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Day 622: JEE Mains Maths

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

  1. Question 1

    Conic Sections. Asked once: 1 Feb 2024

    86
    Let the line L:2x+y=αL:\sqrt{2}x + y =\alpha pass through the point of intersection PP (in the first quadrant) of the circle x2+y2=3x^{2}+y^{2}= 3 and the parabola x2=2yx^{2}= 2y. Let the line LL touch two circles C1C_{1} and C2C_{2} of equal radius 232\sqrt{3}. If the centres Q1Q_{1} and Q2Q_{2} of the circles C1C_{1} and C2C_{2} lie on the yy-axis, then the square of the area of the triangle PQ1Q2PQ_{1}Q_{2} is equal to

  2. Question 2

    Definite Integration. Asked once: 28 Jul 2022

    84
    If ∫0315x31+x2+(1+x2)3dx=α2+β3\int_{0}^{\sqrt{3}} \frac{15x^{3}}{\sqrt{1 +x^{2}+\sqrt{\left( 1 +x^{2} \right)^{3}}}}dx = \alpha\sqrt{2}+ \beta\sqrt{3}, where α,β\alpha,\beta are integers, then α+β\alpha + \beta is equal to

  3. Question 3

    Sequences and Series. Asked once: 28 Jun 2022

    175
    Let for n=1,2n = 1,2, 50, Sn50,{\text{ }S}_{n} be the sum of the infinite geometric progression whose first term is n2n^{2} and whose common ratio is 1(n+1)2\frac{1}{(n + 1)^{2}}. Then the value of 126+∑n=150(Sn+2n+1−n−1)\frac{1}{26}+\sum_{n= 1}^{50} \left( S_{n}+\frac{2}{n+ 1}-n- 1 \right) is equal to

  4. Question 4

    Conic Sections. Asked once: 1 Feb 2024

    155
    Let PP be a point on the ellipse x29+y24=1\frac{x^{2}}{9}+\frac{y^{2}}{4}= 1. Let the line passing through PP and parallel to yy-axis meet the circle x2+y2=9x^{2}+y^{2}= 9 at point QQ such that PP and QQ are on the same side of the xx-axis. Then, the eccentricity of the locus of the point RR on PQPQ such that PR:RQ=4:3PR:RQ = 4:3 as PP moves on the ellipse, is :

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  5. Question 5

    Definite Integration. Asked once: 28 Jul 2022

    180
    The value of the integral ∫0π260sin⁡(6x)sin⁡xdx\int_{0}^{\frac{\pi}{2}} 60\frac{\sin(6x)}{\sin x}dx is equal to