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Day 626: JEE Mains Maths

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

  1. Question 1

    Sequences and Series. Asked once: 25 Jun 2022

    86
    For a natural number nn, let an=19n−12na_{n}=19^{n}-12^{n}. Then, the value of 31α9−α1057α8\frac{31\alpha_{9}-\alpha_{10}}{57\alpha_{8}} is

  2. Question 2

    Conic Sections. Asked once: 13 Apr 2023

    69
    Let the tangent and normal at the point (33,1)(3\sqrt{3},1) on the ellipse x236+y24=1\frac{x^{2}}{36}+\frac{y^{2}}{4}= 1 meet the yy-axis at the points AA and BB respectively. Let the circle CC be drawn taking ABAB as a diameter and the line X=25X = 2\sqrt{5} intersect CC at the points PP and QQ. If the tangents at the points PP and QQ on the circle intersect at the point (α,β)(\alpha,\beta), then α2−β2\alpha^{2}-\beta^{2} is equal to

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  3. Question 3

    Definite Integration. Asked once: 25 Jul 2022

    69
    For any real number xx, let [x]\lbrack x\rbrack denote the largest integer less than equal to xx. Let ff be a real valued function defined on the interval [−10,10]\lbrack - 10,10\rbrack by f(x)={x−[x], if [x] is odd 1+[x]−x if [x] is even f(x) =\left\{ \begin{matrix} x - \lbrack x\rbrack, & \text{~if~}\lbrack x\rbrack\text{~is odd~} \\ 1 + \lbrack x\rbrack - x & \text{~if~}\lbrack x\rbrack\text{~is even~} \end{matrix} \right. Then the value of π210∫−1010f(x)cos⁡πxdx\frac{\pi^{2}}{10}\int_{- 10}^{10} f(x)\cos\pi xdx is :

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  4. Question 4

    Sequences and Series. Asked once: 25 Jun 2022

    90
    The greatest integer less than or equal to the sum of first 100 terms of the sequence 13,59,1927,6581,……\frac{1}{3},\frac{5}{9},\frac{19}{27},\frac{65}{81},\ldots\ldots. is equal to

  5. Question 5

    Conic Sections. Asked once: 13 Apr 2023

    87
    Let m1m_{1} and m2m_{2} be the slopes of the tangents drawn from the point P(4,1)P(4,1) to the hyperbola H:y225−x216=1H:\frac{y^{2}}{25}-\frac{x^{2}}{16}= 1. If QQ is the point from which the tangents drawn to HH have slopes ∣m1∣\left| m_{1} \right| and ∣m2∣\left| m_{2} \right| and they make positive intercepts α\alpha and β\beta on the xx axis, then (PQ)2αβ\frac{(PQ)^{2}}{\alpha\beta} is equal to