Q7
(a) A solid sphere rests inside a fixed rough and hemispherical bowl of twice its radius. If a large amount of weight, whatsoever, is attached to the highest point of the sphere, then show that the equilibrium is stable. (15 marks) (b) Verify Green's theorem in the plane for $\oint\limits_{\mathrm{C}}\left[\left(x y+y^{2}\right) d x+x^{2} d y\right]$, where C is the boundary of the region bounded by the curves $y=x$ and $y=x^{2}$. (15 marks) (c) (i) Find the general solution and singular solution of the differential equation $\left(1+\frac{d y}{d x}\right)^{3}=\frac{27}{8 a}(x+y)\left(1-\frac{d y}{d x}\right)^{3}$. (10 marks) (ii) Find the complete solution of $x^{3} \frac{d^{3} y}{d x^{3}}+3 x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=x \log x$. (10 marks)
हिंदी में प्रश्न पढ़ें
(a) एक ठोस गोला अपनी त्रिज्या से दुगुनी त्रिज्या के स्थिर रूक्ष अर्धगोलीय कटोरे में रखा हुआ है। यदि एक बड़ा भार, कितना भी हो, गोले के सबसे ऊँचे बिंदु पर जुड़ा है, तो दर्शाइए कि संतुलन स्थिर है। (15 अंक) (b) $\oint\limits_{\mathrm{C}}\left[\left(x y+y^{2}\right) d x+x^{2} d y\right]$, जहाँ C, वक्रों $y=x$ और $y=x^{2}$ द्वारा परिबद्ध क्षेत्र की परिसीमा है, के लिए समतल में ग्रीन का प्रमेय सत्यापित कीजिए। (15 अंक) (c) (i) अवकल समीकरण $\left(1+\frac{d y}{d x}\right)^{3}=\frac{27}{8 a}(x+y)\left(1-\frac{d y}{d x}\right)^{3}$ के व्यापक हल और विचित्र हल ज्ञात कीजिए। (10 अंक) (ii) $x^{3} \frac{d^{3} y}{d x^{3}}+3 x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=x \log x$ का पूर्ण हल ज्ञात कीजिए। (10 अंक)
Directive word: Prove
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How this answer will be evaluated
Approach
Prove the stability condition in part (a) by analyzing the potential energy and center of mass displacement; verify Green's theorem in part (b) by computing both line integral and double integral separately; solve the Clairaut-type equation in (c)(i) for general and singular solutions; and apply the substitution x = e^t to reduce (c)(ii) to a linear equation with constant coefficients. Allocate approximately 30% time to (a), 25% to (b), 25% to (c)(i), and 20% to (c)(ii) based on mark distribution.
Key points expected
- Part (a): Correct geometric setup with sphere radius r and bowl radius 2r, identification of contact point and angle θ, calculation of new center of mass position after adding weight W, and proof that potential energy minimum exists for any W
- Part (b): Proper identification of region bounded by y=x and y=x² with intersection points (0,0) and (1,1), correct application of Green's theorem with P=xy+y² and Q=x², accurate computation of both ∮(Pdx+Qdy) and ∬(∂Q/∂x-∂P/∂y)dA
- Part (c)(i): Substitution u=x+y to transform equation, recognition of Clairaut's equation form, derivation of general solution (x+y-c)³(1+8a/27c³)=0 and singular solution envelope
- Part (c)(ii): Substitution x=e^t to convert to Cauchy-Euler form, reduction to linear ODE with constant coefficients, finding complementary function and particular integral for RHS e^t·t
- Correct handling of rough constraint in (a) ensuring no slipping condition is satisfied
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correct geometry with angles and distances labeled, proper free-body diagram; for (b): accurate region sketch with correct orientation and intersection points; for (c): valid substitutions identified and transformed equations written correctly | Basic geometric setup present but missing some labels or incorrect limits in (b); substitutions attempted but with minor errors in transformation | Missing or incorrect geometric setup, wrong region identification in (b), no valid substitution attempted in (c) |
| Method choice | 20% | 10 | Uses potential energy method for stability in (a), applies Green's theorem correctly in (b), recognizes Clairaut equation in (c)(i) and Cauchy-Euler reduction in (c)(ii), with optimal method selection throughout | Correct broad approach chosen but suboptimal methods used (e.g., direct integration instead of Green's theorem in b); recognizes equation types but applies standard methods mechanically | Wrong method selected (e.g., force balance instead of energy in a), fails to identify Green's theorem applicability, no recognition of special equation forms in (c) |
| Computation accuracy | 20% | 10 | Error-free calculations: correct derivatives in (a), exact match of both integrals in (b), accurate algebraic manipulation for envelope in (c)(i), correct particular integral via operator methods in (c)(ii) | Minor arithmetic errors present but method sound; one integral in (b) computed correctly; partial solution for CF or PI in (c)(ii) | Major computational errors, integrals evaluated incorrectly, wrong envelope derivation, incorrect PI calculation leading to wrong final answer |
| Step justification | 20% | 10 | Each step explicitly justified: why potential energy minimum implies stability in (a), verification that ∂Q/∂x-∂P/∂y is continuous in (b), explanation of envelope condition for singular solution, justification of x=e^t substitution | Some steps justified but gaps present; assumes continuity without check, skips explanation of why particular method works | Minimal or no justification, jumps between steps without explanation, treats theorems as black boxes |
| Final answer & units | 20% | 10 | Clear statement of stability proved for any weight W in (a), explicit verification that both sides equal -1/20 in (b), complete general+singular solutions in (c)(i), full particular solution with arbitrary constants in (c)(ii) | Final answers present but incompletely stated; missing singular solution or particular integral; verification in (b) stated without numerical value | Missing final answers, incorrect conclusions, no verification statement in (b), incomplete solutions without arbitrary constants |
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