CUET · MATHS · PYQ PAPER 2025

If \(y=\sin ^{-1} x\), then \(\left(1-x^2\right) \frac{d^2 y}{d x^2}\) is equal to

A \(-x \frac{d y}{d x}\)
B \(x \frac{d y}{d x}\)
C \(\frac{y}{x} \frac{d y}{d x}\)
D \(\frac{-y}{x} \frac{d y}{d x}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(x \frac{d y}{d x}\)

Step-by-step Solution

Detailed explanation

\(y=\sin ^{-1} x\) \(\frac{d y}{d x} = \frac{1}{\sqrt{1-x^2}}\) \(\frac{d^2 y}{d x^2} = \frac{d}{dx} (1-x^2)^{-1/2} = -\frac{1}{2}(1-x^2)^{-3/2}(-2x) = x(1-x^2)^{-3/2}\) \(\left(1-x^2\right) \frac{d^2 y}{d x^2} = (1-x^2) x(1-x^2)^{-3/2} = x(1-x^2)^{-1/2}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let \(\theta\) be the angle between two vectors \(\vec{a}\) and \(\vec{b}\). Then match List-I with List-II

List-I	List-II
(A) \(\sin \theta\)	(I) \(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|\|\vec{b}\|}\)
(B) \(\cos \theta\)	(II) \(\|\vec{a} \times \vec{b}\|\)
(C) Area of the parallelogram with adjacent sides represented by \(\vec{a}\) and \(\vec{b}\)	(III) \(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|}\)
(D) Projection of \(\vec{a}\) on \(\vec{b}\)	(IV) \(\frac{\|\vec{a} \times \vec{b}\|}{\|\vec{a}\|\|\vec{b}\|}\)

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List-I	List-II
(A) Projection of \(\vec{a}\) on \(\vec{b}\) is	(I) \(-7 \hat{i}+4 \hat{j}+6 \hat{k}\)
(B) \(\vec{a} \times \vec{b}\) is	(II) \(\frac{1}{\sqrt{101}}(-7 \hat{i}+4 \hat{j}+6 \hat{k})\)
(C) Unit vector along \(\vec{a}+\vec{b}\) is	(III) \(\frac{5}{3}\)
(D) Unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) is	(IV) \(\frac{1}{\sqrt{33}}(4 \hat{i}+\hat{j}+4 \hat{k})\)

Choose the correct answer from the options given below :

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List I	List II
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(C) Force experienced by a moving charged particle in the magnetic field	(III) Fleming's right hand rule
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