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Day 631: JEE Mains Maths

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

  1. Question 1

    Conic Sections. Asked once: 8 Apr 2023

    165
    Let OO be the origin and OPOP and OQOQ be the tangents to the circle x2+y2−6x+4y+8=0x^{2}+y^{2}- 6x + 4y + 8 = 0 at the point PP and QQ on it. If the circumcircle of the triangle OPQ passes through the point (α,12)\left( \alpha,\frac{1}{2} \right), then a value of α\alpha is

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  2. Question 2

    Definite Integration. Asked once: 25 Jun 2022

    173
    The value of b>3b> 3 for which 12∫3b1(x2−1)(x2−4)dx=log⁡e(4940)12\int_{3}^{b} \frac{1}{\left( x^{2}- 1 \right)\left( x^{2}- 4 \right)}dx =\log_{e}\left( \frac{49}{40} \right), is equal to

  3. Question 3

    Sequences and Series. Asked once: 2021 Paper 14

    52
    1a−b+1a−2b+1a−3b+…1a−nb=α⋅n+β⋅n2,ba\frac{1}{a-b}+\frac{1}{a- 2b}+\frac{1}{a- 3b}+ \ldots\frac{1}{a-nb}=\alpha\cdot n+\beta\cdot n^{2},\frac{b}{a} is small such that (ba)3&\left( \frac{b}{a} \right)^{3}\& other higher powers are neglected. Then find β\beta.

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  4. Question 4

    Conic Sections. Asked once: 8 Apr 2023

    169
    Let A(0,1),B(1,1)A(0,1),B(1,1) and C(1,0)C(1,0) be the mid - points of the sides of a triangle with incentre at the point D. If the focus of the parabola y2=4axy^{2}= 4ax passing through DD is (α+β2,0)(\alpha + \beta\sqrt{2},0), where α\alpha and β\beta are rational numbers, then αβ2\frac{\alpha}{\beta^{2}} is equal to

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  5. Question 5

    Definite Integration. Asked once: 25 Jun 2022

    62
    The value of ∫0πecos⁡xsin⁡x(1+cos⁡2x)(ecos⁡x+e−cos⁡x)dx\int_{0}^{\pi} \frac{e^{\cos x}\sin x}{\left( 1 +\cos^{2}x \right)\left( e^{\cos x}+e^{- \cos x} \right)}dx is equal to

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