Mathbee CBSE Mathbee · CBSE Class 12 Maths
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q345 Markscontinuity_and_differentiability

CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Question 34 (Continuity And Differentiability)

If x = cos t, y = cos mt, prove that (1 − x²) d²y/dx² − x dy/dx + m²y = 0.

✅ Step-by-Step Solution

Final Answer: Proved.
  1. dx/dt = −sin t, dy/dt = −m sin(mt).
  2. dy/dx = (dy/dt)/(dx/dt) = m sin(mt)/sin t, i.e. sint · dy/dx = m sin(mt). Differentiate both sides w.r.t. x: sint·d²y/dx² + (dy/dx)·cost·(dt/dx) = m²cos(mt)·(dt/dx), where dt/dx = −1/sint. Substituting and simplifying: (1−cos²t)d²y/dx² − x dy/dx = −m²cos(mt). Since (1−x²) = sin²t = 1−cos²t and y = cos(mt), we get (1−x²)y'' − x y' + m²y = 0.
  3. Proved.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] dx/dt = −sin t, dy/dt = −m sin(mt).
  • [3 marks] dy/dx = (dy/dt)/(dx/dt) = m sin(mt)/sin t, i.e. sint · dy/dx = m sin(mt). Differentiate both sides w.r.t. x: sint·d²y/dx² + (dy/dx)·cost·(dt/dx) = m²cos(mt)·(dt/dx), where dt/dx = −1/sint. Substituting and simplifying: (1−cos²t)d²y/dx² − x dy/dx = −m²cos(mt). Since (1−x²) = sin²t = 1−cos²t and y = cos(mt), we get (1−x²)y'' − x y' + m²y = 0.
  • [1 mark] Proved.

Master Chapter 5: Continuity And Differentiability

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Continuity And Differentiability.
📘 Chapter Notes & All Exercises 🚀 Open in Interactive Workspace →
← Previous Question
If A = [[0,2,1],[-2,-1,-2],[1,-1,0]], find A⁻¹ and use it to solve…
Next Question →
Check whether the lines given by (x−1)/2 = (y−2)/3 = (z−3)/4 and…

All Questions in CBSE Class 12 Mathematics 2026 Main Paper 65/1/1

Q1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q12Q13Q14Q15Q16Q17Q18Q19Q20Q21Q22Q23Q24Q25Q26Q27Q28Q29Q30Q31Q32Q33Q34Q35Q36Q37Q38

Related Questions in Continuity And Differentiability

Class 12 Continuity And Differentiability · Mcq Single Correct
For k<0, define f(x)=(1−cos(kx))/(x sin x) for nonzero x near 0, and f(0)=1/2. Find k for…
Class 12 Continuity And Differentiability · Mcq Single Correct
Near π/2, f(x)=k cos x/(π−2x) for x≠π/2 and f(π/2)=3. Find k for continuity.
Class 12 Continuity And Differentiability · Mcq Single Correct
Find the full set of real inputs excluded from the natural domain of cot x (its poles).
Class 12 Continuity And Differentiability · Mcq Single Correct
How many real inputs are excluded from the original formula f(x)=(x−1)/[x(x²−1)]?
Class 12 Continuity And Differentiability · Mcq Single Correct
For x≠0, f(x)=(e^(3x)−e^(−5x))/x, and f(0)=k. Find k for continuity at 0.
Class 12 Continuity And Differentiability · Mcq Single Correct
Let f(x)=x/|x| for x<0 and f(x)=−1 for x≥0. What is its complete continuity set?