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Free Maths shorts

गणित · ریاضی

28 topics, each a handful of cards plus practice questions with explanations. Free to read in full — no sign-up. Counts for TRE-4, BPSC PCS, CTET, RRB NTPC, SSC CGL, SSC CHSL, UP TET.

28 shorts · 224 practice questions

• Age Problems — One Variable, One Table, and the Four Formulas Worth Memorisingआयु संबंधी प्रश्न — एक चर, एक सारणी, और याद रखने योग्य चार सूत्रAge questions are easy algebra wearing a confusing costume. The confusion is entirely in the tense words — 'ago', 'hence', 'was', 'will be'. Fix the present age as x, shift everybody by the same number of years, and the problem solves itself.6 CARDS · 8 QUESTIONS
• Algebra — VBODMAS, Surds and Indices, and the Identities That Do the Work For Youबीजगणित — VBODMAS, करणी एवं घातांक, और वे सर्वसमिकाएँ जो आपका काम कर देती हैंAlgebra sits underneath every other chapter of the quantitative paper, and marks leak there for one reason: order of operations, not difficulty. VBODMAS puts the bar first and settles 'of' before division — the single rule that decides more marks than any identity. Five index laws cover every power question, and rationalising by the conjugate turns any surd fraction into a difference of squares. Nine identities plus the x + 1/x family answer almost every algebra item in one line: 127² + 73² + 2·127·73 is not a calculation, it is 200². Worked examples throughout, with the tempting wrong route named each time.6 CARDS · 8 QUESTIONS
• Average and Alligation — The Weighted Average Shortcut and the Cross Methodऔसत एवं एलीगेशन — भारित औसत का शॉर्टकट और क्रॉस विधिSum equals average times count, first n natural numbers average (n+1)/2, equal-distance average speed is 2xy/(x+y) and never (x+y)/2, weighted average always lands between the two groups, and alligation's cross method gives cheaper : dearer = (dearer − mean) : (mean − cheaper). Six formulas that close the entire chapter, plus the repeated-replacement rule P(1 − x/P)ⁿ that SSC asks every year.6 CARDS · 8 QUESTIONS
• Coordinate Geometry — Six Formulas That Answer Almost Every Question in the Chapterनिर्देशांक ज्यामिति — छह सूत्र जो इस अध्याय के लगभग हर प्रश्न का उत्तर दे देते हैंCoordinate geometry is the one chapter where the formula sheet really is the syllabus. Six formulas — distance, midpoint, section, centroid, area and slope — and the only skill left is reading which one a question is asking for.6 CARDS · 8 QUESTIONS
• Data Interpretation — Tables, Bar Graphs, Pie Charts and Line Graphs in Three Calculationsआँकड़ा निर्वचन — सारणी, दंड आलेख, वृत्त आरेख और रैखिक आलेख केवल तीन गणनाओं मेंData interpretation is the highest marks-per-minute block in any quantitative paper, because one reading of the table answers four or five questions. Almost every question is share of total, percentage change, or ratio — and a 45-degree sector is never 45 per cent.6 CARDS · 8 QUESTIONS
• Geometry — Angle Rules, Pythagorean Triplets, the Four Centres and the Circle Facts Papers Reuseज्यामिति — कोणों के नियम, पाइथागोरस त्रिक, चार केंद्र और वृत्त के वे तथ्य जो प्रश्नपत्र दोहराते रहते हैंMensuration says how much a shape holds; geometry says why the shape is what it is — and that is the part papers actually ask. About a dozen results cover everything: complementary and supplementary pairs, the transversal trio (corresponding equal, alternate equal, co-interior supplementary), angle sum 180 degrees and the exterior-angle rule, the triplets 3-4-5 / 5-12-13 / 7-24-25 / 8-15-17, congruence by SSS SAS ASA AAS RHS against similarity by AA, the k / k-squared / k-cubed ratio rule, the centroid's 2:1 median cut, and the circle set — centre angle twice the arc angle, semicircle 90 degrees, cyclic quadrilateral opposite angles 180 degrees, tangent perpendicular to radius. Three traps repeat: co-interior offered as equal, area ratio offered as the side ratio, orthocentre and circumcentre swapped in a right triangle.6 CARDS · 8 QUESTIONS
• Mensuration of Solids: Melting, Recasting and the Volume-Conserved Rule for Cylinder, Cone and Sphereठोसों का क्षेत्रमिति: बेलन, शंकु और गोले को पिघलाना, ढालना और 'आयतन अपरिवर्तित' नियमMelt a sphere into small balls or drop one into a tank: volume before equals volume after. Formulas, worked examples and the five traps.6 CARDS · 8 QUESTIONS
• Mensuration — 2D Areas, 3D Volumes and the Surface-Area Trapsक्षेत्रमिति — 2D क्षेत्रफल, 3D आयतन और पृष्ठीय क्षेत्रफल के जालOne card of 2D formulas, one of 3D formulas, and the three rules that actually decide the marks: curved vs total surface area, volume is conserved when a solid is melted and recast, and a k-times change in length is k² in area and k³ in volume. With the exact traps SSC CGL/CHSL, Railway NTPC, BPSC TRE-4, CTET and UPSC CSAT set on each.6 CARDS · 8 QUESTIONS
• Number System — Divisibility Rules, HCF × LCM, Unit Digit and the Factor-Count Formulaसंख्या पद्धति — विभाज्यता के नियम, HCF × LCM, इकाई अंक और गुणनखंड गिनने का सूत्रEight divisibility rules, one product rule (HCF × LCM = product of the two numbers), the cycle-of-4 trick for unit digits, and (p+1)(q+1)(r+1) for counting factors. Together these solve most of what SSC CGL/CHSL, Railway NTPC and Group D, BPSC TRE-4 and CTET ask from number system — and they solve it without long division.6 CARDS · 8 QUESTIONS
• Pair of Linear Equations: Consistency Conditions, Elimination and the Digit, Fraction and Age Setupsरैखिक समीकरण युग्म: संगति की शर्तें, विलोपन विधि तथा अंक, भिन्न और आयु के प्रश्नों की रचनाTwo lines, three outcomes: learn the ratio test for one, none or infinite solutions, the cross-multiplication formula and the standard setups for digit, fraction and age word problems.6 CARDS · 8 QUESTIONS
• Percentage — The 5 Shortcuts That Cover Most Questionsप्रतिशत — 5 शॉर्टकट जो ज़्यादातर सवाल हल कर देते हैंPercentage questions appear in every exam. Learn the fraction table and you solve them in your head, not on paper.6 CARDS · 8 QUESTIONS
• Probability, Permutation and Combination — nPr, nCr and the Favourable-over-Total Ruleप्रायिकता, क्रमचय और संचय — nPr, nCr तथा अनुकूल/कुल का नियमOne question decides everything here: does order matter? Arrange with nPr, select with nCr, and probability is only ever favourable divided by total.6 CARDS · 8 QUESTIONS
• Profit, Loss and Discount — Every Formula and the Successive-Change Trapलाभ, हानि और छूट — हर सूत्र और क्रमिक परिवर्तन का जालCP, SP and MP untangled; the profit and loss formulas with worked numbers; the a + b + ab/100 shortcut for successive discounts and mark-ups; the false-weight gain formula; and why selling two articles at the same price with equal gain and loss always ends in a loss. Written for the arithmetic sections of BPSC TRE-4, CTET, SSC CGL, Railway NTPC and state exams.6 CARDS · 8 QUESTIONS
• Progressions — Six Formulas That Answer the Whole AP and GP Chapterश्रेढ़ी (Progressions) — छह सूत्र जो समांतर और गुणोत्तर श्रेढ़ी का पूरा अध्याय हल कर देते हैंAn AP adds the same number every time; a GP multiplies by the same number every time. That one difference decides every formula in the chapter — six of them, and the whole topic is done.6 CARDS · 8 QUESTIONS
• Quadratic Equations: Roots, Discriminant and the Sum-Product Shortcutsद्विघात समीकरण: मूल, विविक्तकर और योग-गुणनफल के शॉर्टकटThree tools solve every quadratic: factorisation, the formula, and the sum and product of roots. The discriminant D = b² − 4ac tells you the nature of the roots before you solve, and −b/a and c/a answer most expression questions in two lines.6 CARDS · 8 QUESTIONS
• Races and Circular Tracks: Head Starts, Meeting Points and the Lap Formulasदौड़ और वृत्ताकार ट्रैक: हेड स्टार्ट, मिलन-बिंदु और चक्कर के सूत्रRaces and circular tracks reduce to two ideas: distances run in equal time give the speed ratio, and a track length divided by relative speed gives the meeting time. Learn the four formulas that SSC, Railway and TRE-4 papers keep recycling.6 CARDS · 8 QUESTIONS
• Ratio and Proportion — One Idea Behind Sharing, Partnership and Agesअनुपात और समानुपात — बँटवारा, साझेदारी और आयु के पीछे एक ही विचारCombining ratios, dividing money, mean and fourth proportional, capital × time in partnership, and the age equation — five moves that clear the ratio questions in SSC, Railway, CTET and teacher-recruitment papers.6 CARDS · 8 QUESTIONS
• Remainders, the Minus-One Trick, Casting Out Nines and Binary-Decimal Conversionशेषफल, माइनस-वन ट्रिक, नौ का शेष और बाइनरी-दशमलव रूपांतरणFind the remainder of huge products and powers in seconds, check any number by its digit sum, and convert between binary and decimal. Formulas, worked examples and the traps.6 CARDS · 8 QUESTIONS
• Replacement and Repeated Dilution: The (1 − x/n)^k Formulaप्रतिस्थापन और बार-बार तनुकरण: (1 − x/n)^k सूत्रRemove x litres from n litres of pure milk, top up with water, and repeat k times: pure milk left = n(1 − x/n)^k. One formula settles every 'replace and repeat' question in SSC, Railway and TRE-4 mixture problems.6 CARDS · 8 QUESTIONS
• Simple and Compound Interest: SI–CI Difference, Half-Yearly Compounding and Doubling Shortcutsसाधारण और चक्रवृद्धि ब्याज: SI–CI अंतर, अर्धवार्षिक संयोजन और दोगुना होने के शॉर्टकटSI = PRT/100 and CI amount = P(1 + R/100)^n are all you need. Add the 2-year difference P(R/100)², half-yearly (R/2, 2n) and the doubling rule R×T = 100, and SSC, Railway and TRE-4 interest questions finish in seconds.6 CARDS · 8 QUESTIONS
• Simple vs Compound Interest — Every Formula and the Difference Trickसाधारण बनाम चक्रवृद्धि ब्याज — हर सूत्र और अंतर वाली तरकीबSI = PRT/100, Amount = P(1+R/100)^n, the two-year and three-year CI-minus-SI shortcuts, half-yearly compounding and the R x T = 100 doubling rule — the four formulas that cover every interest question in SSC, Railway, CTET and teacher-recruitment papers.6 CARDS · 8 QUESTIONS
• Simplification and Approximation — The Calculation Shortcuts That Buy You Ten Extra Minutesसरलीकरण एवं सन्निकटन — वे गणना-शॉर्टकट जो आपको दस मिनट अतिरिक्त देते हैंNobody fails an exam because they did not know how to multiply. They fail because they multiplied slowly. Squares to 30, the ending-in-5 rule, the base method, the times-eleven trick and the fraction-to-percentage table together turn a two-minute calculation into a ten-second one.6 CARDS · 8 QUESTIONS
• Speed, Time and Distance — One Formula, Six Question Typesचाल, समय और दूरी — एक सूत्र, छह प्रकार के प्रश्नD = S × T, the 5/18 conversion, the inverse ratio trick, relative speed for trains and the boat formula — the five ideas that clear almost every speed question in SSC, Railway and teacher-recruitment papers.6 CARDS · 8 QUESTIONS
• Squares, Cubes and Square Roots — The Shortcut Methods That Save Calculation Timeवर्ग, घन और वर्गमूल — वे शॉर्टकट तरीके जो गणना का समय बचाते हैं97² or the cube root of 3375 should take five seconds, not fifty. A handful of algebraic shortcuts — the base method, the ends-in-5 trick, and the last-digit rule for roots — turn slow multiplication into instant recall, and exams are timed precisely to reward students who know them.6 CARDS · 8 QUESTIONS
• Statistics — Mean, Median, Mode and Range, with the Correction Formulasसांख्यिकी — माध्य, माध्यिका, बहुलक एवं परिसर, संशोधन सूत्रों सहितThree averages, three different questions. Mean of first n naturals is (n+1)/2, a wrongly copied value costs (correct − wrong)/n, and the combined average is never the average of the two averages.6 CARDS · 8 QUESTIONS
• Time & Work — The LCM Method, Efficiency and Pipes & Cisternsसमय और कार्य — LCM विधि, दक्षता और नल एवं टंकीStop solving Time & Work with fractions. Set total work = LCM of the given days and every rate becomes a whole number: the combined-time formula xy/(x+y), efficiency as the inverse of time, man-days, negative work, and pipes-and-cisterns leak sums — with the exact traps SSC CGL/CHSL, Railway NTPC, BPSC TRE-4, CTET and UPSC CSAT set on each.6 CARDS · 8 QUESTIONS
• Trains, Boats & Streams: The Add-or-Subtract Rule for Relative Speedट्रेन, नाव और धारा: सापेक्ष चाल का जोड़-या-घटाव नियमEvery train, boat and stream problem is one idea in three costumes: speeds add when two things move against each other and subtract when they move the same way. Convert km/h to m/s with 5/18, decide what distance must be covered, and the answer takes two lines.6 CARDS · 8 QUESTIONS
• Trigonometry — Ratios, Identities and Heights & Distances: One Right Triangle Answers Everythingत्रिकोणमिति — अनुपात, सर्वसमिकाएँ और ऊँचाई-दूरी: एक समकोण त्रिभुज ही सब कुछ हल कर देता हैTrigonometry is not a new subject — it is bookkeeping for one right triangle. Six ratios, one standard-angle table, three identities and three Pythagorean triads, and every SSC, Railway and TRE-4 trigonometry question reduces to the same two lines of work.6 CARDS · 8 QUESTIONS