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Day 596: JEE Mains Maths

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

  1. Question 1

    Application of Derivatives. Asked once: 25 Jun 2022

    73
    Let f:R→Rf:R \rightarrow R and g:R→Rg:R \rightarrow R be two functions defined by f(x)=log⁡e(x2+1)−e−x+1f(x) =\log_{e}\left( x^{2}+ 1 \right)-e^{- x}+ 1 and g(x)=1−2e2xexg(x) =\frac{1 - 2e^{2x}}{e^{x}}. Then, for which of the following range of α\alpha, the inequality f(g((α−1)23))>f(g(α−53))f\left( g\left( \frac{(\alpha- 1)^{2}}{3} \right) \right)>f\left( g\left( \alpha-\frac{5}{3} \right) \right) holds?

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  2. Question 2

    Application of Integrals. Asked once: 2021 Paper 8

    83
    Let f:[−3,1]→Rf:\lbrack - 3,1\rbrack \rightarrow R be given as
    f(x)={min⁡{(x+6),x2},−3≤x≤0max⁡{x,x2},0≤x≤1f(x) =\left\{ \begin{matrix} \min\left\{ (x+ 6),x^{2} \right\}, & - 3 \leq x\leq 0 \\ \max\left\{ \sqrt{x},x^{2} \right\}, & 0 \leq x\leq 1 \end{matrix} \right.
    If the area bounded by y=f(x)y=f(x) and xx-axis is AA, then the value of 6 A6\text{ }A is equal to

  3. Question 3

    Complex Numbers. Asked once: 25 Jun 2022

    153
    Let z1z_{1} and z2z_{2} be two complex numbers such that z‾1=iZ‾2{\overline{z}}_{1}= i{\overline{Z}}_{2} and arg(z1z‾2)=πarg\left( \frac{z_{1}}{{\overline{z}}_{2}} \right)=\pi. Then

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  4. Question 4

    Conic Sections. Asked once: 23 Jan 2025

    73
    Let the circle CC touch the line x−y+1=0x-y+ 1 = 0, have the centre on the positive xx-axis, and cut off a chord of length 413\frac{4}{\sqrt{13}} along the line −3x+2y=1- 3x+ 2y= 1. Let HH be the hyperbola x2α2−y2β2=1\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}= 1, whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then 2α2+3β22\alpha^{2}+ 3\beta^{2} is equal to ___\_\_\_

  5. Question 5

    Definite Integration. Asked once: 11 Apr 2023

    83
    For m,n>0m,n > 0, let α(m,n)=∫02tm(1+3t)ndt\alpha(m,n) =\int_{0}^{2} t^{m}(1 + 3t)^{n}dt. If 11α(10,6)+18α(11,5)=p(14)611\alpha(10,6) + 18\alpha(11,5) = p(14)^{6}, then pp is equal to