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Day 589: JEE Mains Maths

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

  1. Question 1

    Sequences and Series. Asked once: 15 Apr 2023

    70
    Let A1A_{1} and A2A_{2} be two arithmetic means and G1,G2G_{1},G_{2}, G3G_{3} be three geometric means of two distinct positive numbers. The G14+G24+G34+G12G32G_{1}^{4}+G_{2}^{4}+G_{3}^{4}+G_{1}^{2}G_{3}^{2} is equal to

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  2. Question 2

    Straight Lines. Asked once: 28 Jul 2022

    67
    For t∈(0,2π)t \in (0,2\pi), if ABCABC is an equilateral triangle with vertices A(sin⁡t,−cos⁡t),B(cos⁡t,sin⁡t)A(\sin t, - \cos t),B(\cos t,\sin t) and C(a,b)C(a,b) such that its orthocentre lies on a circle with centre (1,13)\left( 1,\frac{1}{3} \right), then (a2−b2)\left( a^{2}-b^{2} \right) is equal to:

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  3. Question 3

    Vector Algebra. Asked once: 2021 Paper 5

    62
    Let a vector αi^+βj^\alpha\widehat{i}+ \beta\widehat{j} be obtained by rotating the vector 3i^+j^\sqrt{3}\widehat{i}+\widehat{j} by an angle 45∘45^{\circ} about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices (α,β),(0,β)(\alpha,\beta),(0,\beta) and (0,0)(0,0) is equal to

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  4. Question 4

    Application of Derivatives. Asked once: 26 Jun 2022

    69
    Let SS be the set of all the natural numbers, for which the line xa+yb=2\frac{x}{a}+\frac{y}{b}= 2 is a tangent to the curve (xa)n+(yb)n=2\left( \frac{x}{a} \right)^{n}+\left( \frac{y}{b} \right)^{n}= 2 at the point (a,b),ab≠0(a,b),ab \neq 0. Then:

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  5. Question 5

    Application of Integrals. Asked once: 29 Jun 2022

    175
    For real numbers a, b(a>b>0)a,\text{ }b(a > b > 0), let Area{(x,y):x2+y2≤a2Area\left\{ (x,y):x^{2}+y^{2}\leq a^{2} \right. and  x2a2+y2b2≥1}=30π\left. \ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\geq 1 \right\}= 30\pi and Area{(x,y):x2+y2≥b2Area\left\{ (x,y):x^{2}+y^{2}\geq b^{2} \right. and  x2a2+y2b2≤1}=18π\left. \ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\leq 1 \right\}= 18\pi Then the value of (a−b)2(a - b)^{2} is equal to