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Day 472: JEE Mains Maths

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

  1. Question 1

    Differential Equations. Asked once: 1 Feb 2023

    85
    Let f:R→Rf:R \rightarrow R be a differentiable function such that f′(x)+f(x)=∫02f(t)dtf^{'}(x) + f(x) =\int_{0}^{2} f(t)dt. If f(0)=e−2f(0) =e^{- 2}, then 2f(0)−f(2)2f(0) - f(2) is equal to

  2. Question 2

    Differentiation. Asked once: 1 Feb 2024

    75
    Let f:R→Rf:R \rightarrow R be defined as
    f(x)={a−bcos⁡2xx2;x<0x2+cx+2;0≤x≤12x+1;x>1f(x) =\left\{ \begin{matrix} \frac{a - b\cos2x}{x^{2}} & ; & x < 0 \\ x^{2}+ cx + 2 & ; & 0 \leq x \leq 1 \\ 2x + 1 & ; & x > 1 \end{matrix} \right.
    If ff is continuous everywhere in RR and mm is the number of points where ff is NOT differentiable then m+a+b+cm + a + b + c equals:

    Tap an option to check your answer.

  3. Question 3

    Indefinite Integration. Asked once: 24 Jan 2025

    150
    If ∫2x2+5x+9x2+x+1dx=xx2+x+1+αx2+x+1+\int\frac{2x^{2}+ 5x+ 9}{\sqrt{x^{2}+x+ 1}}dx=x\sqrt{x^{2}+x+ 1}+\alpha\sqrt{x^{2}+x+ 1}+ βlog⁡e∣x+12+x2+x+1∣+C\beta\log_{e}\left| x +\frac{1}{2}+\sqrt{x^{2}+ x + 1} \right|+C, where C is the constant of integration, then α+2β\alpha+ 2\beta is equal to ____\_\_\_\_

  4. Question 4

    Inverse Trigonometric Functions. Asked once: 13 Apr 2023

    82
    If S={x∈R:sin⁡−1(x+1x2+2x+2)−sin⁡−1(xx2+1)=π4}S =\left\{ x\in R:\sin^{- 1}\left( \frac{x + 1}{\sqrt{x^{2}+ 2x + 2}} \right)-\sin^{- 1}\left( \frac{x}{\sqrt{x^{2}+ 1}} \right)=\frac{\pi}{4} \right\}, Then ∑x∈S(sin⁡((x2+x+5)π2)−cos⁡((x2+x+5)π))\sum_{x \in S} \left( \sin\left( \left( x^{2}+ x + 5 \right)\frac{\pi}{2} \right)- \cos\left( \left( x^{2}+ x + 5 \right)\pi \right) \right) is equal to

  5. Question 5

    Limits and Continuity. Asked once: 28 Jan 2025

    147
    Let f(x)=lim⁡n→∞∑r=0n(tan⁡(x/2r+1)+tan⁡3(x/2r+1)1−tan⁡2(x/2r+1))f(x) =\lim_{n \rightarrow \infty} \sum_{r = 0}^{n} \left( \frac{\tan\left( x/2^{r + 1} \right)+\tan^{3}\left( x/2^{r + 1} \right)}{1 -\tan^{2}\left( x/2^{r + 1} \right)} \right). Then lim⁡x→0ex−ef(x)(x−f(x))\lim_{x \rightarrow 0} \frac{e^{x}-e^{f(x)}}{(x - f(x))} is equal to ______\_\_\_\_\_\_ .