CBSE 2026 · Region 1 · Set 1 · Q34 · 5 marks
If $\displaystyle x=\cos \mathrm{t}, \mathrm{y}=\cos \mathrm{mt}$, prove that $\displaystyle \left(1-x^{2}\right) \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} x^{2}}-x \frac{\mathrm{dy}}{\mathrm{d} x}+\mathrm{m}^{2} \mathrm{y}=0$.
Official answer
From CBSE’s own marking scheme for this paper.
Proof verified: Starting with x = cos t and y = cos(mt), we find dy/dx = -m·sin(mt)/(-sin t) = m·sin(mt)/sin t, and d²y/dx² = m(m·cos(mt)·sin t - sin(mt)·cos t)/sin³t. Substituting into ($\displaystyle 1$-x²)d²y/dx² - x·dy/dx + m²y yields $\displaystyle 0$ after algebraic simplification.
Marking-scheme solution
$\displaystyle \dfrac{dx}{dt}=-\sin \mathrm{t}$, $\displaystyle \dfrac{dy}{dt}=-\mathrm{m}\sin(mt)$,
$\displaystyle \Rightarrow\dfrac{dy}{dx}=\dfrac{\mathrm{m}\sin(mt)}{\sin \mathrm{t}}$ or $\displaystyle \sin \mathrm{t}\dfrac{dy}{dx} = \mathrm{m}\sin(mt)$
differentiating w.r.t. $\displaystyle \mathrm{t}$ and dividing by $\displaystyle dx/dt$:
$\displaystyle \Rightarrow \sin \mathrm{t}\dfrac{\mathrm{d}^2\mathrm{y}}{dx^2} + \cos \mathrm{t}\times\dfrac{dy}{dx}\times\dfrac{dt}{dx} = \mathrm{m}^2\cos(mt)\times\dfrac{dt}{dx}$
$\displaystyle \Rightarrow(1-\cos^2\mathrm{t})\dfrac{\mathrm{d}^2\mathrm{y}}{dx^2} - x\dfrac{dy}{dx} = -\mathrm{m}^2\cos(mt)$
Since $\displaystyle x=\cos \mathrm{t}$, $\displaystyle \mathrm{y}=\cos mt$: $\displaystyle \Rightarrow(1-x^2)\dfrac{\mathrm{d}^2\mathrm{y}}{dx^2} - x\dfrac{dy}{dx} + \mathrm{m}^2\mathrm{y} = 0$ (proved)
Continuity and DifferentiabilitySecond Order DerivativeApplylong_answermedium
More from Continuity and Differentiability
- If f(x)=. is continuous at x=0, then the value of a is:2025 · asked 4×
- The value of k for which function f(x)=. is differentiable at x=0 is:2023 · asked 3×
- If y=( cos x- sin x)/( cos x+ sin x), then (d y)/(d x) is:2023 · asked 3×
- If f(x)=. is continuous at x=0, then the value of k is:2025 · asked 3×
- Differentiate sec^-1( 1√1-x^2) w.r.t. sin^-1(2 x √1-x^2). OR If y= tan x+ sec x, then prove that (d^2 y)/(d…2023 · asked 3×
- If f(x)=. is continuous at x=-1, then the value of k is:2026 · asked 3×
- The value of k for which the function f(x)= casesx^2 sin 1/x, & x ≠ 0 k(x+1), & x=0 cases is a continuous…2026 · asked 3×
- If tan ((x+y)/(x-y))=k, then (dy)/(dx) is equal to2023 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2026board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.