JEE Mains · Maths · STD 11 - 8. sequence and series
If \(\alpha, \beta\) are natural numbers such that \(100^{\alpha}-199 \beta=(100)(100)+(99)(101)+(98)(102)\) \(+\ldots .+(1)(199),\) then the slope of the line passing through \((\alpha, \beta)\) and origin is
- A \(540\)
- B \(550\)
- C \(530\)
- D \(510\)
Answer & Solution
Correct Answer
(B) \(550\)
Step-by-step Solution
Detailed explanation
\(S =(100)(100)+(99)(101)+(98)(102) \ldots .\)\(\ldots(2)(198)+(1)(199)\) \(S =\sum_{ x =0}^{99}(100- x )(100+ x )=\sum 100^{2}- x ^{2}\) \(=100^{3}-\frac{99 \times 100 \times 199}{6}\) \(\alpha=3 \quad \beta=1650\) slope \(=\frac{1650}{3}=550\)
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
- Let \(a_1, a_2, a_3, \ldots \ldots\) be an A.P. If \(a_7=3\), the product \(a_1 a_4\) is minimum and the sum of its first \(n\) terms is zero, then \(n !-4 a_{n(n+2)}\) is equal to :JEE Mains 2023 Hard
- Two number \(\mathrm{k}_1\) and \(\mathrm{k}_2\) are randomly chosen from the set of natural numbers. Then, the probability that the value of \(\mathrm{i}^{\mathrm{k}_1}+\mathrm{i}^{\mathrm{k}_2},(\mathrm{i}=\sqrt{-1})\) is non-zero, equalsJEE Mains 2025 Medium
- A box contains \(5\) blue, \(6\) yellow and \(4\) red balls. The number of ways, of drawing \(8\) balls containing at least two balls of each colour, is :JEE Mains 2026 Easy
- If \({e^y} + xy = e\), the ordered pair \(\left( {\frac{{dy}}{{dx}},\frac{{{d^2}y}}{{d{x^2}}}} \right)\) at \(x = 0\) is equal toJEE Mains 2019 Hard
- The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , isJEE Mains 2025 Medium
- Let \(P = \left\{ {\theta :\sin \,\theta - \cos \,\theta = \sqrt 2 \,\cos \,\theta } \right\}\) and \(Q = \left\{ {\theta :\sin \,\theta + \cos \,\theta = \sqrt {2\,} \sin \,\theta } \right\}\) be two sets. ThenJEE Mains 2016 Hard
More PYQs from JEE Mains
- if the curves \({y^2} = 6x,9{x^2} + b{y^2} = 16\) intersect each other at right angles , then the value of \(b\) is :JEE Mains 2018 Hard
- If \((2021)^{3762}\) is divided by \(17\), then the remainder is ........JEE Mains 2021 Hard
- Let the direction cosines of two lines satisfy the equations: \( 4l+m-n=0 \) and \( 2mn+10nl+3lm=0 \). Then the cosine of the acute angle between these lines is:JEE Mains 2026 Easy
- A wire of length \(20 m\) is to be cut into two pieces. A piece of length \(\ell_1\) is bent to make a square of area \(A_1\) and the other piece of length \(\ell_2\) is made into a circle of area \(A _2\). If \(2 A _1+3 A _2\) is minimum then \(\left(\pi \ell_1\right): \ell_2\) is equal to:JEE Mains 2023 Hard
- Let \(a, b, c\) and \(d\) be non-zero numbers. If the point of intersection of the lines \(4ax + 2ay + c = 0\) and \(5bx + 2by + d =0\) lies in the fourth quadrant and is equidistant from the two axes thenJEE Mains 2014 Hard
- The shortest distance between the lines \(\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-6}{2}\) and \(\frac{x-6}{3}=\frac{1-y}{2}=\frac{z+8}{0}\) is equal to \(............\)JEE Mains 2023 Hard