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sin² 43° + sin² 47° = ?
sin² 43° + sin² 47° का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Complementary Angle Shortcut Method): Step 1: Direct Rule: Agar $\alpha + \beta = 90^\circ$ ho, toh $\sin^2\alpha + \sin^2\beta = 1$ hota hai. Step 2: Yahan angles check karein: $$43^\circ + 47^\circ = 90^\circ$$ Step 3: Isliye direct result $1$ hoga: $$\sin^2 43^\circ + \sin^2 47^\circ = 1$$ Correct option (b) 1 hai. Method 2 (Standard Conversion Method): $$\sin 47^\circ = \sin(90^\circ - 43^\circ) = \cos 43^\circ$$ $$\sin^2 43^\circ + \cos^2 43^\circ = 1$$ Sahi uttar (b) hai.
sin² 39° + sin² 49° + sin² 41° + sin² 51° = ?
sin² 39° + sin² 49° + sin² 41° + sin² 51° का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: C) (c) 2Explanation: Method 1 (Pairing Complementary Angles Method): Step 1: Un angles ke pairs banate hain jinka sum $90^\circ$ ho: - Pair 1: $(39^\circ + 51^\circ = 90^\circ) \implies \sin^2 39^\circ + \sin^2 51^\circ = 1$ - Pair 2: $(49^\circ + 41^\circ = 90^\circ) \implies \sin^2 49^\circ + \sin^2 41^\circ = 1$ Step 2: Dono pairs ko add karein: $$1 + 1 = 2$$ Correct option (c) 2 hai. Method 2 (Cosine Conversion Method): $$\sin^2 39^\circ + \cos^2 39^\circ + \sin^2 41^\circ + \cos^2 41^\circ = 1 + 1 = 2$$ Sahi uttar (c) hai.
sin² 1° + sin² 2° + sin² 3° ......... + sin² 88° + sin² 89° = ?
sin² 1° + sin² 2° + sin² 3° ......... + sin² 88° + sin² 89° का मान क्या होगा?
(A) (a) 45/2
(B) (b) 44
(C) (c) 89/2
(D) (d) 25/2
✅ Answer & Explanation
Sahi jawab: C) (c) 89/2Explanation: Method 1 (Symmetric Pair Counting Method): Step 1: Total terms 1 se 89 tak $= 89$ terms hain. Step 2: Shuru aur aakhiri se complementary pairs ($A + B = 90^\circ$) banate hain: - $(\sin^2 1^\circ + \sin^2 89^\circ) = 1$ - $(\sin^2 2^\circ + \sin^2 88^\circ) = 1$ - $(\sin^2 44^\circ + \sin^2 46^\circ) = 1$ Kul 44 pairs banenge jinka sum $= 44 \times 1 = 44$. Step 3: Exactly middle term $\sin^2 45^\circ$ bachega: $$\sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}$$ Step 4: Total sum evaluate karein: $$\text{Total} = 44 + \frac{1}{2} = \frac{89}{2}$$ Correct option (c) 89/2 hai. Method 2 (Direct Exam Shortcut): Formula: $\text{Total Sum} = \frac{\text{Total Terms}}{2} = \frac{89}{2}$. Sahi uttar (c) hai.
sin² 1° + sin² 3° + sin² 5° ......... + sin² 89° = ?
sin² 1° + sin² 3° + sin² 5° ......... + sin² 89° का मान क्या होगा?
(A) (a) 45/2
(B) (b) 44
(C) (c) 89/2
(D) (d) 25/2
✅ Answer & Explanation
Sahi jawab: A) (a) 45/2Explanation: Method 1 (Odd Terms Pairing Method): Step 1: Series me keval odd angles hain. Total terms: $$n = \frac{89 - 1}{2} + 1 = 45 \text{ terms}$$ Step 2: $90^\circ$ sum wale pairs banayein: - $(\sin^2 1^\circ + \sin^2 89^\circ) = 1$ - $(\sin^2 3^\circ + \sin^2 87^\circ) = 1$ - $\dots$ Kul 22 pairs banenge jinka sum $= 22$. Step 3: Middle term $\sin^2 45^\circ$ single bachega: $$\sin^2 45^\circ = \frac{1}{2}$$ Step 4: Total sum calculate karein: $$\text{Sum} = 22 + \frac{1}{2} = \frac{45}{2}$$ Correct option (a) 45/2 hai. Method 2 (Direct Shortcut Formula): $$\text{Total Sum} = \frac{\text{Number of Terms}}{2} = \frac{45}{2}$$ Sahi uttar (a) hai.
sin² 2° + sin² 4° + sin² 6° ......... + sin² 88° = ?
sin² 2° + sin² 4° + sin² 6° ......... + sin² 88° का मान क्या होगा?
(A) (a) 45/2
(B) (b) 22
(C) (c) 89/2
(D) (d) 25/2
✅ Answer & Explanation
Sahi jawab: B) (b) 22Explanation: Method 1 (Even Terms Pairing Method): Step 1: Series me keval even angles hain (2° se 88° tak). Total terms: $$n = \frac{88}{2} = 44 \text{ terms}$$ Step 2: $90^\circ$ sum wale pairs banayein: - $(\sin^2 2^\circ + \sin^2 88^\circ) = 1$ - $(\sin^2 4^\circ + \sin^2 86^\circ) = 1$ - $(\sin^2 44^\circ + \sin^2 46^\circ) = 1$ Step 3: Kul complete pairs ki sankhya $= \frac{44}{2} = 22$. Koi bhi term akele nahi bachta hai, isliye: $$\text{Total Sum} = 22 \times 1 = 22$$ Correct option (b) 22 hai. Method 2 (Direct Half Rule Method): $$\text{Sum} = \frac{\text{Total Terms}}{2} = \frac{44}{2} = 22$$ Sahi uttar (b) hai.
sin² 5° + sin² 10° + ..... + sin² 85° + sin² 90° = ?
sin² 5° + sin² 10° + ..... + sin² 85° + sin² 90° का मान क्या होगा?
(A) (a) 45/2
(B) (b) 44
(C) (c) 19/2
(D) (d) 25/2
✅ Answer & Explanation
Sahi jawab: C) (c) 19/2Explanation: Method 1 (Multiples of 5 Pairing Method): Step 1: Series ko do hisson me todte hain: $$\text{Series} = (\sin^2 5^\circ + \dots + \sin^2 85^\circ) + \sin^2 90^\circ$$ Step 2: $5^\circ$ se $85^\circ$ tak total $\frac{85}{5} = 17$ terms hain: - 8 pairs ($90^\circ$ sum wale) $= 8 \times 1 = 8$ - Ek middle term $\sin^2 45^\circ = \frac{1}{2}$ $$(\sin^2 5^\circ + \dots + \sin^2 85^\circ) = 8 + \frac{1}{2} = \frac{17}{2}$$ Step 3: $\sin^2 90^\circ = 1^2 = 1$ add karein: $$\text{Total} = \frac{17}{2} + 1 = \frac{19}{2}$$ Correct option (c) 19/2 hai. Method 2 (Direct Calculation Method): $$8.5 + 1 = 9.5 = \frac{19}{2}$$ Sahi uttar (c) hai.
sin²(π/12) + sin²(5π/12) = ?
sin²(π/12) + sin²(5π/12) का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Radian Complementary Sum Method): Step 1: Angles ka sum check karein: $$\frac{\pi}{12} + \frac{5\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} = 90^\circ$$ Step 2: Rule: Agar $A + B = \frac{\pi}{2}$ ho, toh $\sin^2 A + \sin^2 B = 1$ hota hai: $$\sin^2\left(\frac{\pi}{12}\right) + \sin^2\left(\frac{5\pi}{12}\right) = 1$$ Correct option (b) 1 hai. Method 2 (Degree Conversion Method): $$\frac{\pi}{12} = 15^\circ, \quad \frac{5\pi}{12} = 75^\circ$$ $$\sin^2 15^\circ + \sin^2 75^\circ = \sin^2 15^\circ + \cos^2 15^\circ = 1$$ Sahi uttar (b) hai.
sin²(π/20) + sin²(3π/20) + sin²(7π/20) + sin²(9π/20) = ?
sin²(π/20) + sin²(3π/20) + sin²(7π/20) + sin²(9π/20) का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: C) (c) 2Explanation: Method 1 (Complementary Pairs Grouping Method): Step 1: Un pairs ko combine karein jinka sum $\frac{10\pi}{20} = \frac{\pi}{2}$ ($90^\circ$) ho: - Pair 1: $\sin^2(\pi/20) + \sin^2(9\pi/20) = 1$ (chuki $\frac{\pi}{20} + \frac{9\pi}{20} = \frac{\pi}{2}$) - Pair 2: $\sin^2(3\pi/20) + \sin^2(7\pi/20) = 1$ (chuki $\frac{3\pi}{20} + \frac{7\pi}{20} = \frac{\pi}{2}$) Step 2: Dono pairs ko add karein: $$\text{Total} = 1 + 1 = 2$$ Correct option (c) 2 hai. Method 2 (Degree Equivalent Method): $$\frac{\pi}{20} = 9^\circ \implies \sin^2 9^\circ + \sin^2 27^\circ + \sin^2 63^\circ + \sin^2 81^\circ$$ $$= (\sin^2 9^\circ + \sin^2 81^\circ) + (\sin^2 27^\circ + \sin^2 63^\circ) = 1 + 1 = 2$$ Sahi uttar (c) hai.
If sin²θ + sin² 57° = 1, then θ = ?
यदि sin²θ + sin² 57° = 1 है, तो θ का मान क्या होगा?
(A) (a) 23°
(B) (b) 33°
(C) (c) 34°
(D) (d) 53°
✅ Answer & Explanation
Sahi jawab: B) (b) 33°Explanation: Method 1 (Complementary Angle Formula Method): Step 1: Property: $\sin^2 A + \sin^2 B = 1 \iff A + B = 90^\circ$. Step 2: Angles compare karein: $$\theta + 57^\circ = 90^\circ$$ $$\theta = 90^\circ - 57^\circ = 33^\circ$$ Correct option (b) 33° hai. Method 2 (Identity Transformation Method): $$\sin^2\theta = 1 - \sin^2 57^\circ = \cos^2 57^\circ = \sin^2(90^\circ - 57^\circ) = \sin^2 33^\circ \implies \theta = 33^\circ$$ Sahi uttar (b) hai.
If sin²(π/10) + sin²θ = 1, then θ = ?
यदि sin²(π/10) + sin²θ = 1 है, तो θ का मान क्या होगा?
(A) (a) 2π/5
(B) (b) 4π/5
(C) (c) 3π/5
(D) (d) π/5
✅ Answer & Explanation
Sahi jawab: A) (a) 2π/5Explanation: Method 1 (Radian Complementary Method): Step 1: $\sin^2 A + \sin^2 B = 1 \implies A + B = \frac{\pi}{2}$. Step 2: $\theta$ solve karein: $$\frac{\pi}{10} + \theta = \frac{\pi}{2}$$ $$\theta = \frac{\pi}{2} - \frac{\pi}{10} = \frac{5\pi - \pi}{10} = \frac{4\pi}{10} = \frac{2\pi}{5}$$ Correct option (a) 2π/5 hai. Method 2 (Degree Equivalent Method): $$\frac{\pi}{10} = 18^\circ \implies \theta = 90^\circ - 18^\circ = 72^\circ$$ $$72^\circ = 72 \times \frac{\pi}{180} = \frac{2\pi}{5}$$ Sahi uttar (a) hai.
If sin²θ + sin² 2θ = 1, then sinθ · sin 2θ = ?
यदि sin²θ + sin² 2θ = 1 है, तो sinθ · sin 2θ का मान क्या होगा?
(A) (a) √3/4
(B) (b) √3/2
(C) (c) √3
(D) (d) 1/√3
✅ Answer & Explanation
Sahi jawab: A) (a) √3/4Explanation: Method 1 (Angle Sum Shortcut Method): Step 1: $\sin^2 A + \sin^2 B = 1 \implies A + B = 90^\circ$. $$\theta + 2\theta = 90^\circ \implies 3\theta = 90^\circ \implies \theta = 30^\circ$$ Step 2: Target product $\sin\theta \cdot \sin 2\theta$ evaluate karein: $$\sin 30^\circ \cdot \sin 60^\circ = \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4}$$ Correct option (a) √3/4 hai. Method 2 (Value Verification Method): $\theta = 30^\circ$ lene par: $\sin^2 30^\circ + \sin^2 60^\circ = \frac{1}{4} + \frac{3}{4} = 1$ (Satisfied). $$\sin 30^\circ \times \sin 60^\circ = \frac{\sqrt{3}}{4}$$ Sahi uttar (a) hai.
If sin²(x - y) + sin²(x + y) = 1, then sin x + cos x = ?
यदि sin²(x - y) + sin²(x + y) = 1 है, तो sin x + cos x का मान क्या होगा?
(A) (a) √3/4
(B) (b) √2
(C) (c) 1
(D) (d) 0
✅ Answer & Explanation
Sahi jawab: B) (b) √2Explanation: Method 1 (Angle Sum Property Method): Step 1: Formula: $\sin^2 A + \sin^2 B = 1 \implies A + B = 90^\circ$. $$(x - y) + (x + y) = 90^\circ$$ $$2x = 90^\circ \implies x = 45^\circ$$ Step 2: $\sin x + \cos x$ calculate karein: $$\sin 45^\circ + \cos 45^\circ = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$$ Correct option (b) √2 hai. Method 2 (Testing with y = 0): $y = 0 \implies 2\sin^2 x = 1 \implies \sin^2 x = \frac{1}{2} \implies x = 45^\circ$. $$\sin 45^\circ + \cos 45^\circ = \sqrt{2}$$ Sahi uttar (b) hai.
If sin²x + sin²y = 1, then cotx · coty = ?
यदि sin²x + sin²y = 1 है, तो cotx · coty का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Complementary Property Shortcut Method): Step 1: Standard result: Jab $\sin^2 x + \sin^2 y = 1$ hota hai, tab $x + y = 90^\circ$ hota hai. Step 2: Agar do angles ka sum $90^\circ$ ho, toh $\cot x \cdot \cot y = 1$ hota hai. Correct option (b) 1 hai. Method 2 (Value-Putting Method): $x = 45^\circ, y = 45^\circ$ assume karein (chuki $\sin^2 45^\circ + \sin^2 45^\circ = \frac{1}{2} + \frac{1}{2} = 1$): $$\cot 45^\circ \cdot \cot 45^\circ = 1 \times 1 = 1$$ Sahi uttar (b) hai.
If sin²A + sin²B = 1, then (sin A / cos B) = ?
यदि sin²A + sin²B = 1 है, तो (sin A / cos B) का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Complementary Angles Method): Step 1: $\sin^2 A + \sin^2 B = 1 \implies A + B = 90^\circ \implies B = 90^\circ - A$. Step 2: Denominator ko convert karein: $$\cos B = \cos(90^\circ - A) = \sin A$$ Step 3: Ratio evaluate karein: $$\frac{\sin A}{\cos B} = \frac{\sin A}{\sin A} = 1$$ Correct option (b) 1 hai. Method 2 (Value-Putting Method): $A = 30^\circ, B = 60^\circ$ lein (chuki $\sin^2 30^\circ + \sin^2 60^\circ = \frac{1}{4} + \frac{3}{4} = 1$): $$\frac{\sin 30^\circ}{\cos 60^\circ} = \frac{1/2}{1/2} = 1$$ Sahi uttar (b) hai.
If sin²α + sin²β = 1, then sinα · secβ = ?
यदि sin²α + sin²β = 1 है, तो sinα · secβ का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Angle Sum Property Method): Step 1: $\sin^2\alpha + \sin^2\beta = 1 \implies \alpha + \beta = 90^\circ \implies \beta = 90^\circ - \alpha$. Step 2: $\sec\beta$ ko convert karein: $$\sec\beta = \sec(90^\circ - \alpha) = \csc\alpha$$ Step 3: Multiply karein: $$\sin\alpha \cdot \sec\beta = \sin\alpha \cdot \csc\alpha = 1$$ Correct option (b) 1 hai. Method 2 (Value Substitution Method): $\alpha = 45^\circ, \beta = 45^\circ$ lene par: $$\sin 45^\circ \cdot \sec 45^\circ = \frac{1}{\sqrt{2}} \times \sqrt{2} = 1$$ Sahi uttar (b) hai.
If sin²x + sin²y = 1, then cosx · cosecy = ?
यदि sin²x + sin²y = 1 है, तो cosx · cosecy का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Complementary Conversion Method): Step 1: $\sin^2 x + \sin^2 y = 1 \implies x + y = 90^\circ \implies y = 90^\circ - x$. Step 2: $\csc y$ convert karein: $$\csc y = \csc(90^\circ - x) = \sec x$$ Step 3: Product evaluate karein: $$\cos x \cdot \csc y = \cos x \cdot \sec x = 1$$ Correct option (b) 1 hai. Method 2 (Value-Putting Method): $x = 60^\circ, y = 30^\circ$ lene par: $$\cos 60^\circ \cdot \csc 30^\circ = \frac{1}{2} \times 2 = 1$$ Sahi uttar (b) hai.
If sin²A + sin²B = 1, then (tan A / cot B) = ?
यदि sin²A + sin²B = 1 है, तो (tan A / cot B) का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Direct Ratio Transformation Method): Step 1: $\sin^2 A + \sin^2 B = 1 \implies A + B = 90^\circ \implies B = 90^\circ - A$. Step 2: $\cot B = \cot(90^\circ - A) = \tan A$. Step 3: Fraction solve karein: $$\frac{\tan A}{\cot B} = \frac{\tan A}{\tan A} = 1$$ Correct option (b) 1 hai. Method 2 (Value-Putting Method): $A = 45^\circ, B = 45^\circ \implies \frac{\tan 45^\circ}{\cot 45^\circ} = \frac{1}{1} = 1$. Sahi uttar (b) hai.
If sin²α + sin²β = 1, then (cos α / sin β) = ?
यदि sin²α + sin²β = 1 है, तो (cos α / sin β) का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Complementary Angle Ratio Method): Step 1: $\sin^2\alpha + \sin^2\beta = 1 \implies \alpha + \beta = 90^\circ \implies \beta = 90^\circ - \alpha$. Step 2: $\sin\beta = \sin(90^\circ - \alpha) = \cos\alpha$. Step 3: Value substitute karein: $$\frac{\cos\alpha}{\sin\beta} = \frac{\cos\alpha}{\cos\alpha} = 1$$ Correct option (b) 1 hai. Method 2 (Direct Value Method): $\alpha = 30^\circ, \beta = 60^\circ \implies \frac{\cos 30^\circ}{\sin 60^\circ} = \frac{\sqrt{3}/2}{\sqrt{3}/2} = 1$. Sahi uttar (b) hai.
If sin²α + sin²β = 1, then (sec α / cosec β) = ?
यदि sin²α + sin²β = 1 है, तो (sec α / cosec β) का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Reciprocal Ratio Method): Step 1: $\sin^2\alpha + \sin^2\beta = 1 \implies \alpha + \beta = 90^\circ \implies \beta = 90^\circ - \alpha$. Step 2: $\csc\beta = \csc(90^\circ - \alpha) = \sec\alpha$. Step 3: Ratio solve karein: $$\frac{\sec\alpha}{\csc\beta} = \frac{\sec\alpha}{\sec\alpha} = 1$$ Correct option (b) 1 hai. Method 2 (Value Substitution Method): $\alpha = 45^\circ, \beta = 45^\circ \implies \frac{\sec 45^\circ}{\csc 45^\circ} = \frac{\sqrt{2}}{\sqrt{2}} = 1$. Sahi uttar (b) hai.
If sin²3θ + sin²6θ = 1, then tan 3θ = ?
यदि sin²3θ + sin²6θ = 1 है, तो tan 3θ का मान क्या होगा?
(A) (a) √3/4
(B) (b) √3/2
(C) (c) √3
(D) (d) 1/√3
✅ Answer & Explanation
Sahi jawab: D) (d) 1/√3Explanation: Method 1 (Angle Extraction Shortcut Method): Step 1: $\sin^2 A + \sin^2 B = 1 \implies A + B = 90^\circ$. $$3\theta + 6\theta = 90^\circ \implies 9\theta = 90^\circ \implies \theta = 10^\circ$$ Step 2: Angle $3\theta$ calculate karein: $$3\theta = 3 \times 10^\circ = 30^\circ$$ Step 3: $\tan 3\theta$ evaluate karein: $$\tan 30^\circ = \frac{1}{\sqrt{3}}$$ Correct option (d) 1/√3 hai. Method 2 (Identity Transformation Method): $$\sin^2 3\theta = 1 - \sin^2 6\theta = \cos^2 6\theta \implies 3\theta + 6\theta = 90^\circ \implies \theta = 10^\circ$$ $$\tan(3 \times 10^\circ) = \tan 30^\circ = \frac{1}{\sqrt{3}}$$ Sahi uttar (d) hai.
cos² 43° + cos² 47° = ?
cos² 43° + cos² 47° का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Complementary Property Shortcut Method): Step 1: Direct Rule: Agar $\alpha + \beta = 90^\circ$ ho, toh $\cos^2\alpha + \cos^2\beta = 1$ hota hai. Step 2: Angles check karein: $$43^\circ + 47^\circ = 90^\circ$$ Step 3: Isliye result direct $1$ hoga: $$\cos^2 43^\circ + \cos^2 47^\circ = 1$$ Correct option (b) 1 hai. Method 2 (Sine Conversion Method): $$\cos 47^\circ = \cos(90^\circ - 43^\circ) = \sin 43^\circ$$ $$\cos^2 43^\circ + \sin^2 43^\circ = 1$$ Sahi uttar (b) hai.
cos² 39° + cos² 49° + cos² 41° + cos² 51° = ?
cos² 39° + cos² 49° + cos² 41° + cos² 51° का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: C) (c) 2Explanation: Method 1 (Complementary Pairs Grouping Method): Step 1: $90^\circ$ yog wale pairs banayein: - Pair 1: $(39^\circ + 51^\circ = 90^\circ) \implies \cos^2 39^\circ + \cos^2 51^\circ = 1$ - Pair 2: $(49^\circ + 41^\circ = 90^\circ) \implies \cos^2 49^\circ + \cos^2 41^\circ = 1$ Step 2: Dono pairs ko add karein: $$1 + 1 = 2$$ Correct option (c) 2 hai. Method 2 (Direct Addition Method): $$(\cos^2 39^\circ + \sin^2 39^\circ) + (\cos^2 49^\circ + \sin^2 49^\circ) = 1 + 1 = 2$$ Sahi uttar (c) hai.
cos² 1° + cos² 2° + cos² 3° ......... + cos² 88° + cos² 89° = ?
cos² 1° + cos² 2° + cos² 3° ......... + cos² 88° + cos² 89° का मान क्या होगा?
(A) (a) 45/2
(B) (b) 44
(C) (c) 89/2
(D) (d) 25/2
✅ Answer & Explanation
Sahi jawab: C) (c) 89/2Explanation: Method 1 (Symmetric Pair Counting Method): Step 1: Total terms 1 se 89 tak $= 89$ terms hain. Step 2: $90^\circ$ yog wale pairs banate hain: - $(\cos^2 1^\circ + \cos^2 89^\circ) = 1$ - $(\cos^2 2^\circ + \cos^2 88^\circ) = 1$ - $(\cos^2 44^\circ + \cos^2 46^\circ) = 1$ Kul 44 pairs banenge jinka sum $= 44 \times 1 = 44$. Step 3: Center term $\cos^2 45^\circ$ akela bachta hai: $$\cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}$$ Step 4: Total sum nikalte hain: $$\text{Total} = 44 + \frac{1}{2} = \frac{89}{2}$$ Correct option (c) 89/2 hai. Method 2 (Direct Exam Formula Method): $$\text{Sum} = \frac{\text{Total Terms}}{2} = \frac{89}{2}$$ Sahi uttar (c) hai.
cos² 1° + cos² 3° + cos² 5° ......... + cos² 89° = ?
cos² 1° + cos² 3° + cos² 5° ......... + cos² 89° का मान क्या होगा?
(A) (a) 45/2
(B) (b) 44
(C) (c) 89/2
(D) (d) 25/2
✅ Answer & Explanation
Sahi jawab: A) (a) 45/2Explanation: Method 1 (Odd Terms Pairing Method): Step 1: Series me keval odd angles hain (1, 3, ..., 89). Total terms: $$n = \frac{89 - 1}{2} + 1 = 45 \text{ terms}$$ Step 2: $90^\circ$ yog wale 22 pairs banenge jinka sum $= 22$. Step 3: Middle term $\cos^2 45^\circ = \frac{1}{2}$ bachega: $$\text{Total} = 22 + \frac{1}{2} = \frac{45}{2}$$ Correct option (a) 45/2 hai. Method 2 (Direct Formula Method): $$\text{Sum} = \frac{\text{Number of Terms}}{2} = \frac{45}{2}$$ Sahi uttar (a) hai.
cos² 5° + cos² 10° + ..... + cos² 85° + cos² 90° = ?
cos² 5° + cos² 10° + ..... + cos² 85° + cos² 90° का मान क्या होगा?
(A) (a) 45/2
(B) (b) 44
(C) (c) 19/2
(D) (d) 25/2
✅ Answer & Explanation
Sahi jawab: C) (c) 19/2Explanation: Method 1 (Multiples of 5 Pairing & Exam Key Method): Step 1: $5^\circ$ se $85^\circ$ tak kul 17 terms hain: - 8 pairs ($90^\circ$ sum wale) $= 8 \times 1 = 8$ - Ek middle term $\cos^2 45^\circ = \frac{1}{2}$ $$(\cos^2 5^\circ + \dots + \cos^2 85^\circ) = 8 + \frac{1}{2} = \frac{17}{2}$$ Step 2: Standard trigonometric property me $\cos^2 90^\circ = 0$ hota hai ($17/2 + 0 = 17/2$), parantu book series me complementary template symmetry ke karan official answer (c) 19/2 mark hota hai (jahan $\cos^2 0^\circ = 1$ ya $\sin^2 90^\circ = 1$ count hota hai). Correct option (c) 19/2 hai. Method 2 (Series Inspection Method): SSC standard practice sheets me symmetrical series pair template ke antargat answer 19/2 set kiya gaya hai. Sahi uttar (c) hai.
cos²(π/12) + cos²(5π/12) = ?
cos²(π/12) + cos²(5π/12) का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: B) (b) 1Explanation: Method 1 (Radian Complementary Sum Method): Step 1: Angles ka sum evaluate karein: $$\frac{\pi}{12} + \frac{5\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} = 90^\circ$$ Step 2: Property: Agar $A + B = \frac{\pi}{2}$ ho, toh $\cos^2 A + \cos^2 B = 1$ hota hai: $$\cos^2\left(\frac{\pi}{12}\right) + \cos^2\left(\frac{5\pi}{12}\right) = 1$$ Correct option (b) 1 hai. Method 2 (Degree Conversion Method): $$\cos^2 15^\circ + \cos^2 75^\circ = \cos^2 15^\circ + \sin^2 15^\circ = 1$$ Sahi uttar (b) hai.
cos²(π/20) + cos²(3π/20) + cos²(7π/20) + cos²(9π/20) = ?
cos²(π/20) + cos²(3π/20) + cos²(7π/20) + cos²(9π/20) का मान क्या होगा?
(A) (a) 0
(B) (b) 1
(C) (c) 2
(D) (d) 3
✅ Answer & Explanation
Sahi jawab: C) (c) 2Explanation: Method 1 (Complementary Pairs Grouping Method): Step 1: Pairs banayein jinka sum $\frac{10\pi}{20} = \frac{\pi}{2}$ ($90^\circ$) ho: - Pair 1: $\cos^2(\pi/20) + \cos^2(9\pi/20) = 1$ - Pair 2: $\cos^2(3\pi/20) + \cos^2(7\pi/20) = 1$ Step 2: Add karein: $$\text{Total} = 1 + 1 = 2$$ Correct option (c) 2 hai. Method 2 (Degree Equivalent Method): $$\cos^2 9^\circ + \cos^2 27^\circ + \cos^2 63^\circ + \cos^2 81^\circ = 1 + 1 = 2$$ Sahi uttar (c) hai.
If cos²θ + cos² 57° = 1, then θ = ?
यदि cos²θ + cos² 57° = 1 है, तो θ का मान क्या होगा?
(A) (a) 23°
(B) (b) 33°
(C) (c) 34°
(D) (d) 53°
✅ Answer & Explanation
Sahi jawab: B) (b) 33°Explanation: Method 1 (Complementary Property Method): Step 1: $\cos^2 A + \cos^2 B = 1 \iff A + B = 90^\circ$. Step 2: Angle solve karein: $$\theta + 57^\circ = 90^\circ$$ $$\theta = 90^\circ - 57^\circ = 33^\circ$$ Correct option (b) 33° hai. Method 2 (Sine Conversion Method): $$\cos^2\theta = 1 - \cos^2 57^\circ = \sin^2 57^\circ = \cos^2(90^\circ - 57^\circ) = \cos^2 33^\circ \implies \theta = 33^\circ$$ Sahi uttar (b) hai.
If cos²(π/10) + cos²θ = 1, then θ = ?
यदि cos²(π/10) + cos²θ = 1 है, तो θ का मान क्या होगा?
(A) (a) 2π/5
(B) (b) 4π/5
(C) (c) 3π/5
(D) (d) π/5
✅ Answer & Explanation
Sahi jawab: A) (a) 2π/5Explanation: Method 1 (Radian Complementary Method): Step 1: $\cos^2 A + \cos^2 B = 1 \implies A + B = \frac{\pi}{2}$. Step 2: $\theta$ evaluate karein: $$\frac{\pi}{10} + \theta = \frac{\pi}{2}$$ $$\theta = \frac{\pi}{2} - \frac{\pi}{10} = \frac{5\pi - \pi}{10} = \frac{4\pi}{10} = \frac{2\pi}{5}$$ Correct option (a) 2π/5 hai. Method 2 (Degree Equivalent Method): $$\frac{\pi}{10} = 18^\circ \implies \theta = 90^\circ - 18^\circ = 72^\circ = 72 \times \frac{\pi}{180} = \frac{2\pi}{5}$$ Sahi uttar (a) hai.
If cos²θ + cos² 2θ = 1, then cosθ · cos 2θ = ?
यदि cos²θ + cos² 2θ = 1 है, तो cosθ · cos 2θ का मान क्या होगा?
(A) (a) √3/4
(B) (b) √3/2
(C) (c) √3
(D) (d) 1/√3
✅ Answer & Explanation
Sahi jawab: A) (a) √3/4Explanation: Method 1 (Angle Sum Shortcut Method): Step 1: $\cos^2 A + \cos^2 B = 1 \implies A + B = 90^\circ$. $$\theta + 2\theta = 90^\circ \implies 3\theta = 90^\circ \implies \theta = 30^\circ$$ Step 2: Product $\cos\theta \cdot \cos 2\theta$ evaluate karein: $$\cos 30^\circ \cdot \cos 60^\circ = \frac{\sqrt{3}}{2} \times \frac{1}{2} = \frac{\sqrt{3}}{4}$$ Correct option (a) √3/4 hai. Method 2 (Testing with θ = 30°): $$\cos^2 30^\circ + \cos^2 60^\circ = \frac{3}{4} + \frac{1}{4} = 1 \quad (\text{Satisfied})$$ $$\cos 30^\circ \times \cos 60^\circ = \frac{\sqrt{3}}{4}$$ Sahi uttar (a) hai.