common terms are 9, 24, 39, 54, 69, 84, 99 Conic section

 Q. All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" IS                                                         (jee mains 2024) Show Answer (553)  Words starting with E = 360  Words starting with GE = 60  Words starting with GN = 60  Words starting with GTE = 24  Words starting with GTN = 24  Words starting with GTT = 24  GTWENTY = 1  Total = 553 Physics                  Chemistry