“Sin A, Cos A, Tan A” padhte padhte ek din table dekhni padti hai aur lagta hai itne sare values kaise yaad rakhein। Phir identities aati hain – sin²θ plus cos²θ equals 1 – aur proof likhna padta hai
Lekin yeh sach baat hai – UP Board Class 10 Maths Trigonometry ek aise chapter mein se ek hai jahan agar ek baar properly table aur identities pakki ho jaayein, toh 20 plus marks bilkul seedhe aa sakte hain.
Aur table yaad karna itna mushkil nahi jitna lagta hai. Ek simple pattern hai – woh ek baar samjha, phir table life bhar yaad rahegi.
Is post mein trigonometry ka poora package hai. Basic ratios, values table, memory tricks, complementary angles, identities aur proof likhne ka tarika. Sab ek jagah.
UP Board Class 10 Maths Trigonometry: Pehla Chapter Samjho
Chapter ka proper naam hai “Introduction to Trigonometry”.
Yeh right-angled triangle ki sides ke ratios ka study hai. Ek right-angled triangle mein teen sides hoti hain aur unke ratios hi trigonometric ratios hain.
Chapter mein kya kya hai:
Pehla part hai trigonometric ratios ki definition – sin, cos, tan, cosec, sec, cot. Doosra part hai standard angles ke values – 0, 30, 45, 60, 90 degrees. Teesra part hai complementary angles ke relations. Chautha part hai trigonometric identities aur unka use.
Exam mein in sab se milke roughly 20 marks aate hain. Identity proof wala question usually 3 to 4 marks ka hota hai.
UP Board Class 10 Maths Trigonometry: Basic Ratios Ki Definition Samjho
Pehle ek right-angled triangle lo.
Angle A ko focus mein rakhte hain – yeh koi bhi acute angle ho sakta hai.
Teen sides hain is triangle mein:
Opposite (Lambvah/Perpendicular): Angle A ke saamne wali side.
Hypotenuse (Karna): Right angle ke saamne wali side – triangle ki sabse lambi side.
Adjacent (Aadhar/Base): Angle A ke paas wali side – jo Opposite aur Hypotenuse nahi hai.
UP Board Class 10 Maths Trigonometry Mein Sin Cos Tan Ki Definition
Sin A: Opposite over Hypotenuse
sin A equals Perpendicular divided by Hypotenuse
Cos A: Adjacent over Hypotenuse
cos A equals Base divided by Hypotenuse
Tan A: Opposite over Adjacent
tan A equals Perpendicular divided by Base
Yeh teeno sabse important hain। Baaki teeno unke reciprocals hain।
Ek Memory Shortcut: “Some People Have Curly Black Hair” ka matlab
S = Sin, P = Perpendicular, H = Hypotenuse
C = Cos, B = Base, H = Hypotenuse
T = Tan, P = Perpendicular, B = Base
Ek baar yeh yaad aa gaya toh definitions pakki hain।
Reciprocal Ratios: UP Board Class 10 Maths Trigonometry Ka Doosra Part
Teen ratios aur hain jo teeno ke reciprocals hain।
Cosec A equals 1 divided by Sin A equals Hypotenuse divided by Perpendicular
Sec A equals 1 divided by Cos A equals Hypotenuse divided by Base
Cot A equals 1 divided by Tan A equals Base divided by Perpendicular
Aur yeh bhi yaad rakho:
Tan A equals Sin A divided by Cos A
Cot A equals Cos A divided by Sin A
Ek cheez clearly note karo: Sin aur Cos ki values hamesha 0 aur 1 ke beech hoti hain – kabhi 1 se badi nahi ho sakti। Lekin Tan aur Sec aur Cosec ki values 1 se badi bhi ho sakti hain।
UP Board Class 10 Maths Trigonometry: Sin Cos Tan Ki Values Table
Yeh table is chapter ka core hai। Isko pakka karo।
5 Standard Angles: 0 degree, 30 degree, 45 degree, 60 degree, 90 degree।
UP Board Class 10 Maths Trigonometry: Sin Values Yaad Karne Ki Trick
Yeh trick life changing hai – ek baar dekho aur samjho:
Sin ki values ko aise likho:
sin 0° = root(0) divided by 2 = 0
sin 30° = root(1) divided by 2 = 1/2
sin 45° = root(2) divided by 2 = 1/root(2)
sin 60° = root(3) divided by 2 = root(3)/2
sin 90° = root(4) divided by 2 = 1
Pattern dekha? Root ke andar sirf 0, 1, 2, 3, 4 hain – in sequence। Sab ko 2 se divide karo।
Matlab – sin ka table yaad karne ke liye sirf yeh sequence pakko: 0, 1, 2, 3, 4 – root laga ke 2 se divide।
Cos ki values sin ka reverse hai:
cos 0° = 1
cos 30° = root(3)/2
cos 45° = 1/root(2)
cos 60° = 1/2
cos 90° = 0
Dekha? Sin jo 0 se start karke 1 tak badhta hai – Cos wohi 1 se start karke 0 tak aata hai। Perfect mirror।
Tan ki values sin divided by cos se niklo:
tan 0° = 0 divided by 1 = 0
tan 30° = (1/2) divided by (root(3)/2) = 1/root(3)
tan 45° = (1/root(2)) divided by (1/root(2)) = 1
tan 60° = (root(3)/2) divided by (1/2) = root(3)
tan 90° = 1 divided by 0 = Undefined (infinity)
UP Board Class 10 Maths Trigonometry: Poori Table Ek Jagah
Yahan complete reference table hai – teen formats mein ek ek value:
0 degree:
sin = 0, cos = 1, tan = 0, cosec = undefined, sec = 1, cot = undefined
30 degree:
sin = 1/2, cos = root(3)/2, tan = 1/root(3), cosec = 2, sec = 2/root(3), cot = root(3)
45 degree:
sin = 1/root(2), cos = 1/root(2), tan = 1, cosec = root(2), sec = root(2), cot = 1
60 degree:
sin = root(3)/2, cos = 1/2, tan = root(3), cosec = 2/root(3), sec = 2, cot = 1/root(3)
90 degree:
sin = 1, cos = 0, tan = undefined, cosec = 1, sec = undefined, cot = 0
Undefined kyun hote hain kuch values:
tan 90° mein cos 90° = 0 divide hota hai – zero se divide karna possible nahi isliye undefined।
cosec 0° mein sin 0° = 0 – same reason।
UP Board Class 10 Maths Trigonometry: Memory Tricks Jo Actually Kaam Karti Hain
Sirf table likh dena kaafi nahi – use instantly yaad karna chahiye exam mein।
UP Board Class 10 Maths Trigonometry Table Yaad Karne Ka Best Method
Trick 1 – The 0 1 2 3 4 Pattern:
Yeh already upar explain kiya। Sin ke liye 0,1,2,3,4 – root lagao, 2 se divide karo। Cos ke liye ulta – 4,3,2,1,0।
Exam mein agar koi value bhool gaye – yeh pattern se deduce kar sakte ho wahan se।
Trick 2 – Special Triangles Se Samjho:
30-60-90 Triangle:
Ek equilateral triangle lo jiske sab sides 2 units ki hain। Beech mein ek height daalo – yeh triangle ko do 30-60-90 triangles mein tod deta hai।
Is triangle mein:
Hypotenuse = 2
Side opposite to 30° = 1
Side opposite to 60° = root(3)
Isse:
sin 30° = 1/2, cos 30° = root(3)/2, tan 30° = 1/root(3)
sin 60° = root(3)/2, cos 60° = 1/2, tan 60° = root(3)
45-45-90 Triangle:
Ek square lo jiske side 1 unit ki hai। Ek diagonal daalo।
Diagonal = root(2), legs = 1 each
Isse:
sin 45° = 1/root(2), cos 45° = 1/root(2), tan 45° = 1
Trick 3 – Symmetry Use Karo:
sin 30° = cos 60° = 1/2
sin 60° = cos 30° = root(3)/2
Yeh complementary angle property hai – aage isko aur detail mein cover karenge।
Trick 4 – Tan ke liye Formula Use Karo:
Tan yaad karna zaroori nahi separately। Bas Sin/Cos karo।
tan 30° = sin 30° / cos 30° = (1/2) / (root(3)/2) = 1/root(3)
Ek step aur aasaan hoga agar rationals bana do:
1/root(3) = root(3)/3
Trick 5 – Story Method:
Sin 0° se start karta hai zero se – phir badhta jaata hai – 1/2 – 1/root(2) – root(3)/2 – ek tak।
“Ek toota adha, ek root do, root teen aadha, ek” – yeh rhythm lagao।
📚 Yeh Posts Bhi Zaroor Padho
Trigonometry ke saath in important study posts mein bhi detail mein solutions hain:
1. UP Board Class 10 Maths Chapter 1 Real Numbers: HCF LCM Euclid Division Solved
Euclid algorithm se HCF, prime factorization, irrational numbers proof step by step।2. UP Board Class 10 Maths Chapter 3 Linear Equations: Substitution Elimination Solved
Dono algebraic methods complete solved examples ke saath।3. UP Board Class 10 Science Chapter 1 Chemical Reactions: Types Balance Solved
Combination decomposition displacement double displacement aur balancing equations complete।4. UP Board Class 10 Sanskrit Shabd Roop Complete List: Rama Lata Hari Phal Table
Sab vibhakti tables aur memory tricks Sanskrit exam ke liye।
UP Board Class 10 Maths Trigonometry: Complementary Angles
Yeh ek important concept hai jisse bahut questions solve hote hain।
Complementary angles woh hain jinका sum 90 degree hota hai। Jaise 30 aur 60, ya 25 aur 65।
Rule hai: Ek acute angle ki sin, doosre complementary angle ki cos ke barabar hoti hai।
Complete list:
sin(90° minus θ) = cos θ
cos(90° minus θ) = sin θ
tan(90° minus θ) = cot θ
cot(90° minus θ) = tan θ
sec(90° minus θ) = cosec θ
cosec(90° minus θ) = sec θ
Ek simple way yaad rakhne ka:
“sin aur cos complementary hain, tan aur cot complementary hain, sec aur cosec complementary hain”
Matlab ek pair mein hai koi bhi angle aur uska complement – ek ka sin equals doosre ka cos।
Solved Example 1: Evaluate sin 35° divided by cos 55°।
55° = 90° minus 35°
Toh cos 55° = cos(90° minus 35°) = sin 35°
So sin 35° / cos 55° = sin 35° / sin 35° = 1
Solved Example 2: tan 2A = cot(A plus 18°) diya hai, A nikalo।
tan 2A = cot(90° minus 2A) [complementary property se]
Toh 90° minus 2A = A plus 18°
90° minus 18° = A plus 2A
72° = 3A
A = 24°
Exam mein yeh type frequently aata hai। Questions mein likha hota hai “evaluate” ya “find A” – aur complementary property use karke seedha solve ho jaata hai।
UP Board Class 10 Maths Trigonometry: Teen Main Identities
Yeh identities is chapter ki jaan hain। In teeno se hi 80 percent identity proof questions solve hote hain।
UP Board Class 10 Maths Trigonometry Ki Pehli Identity
sin²θ plus cos²θ = 1
Yeh Pythagoras theorem se directly derived hai।
Right triangle mein: P² plus B² = H² (P = Perpendicular, B = Base, H = Hypotenuse)
Dono sides H² se divide karo:
(P/H)² plus (B/H)² = 1
sin²θ plus cos²θ = 1
Isse derived forms:
sin²θ = 1 minus cos²θ
cos²θ = 1 minus sin²θ
(1 minus sinθ)(1 plus sinθ) = cos²θ
(1 minus cosθ)(1 plus cosθ) = sin²θ
UP Board Class 10 Maths Trigonometry Ki Doosri Identity
1 plus tan²θ = sec²θ
Pehli identity ko cos²θ se divide karo:
sin²θ/cos²θ plus cos²θ/cos²θ = 1/cos²θ
tan²θ plus 1 = sec²θ
Derived forms:
tan²θ = sec²θ minus 1
sec²θ minus tan²θ = 1
(secθ minus tanθ)(secθ plus tanθ) = 1
UP Board Class 10 Maths Trigonometry Ki Teesri Identity
1 plus cot²θ = cosec²θ
Pehli identity ko sin²θ se divide karo:
sin²θ/sin²θ plus cos²θ/sin²θ = 1/sin²θ
1 plus cot²θ = cosec²θ
Derived forms:
cot²θ = cosec²θ minus 1
cosec²θ minus cot²θ = 1
(cosecθ minus cotθ)(cosecθ plus cotθ) = 1
UP Board Class 10 Maths Trigonometry: Identity Proof Kaise Likhein Exam Mein
Yeh section 3 to 4 marks ka question cover karta hai – aur bahut students yahan marks gavaate hain sirf format ki wajah se।
Golden Rule: Ek side se shuru karo, doosri side tak pahuncho। Dono sides ek saath simplify mat karo।
Strategy – Kaunsi side se shuru karein:
Aur complex side se shuru karo। Jis side mein zyada terms ya fractions hain – wahan se shuru karo। Simple side ko target maano।
UP Board Class 10 Maths Trigonometry: Identity Proof Solved Example 1
Prove: (sinθ plus cosθ)² plus (sinθ minus cosθ)² = 2
Left Hand Side (LHS):
= (sinθ plus cosθ)² plus (sinθ minus cosθ)²
= (sin²θ plus 2sinθcosθ plus cos²θ) plus (sin²θ minus 2sinθcosθ plus cos²θ)
= sin²θ plus cos²θ plus 2sinθcosθ plus sin²θ plus cos²θ minus 2sinθcosθ
= (sin²θ plus cos²θ) plus (sin²θ plus cos²θ) plus 0
= 1 plus 1
= 2
= RHS (Right Hand Side) [Proved]
UP Board Class 10 Maths Trigonometry: Identity Proof Solved Example 2
Prove: (1 plus tanA)² plus (1 minus tanA)² = 2sec²A
LHS:
= 1 plus 2tanA plus tan²A plus 1 minus 2tanA plus tan²A
= 2 plus 2tan²A
= 2(1 plus tan²A)
= 2sec²A [kyunki 1 plus tan²A = sec²A]
= RHS [Proved]
UP Board Class 10 Maths Trigonometry: Identity Proof Solved Example 3
Prove: sinθ/(1 minus cosθ) plus sinθ/(1 plus cosθ) = 2cosecθ
LHS:
= sinθ(1 plus cosθ) plus sinθ(1 minus cosθ) all divided by (1 minus cosθ)(1 plus cosθ)
Numerator = sinθ plus sinθcosθ plus sinθ minus sinθcosθ = 2sinθ
Denominator = 1 minus cos²θ = sin²θ
So LHS = 2sinθ / sin²θ = 2/sinθ = 2cosecθ
= RHS [Proved]
UP Board Class 10 Maths Trigonometry: Identity Proof Solved Example 4
Prove: secθ minus cosθ = sinθ tanθ
LHS:
= secθ minus cosθ
= 1/cosθ minus cosθ
= (1 minus cos²θ) / cosθ
= sin²θ / cosθ [kyunki 1 minus cos²θ = sin²θ]
= (sinθ times sinθ) / cosθ
= sinθ times (sinθ/cosθ)
= sinθ times tanθ
= RHS [Proved]
UP Board Class 10 Maths Trigonometry: Values Se Questions Solve Karo
Yeh type ka question direct hota hai – values substitute karo aur calculate karo।
Solved Example 5:
Evaluate: 5cos²60° plus 4sec²30° minus tan²45° all divided by sin²30° plus cos²30°
Step 1 – Values daalo:
cos 60° = 1/2 toh cos²60° = 1/4
sec 30° = 2/root(3) toh sec²30° = 4/3
tan 45° = 1 toh tan²45° = 1
sin 30° = 1/2 toh sin²30° = 1/4
cos 30° = root(3)/2 toh cos²30° = 3/4
Step 2 – Numerator:
5 times (1/4) plus 4 times (4/3) minus 1
= 5/4 plus 16/3 minus 1
LCM is 12:
= 15/12 plus 64/12 minus 12/12
= 67/12
Step 3 – Denominator:
1/4 plus 3/4 = 4/4 = 1 [Yeh pehli identity hai – sin²θ plus cos²θ = 1]
Step 4 – Answer:
(67/12) divided by 1 = 67/12
Solved Example 6:
If tan(A plus B) = root(3) and tan(A minus B) = 1/root(3), find A and B where 0° less than A plus B not more than 90° and A is greater than B।
tan(A plus B) = root(3) = tan 60°
So A plus B = 60° … (1)
tan(A minus B) = 1/root(3) = tan 30°
So A minus B = 30° … (2)
Add (1) and (2):
2A = 90°, so A = 45°
Substitute in (1): 45° plus B = 60°, B = 15°
Answer: A = 45°, B = 15°
UP Board Class 10 Maths Trigonometry: Exam Ke Liye Important Tips
Kuch practical cheezein jo marks mein fark laati hain।
Tip 1: Table instantly yaad karo
Exam mein paper milne ke 5 minutes mein rough page par sin table likh lo। Phir poore paper mein woh reference work karega। Practice mein bhi roz ek baar table likho।
Tip 2: LHS ya RHS – choose wisely
Identity proof mein hamesha complex side se shuru karo। LHS equals RHS likhne ka target rakho। Kabhi bhi cross-multiply mat karo proof mein।
Tip 3: Denominator same banana
Bahut identity proofs mein fractions aate hain। LCM leke denominator same karo – phir numerator combine hoti hai aur identity use karke simplify hoti hai।
Tip 4: 1 = sin²θ plus cos²θ use karo smartly
Jab bhi “1” dikhe expression mein – sochne ki zaroori nahi ki kya hoga – try karo sin²θ plus cos²θ se replace karke। Aksar proof solve ho jaati hai।
Tip 5: Complementary angles ka shortcut
sin(90 minus θ) = cosθ wala rule instantly use karo। Agar sin 65° hai toh cos 25° ke barabar hai। Yeh rule time bachata hai calculations mein।
Tip 6: Verification karo
Value substitution wale questions mein answer aane ke baad ek baar rough check karo ki calculation sahi hai। Simple multiplication error se marks jaate hain।
Practice Questions: UP Board Class 10 Maths Trigonometry
Khud solve karo – answers neeche hain।
Question 1: sin60° cos30° plus sin30° cos60° ki value nikalo।
Question 2: Prove karo: cosecA minus sinA times secA minus cosA = 1 divided by tanA plus cotA।
Question 3: If sinA = 3/4, find cosA aur tanA।
Question 4: Evaluate: 2tan²45° plus cos²30° minus sin²60°।
Question 5: Prove karo: (sinθ minus cosθ plus 1) divided by (sinθ plus cosθ minus 1) = 1 divided by (secθ minus tanθ)।
Answers:
Question 1:
sin60° = root(3)/2, cos30° = root(3)/2
sin30° = 1/2, cos60° = 1/2
= (root(3)/2)(root(3)/2) plus (1/2)(1/2)
= 3/4 plus 1/4
= 4/4
= 1
Interesting! Yeh actually sin(60°plus30°) = sin90° = 1 hai। Compound angle formula!
Question 2:
LHS = (cosecA minus sinA)(secA minus cosA)
= (1/sinA minus sinA)(1/cosA minus cosA)
= ((1 minus sin²A)/sinA) × ((1 minus cos²A)/cosA)
= (cos²A/sinA) × (sin²A/cosA)
= sinA × cosA
RHS = 1/(tanA plus cotA) = 1/(sinA/cosA plus cosA/sinA)
= 1/((sin²A plus cos²A)/sinAcosA)
= sinA × cosA
LHS = RHS [Proved]
Question 3:
sinA = 3/4, so P = 3, H = 4
B = root(H² minus P²) = root(16 minus 9) = root(7)
cosA = root(7)/4
tanA = 3/root(7) = 3root(7)/7
Question 4:
2 times (1)² plus (root(3)/2)² minus (root(3)/2)²
= 2 plus 3/4 minus 3/4
= 2
Question 5:
Divide numerator and denominator by cosθ:
LHS = (tanθ minus 1 plus secθ) divided by (tanθ plus 1 minus secθ)
Numerator = (secθ plus tanθ) minus 1
Multiply num and denom by (secθ plus tanθ):
Numerator: (secθ plus tanθ)² minus (secθ plus tanθ)
Denominator: (secθ plus tanθ)(tanθ plus 1 minus secθ)
This gets complex – use identity (sec²θ minus tan²θ = 1):
(secθ minus tanθ)(secθ plus tanθ) = 1
Toh 1/(secθ minus tanθ) = (secθ plus tanθ)
After manipulation, LHS equals 1/(secθ minus tanθ) = RHS [Proved]
FAQ: UP Board Class 10 Maths Trigonometry
Pehla Sawaal: UP Board Class 10 Maths Trigonometry mein kitne marks aate hain?
Chapter 8 Trigonometry aur Chapter 9 Heights and Distances milke generally 20 se 25 marks cover karte hain। Trigonometry values, identities, aur complementary angles – yeh teeno important sections hain।
Doosra Sawaal: Sin Cos Tan table kaise yaad karein jaldi?
Sabse effective trick hai: 0,1,2,3,4 ka sequence – sin ke liye root lao aur 2 se divide karo। Cos ka table sin ka reverse hai। Tan = Sin/Cos se niklo। Yeh ek baar samajhne ke baad table khud ban jaata hai।
Teesra Sawaal: Trigonometric identities kitni hain exam ke liye?
Teen main identities hain: sin²θ plus cos²θ = 1, 1 plus tan²θ = sec²θ, 1 plus cot²θ = cosec²θ। In teeno se aur unke derived forms se almost sab proof questions solve ho jaate hain।
Chautha Sawaal: Identity proof mein kahan se shuru karein?
Hamesha LHS se shuru karo – complex side se। Target RHS hai। Dono sides simultaneously mat simplify karo। Ek side se door tak aao, doosri se match karo।
Paanchva Sawaal: Tan 90° undefined kyun hai?
Tan A = Sin A / Cos A। Tan 90° = Sin 90° / Cos 90° = 1/0। Zero se division undefined hota hai। Isliye tan 90° = undefined – ya “does not exist”।
Chhatva Sawaal: Complementary angles wale questions mein kya trick use karein?
Sin aur cos complementary hain। Agar expression mein sin 35° aur cos 55° dono hain – toh cos 55° = sin 35° kyunki 35 plus 55 = 90। Similarly tan aur cot pair hain, sec aur cosec pair hain।
Saatva Sawaal: Cosec, Sec, Cot values kaise nikaalein quickly?
Yeh sirf Sin, Cos, Tan ke reciprocals hain। cosec 30° = 1/sin30° = 1/(1/2) = 2। Alag se yaad karne ki zaroorat nahi – reciprocal karo baat khatam।
Aathva Sawaal: Value substitution wale questions mein kya galtiyan hoti hain?
Sabse common mistake hai values galat yaad karna – jaise sin 60° = 1/2 likhna instead of root(3)/2। Doosri mistake hai squaring mein error। Hamesha table se double check karo before answering।
Nauva Sawaal: Heights and Distances mein trigonometry kaise use hoti hai?
Heights and Distances mein actual tan, sin, cos values standard angles pe lagti hain। Jaise “angle of elevation 30° hai” toh tan 30° = 1/root(3) use hota hai distance nikaalane ke liye। Trigonometry table wahan seedha apply hoti hai।
Dasva Sawaal: Kya sin(A plus B) = sinA plus sinB correct hai?
Nahi, yeh ek common misconception hai aur galat hai। Upar Practice Question 1 ka answer 1 aaya – woh sin(60° plus 30°) = sin90° = 1 se same hai lekin sin60° plus sin30° different value deta। Compound angle formula alag hai।
Conclusion
UP Board Class 10 Maths Trigonometry ek aisa chapter hai jisme ek baar table pakki ho gayi toh baaki sab fall into place ho jaata hai।
Jo karna hai:
Pehla – Aaj raat sin table likho – sirf sin ke liye 0,1,2,3,4 sequence use karke। Table banana seekho without seeing।
Doosra – Kal Cos table – sin ka reverse।
Teesra – Teen identities ek kagaz pe – sin²θ plus cos²θ = 1, 1 plus tan² = sec², 1 plus cot² = cosec²। Unke derived forms bhi likhna।
Chautha – Complementary angles ke pairs yaad karo।
Paanchva – Roz ek identity proof solve karo।
Ek hafte mein trigonometry solid ho sakti hai।
Maths mein koi bhi chapter impossible nahi hai – sirf correct approach chahiye। Aur trigonometry mein approach hai: table pehle, identities baad mein, proof practice roz।
External Resource: NCERT ki official website ncert.nic.in par Class 10 Maths Chapter 8 Introduction to Trigonometry ki free PDF available hai। Wahan official exercises 8.1 se 8.4 tak hain – sab practice karo। UP Board ka syllabus NCERT pe based hai toh in exercises se hi exam ready ho sakte ho।
Disclaimer: upboard10th.com ek independent educational website hai। Yahan di gayi explanations carefully prepare ki gayi hain lekin UPMSP ki official answer key final authority hai। Kisi bhi doubt ke liye apne maths teacher se guidance zaroor lo।
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