JEE Mains · Maths · STD 12 - 9. differential equations
Let \(f : R \rightarrow R\) be a differentiable function such that \(f^{\prime}(x)+f(x)=\int \limits_0^2 f(t) d t\). If \(f(0)=e^{-2}\), then \(2 f (0)- f (2)\) is equal to \(.........\).
- A \(2\)
- B \(3\)
- C \(1\)
- D \(4\)
Answer & Solution
Correct Answer
(C) \(1\)
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x}+y=k\) \(y \cdot e^x=k \cdot e^x+c\) \(f(0)=e^{-2}\) \(c=e^{-2}-k\) \(y=k+\left(e^{-2}-k\right) e^{-x}\) \(\text { now } k=\int \limits_0^2\left(k+\left(e^{-2}-k\right) e^{-x}\right) d x\) \(k=e^{-2}-1\) \(y=\left(e^{-2}-1\right)+e^{-x}\)…
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 circle passing through the intersection of the circles, \(x^{2}+y^{2}-6 x=0\) and \(x^{2}+y^{2}-4 y=0\) having its centre on the line, \(2 x-3 y+12=0\), also passes through the pointJEE Mains 2020 Hard
- For \(x \in R\), let \([x]\) denote the greatest integer \( \le x\), then the sum of the series \(\left[ { - \frac{1}{3}} \right] + \left[ { - \frac{1}{3} - \frac{1}{{100}}} \right] + \left[ { - \frac{1}{3} - \frac{2}{{100}}} \right] + .....+\left[ { - \frac{1}{3} - \frac{{99}}{{100}}} \right]\)JEE Mains 2019 Hard
- The value of
\(\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}}{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}+e^{\left(\left(6-\log _e x\right)^2+1\right)^{-1}}}\right) d x\) isJEE Mains 2025 Medium - The value of \(\sum_{r=0}^{22}{ }^{22} C_r \cdot{ }^{23} C_r\) isJEE Mains 2023 Medium
- A \(10\, inches\) long pencil \(\mathrm{AB}\) with mid point \(\mathrm{C}\) and a small eraser \(\mathrm{P}\) are placed on the horizontal top of a table such that \(\mathrm{PC}=\sqrt{5}\) inches and \(\angle \mathrm{PCB}=\tan ^{-1}(2)\). The acute angle through which the pencil must be rotated about \(\mathrm{C}\) so that the perpendicular distance between eraser and pencil becomes exactly \(1\, inch\) is:
JEE Mains 2021 Hard - If \(A\) and \(B\) are two events such that \(P ( A )=\frac{1}{3}, P ( B )=\frac{1}{5} \) and \(P ( A \cup B )=\frac{1}{2}\), then \(P \left( A \mid B ^{\prime}\right)+ P \left( B \mid A ^{\prime}\right)\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- If \(\left| {\begin{array}{*{20}{c}}
{a - b - c}&{2a}&{2a}\\
{2b}&{b - c - a}&{2b}\\
{2c}&{2c}&{c - a - b}
\end{array}} \right|\) \( = \left( {a + b + c} \right)\,{\left( {x + a + b + c} \right)^2}\) , \(x \ne 0\) and \(a + b + c \ne 0\), then \(x\) is equal toJEE Mains 2019 Hard - Let a relation \(R\) on \(\mathbb{N} \times \mathbb{N}\) be defined as : \(\left(\mathrm{x}_1, \mathrm{y}_1\right) \mathrm{R}\left(\mathrm{x}_2, \mathrm{y}_2\right)\) if and only if \(\mathrm{x}_1 \leq \mathrm{x}_2\) or \(\mathrm{y}_1 \leq \mathrm{y}_2\) Consider the two statements : (\(I\)) \(\mathrm{R}\) is reflexive but not symmetric. (\(II\)) \(\mathrm{R}\) is transitive Then which one of the following is true?JEE Mains 2024 Medium
- Let \(\vec{c}\) be the projection vector of \(\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda\gt0\), on the vector \(\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}\). If \(|\vec{a}+\vec{c}|=7\), then the area of the parallelogram formed by the vectors \(\vec{b}\) and \(\vec{c}\) is ________JEE Mains 2025 Medium
- If the distance of the point \(P(43, \alpha, \beta), \beta<0\), from the line \(\vec{r}=4\hat{i}-\hat{k}+\mu(2\hat{i}+3\hat{k}), \mu\in R\) along a line with direction ratios \(3, -1, 0\) is \(13\sqrt{10}\), then \(\alpha^{2}+\beta^{2}\) is equal to ___ .JEE Mains 2026 Medium
- The number of ways, in which the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}\), D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :
JEE Mains 2025 Hard - Let \(z\) be a complex number such that \(|z+2| = |z-2|\) and \(\arg\left(\dfrac{z+3}{z-i}\right) = \dfrac{\pi}{4}\). Then \(|z|^2\) is equal to:JEE Mains 2026 Medium