JEE Mains · Maths · STD 12 - 9. differential equations
Let \(x=x(y)\) be the solution of the differential equation \(y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y\gt0\) and \(x(1)=\frac{\pi}{2}\). Then \(\cos (x(2))\) is equal to :
- A \(1-2\left(\log _e 2\right)^2\)
- B \(1-2\left(\log _{\mathrm{e}} 2\right)\)
- C \(2\left(\log _e 2\right)-1\)
- D \(2\left(\log _e 2\right)^2-1\)
Answer & Solution
Correct Answer
(D) \(2\left(\log _e 2\right)^2-1\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & y d y=(x d y-y d x) \sin \left(\frac{x}{y}\right) \\ & \frac{d y}{y}=\left(\frac{x d y-y d x}{y^2}\right) \sin \left(\frac{x}{y}\right) \\ & \frac{d y}{y}=\sin \left(\frac{x}{y}\right) d\left(-\frac{x}{y}\right) \\ & \rightarrow \ell n y=\cos…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The area of the region, inside the ellipse \( x^{2}+4y^{2}=4 \) and outside the region bounded by the curves \( y=|x|-1 \) and \( y=1-|x| \), is:JEE Mains 2026 Medium
- Let \(\vec a = 3\hat i + 2\hat j + 2\hat k\) and \(\vec b = \hat i + 2\hat j - 2\hat k\) be two vectors. If a vector perpendicular to both the vectors \(\vec a + \vec b\) and \(\vec a - \vec b\) has the magnitude \(12\) then one such vector isJEE Mains 2019 Hard
- Let P be the foot of the perpendicular from the point \((1,2,2)\) on the line \(\mathrm{L}: \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}\). Let the line \(\vec{r}=(-\hat{i}+\hat{j}-2 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}), \lambda \in \mathbf{R}\), intersect the line L at Q . Then \(2(\mathrm{PQ})^2\) is equal to:JEE Mains 2025 Easy
- Let the tan gents drawn to the circle, \(x^2 + y^2 = 16\) from the point \(P(0, h)\) meet the \(x-\) axis at point \(A\) and \(B.\) If the area of \(\Delta APB\) is minimum, then \(h\) is equal toJEE Mains 2015 Hard
- Consider the system of linear equation \(x+y+z=\) \(4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15\), where \(\lambda, \mu \in R\). Which one of the following statements is \(NOT\) correct?JEE Mains 2024 Hard
- Let \(A=\left[a_{i j}\right]\) be a square matrix of order \(3\) such that \(a_{i j}=2^{j-i}\), for all \(i, j=1,2,3\). Then, the matrix \(A ^{2}+ A ^{3}+\ldots+ A ^{10}\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- If \(\mathrm{U}_{\mathrm{n}}=\left(1+\frac{1}{\mathrm{n}^{2}}\right)\left(1+\frac{2^{2}}{\mathrm{n}^{2}}\right)^{2} \ldots\left(1+\frac{\mathrm{n}^{2}}{\mathrm{n}^{2}}\right)^{\mathrm{n}}\), then \(\lim _{n \rightarrow \infty}\left(U_{n}\right)^{\frac{-4}{n^{2}}}\) is equal to :JEE Mains 2021 Hard
- If the \(x-\) intercept of some line \(L\) is double as that of the line, \(3x + 4y = 12\) and the \(y-\) intercept of \(L\) is half as that of the same line, then the slope of \(L\) isJEE Mains 2013 Hard
- If for some \(x \in R\), the frequency distribution of the marks obtained by \(20\) students in a test is
Marks \(2\) \(3\) \(5\) \(7\)
Frequency \((x+1)^2\) \(2x -5\) \(x^2 -3x\) \(x\)
Then the mean of the marks isJEE Mains 2019 Medium - For \(0<\mathrm{c}<\mathrm{b}<\mathrm{a}\), let \((\mathrm{a}+\mathrm{b}-2 \mathrm{c}) \mathrm{x}^2+(\mathrm{b}+\mathrm{c}-2 \mathrm{a}) \mathrm{x}\) \(+(c+a-2 b)=0\) and \(\alpha \neq 1\) be one of its root. Then, among the two statements \((I)\) If \(\alpha \in(-1,0)\), then \(\mathrm{b}\) cannot be the geometric mean of \(\mathrm{a}\) and \(\mathrm{c}\) \((II)\) If \(\alpha \in(0,1)\), then \(\mathrm{b}\) may be the geometric mean of \(a\) and \(c\)JEE Mains 2024 Hard
- Let \(C: x^2+y^2=4\) and \(C^{\prime}: x^2+y^2-4 \lambda x+9=0\) be two circles. If the set of all values of \(\lambda\) so that the circles \(\mathrm{C}\) and \(\mathrm{C}^{\prime}\) intersect at two distinct points, is \({R}-[a, b]\), then the point \((8 a+12,16 b-20)\) lies on the curve :JEE Mains 2024 Hard
- Let \(f(x) = log_e\,(sin\,x),\) \((0\,<\,x\,< \pi )\) and \(g(x) = sin^{-1}\,(e^{-x}),\) \((x\, \ge \,0)\). If \(\alpha \) is a positive real number such that \(a\) \( = (fog)’(\alpha )\) and \(b = (fog)(\alpha ),\) thenJEE Mains 2019 Hard