CUET · MATHS · PYQ PAPER 2025
Consider the curve which is represented by the differential equation \(\frac{d y}{d x}=1+x+y+x y\). If it passes through the point \((0,0)\), then which of the following is/are true?
(A) it is a straight line.
(B) it is a parabola.
(C) it also passes through the point \(\left(-1, \frac{1}{\sqrt{e}}-1\right)\)
(D) Its equation is \(x y(x+1)\left(y-\frac{1}{\sqrt{e}}+1\right)=0\)
Choose the correct answer from the options given below :
- A (C) and (D) only
- B (A) only
- C (B) only
- D (C) only
Answer & Solution
Correct Answer
(D) (C) only
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x}=(1+x)(1+y)\) \(\int \frac{d y}{1+y}=\int (1+x) d x\) \(\ln |1+y|=x+\frac{x^2}{2}+C\) \(0=\ln|1+0|=0+\frac{0^2}{2}+C \Rightarrow C=0\) \(y=e^{x+\frac{x^2}{2}}-1\) Substitute \(x=-1\): \(y(-1)=e^{-1+\frac{(-1)^2}{2}}-1\) \(y(-1)=e^{-1+\frac{1}{2}}-1\)…
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