Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced 2021Paper 2
Exam
JEE Advanced
Year
2021
Paper
Paper 2
Subject
Mathematics
Chapter
Sequences and Series

JEE Advanced 2021 · Paper 2 · Mathematics · Sequences and Series

Paragraph: Let M=\ (x, y) ∈ ℝ × ℝ: x^ 2 +y^ 2 ≤ r^ 2 \ , where r>0...

Paragraph: Let M={(x,y)∈R×R:x2+y2≤r2}M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2} \leq r^{2}\right\}, where r>0.r>0 . Consider the geometric progression an=12n−1,n=1,2,3,….a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots . Let S0=0S_{0}=0 and, for n≥1n \geq 1, let SnS_{n} denote the sum of the first nn terms of this progression. For n≥1n \geq 1, let CnC_{n} denote the circle with center \left(S_{n-1}, 0\right)andradiusand radiusa_{n},and, andD_{n}denotethecirclewithcenter(Sn−1,Sn−1)denote the circle with center \left(S_{n-1}, S_{n-1}\right) and radius ana_{n}.
Question: Consider MM with r=(2199−1)22198.r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}} . The number of all those circles DnD_{n} that are inside MM is

Options

  1. A198198
  2. B199199
  3. C200200
  4. D201201

Correct answer

B. 199199

Step-by-step solution

2Sn−1+an<(2199−12198)2\sqrt{2}{S}_{n - 1} + {a}_{n} < \left(\dfrac{{2}^{199} - 1}{{2}^{198}}\right)\sqrt{2}

2(2−12n−2)+12n−1<(2199−12198)2\sqrt{2}\left(2 - \dfrac{1}{{2}^{n - 2}}\right) + \dfrac{1}{{2}^{n - 1}} < \left(\dfrac{{2}^{199} - 1}{{2}^{198}}\right)\sqrt{2}

22−12n−22+12n−1<22−221982\sqrt{2} - \dfrac{1}{{2}^{n - 2}}\sqrt{2} + \dfrac{1}{{2}^{n - 1}} < 2\sqrt{2} - \dfrac{\sqrt{2}}{{2}^{198}}

12n−2(12−2)<−22198\dfrac{1}{{2}^{n - 2}}\left(\dfrac{1}{2} - \sqrt{2}\right) < - \dfrac{\sqrt{2}}{{2}^{198}}

12n−2(22−1)2>22198\dfrac{1}{{2}^{n - 2}}\dfrac{( 2\sqrt{2} - 1 )}{2} > \dfrac{\sqrt{2}}{{2}^{198}}

2n−2<(2−12)2197{2}^{n - 2} < \left(2 - \dfrac{1}{\sqrt{2}}\right){2}^{197}

n−2≤197n - 2 \le 197

number of circle=199= 199

Practice Sequences and Series on Quantrex Academy →

More from Sequences and Series

Let α = 3+4+8+9+13+14+… upto 40 terms. If ( β)^ (α)/(1020) is a root of the equation… JEE Main 2026The sum 1 + (1)/(2)(1^2 + 2^2) + (1)/(3)(1^2 + 2^2 + 3^2) + … upto 10 terms is equal to : JEE Main 2026The value of 1^3 - 2^3 + 3^3 - … + 15^3 is: JEE Main 2026The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a… JEE Main 2026For the functions f(θ) = α ^2θ + β ^2θ , and g(θ) = α ^2θ + β ^2θ , α > β > 0 , let _ 0 <… JEE Main 2026If the sum of the first 10 terms of the series (1)/(1 + 1^4 × 4) + (2)/(1 + 2^4 × 4) +… JEE Main 2026Let A_1, A_2, A_3, …, A_ 39 be 39 arithmetic means between the numbers 59 and 159 . Then… JEE Main 2026Let the sum of the first n terms of an A.P. be 3n^2 + 5n . Then the sum of squares of the… JEE Main 2026∑_ n=1 ^ 10 ( (528)/(n(n+1)(n+2)) ) is equal to: JEE Main 2026Let α=(1)/(4)+(1)/(8)+(1)/(16)+…∞ and β=(1)/(3)+(1)/(9)+(1)/(27)+…∞ . Then the value of… JEE Main 2026The first term of an A.P. of 30 non-negative terms is (10)/(3) . If the sum of this A.P.… JEE Main 2026Let a_1, a_2, a_3, … be an A.P. and g_1 = a_1, g_2, g_3, … be an increasing G.P. If a_1 =… JEE Main 2026The sum (1^3)/(1) + (1^3 + 2^3)/(1 + 3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) + ⋯ up to 8… JEE Main 2026Let α, α + 2, α ∈ ℤ , be the roots of the quadratic equation x(x+2) + (x+1)(x+3) +… JEE Main 2026Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common… JEE Main 2026If ∑_ k=1 ^ n a_k = 6n^3 , then ∑_ k=1 ^ 6 ( a_ k+1 - a_k 36 )^2 is equal to _______. JEE Main 2026 Full Sequences and Series list Sequences and Series JEE Maths PYQsJEE Mathematics PYQs All JEE Advanced PYQs