Chapter 1 of 19

Andhra Pradesh (BIEAP)Class 10 Mathematics★ AP Board (BSEAP) SSC Mathematics · ~40 marks · Board: AP Board

Mathematics — Quadratics, Progressions, Trigonometry & Mensuration (SSC)

Complete AP SSC Mathematics: QUADRATIC EQUATIONS (formula, discriminant, nature of roots). ARITHMETIC and GEOMETRIC PROGRESSIONS. TRIGONOMETRY (ratios, identities, heights and distances). COORDINATE GEOMETRY. MENSURATION of 3D solids. STATISTICS (mean, median, mode, ogive). PROBABILITY. BSEAP Class 10 Mathematics.

⏱ 20 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Solve quadratic equations by factorisation and formula; solve word problems
2Find nth terms and sums for AP and GP; solve AP word problems
3Evaluate trigonometric expressions using standard angle values; prove identities
4Solve heights-and-distances problems using angles of elevation and depression
5Calculate surface areas and volumes of 3D solids and their combinations

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Why this chapter matters

This is the MASTER REVISION FILE for AP SSC Mathematics — consolidating the four most tested topic groups: Quadratic Equations, Progressions (AP and GP), Trigonometry (ratios, identities, heights/distances), and Mensuration (surface areas and volumes). Together these account for approximately 40–45 marks in the AP SSC Mathematics paper. Students who master these four areas and combine them with coordinate geometry, linear equations, and similar triangles have a very high chance of scoring 80+ out of 100.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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AP SSC Mathematics — Complete Examination Guide

1. Quadratic Equations

Standard Form: ax² + bx + c = 0 (a ≠ 0)

The Quadratic Formula — Your Most Important Tool

x = [−b ± √(b² − 4ac)] / 2a. 'This formula solves ANY quadratic — even when factorisation is difficult or IMPOSSIBLE.'

The Discriminant (Δ = b² − 4ac)

Δ > 0: TWO DISTINCT real roots. The graph crosses the x-axis at TWO points.
Δ = 0: ONE REAL root (repeated). The vertex of the parabola TOUCHES the x-axis.
Δ < 0: NO real roots. The graph does NOT cross the x-axis. 'The roots are COMPLEX CONJUGATES — you'll study these in Intermediate (Class 11-12).'
For roots to be RATIONAL: Δ must be a PERFECT SQUARE.

Sum and Product of Roots (if α and β are roots)

α + β = −b/a. αβ = c/a. 'These shortcuts save TIME. When you're asked to form a quadratic equation from given roots: x² − (sum)x + (product) = 0.'

Word Problems — Common Types

Type	Approach
Numbers	Let the number be x. Frame based on given conditions.
Speed-Distance-Time	Speed = Distance/Time. Use quadratic when speed is unknown.
Work	Rate × Time = Work done.
Area	Rectangle: A = lw. Triangle: A = ½bh. When dimensions are changed, frame quadratic.
Ages	Let present age be x. Express other ages in terms of x using given relationships.

2. Arithmetic Progression (AP)

Definition: A sequence where the DIFFERENCE between consecutive terms is CONSTANT.

Common Difference: d = a₂ − a₁ = a₃ − a₂ = ... (can be positive, negative, or zero).

nth Term: aₙ = a + (n−1)d

Example: Find the 20th term of the AP: 3, 7, 11, 15... a=3, d=4. a₂₀ = 3 + (20−1)(4) = 3 + 76 = 79.

Sum of First n Terms (Two Formulas — Use the Appropriate One)

Sₙ = n/2[2a + (n−1)d] — Use when you know a and d.
Sₙ = n/2(a + l) — Use when you know the first AND last terms. 'This formula is BEAUTIFUL — the sum equals the AVERAGE of first and last terms multiplied by the number of terms.'
Sₙ − Sₙ₋₁ = aₙ — The difference between consecutive sums gives the nth term.

3. Geometric Progression (GP)

Common Ratio: r = a₂/a₁ = a₃/a₂ = ...

nth Term: aₙ = arⁿ⁻¹

Sum of First n Terms: Sₙ = a(rⁿ − 1)/(r − 1) for r > 1. Sₙ = a(1 − rⁿ)/(1 − r) for r < 1.

4. Trigonometry

Ratios in a Right Triangle

sin θ = Opposite/Hypotenuse. cos θ = Adjacent/Hypotenuse. tan θ = Opposite/Adjacent = sin θ/cos θ. cosec θ = 1/sin θ. sec θ = 1/cos θ. cot θ = 1/tan θ = cos θ/sin θ.

Standard Angle Values — Must Memorise

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	ND

Fundamental Identity: sin²θ + cos²θ = 1

Derived: 1 + tan²θ = sec²θ. 1 + cot²θ = cosec²θ.

Complementary Angle Relationships

sin(90°−θ) = cos θ. cos(90°−θ) = sin θ. tan(90°−θ) = cot θ. 'These are FREQUENTLY tested — especially in simplification problems.'

Heights and Distances — Two Common Problem Types

Single Observation: One right triangle. Use tan (most common), sin, or cos depending on what you know.
Two Observations: 'A person observes an object from TWO different positions.' These problems involve TWO right triangles sharing a common height. Set up equations using tan. Solve simultaneously. 'ALWAYS draw a clear diagram. Label everything. The diagram is HALF the solution.'

5. Coordinate Geometry

Distance Formula: d = √[(x₂−x₁)² + (y₂−y₁)²]

Section Formula (Internal Division): Point dividing P and Q in ratio m:n = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2).

Area of Triangle = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. 'If area = 0 → points are COLLINEAR.'

6. Mensuration — Surface Areas and Volumes

Solid	CSA	TSA	Volume
Cube (side s)	4s²	6s²	s³
Cuboid (l,b,h)	2(l+b)h	2(lb+bh+hl)	lbh
Cylinder	2πrh	2πr(r+h)	πr²h
Cone (l=√(r²+h²))	πrl	πr(r+l)	⅓πr²h
Sphere	4πr²	4πr²	(4/3)πr³
Hemisphere	2πr²	3πr²	(2/3)πr³

Frustum of a Cone

When cut by a plane parallel to base. Volume = ⅓πh(r₁² + r₂² + r₁r₂). CSA = π(r₁+r₂)l.

7. Statistics

Mean — Three Methods

Direct: X̄ = Σfx/Σf.
Assumed Mean (Shortcut) : X̄ = A + (Σfd/Σf), where d = x − A.
Step Deviation: X̄ = A + h(Σfu/Σf), where u = (x−A)/h.

Median (Grouped Data): Median = L + [(N/2 − CF)/f] × h

L = lower limit of median class. N = Σf. CF = cumulative frequency BEFORE median class. f = frequency of median class. h = class width.

Mode: Mode = L + [(f₁−f₀)/(2f₁−f₀−f₂)] × h

f₁ = frequency of modal class. f₀ = frequency before. f₂ = frequency after.

Empirical Relationship: 3 Median = Mode + 2 Mean (approximately, for moderately skewed data)

Ogive (Cumulative Frequency Curve)

Plot upper limits vs. cumulative frequencies. 'Less than' ogive. Draw a horizontal line from N/2 on the y-axis to the curve. Drop a vertical to the x-axis. The x-value is the MEDIAN.

8. Probability

Classical Definition: P(E) = n(E)/n(S). 0 ≤ P(E) ≤ 1.

Playing Cards (52 Cards)

4 suits. 26 RED (Hearts, Diamonds). 26 BLACK (Spades, Clubs). Face cards: J, Q, K (12 total). Aces: 4 total. 'Know the deck COLD. Card problems appear in almost EVERY AP SSC exam.'

Key Formulas

P(not E) = 1 − P(E). P(E or F) = P(E) + P(F) − P(E and F).
For MUTUALLY EXCLUSIVE events: P(E and F) = 0.

Exam Strategy for AP SSC Mathematics

Unit	Approx. Marks	Key Topics
Algebra (Quadratics, Progressions)	20-22	Quadratic formula. AP nth term and sum.
Trigonometry	12-14	Standard angles. Identities. Heights/Distances.
Geometry/Mensuration	14-16	SA and Volume formulas. Frustum.
Coordinate Geometry	8-10	Distance. Section formula. Area of triangle.
Statistics/Probability	10-12	Mean/Median/Mode. Cards. Dice.

Common Mistakes to Avoid

Forgetting the ± in the quadratic formula. 'The "±" means TWO roots. ALWAYS write BOTH.'
Confusing AP and GP formulas. AP: common DIFFERENCE (d). GP: common RATIO (r). 'Know which formula belongs to which.'
Not drawing a DIAGRAM for heights and distances. 'A clear diagram prevents sign errors and confusion about which angle is elevation vs depression.'
Forgetting units in mensuration. 'If dimensions are in cm, area is in cm² and volume in cm³.'

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Quick Reference — All Major Formulas

QUADRATIC: x = [−b ± √(b²−4ac)] / 2a. Δ=b²−4ac (>0:two roots, =0:equal, <0:none). AP PROGRESSION: aₙ=a+(n−1)d. Sₙ=n/2[2a+(n−1)d]. GP PROGRESSION: aₙ=arⁿ⁻¹. Sₙ=a(rⁿ−1)/(r−1). TRIG STANDARD: sin30=1/2, cos30=√3/2, tan30=1/√3. sin45=cos45=1/√2. sin60=√3/2, cos60=1/2, tan60=√3. TRIG IDENTITY: sin²+cos²=1. HEIGHTS: tan θ = height/distance. MENSURATION: Cylinder V=πr²h. Cone V=πr²h/3. Sphere V=(4/3)πr³. Frustum V=(πh/3)(r₁²+r₂²+r₁r₂).

For chapter-specific detailed explanations, practice problems, and common mistakes, refer to the individual chapter files: quadratic-equations, arithmetic-progression, progressions, trigonometry, applications-of-trigonometry, mensuration.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Not rejecting negative roots in quadratic word problems

✓ Physical quantities cannot be negative. Always explicitly reject negative roots and state why. AP Board requires this step for full marks.

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Mathematics — Quadratics, Progressions, Trigonometry & Mensuration (SSC)?

1 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

1 questions~2 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•QUADRATICS: x = [−b ± √(b²−4ac)] / 2a. Discriminant Δ = b²−4ac: Δ>0 → two distinct real roots. Δ=0 → two equal roots. Δ<0 → no real roots. SUM of roots = −b/a. PRODUCT of roots = c/a. WORD PROBLEMS: form the quadratic from the given conditions, solve, REJECT NEGATIVE/FRACTIONAL ROOTS that don't fit the physical context.
•ARITHMETIC PROGRESSION: aₙ = a+(n−1)d. Sₙ = n/2[2a+(n−1)d]. Three-term trick: take as (a−d), a, (a+d), sum = 3a. Two conditions to find a and d → two equations, solve simultaneously.
•GEOMETRIC PROGRESSION: aₙ = arⁿ⁻¹. Sₙ = a(rⁿ−1)/(r−1) for r≠1. Three-term trick: take as a/r, a, ar, product = a³. AP has constant DIFFERENCE; GP has constant RATIO. To check: if subtracting gives constant → AP. If dividing gives constant → GP.
•COORDINATE GEOMETRY: Distance = √[(x₂−x₁)²+(y₂−y₁)²]. Section (internal, m:n): P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)). Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Area of triangle: ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|. Area=0 → collinear.
•TRIGONOMETRY STANDARD TABLE: sin 0°/30°/45°/60°/90° = 0, ½, 1/√2, √3/2, 1. cos is REVERSE of sin. tan: 0, 1/√3, 1, √3, undefined. THREE IDENTITIES: sin²+cos²=1. 1+tan²=sec². 1+cot²=cosec². Complementary: sin(90°−θ)=cosθ, tan(90°−θ)=cotθ.
•HEIGHTS AND DISTANCES: DRAW DIAGRAM FIRST. Angle of elevation = from ground looking UP. Angle of depression = from height looking DOWN. Key: angle of elevation from A = angle of depression from B (alternate angles). tan θ = height/horizontal distance.
•MENSURATION FORMULAS: Cylinder: V=πr²h, CSA=2πrh. Cone: V=(1/3)πr²h, CSA=πrl, l=√(r²+h²). Sphere: V=(4/3)πr³, SA=4πr². Hemisphere: V=(2/3)πr³, CSA=2πr², TSA=3πr². Frustum: V=(πh/3)(r₁²+r₂²+r₁r₂), l=√[h²+(r₂−r₁)²].
•COMBINATION SOLIDS: TSA = count EXPOSED surfaces ONLY (hidden junction faces not counted). Volume = add all volumes. RECASTING: Volume₁ = Volume₂ (conservation). WATER DISPLACED = Volume of immersed solid.
•STATISTICS: Mean by assumed mean: X̄ = A + Σfd/Σf. MEDIAN: Median class = first class where CF ≥ N/2. Formula: L + [(N/2−CF)/f]×h. MODE: Modal class = highest freq. Formula: L + [(f₁−f₀)/(2f₁−f₀−f₂)]×h. EMPIRICAL: 3 Median = Mode + 2 Mean.
•PROBABILITY: P(E) = n(E)/n(S). P(not E) = 1−P(E). Two dice: n(S)=36. Cards: n(S)=52 (4 suits × 13; face cards = 12; aces = 4). Mutually exclusive: P(A∪B) = P(A)+P(B). Non-mutually exclusive: P(A∪B) = P(A)+P(B)−P(A∩B).

Andhra Pradesh (BIEAP) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

AP SSC Mathematics connects to every STEM career

AP SSC Mathematics is not just a board exam — it is the foundation for every STEM path available to AP students. Quadratic equations appear in physics (projectile motion), chemistry (equilibrium calculations), and engineering economics (break-even analysis). Progressions appear in computer science (algorithmic complexity, loop iteration counts), finance (compound interest, loan amortisation), and music (equal temperament tuning). Trigonometry is essential for any engineering branch (electrical, mechanical, civil). Mensuration is used daily by civil engineers designing water tanks, industrial structures, and road surfaces. Students who master Class 10 Mathematics have the conceptual toolkit for Class 11–12 and beyond.

Data science and analytics careers starting with statistics

India's data science job market is growing rapidly — Hyderabad and Bengaluru have major data analytics centres. Every data scientist works with frequency distributions, mean/median/mode, and probability daily. AP SSC Statistics is the literal foundation: class 10 frequency tables → class 11 standard deviation → class 12 probability distributions → JEE/EAPCET → engineering → machine learning. The ogive in class 10 is the same concept as a CDF (cumulative distribution function) used in every statistics textbook. Students who understand this chapter deeply have a head start in data analytics.

AP's construction and infrastructure boom (trigonometry + mensuration)

AP is building new infrastructure — the Amaravati capital (though scaled back), Vizag-Chennai industrial corridor, Polavaram dam, and new ports. Every structure requires trigonometry (angles for structural design, heights of buildings, slopes of embankments) and mensuration (volumes of concrete needed, surface areas for waterproofing, pipe capacities). Civil engineers in AP applying for tenders must submit detailed quantity estimates — all computed from mensuration formulas. AP's ongoing development directly demands the mathematical skills from this chapter.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

FORMULA SHEET STRATEGY: In the first 5 minutes of the exam, write a quick reference at the top of your rough work page — quadratic formula, aₙ, Sₙ (AP), trig standard values, mensuration formulas. This prevents mid-exam formula blanks and saves time across the entire paper.
MARK MAXIMISATION STRATEGY: Start with the questions you're most confident about (probability, mensuration if formulas are memorised, AP formula problems). Build momentum. Leave identity proofs and complex word problems for after you've secured easy marks.
REJECT NEGATIVE ROOTS: In every quadratic word problem, both roots come from the formula. Always evaluate both, then explicitly reject the physically impossible one (e.g., negative length, negative time, negative number of people). Write: 'x = [negative value] is rejected since [physical quantity] cannot be negative.' This step earns 1 mark specifically in AP SSC.
DIAGRAMS IN TRIGONOMETRY AND HEIGHTS: For every heights/distances problem, draw a labelled diagram FIRST (right triangle, observer, object, angle). For every trig identity proof, write 'LHS =' or 'RHS =' and work that side only. Examiners award marks for diagrams and method even if the final answer has arithmetic errors.
TIME CHECKS: After 50 minutes, you should have completed Part A (objective) and at least 3 long answers. After 100 minutes, all main questions should be answered (possibly with gaps). The final 50 minutes: complete remaining questions, do the ogive graph, and review. Never spend more than 15 minutes on a single question — move on and return.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Research the Ramanujan Sum — a strange formula by Indian mathematical genius Srinivasa Ramanujan (1887–1920): 1+2+3+4+... = −1/12. This is NOT a standard infinite sum (which diverges). It is the result of analytic continuation of the Riemann zeta function ζ(s) evaluated at s=−1. This result is used in string theory and quantum field theory. Research: how can the sum of all positive integers be negative? What does 'analytic continuation' mean? Ramanujan was a self-taught mathematician from Tamil Nadu who corresponded with Hardy at Cambridge — one of the most remarkable stories in mathematical history.
Investigate Bézier curves — the smooth curves used in computer graphics, font design (the curves in the letters you're reading), and CAD software. A quadratic Bézier curve is defined by three control points P₀, P₁, P₂: B(t) = (1−t)²P₀ + 2t(1−t)P₁ + t²P₂ for t in [0,1]. The 't²' terms are exactly from a quadratic function. A cubic Bézier has t³ terms. Research how changing control points changes the curve's shape — font designers use this to create every letter in every digital typeface.
Explore non-Euclidean mensuration — on the surface of a sphere (like the Earth), the area of a triangle is not base × height / 2. The area of a spherical triangle = R²(A+B+C − π) where A, B, C are the interior angles in radians and R is the sphere's radius. In spherical geometry, the angles of a triangle SUM TO MORE than 180°! Research the 'spherical excess' and how this formula is used to calculate the area of countries and continents on maps.
Research P vs NP — one of the seven Millennium Prize Problems (each worth $1 million). P = problems solvable in polynomial time. NP = problems whose solutions can be VERIFIED in polynomial time. The question: is P = NP? If so, every problem whose answer can be checked quickly can also be FOUND quickly — which would break all modern cryptography (RSA encryption, used by every bank, relies on the hardness of factoring large numbers, which is in NP). Research why this problem is so hard and what solving it would mean for cybersecurity and mathematics.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

AP Board SSC (Class 10)Very High — this umbrella file covers the four most heavily tested topics in AP SSC Mathematics, accounting for 40–45 marks of the 100-mark paper

JEE Main and AdvancedVery High — all four topic groups (Algebra/Quadratics, Sequences/Series, Trigonometry, 3D Geometry) are among the most tested in JEE; Class 10 is the foundation

NTSE (Mathematics)High — all major areas are covered in NTSE Stage I and II mathematics

AP EAPCET (Engineering entrance)Very High — Trigonometry, Algebra, Coordinate Geometry, and 3D Mensuration are major sections in EAPCET Mathematics; mastering Class 10 is the essential prerequisite

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

AP SSC Mathematics: 150 minutes, 100 marks. SUGGESTED ALLOCATION: (1) Very short answers / objective (Part A): ~20 minutes for 10–15 marks. Answer confidently and quickly — these are direct formula or concept questions. (2) Short answer questions (Part B, 2-mark): ~30 minutes for 20–30 marks. Most should take 2–3 minutes each if well-prepared. (3) Long answer questions (Part C, 4–5 mark): ~70 minutes for 40–50 marks. Budget 10–12 minutes per question — draw diagrams, show all steps. (4) Review: ~20 minutes. PRIORITY ORDER for Part C: Do trigonometry identity proofs first (most marks, most predictable). Then mensuration combination problems. Then quadratic word problems. Then statistics (ogive takes time to draw — leave till you've done others). Leave probability last (quick, 2 marks).

HIGHEST MARKS PER HOUR OF PREP: (1) PROBABILITY: 2–4 marks for 2 hours of prep. The formula is simple, the card/dice structure just needs memorisation. Very high ROI. (2) MENSURATION: 6–10 marks for 3–4 hours. Pure formula substitution — no proofs, no deduction. Once you memorise the formula table, questions are mechanical. (3) ARITHMETIC PROGRESSION: 4–6 marks for 3 hours. Just two formulas (nth term and sum), and the 3-term trick. Word problems follow predictable patterns. In contrast, SIMILAR TRIANGLES and TRIGONOMETRIC IDENTITY PROOFS require more understanding but give 6–10 marks — they are high-mark but need more preparation time.

FIVE-STEP QUADRATIC WORD PROBLEM APPROACH: (1) READ CAREFULLY: identify what unknown you are solving for. Call it x. (2) FORMULATE: write the relationship given in the problem as an equation involving x. Usually it comes from 'product = value' or 'sum = value' or two conditions. (3) REARRANGE: get the equation into standard form ax²+bx+c = 0. (4) SOLVE: by factorisation (preferred if factorisable) or quadratic formula. (5) CHECK AND REJECT: verify both roots in the original problem context. Physical quantities cannot be negative — explicitly state 'x = −7 is rejected since [measurement] cannot be negative.' AP Board gives marks for this rejection step — never skip it.

EXACT STEPS FOR LESS-THAN OGIVE: (1) Make a table: Column 1 = 'Upper Class Boundary', Column 2 = 'Cumulative Frequency (CF)'. Start with (lower boundary of first class, 0) as the starting point. Then: (upper boundary of first class, CF₁), (upper boundary of second class, CF₂), etc. Last point: (upper boundary of last class, N = total frequency). (2) Take graph paper or plain paper. x-axis = marks/values (use upper class boundaries). y-axis = cumulative frequency (max = N). Choose appropriate scale for both axes. (3) Plot each (upper boundary, CF) point as a dot. Connect with smooth S-curve (not zigzag lines). (4) TO FIND MEDIAN: Draw horizontal dotted line at y = N/2. Where this line meets the curve, draw vertical dotted line to x-axis. Read the x-value — that is the median. Circle the median value and label it.

DECISION TREE: (1) 'Find SURFACE AREA / TSA of the combination' → IDENTIFY all EXPOSED surfaces. For each exposed surface, identify what solid's formula it belongs to. Add them up. DO NOT count hidden faces. (2) 'Find VOLUME of the combination' → ADD all individual volumes. No adjustment for the joining surface (area has no volume). (3) 'Find how many small solids from a large one (recasting)' → set Volume(large) = n × Volume(small). Cancel π and solve for n. (4) 'Find how much water spills / rises' → Volume spilled = Volume of immersed solid. Water rise in cylinder = Volume of solid / Base area of cylinder. Always identify which of these four categories the problem belongs to before selecting formulas.

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