Limits, Continuity & Differentiability · JEE Advanced previous year questions

2024 4 questions

1. Q12024 · 26 May 2024 · Paper 1

   Let $f(x)$ be a continuously differentiable function on the interval $(0,\infty )$ such that $f(1)=2$ and $$\underset{t\to x}{\lim }\frac{t^{10}f(x)-x^{10}f(t)}{t^9-x^9}=1$$ for each $x>0$. Then, for all $x>0$, $f(x)$ is equal to

   1. A.$\frac{31}{11x}-\frac{9}{11}{x}^{10}$
   2. B.$\frac{9}{11x}+\frac{13}{11}{x}^{10}$
   3. C.$\frac{-9}{11x}+\frac{31}{11}{x}^{10}$
   4. D.$\frac{13}{11x}+\frac{9}{11}{x}^{10}$
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   Answer: B

2. Q32024 · 26 May 2024 · Paper 2

   Let $k\in \mathbb{R}$. If ${\lim }_{x\to 0+}(\sin (\sin kx)+\cos x+x{)}^{\frac{2}{x}}=e^6$, then the value of $k$ is

   1. A.1
   2. B.2
   3. C.3
   4. D.4
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   Answer: B

3. Q52024 · 26 May 2024 · Paper 2· One or more correct

   Let $S$ be the set of all $(\alpha ,\beta )\in \mathbb{R}\times \mathbb{R}$ such that $$\underset{x\to \infty }{\lim }\frac{\sin (x^2)({\log }_{e}x{)}^{\alpha }\sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta }({\log }_e(1+x){)}^{\beta }}=0.$$ Then which of the following is (are) correct?

   1. A.$(-1,3)\in S$
   2. B.$(-1,1)\in S$
   3. C.$(1,-1)\in S$
   4. D.$(1,-2)\in S$
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   Answer: B, C

4. Q132024 · 26 May 2024 · Paper 2· Numerical answer

   Let the function $f:[1,\infty )\to \mathbb{R}$ be defined by $$f(t)=\{\begin{array}{ll}(-1{)}^{n+1}2,& \text{if }t=2n-1,n\in \mathbb{N},\\ \frac{(2n+1-t)}{2}f(2n-1)+\frac{(t-(2n-1))}{2}f(2n+1),& \text{if }2n-1<t<2n+1,n\in \mathbb{N}.\end{array}$$ Define $g(x)={\int }_1^{x}f(t)dt,x\in (1,\infty )$. Let $\alpha$ denote the number of solutions of the equation $g(x)=0$ in the interval $(1,8]$ and $\beta ={\lim }_{x\to 1+}\frac{g(x)}{x-1}$. Then the value of $\alpha +\beta$ is equal to ____.

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   Answer: 5