JEE Mains · Maths · STD 11 - 12. limits
The value of \(\lim _{n \rightarrow \infty} \frac{[ r ]+[2 r ]+\ldots . .+[ nr ]}{ n ^{2}},\) where
is non-zero real number and \([r]\) denotes the greatest integer less than or equal to \(r\), is equal to ...... .
- A \(\frac{ r }{2}\)
- B \(r\)
- C \(2r\)
- D \(0\)
Answer & Solution
Correct Answer
(A) \(\frac{ r }{2}\)
Step-by-step Solution
Detailed explanation
We know that and \(\begin{array}{c} r \leq[ r ]< r +1 \\ 2 r \leq[2 r ]<2 r +1 \end{array} \)\( \)\( \begin{array}{ccc} 3 r & \leq[3 r ] & <3 r +1 \\ \vdots & \vdots & \vdots \\ nr & \leq[ nr ] & < nr +1 \end{array}\)…
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
- If the standard deviation of the numbers \( 2,3,a \) and \(11\) is \(3.5\) then which of the following is true ?JEE Mains 2016 Medium
- In a class of \(140\) students numbered \(1\) to \(140\), all even numbered students opted Mathematics course, those whose number is divisible by \(3\) opted Physics course and those whose number is divisible by \(5\) opted Chemistry course. Then the number of students who did not opt for any of the three courses isJEE Mains 2019 Hard
- If the mean of the following probability distribution of a random variable \(X\);
is \(\frac{46}{9}\) , then the variance of the distribution is\(X\) \(0\) \(2\) \(4\) \(6\) \(8\) \(P(X)\) \(a\) \(2a\) \(a+b\) \(2b\) \(3b\) JEE Mains 2024 Hard - The angles \(A, B\) and \(C\) of a triangle \(ABC\) are in \(A.P\) and \(a : b = 1 : \sqrt 3 .\) If \(c = 4\, cm,\) then the area (in sq. cm) of this triangle isJEE Mains 2019 Hard
- Let \(\mathrm{A}\) be the set of all points \((\alpha, \beta)\) such that the area of triangle formed by the points \((5,6),(3,2)\) and \((\alpha, \beta)\) is \(12\, square\, units.\) Then the least possible length of a line segment joining the origin to a point in \(A,\) is :JEE Mains 2021 Hard
- If \(n \geq 2\) is a positive integer, then the sum of the series \({ }^{ n +1} C _{2}+2\left({ }^{2} C _{2}+{ }^{3} C _{2}+{ }^{4} C _{2}+\ldots+{ }^{ n } C _{2}\right)\) is ...... .JEE Mains 2021 Hard
More PYQs from JEE Mains
- If \(\alpha = \displaystyle\int_0^{2\sqrt{3}} \log_2(x^2 + 4)\,dx + \displaystyle\int_2^4 \sqrt{2^x - 4}\,dx\), then \(\alpha^2\) is equal to _______.JEE Mains 2026 Hard
- An integer is chosen at random from the integers \(\{1,2,3, \ldots \ldots . .50\}\). The probability that the chosen integer is a multiple of atleast one of \(4,6\) and \(7\) isJEE Mains 2024 Medium
- Suppose that the points \((h, k), (1, 2)\) and \((-3, 4)\) lie on the line \(L_1\). If a line \(L_2\) passing through the points \((h, k)\) and \((4, 3)\) is perpendicular to \(L_1\), then \(\frac{k}{h}\) equalsJEE Mains 2019 Hard
- The integral \(\int_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} d x\) is equal to.JEE Mains 2022 Medium
- Let \( \alpha \) and \( \beta \) respectively be the maximum and the minimum values of the function \( f(\theta)=4(\sin^{4}(\frac{7\pi}{2}-\theta)+\sin^{4}(11\pi+\theta)) - 2(\sin^{6}(\frac{3\pi}{2}-\theta)+\sin^{6}(9\pi-\theta)) \), \(\theta \in R\). Then \( \alpha+2\beta \) is equal to:JEE Mains 2026 Medium
- The angle of elevation of the top \(P\) of a vertical tower \(PQ\) of height \(10\) from a point \(A\) on the horizontal ground is \(45^{\circ}\). Let \(R\) be a point on \(AQ\) and from a point \(B\), vertically above \(R\), the angle of elevation of \(P\) is \(60^{\circ}\). If \(\angle BAQ =30^{\circ}, AB = d\) and the area of the trapezium \(PQRB\) is \(\alpha\), then the ordered pair \(( d , \alpha)\) is.JEE Mains 2022 Hard