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Class 12 Continuity And Differentiability · Long Answer3 Marksimplicit_differentiation

Class 12 Maths · Continuity And Differentiability · Long Answer #103

If cos y=x cos(a+y), sin a≠0, prove y′=cos²(a+y)/sin a and hence sin a·y″+sin(2(a+y))y′=0.

✅ Step-by-Step Solution

Final Answer: Both identities proved.
  1. The relation gives x=cos y/cos(a+y). The denominator cannot be zero on a solution when sin a≠0.
  2. Differentiate in y: dx/dy=[−sin y cos(a+y)+cos y sin(a+y)]/cos²(a+y)=sin a/cos²(a+y). Inverting gives y′=cos²(a+y)/sin a.
  3. Differentiate in x: y″=[−2cos(a+y)sin(a+y)/sin a]y′=−sin(2(a+y))y′/sin a.
  4. Therefore sin a·y″+sin(2(a+y))y′=0, proving both statements.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] The relation gives x=cos y/cos(a+y). The denominator cannot be zero on a solution when sin a≠0.
  • [1 mark] Differentiate in y: dx/dy=[−sin y cos(a+y)+cos y sin(a+y)]/cos²(a+y)=sin a/cos²(a+y). Inverting gives y′=cos²(a+y)/sin a.
  • [1 mark] Differentiate in x: y″=[−2cos(a+y)sin(a+y)/sin a]y′=−sin(2(a+y))y′/sin a.
  • [0.5 marks] Therefore sin a·y″+sin(2(a+y))y′=0, proving both statements.

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