Q.1: Multiple Choice Questions
(i) Inductive reasoning se related expression → (A) based on repeated experiments
Inductive reasoning mein hum baar baar experiments/observations kar ke general conclusion nikalte hain — jaise agar koi cheez baar baar ho rahi ho to hum ussay general rule bana lete hain.
(ii) Deductive reasoning ko describe karne wala jumla → (D) draw conclusion from well-known facts
Deductive reasoning mein hum pehle se maloom (well-known) facts se ek naya, guaranteed-true conclusion nikalte hain — general se specific ki taraf jate hain.
(iii) Konsi statement sahi hai → (C) 22/7 ∉ Q’
22/7 ek rational number hai (Q ka member), aur Q’ irrational numbers ka set hai. Isliye 22/7 Q’ mein nahi hai — ye sahi statement hai. Baaki options ghalat hain (integers infinite hote hain, quadrilateral ke angles 360° hote hain 180° nahi, aur isosceles triangles equilateral nahi hote).
(iv) “The stove is burning” ki negation → (A) the stove is not burning
Negation ka matlab hai statement ko seedha ulta karna — “burning” ki negation “not burning” hai.
(v) p aur q ki conjunction (AND) sach kab hoti hai → (B) both p and q are true
AND (∧) sirf tab true hoti hai jab dono statements true hon.
(vi) Conditional (if-then) kab false hoti hai → (A) antecedent is true and consequent is false
“If p then q” sirf tab false hota hai jab p (antecedent/pehla hissa) true ho lekin q (consequent/doosra hissa) false ho.
(vii) q → p ka Contrapositive → (B) ~q → p
Wait, let me check — actual formula: Contrapositive of (q→p) is (~p→~q). Answer key shows option (C) ~p → ~q as correct based on standard logic rule.
Contrapositive of q → p → (C) ~p → ~q
Contrapositive banane ka rule hai: order ulta karo aur dono taraf negation lagao. Isliye q→p ka contrapositive ~p→~q banta hai.
(viii) “Every integer greater than 2 is a sum of two prime numbers” → (B) conjecture
Ye statement (Goldbach Conjecture) abhi tak proved nahi hui, sirf believed hai ke sahi hai — isliye ye conjecture hai, theorem nahi.
(ix) “A straight line can be drawn between any two points” → (C) axiom
Ye ek basic accepted fact hai jisay bina proof ke maana jata hai — isliye axiom hai (Euclid’s Axiom).
(x) “Sum of interior angles of a triangle is 180°” → (B) theorem
Ye ek proven statement hai (geometry mein prove hoti hai), isliye theorem hai.
Q.2: Converse, Inverse, aur Contrapositive nikalo
Basic Rules yaad rakho:
- Converse: Order ulta karo (q → p)
- Inverse: Dono taraf negation lagao, order same rakho (~p → ~q)
- Contrapositive: Order ulta karo AUR negation lagao (~q → ~p)
(i) ~p → q: Converse = q → ~p, Inverse = p → ~q, Contrapositive = ~q → p
(ii) q → p: Converse = p → q, Inverse = ~q → ~p, Contrapositive = ~p → ~q
(iii) ~p → ~q: Converse = ~q → ~p, Inverse = p → q, Contrapositive = q → p
(iv) ~q → ~p: Converse = ~p → ~q, Inverse = q → p, Contrapositive = p → q
Trick: Hamesha original statement ko table mein rakh kar teeno formulas ek ek kar ke apply karo — mistake kam hoti hai.
Q.3: Truth Tables
(i) ~(p∨q)∨(~q):
Pehle p∨q (OR) nikalo, phir uski negation lo, phir ~q ke sath OR lagao. Har row mein T/F values calculate kar ke final column tak pohanchte hain.
(ii) ~(~q∨~p):
Pehle ~q aur ~p nikalo, phir unka OR lo, phir poori cheez ki negation lo.
(iii) (p∨q)↔(p∧q):
Ye biconditional hai — sach tab hogi jab dono taraf (p∨q aur p∧q) ki value same ho, chahe dono true hon ya dono false.
Truth table banane ka tareeqa: Hamesha p aur q ke saare 4 combinations (TT, TF, FT, FF) likho, phir step by step column banate jao jab tak final expression tak na pohanch jao.
Q.4: Mathematical Statement aur Proof mein farq
Mathematical Statement: Koi jumla ya expression jo ya to true ho sakta hai ya false, lekin dono nahi — isay statement kehte hain. Misal: “a = b” ya to sach hai ya jhoot.
Mathematical Proof: Ye evidence deti hai ke statement sahi hai, logical steps ke through. Jaise agar koi bacha (Fayyaz) kahe ke “main school gaya tha” to uska father proof mangega (attendance register) — bilkul isi tarah math mein bhi statement ko sach sabit karne ke liye proof chahiye hoti hai. Doosri misal: mobile phone ki warranty claim karne ke liye warranty card proof ki tarah chahiye hota hai.
Q.5: Axiom aur Theorem mein farq
Axiom: Wo mathematical statement jisay bina kisi proof ke sach maan liya jata hai — ye basic facts hain jo aage ki ideas ki bunyad banate hain. Misalein: “Ek point se infinite lines guzar sakti hain”, “Do points ke beech seedhi line draw ki ja sakti hai” (Euclid Axiom), “Har natural number ka successor hota hai” (Peano Axiom).
Theorem: Wo mathematical statement jo already known facts ki madad se prove ki gayi ho. Misalein: “Quadrilateral ke interior angles ka sum 360° hota hai”, Fundamental Theorem of Arithmetic (har integer >1 ko prime numbers ke product ki tarah likha ja sakta hai), Fermat’s Last Theorem.
Q.6: Mathematical proofs mein logical reasoning ki importance
Logic ek systematic tareeqa hai jis se hum statements ke meanings samajh sakte hain, unki sachai check kar sakte hain, aur known facts se nayi information nikaal sakte hain. Ye problem-solving aur decision-making mein zaroori hai. Misal: Agar kisi ko penicillin ka injection lagne ke baad reaction ho, to wo general conclusion nikal leta hai ke usay penicillin se allergy hai — ye logic ki misal hai jo hum daily life mein use karte hain.
Q.7: Axiom, Conjecture, ya Theorem identify karo
(i) “Do points ke beech exactly ek seedhi line hoti hai” → Axiom
Ye geometry ki bunyadi assumption hai jisay bina proof ke sach maana jata hai.
(ii) “Har even number >2 do prime numbers ka sum hota hai” → Conjecture
Ye Goldbach Conjecture hai — abhi tak general tor par proved nahi hui, sirf believed hai.
(iii) “Triangle ke angles ka sum 180° hota hai” → Theorem
Ye proven statement hai, alag alag methods se prove ki ja sakti hai (jaise parallel line kheench kar).
Q.8: Deductive Proofs (Algebraic Expressions)
(i) Prove karo (x−4)² + 9 = x²−8x+25:
LHS ko expand karo: (x²−8x+16)+9 = x²−8x+25. Ye exactly RHS ke barabar hai — proof complete.
(ii) Prove karo (x+1)²−(x−1)² = 4x:
Dono terms expand karo: (x²+2x+1)−(x²−2x+1). Subtract karne se x² cancel ho jate hain, sirf 4x reh jata hai.
(iii) Prove karo (x+5)²−(x−5)² = 20x:
Isi tarah expand kar ke subtract karo: (x²+10x+25)−(x²−10x+25) = 20x.
Pattern: (a+b)²−(a−b)² hamesha 4ab ke barabar hota hai — ye ek useful shortcut hai.
Q.9: Har step ko justify karte hue prove karo
(i) (4+16x)/4 = 1+4x:
Fraction ko factor kar ke Distributive Law, Associative Law, aur Multiplicative Inverse/Identity use karte hain, taake step by step 1+4x tak pohanch sakein.
(ii) (6x²+18x)/(3x²−27) = 2x/(x−3):
Numerator aur denominator dono ko factor karo. Denominator mein a²−b² = (a−b)(a+b) identity use hoti hai. Common factors cancel karne se simplified form milti hai.
(iii) (x²+7x+10)/(x²−3x−10) = (x+5)/(x−5):
Dono quadratics ko factorize karo (middle term break kar ke), common factor (x+2) cancel karo.
Ye sab proofs mein zaroori hai ke har step ke sath justification (kis law/rule ka use hua) likhi jaye.
Q.10-12: Odd/Even Integers Prove Karna
Q10. Agar x odd hai, to 9x+4 bhi odd hai:
x = 2k+1 rakho (odd ki definition). 9x+4 = 18k+13, jo odd hai (kyunke 18k even hai, +13 se odd ban jata hai).
Q11. Agar x odd hai, to 7x+5 even hai:
x = 2k+1 rakho. 7x+5 = 14k+12 = 2(7k+6), jo 2 se divisible hai isliye even hai.
Q12(a). Agar x odd hai, to x²−4x+6 odd hai:
x = 2k+1 rakh kar expand karo: 4k²−4k+3. Chunke 4k²−4k hamesha even hota hai, +3 se poora expression odd ban jata hai.
Q12(b). Agar x even hai, to x²+2x+4 even hai:
x = 2k rakh kar expand karo: 4k²+4k+4 = 4(k²+k+1), jo 4 ka multiple hai isliye even hai.
Ye sab “direct proof” ka style hai: pehle variable ko definition ke mutabiq express karo, phir algebra se result nikalo.
Q.13: Set Theory Proof — (A∩B)’ = A’∪B’
Ye De Morgan’s Law hai. Proof do directions mein hoti hai:
- Pehle dikhao (A∩B)’ ⊆ A’∪B’ (agar x ∈ (A∩B)’ to x ∈ A’∪B’)
- Phir dikhao A’∪B’ ⊆ (A∩B)’ (opposite direction)
- Dono directions sach hone se equality prove ho jati hai.
Q.14: Agar x²<y² aur x,y positive hain, to x<y
Dono taraf square root lo (chunke x, y positive hain, inequality ka direction change nahi hoga): √x² < √y², jo simplify ho kar x < y ban jata hai.
Q.15: Triangle ke interior angles ka sum 180° hota hai
Geometric Proof: Triangle ABC lo, AB ke parallel ek line kheencho C se guzarti hui. Alternate interior angles equal hote hain (∠1=∠4, ∠2=∠5). Chunke straight line par angles ka sum 180° hota hai (∠3+∠4+∠5=180°), isliye substitute karne se ∠1+∠2+∠3 = 180° milta hai — yehi triangle ke angles hain.
Q.16: Fraction Properties Prove Karna
(a) a/b = c/d ⟺ ad = bc: Dono taraf bd se multiply kar ke simplify karo, cross-multiplication rule sabit hota hai.
(b) (a/b)·(c/d) = ac/bd: Fractions ko multiplicative inverse ki tarah likh kar simplify karo.
(c) a/b + c/b = (a+c)/b: Common denominator hone ki wajah se numerators seedhe add ho jate hain.