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Day 54: JEE Mains Maths

5 questions. Try each one first, then open its answer.

  1. Question 1

    Sequences and Series. Asked 3 times: 2021 Paper 23*, 27 Jun 2022*, 29 Jun 2022 (* = numbers or wording changed that year) See all 3 times it was asked

    154
    The sum of the infinite series 1+56+1262+2263+3564+5165+7066+…1 +\frac{5}{6}+\frac{12}{6^{2}}+\frac{22}{6^{3}}+\frac{35}{6^{4}}+\frac{51}{6^{5}}+\frac{70}{6^{6}}+ \ldots. is equal to:

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  2. Question 2

    Quadratic Equations. This type asked 3 times: 2021 Paper 25, 23 Jan 2026 S2, 8 Apr 2026 S2. This one: 8 Apr 2026 S2 See all 3 of this type

    71
    The sum of squares of all real solution of the equation log⁡(x+1)(2x2+5x+3)=4−log⁡(2x+3)(x2+2x+1)\log_{(x + 1)}\left( 2x^{2} + 5x + 3 \right) = 4 - \log_{(2x + 3)}\left( x^{2} + 2x + 1 \right) is equal to ____\_\_\_\_ .

  3. Question 3

    Application of Derivatives. This type asked 3 times: 30 Jan 2024, 22 Jan 2026 S1, 6 Apr 2026 S2. This one: 6 Apr 2026 S2 See all 3 of this type

    67
    Let A=∣13−121α01−1∣A =\left| \begin{matrix} 1 & 3 & - 1 \\ 2 & 1 & \alpha \\ 0 & 1 & - 1 \end{matrix} \right| be a singular matrix. Let f(x)=∫0x(t2+2t+3)dt,x∈[1,α]f(x) =\int_{0}^{x} \left( t^{2}+ 2t+ 3 \right)dt,x\in \lbrack 1,\alpha\rbrack. If MM and mm are respectively the maximum and the minimum values of ff in [1,α]\lbrack 1,\alpha\rbrack then 3(M−m)3(M-m) is equal to :

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  4. Question 4

    Conic Sections. This type asked 3 times: 2021 Paper 17, 28 Jan 2025, 6 Apr 2026 S2. This one: 6 Apr 2026 S2 See all 3 of this type

    60
    Let C be a circle having centre in the first quadrant and touching the x -axis at a distance 3 units from the origin. If the circle C has an intercept of length 636\sqrt{3} on yy-axis, then the length of the chord of the circle CC on the line x−y=3x-y= 3 is :

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  5. Question 5

    Quadratic Equations. This type asked 3 times: 1 Feb 2023, 21 Jan 2026 S2, 6 Apr 2026 S2. This one: 6 Apr 2026 S2 See all 3 of this type

    66
    Let lim⁡x→2(tan⁡(x−2))(rx2+(p−2)x−2p)(x−2)2=5\lim_{x \rightarrow 2} \frac{(\tan(x - 2))\left( rx^{2}+ (p - 2)x - 2p \right)}{(x - 2)^{2}}= 5 for some r,p∈Rr,p \in R. If the set of all possible values of q , such that the roots of the equation rx2−px+q=0rx^{2}- px + q = 0 lie in (0,2)(0,2), be the interval (α,β](\alpha,\beta\rbrack, then 4(α+β)4(\alpha+\beta) equals.

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