Limits, Continuity & Differentiability · JEE Advanced previous year questions

2025 4 questions

1. Q112025 · 18 May 2025 · Paper 1· Numerical answer

   Let $\alpha$ and $\beta$ be the real numbers such that $$\underset{x\to 0}{\lim }\frac{1}{x^3}\left(\frac{\alpha }{2}{\int }_0^x\frac{1}{1-t^2}dt+\beta x\cos x\right)=2.$$ Then the value of $\alpha +\beta$ is ____.

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   Answer: 2.35 to 2.45

2. Q132025 · 18 May 2025 · Paper 1· Numerical answer

   For all $x>0$, let $y_1(x),y_2(x)$, and $y_3(x)$ be the functions satisfying $$\frac{d{y}_1}{dx}-(\sin x{)}^2{y}_1=0,y_1(1)=5,$$ $$\frac{d{y}_2}{dx}-(\cos x{)}^2{y}_2=0,y_2(1)=\frac{1}{3},$$ $$\frac{d{y}_3}{dx}-\left(\frac{2-x^3}{x^3}\right)y_3=0,y_3(1)=\frac{3}{5e},$$ respectively. Then $$\underset{x\to 0^{+}}{\lim }\frac{y_1(x)y_2(x)y_3(x)+2x}{e^{3x}\sin x}$$ is equal to ____.

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   Answer: 2

3. Q152025 · 18 May 2025 · Paper 1

   Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I (P) The minimum value of $n$ for which the function $f(x)=\left[\frac{10x^3-45x^2+60x+35}{n}\right]$ is continuous on the interval $[1,2]$, is (Q) The minimum value of $n$ for which $g(x)=(2n^2-13n-15)(x^3+3x)$, $x\in \mathbb{R}$, is an increasing function on $\mathbb{R}$, is (R) The smallest natural number $n$ which is greater than 5, such that $x=3$ is a point of local minima of $h(x)=(x^2-9{)}^n(x^2+2x+3)$, is (S) Number of $x_0\in \mathbb{R}$ such that $l(x)={\sum }_{k=0}^4\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right)$, $x\in \mathbb{R}$, is $\text{NOT}$ differentiable at $x_0$, is List-II (1) 8 (2) 9 (3) 5 (4) 6 (5) 10

   1. A.(P) → (1) (Q) → (3) (R) → (2) (S) → (5)
   2. B.(P) → (2) (Q) → (1) (R) → (4) (S) → (3)
   3. C.(P) → (5) (Q) → (1) (R) → (4) (S) → (3)
   4. D.(P) → (2) (Q) → (3) (R) → (1) (S) → (5)
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   Answer: B

4. Q12025 · 18 May 2025 · Paper 2

   Let $x_0$ be the real number such that $e^{x_0}+x_0=0$. For a given real number $\alpha$, define $$g(x)=\frac{3x{e}^x+3x-\alpha e^x-\alpha x}{3(e^x+1)}$$ for all real numbers $x$. Then which one of the following statements is TRUE?

   1. A.For $\alpha =2$, ${\lim }_{x\to x_0}\left|\frac{g(x)+e^{x_0}}{x-x_0}\right|=0$
   2. B.For $\alpha =2$, ${\lim }_{x\to x_0}\left|\frac{g(x)+e^{x_0}}{x-x_0}\right|=1$
   3. C.For $\alpha =3$, ${\lim }_{x\to x_0}\left|\frac{g(x)+e^{x_0}}{x-x_0}\right|=0$
   4. D.For $\alpha =3$, ${\lim }_{x\to x_0}\left|\frac{g(x)+e^{x_0}}{x-x_0}\right|=\frac{2}{3}$
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   Answer: C