400 g of apple vinegar has 40% apple… - CUET UG 2025 Mathematics PYQ | PrepForBharat

CUET UG 2025 — Mathematics Arithmetic

2025mcqmedium

400 g of apple vinegar has 40% apple juice in it. The amount of apple juice, which should be added to make it 60% in apple vinegar, is:

Official previous-year question

Held on 2 Jun 2025 · Verified 13 Jul 2026.

Options

A
100 g
B
200 g
C
150 g
D
170 g

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Solution

Given $y = \sin^{-1}x$

The derivative of inverse sine:

$\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}$

To find the second derivative, rewrite the first derivative as:

$\frac{dy}{dx} = (1-x^2)^{-1/2}$

Applying the chain rule:

$\frac{d^2y}{dx^2} = -\frac{1}{2}(1-x^2)^{-3/2} \times (-2x)$

$\frac{d^2y}{dx^2} = \frac{x}{(1-x^2)^{3/2}}$

Computing $(1-x^2)\frac{d^2y}{dx^2}$ :

$(1-x^2)\frac{d^2y}{dx^2} = (1-x^2) \times \frac{x}{(1-x^2)^{3/2}}$

$= \frac{x(1-x^2)}{(1-x^2)^{3/2}}$

$= \frac{x}{(1-x^2)^{1/2}}$

$= \frac{x}{\sqrt{1-x^2}}$

Since $\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}$ :

$\frac{x}{\sqrt{1-x^2}} = x \times \frac{1}{\sqrt{1-x^2}}$

$= x\frac{dy}{dx}$

Therefore, $(1-x^2)\frac{d^2y}{dx^2} = x\frac{dy}{dx}$

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