JEE Advanced · Mathematics · 27. Definite Integration
For any real number , let denotes the largest integer less than or equal to . If , then the value of is ____.
- A 180
- B 189
- C 182
- D 220
Answer & Solution
Correct Answer
(C) 182
Step-by-step Solution
Detailed explanation
Given
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 Mathematics
- If \(f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3) d t\) for all \(x \in(0, \infty)\), thenJEE Advanced 2012 Hard
- Paragraph:
There are \(n\) urns each containing \((n+1)\) balls such that the ith urn contains \(i\) white balls and \((n+1-i)\) red balls. Let \(u_i\) be the event of selecting ith urn, \(i=1,2,3, \ldots, n\) and \(W\) denotes the event of getting a white balls.Question:
If \(P\left(u_i\right)=c\), where \(c\) is a constant, then \(P\left(u_i / W\right)\) is equal toJEE Advanced 2006 Medium - Let be the vertices of a regular octagon that lie on a circle of radius . Let be a point on the circle and let denote the distance between the points and for . If varies over the circle, then the maximum value of the product , isJEE Advanced 2023 Hard
- Two parallel chords of a circle of radius 2 are at a distance \(\sqrt{3}+1\) apart. If the chords subtend at the centre, angles of \(\frac{\pi}{k}\) and \(\frac{2 \pi}{k}\), where \(k>0\), then the value of \([k]\) is
[Note : \([k]\) denotes the largest integer less than or equal to \(k\) ]JEE Advanced 2010 Hard - For a polynomial with real coefficients, let denote the number of distinct real roots of . Suppose is the set of polynomials with real coefficients defined by . For a polynomial , let and denote its first and second order derivatives respectively. Then the minimum possible value of , where , is ____JEE Advanced 2020 Medium
- Paragraph:
Let \(\psi_{1}:[0, \infty) \rightarrow \mathbb{R}, \psi_{2}:[0, \infty) \rightarrow \mathbb{R}, f:[0, \infty) \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow \mathbb{R}\) be functions such that \(f(0)=g(0)=0\),
\(\psi_{1}(x)=e^{-x}+x, \quad x \geq 0\),
\(\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, \quad x \geq 0\),
\(f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t, \quad x>0\)
and \(g(x)=\int_{0}^{x^{2}} \sqrt{t} e^{-t} d t, \quad x>0\)
Question:
Which of the following statements is TRUE ?JEE Advanced 2021 Hard
More PYQs from JEE Advanced
- The total number of cyclic isomers possible for a hydrocarbon with the molecular formula \(\mathrm{C}_4 \mathrm{H}_6\) isJEE Advanced 2010 Medium
- A horizontal circular platform of radius 0.5 m and mass 0.45 kg is free to rotate about its axis. Two massless spring toy-guns, each carrying a steel ball of mass 0.05 kg are attached to the platform at a distance 0.25 m from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform, the balls have horizontal speed of 9 with respect to the ground. The rotational speed of the platform in rad after the balls leave the platform is
JEE Advanced 2014 Medium - Let the function \(f:[1, \infty) \rightarrow \mathbb{R}\) be defined by
\(f(t)=\left\{\begin{array}{cc}(-1)^{n+1} 2, & \text { if } t=2 n-1, n \in \mathbb{N}, \\ \frac{(2 n+1-t)}{2} f(2 n-1)+\frac{(t-(2 n-1))}{2} f(2 n+1), & \text { if } 2 n-1 < t < 2 n+1, n \in \mathbb{N} .\end{array}\right.\)
Define \(g(x)=\int_1^x f(t) d t, x \in(1, \infty)\). Let \(\alpha\) denote the number of solutions of the equation \(g(x)=0\) in the interval \((1,8]\) and \(\beta=\lim _{x \rightarrow 1^+} \frac{g(x)}{x-1}\). Then the value of \(\alpha+\beta\) is equal to ________.JEE Advanced 2024 Medium - Passing \(\mathrm{H}_2 \mathrm{~S}\) gas into a mixture of \(\mathrm{Mn}^{2+}, \mathrm{Ni}^{2+}, \mathrm{Cu}^{2+}\) and \(\mathrm{Hg}^{2+}\) ions in an acidified aqueous solution precipitatesJEE Advanced 2011 Medium
- The total number of diprotic acids among the following is
\(\begin{array}{lll}\mathrm{H}_3 \mathrm{PO}_4 & \mathrm{H}_2 \mathrm{SO}_4 & \mathrm{H}_3 \mathrm{PO}_3 \\ \mathrm{H}_2 \mathrm{CO}_3 & \mathrm{H}_2 \mathrm{~S}_2 \mathrm{O}_7 & \mathrm{H}_3 \mathrm{BO}_3 \\ \mathrm{H}_3 \mathrm{PO}_2 & \mathrm{H}_2 \mathrm{CrO}_4 & \mathrm{H}_2 \mathrm{SO}_3\end{array}\)JEE Advanced 2010 Hard - A small object is placed 50 cm to the left of thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are:
JEE Advanced 2016 Hard